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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf,len=2366
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='00529v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='RT]  2 Jan 2023  HARMONIC ANALYSIS ON THE FOURFOLD COVER OF THE SPACE OF  ORDERED TRIANGLES I: THE INVARIANT DIFFERENTIALS  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Denote by SLn(R) the group of n × n real matrices with determinant one, A the  subgroup consisting of the diagonal matrices with positive entries, and SLn(R)/A the manifold  of left cosets gA, g ∈ SLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In this paper, we will be concerned with the harmonic analysis  on the homogeneous space SLn(R)/A when n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In particular, we provide explicit generators  and their relations for the algebra of the invariant differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then we prove that  some of the non-central generators are essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Structure of the Algebra of the Invariant Differentials  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Generators of the invariant differentials on SLn(R)/A  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Relations among the generators  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The center of the algebra of the invariant differentials  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Essential Self-Adjointness  19  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Essential self-adjointness of the central elements  19  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Reduction to the density of C∞  c (X) in Dom(∆)  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Coordinate charts induced by the Euler angles  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Presence of left derivatives in D12  22  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of the density and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3  28  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Computations in the Euler-Iwasawa Coordinates  36  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Explicit formulas for the generators of the left derivatives on SL3(R)/A  36  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Explicit formulas for the generators of the left-invariant differentials on SL3(R)  42  References  46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Introduction  Denote by SLn(R) the group of n × n real matrices with determinant one, A the subgroup  consisting of the diagonal matrices with positive entries, and SLn(R)/A the manifold of left  cosets gA, g ∈ SLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In this paper, we will be concerned with the harmonic analysis on  the homogeneous space SLn(R)/A when n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In particular, we will restrict attention to  the commutation relations and essential self-adjointness of the invariant differential operators  1  2  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  on SL3(R)/A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' spectral decomposition of the invariant differential operators and its interaction  with Plancherel theorems are left for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The space SL3(R)/A distinguishes itself among homogeneous spaces in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Firstly,  it has a natural interpretation as the fourfold cover of the space of nondegenerate Schubert  triangles in the plane ([Sc]), of which the compactification is well studied as of a homogeneous  space of complexity 1 ([Sem], [Ti]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' One may wish to investigate the Plancherel type formulas  (see [GGV], [GG] for the relation with the Radon transforms), via the Bernstein maps and  the Maass-Selberg relations (see [SV], [DKKS] for the spherical case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Secondly, SL3(R)/A is  one of the simplest examples of non-spherical homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' These spaces exhibit a very  different nature compared with spherical ones, and very little of their harmonic theory is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  For instance, the finite multiplicity theorem for induction does not hold anymore ([KO]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' the  algebra of invariant differential operators is a noncommutative algebra instead of a polynomial  ring ([Kn]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' the representation theory of the ring of bi-invariant functions is mysterious as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For our case, at least one knows, a priori by the systematic work of Benoist-Kobayashi  ([BK1], [BK2], [BK3], [BK4], [BIK]), that the natural unitary representation of SLn(R) in  L2 (SLn(R)/A) is tempered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  From the technical perspective, the investigation of the spectral theory of pseudo-Riemannian  manifolds is challenging, for the traditional elliptic theory is not applicable to the Laplace-  Beltrami operators with mixed signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' To remedy the situation, the analytic methods are always  fused with the peculiar geometry of the underlying manifold related to the representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The considerable efforts have been made mainly on the symmetric spaces, such as the group  manifolds ([HC1], [HC2]), the real hyperboloids ([RLN], [LNR1], [LNR2], [Sh], [St], [Ros],  [Fa], [Sek], [Mo]), special symmetric spaces ([Ma], [Sa], [DP], [KD], [BH], [Ha]), and general  symmetric spaces ([FJ], [OM], [OS], [Ba], [O], [De], [BS1], [BS2]), and, more recently, certain  locally symmetric spaces and spherical varieties ([KK1], [KK2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  In the present situation,  SL3(R)/A has an inherent approachable geometry in spite of the rather involved analysis due  to its non-sphericity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For instance, the underlying manifold is diffeomorphic to SO3(R) × R3,  so that an extensive calculation is possible, just as the hyperboloid case, where the usage of  the spherical coordinates of the underlying product space Sp−1 × Sq−1 × R is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Before describing our results in more detail, we first set certain notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' To treat in a unified  fashion, let G denote SL3(R), g the Lie algebra of G, and U(g) the universal enveloping algebra  of the complexification of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Denote by C∞(G) the space of complex-valued smooth functions  on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then, the infinitesimal action R on C∞(G) induced by the right regular representation  of G, maps U(g) into the algebra of algebraic differentials on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' More precisely, R acts on  u = X1X2 · · · Xk ∈ U(g) by  (Ruf) (g) := (R (X1 · · · Xk) f) (g) :=  ∂  ∂t1  ����  t1=0  · · ∂  ∂tk  ����  tk=0  f(g exp (t1X1) · · · exp (tkXk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (1)  Here X1, X2, · · · , Xk ∈ g, and f ∈ C∞(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' the exponential map is given by exp(X) := γ(1),  where γ : R → G is the one-parameter subgroup of G whose tangent vector at the identity is  equal to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It is easy to verify that Ru, u ∈ U(g), is a left G-invariant differential operators  on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Denote by D(G) the algebra of the left G-invariant differential operators on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  3  For a closed subgroup H ⊂ G, denote by D(G/H) the algebra of G-invariant differential  operators on the homogeneous space G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Denote by π : G → G/H the natural projection,  and h the Lie algebra of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Define  DH(G) := {D ∈ D(G) | D(f ◦ Rh) ◦ R−1  h  = Df, ∀h ∈ H and f ∈ C∞(G)},  (2)  where Rh : g �→ gh is the right translation of G for h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Assuming G and H are reductive,  we have the standard isomorphism (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6 in Chapter 2 of [He])  DH(G)/  �  DH(G) ∩ D(G)h  � ∼= D(G/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (3)  It is induced by the map µ : DH(G) → D(G/H), such that for each D ∈ DH(G), µ(D) is the  element of D(G/H) such that  (µ(D)f) ◦ π = D(f ◦ π) for all smooth functions f on G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (4)  Denote by Eij the 3 × 3 matrix unit with a 1 in the ith row and jth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For distinct  i, j, k ∈ {1, 2, 3}, define differential operators on SL3(R)/A  Dij = µ  �  1  2  �  σ∈S2  R  �  Eσ(i)σ(j)Eσ(j)σ(i)  �  �  ,  (5)  and  Dijk = µ  �  1  6  �  σ∈S3  R  �  Eσ(i)σ(j)Eσ(j)σ(k)Eσ(k)σ(i)  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (6)  Note that Dij = Dji and Dijk = Djki = Dkij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  In the first part of the paper, we prove  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' D (SL3(R)/A) is the noncommutative associative algebra generated over C by  {D12, D13, D23, D123, D213} with relations  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  [D123, D213] = 0,  [Dij, Dik] = Dijk − Dikj,  i, j, k ∈ {1, 2, 3} are distinct,  [Dijk, Dij] = DjkDij − DijDik,  i, j, k ∈ {1, 2, 3} are distinct,  2 (D123D213 + D213D123 − D12D23D31 − D13D32D21) = (D23 − D13 − D12)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (7)  The center of D (SL3(R)/A) is a polynomial ring in D123 + D213 and D12 + D23 + D13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  In the theory of harmonic analysis on homogeneous spaces, one of the central problems is  whether a symmetric invariant differential operator has a unique self-adjoint extension, as it  would enable a simultaneous study of spectral decomposition of both the regular representation  and the invariant differential operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' see [BS1], [BS2] for the case of reductive symmetric  spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' There is considerable literature on essential self-adjointness for natural operators on a  complete Riemannian or Hermitian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' See for instance [Ga1], [Ga2], [Ga3], [Roe], [Co],  [Ri] for the Hodge-Laplace-Beltrami operator, [AV] for the ¯∂-Laplacian, [P], [W] for the Dirac  operator, and [Ch] for certain first order differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For differential operators on  Rn, much more is known ([RS]), and we mention here that certain ellipticity ([IK]), or semi-  boundedness together with temperedness ([Dev]), guarantees the essential self-adjointness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  4  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  For general homogeneous spaces, a classical result shows that the symmetric elements in the  image of the center of the universal enveloping algebra are essentially self-adjoint ([Seg], [NS],  or [Th]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' As it can be shown that the center of D (SL3(R)/A) equals the image of the center of  U(sl3(R)), it follows  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Every symmetric differential operator in the center of D (SL3(R)/A) is es-  sentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  For non-central elements, the most general result is due to Van den Ban [Ba], who established  the essential self-adjointness of the symmetric invariant differential operators for semi-simple  symmetric pairs, even if it is not a generalized Gelfand pair, semiboundedness is absent, or the  underlying manifold is non-Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Beyond that, to the best of our knowledge, there is no  general theory ensuring the essential self-adjointness in the pseudo-Riemannian setting, even  for the Laplacian operators (see [KK1], [KK2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The major part of the paper is to devoted to  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The differential operators D12, D13, D23 on SL3(R)/A are essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We now briefly describe the basic ideas for the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We exploit the universal enveloping  algebra to extract the algebraic structure of D(SL3(R)/A), and the normal form theory in [FH]  to determine the center of D(SL3(R)/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' To study the essential self-adjointness of symmetric  operators, we modify the scheme of [Ba].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The elegant proof in [Ba] is to decompose the  differential operator into a bounded sum of left derivatives so that the wild growth of the  coefficients can be treated as bounded ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Unfortunately, in this non-spherical case, the  left derivatives are too degenerate to span the whole space of invariant differentials in a mild  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We make the observation that by choosing the cutoff functions and the mollifiers in a  compatible way instead of separating the G˚arding type space as the operator core, one may  gain extra decays to balance the extraordinary growth of the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In fact, the chosen  cutoff functions are annihilated by the wildest terms and contribute desired decays thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The organization of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  §2 is devoted to the algebraic structure of  D(SL3(R)/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We study the presence of left derivatives in the coordinate form of D12 in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5, we establish the density of C∞  c (SL3(R)/A) in Dom(D12), and, as a consequence, prove  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The explicit formulas for the generators of the left derivatives and of the left  invariant differentials are given in Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The authors appreciate greatly Professors N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Xu and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Yu for many helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' This research is supported by National Key R& D Pro-  gram of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' 2022YFA1006700).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The first author is partially supported by NSFC grant  (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' 12201012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Structure of the Algebra of the Invariant Differentials  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Generators of the invariant differentials on SLn(R)/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Let S(g) be the symmetric  algebra over g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then for a basis {X1, · · · , Xn} of g, S(g) can be identified with the algebra of  polynomials  �  (k1,··· ,kn)∈Nn  ak1···knXk1  1 · · · Xkn  n , ak1···kn ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (8)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  5  We have the following symmetrizer map λ : S(g) → D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3 in Chapter 2 of [He]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' There is a unique linear bijection λ from  S(g) to D(G) such that λ(Xm) = R(Xm) for X ∈ g and m ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' More precisely,  (λ(P)f)(g) := P  � ∂  ∂t1  , · · · , ∂  ∂tn  �  f (g exp(t1X1 + · · · + tnXn))  ����  t1=···=tn=0  ,  (9)  for P ∈ S(g) and f ∈ C∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In particular (see Page 282 of [He]),  λ(Y1 · · · Yk) = 1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �  σ∈Sk  R  �  Yσ(1) · · · Yσ(k)  �  ,  (10)  where Y1, · · · , Yk ∈ g, and Sk is the symmetric group of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Denote by Eij the n × n matrix unit with a 1 in the ith row and jth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Define  Xij := Eij, 1 ≤ i ̸= j ≤ n,  Xll := Ell − Enn, 1 ≤ l ≤ n − 1,  (11)  which constitute a basis of sln(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The algebra D(SLn(R)/A) is generated by  �  (µ ◦ λ)  �  Ei1i2Ei2i3 · · · Eik−1ikEiki1  � ����  2 ≤ k ≤ n, 1 ≤ i1, i2, · · · , ik ≤ n  i1, i2, · · · , ik are distinct  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (12)  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  By (3), the invariant differential operators on SLn(R)/A are  induced from the left SLn(R) and right A invariant differential operators on SLn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then by  (3), it suffices to prove that DA(SLn(R)) is generated by the elements in (12) and D(SLn(R))a,  where a is the Lie algebra of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Take D ∈ DA(SLn(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' There is a polynomial PD such that λ(PD) = D by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Arrange PD in the following lexicographic order,  PD =  ∞  �  k=1  �  1≤i1≤i2≤···≤it≤···≤ik≤n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  1≤jt≤n and (it,jt)̸=(n,n) for 1≤t≤k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  if 1≤u<v≤k and iu=iv, then ju≤jv  pi1i2···ik  j1j2···jkXi1j1Xi2j2 · · · Xitjt · · · Xikjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (13)  By definition,  (Df)(g) = PD  � ∂  ∂t11  , · · · ,  ∂  ∂tn(n−1)  �  f  \uf8eb  \uf8ec  \uf8ec  \uf8edg exp  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  1≤i,j≤n  (i,j)̸=(n,n)  tij · Xij  \uf8f6  \uf8f7  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f7  \uf8f8  ��������  t=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (14)  6  HANLONG FANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' XIAOCHENG LI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' AND YUNFENG ZHANG  For each a = (aij) ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' we have  (D(f ◦ Ra) ◦ R−1  a  ) (g) = PD  � ∂  ∂t11  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  ∂  ∂tn(n−1)  �  f  \uf8eb  \uf8ec  \uf8ec  \uf8edga−1 exp  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  1≤i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j≤n  (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j)̸=(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='n)  tij · Xij  \uf8f6  \uf8f7  \uf8f7  \uf8f8 a  \uf8f6  \uf8f7  \uf8f7  \uf8f8  ��������  t=0  = PD  � ∂  ∂t11  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  ∂  ∂tn(n−1)  �  f  \uf8eb  \uf8ec  \uf8ec  \uf8edg exp  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  1≤i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j≤n  i̸=j  a−1  ii tijajj · Xij  \uf8f6  \uf8f7  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f7  \uf8f8  ��������  t=0  = PD  �  · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' a−1  ii ajj  ∂  ∂tij  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' · · ·  �  f  \uf8eb  \uf8ec  \uf8ec  \uf8edg exp  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  1≤i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j≤n  (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j)̸=(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='n)  tij · Xij  \uf8f6  \uf8f7  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f7  \uf8f8  ��������  t=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (15)  Since D is right A-invariant, P  �  · · ,  ∂  ∂tij , · · ·  �  = P  �  · · , a−1  ii ajj  ∂  ∂tij , · · ·  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' more precisely,  ∞  �  k=1  �  1≤i1≤i2≤···≤it≤···≤ik≤n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  1≤jt≤n and (it,jt)̸=(n,n) for 1≤t≤k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  if 1≤u<v≤k and iu=iv, then ju≤jv  pi1i2···ik  j1j2···jkXi1j1Xi2j2 · · ·Xitjt · · · Xikjk  =  ∞  �  k=1  �  1≤i1≤i2≤···≤it≤···≤ik≤n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  1≤jt≤n and (it,jt)̸=(n,n) for 1≤t≤k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  if 1≤u<v≤k and iu=iv, then ju≤jv  �  pi1i2···ik  j1j2···jk  k  �  t=1  a−1  ititajtjt  �  Xi1j1Xi2j2 · · · Xitjt · · · Xikjk,  (16)  where �n  l=1 all = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then, for 1 ≤ t ≤ k, 1 ≤ it, jt ≤ n, and (it, jt) ̸= (n, n),  pi1i2···ik  j1j2···jk = pi1i2···ik  j1j2···jk  k  �  t=1  a−1  ititajtjt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (17)  Without loss of generality, we can assume that  PD = pi1i2···ik  j1j2···jk  k  �  t=1  a−1  ititajtjtXi1j1Xi2j2 · · · Xitjt · · · Xikjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (18)  Write Ik := ((i1, j1), (i2, j2), · · · , (ik, jk)), and  k  �  t=1  a−1  ititajtjt =  n  �  l=1  aσl(Ik)  ll  ,  (19)  where for 1 ≤ l ≤ n,  σl(Ik) := cardinality of {1 ≤ t ≤ n | jt = l } − cardinality of {1 ≤ t ≤ n | it = l }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (20)  If pi1i2···ik  j1j2···jk ̸= 0, then σ1(Ik) = σ2(Ik) = · · · = σn(Ik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Moreover, since �n  l=1 σl(Ik) = 0,  σ1(Ik) = σ2(Ik) = · · · = σn(Ik) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (21)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  7  Construct a directed graph (V, A) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Let V := {1, 2, · · · , n} be the set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The set of directed edges A consists of directed edges (it, jt) from it to jt, 1 ≤ t ≤ k, for (it, jt)  appearing in Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It is clear that the indegree equals the outdegree for each vertex in V by (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Now, if the directed graph (V, A) contains a self-loop (a directed edge connects a vertex to  itself), then PD ∈ D(SLn(R))a, that is, PD is the composition of a differential operator in  D(SLn(R)) and a vector field generated by a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Otherwise, we can partition A into simple cycles,  that is,  {(it, jt)|1 ≤ t ≤ k} =  �  ls≥2  �  (is  1, is  2), (is  2, is  3), · · · , (is  ls−1, is  ls), (is  ls, is  1)  ����  is  u ̸= is  v for  1 ≤ u < v ≤ ls  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (22)  Noticing that each simple cycle in (22) corresponds to an element in (12), we can verify that  PD −  �  ls≥2  (µ ◦ λ)  �  Xis  1is  2Xis  2is  3 · · · Xis  ls−1is  lsXis  lsis  1  �  (23)  is a SLn(R)-invariant differential operator with degree strictly less than that of PD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By induc-  tion, we can conclude that PD is generated by the elements in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We complete the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Relations among the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1, the algebra of  the SL3(R)-invariant differential operators on SL3(R)/A are generated by  Dij = µ  �  1  2  �  σ∈S2  R  �  Eσ(i)σ(j)Eσ(j)σ(i)  �  �  , (i, j) = (1, 2), (1, 3), (2, 3),  (24)  and  Dijk = µ  �  1  6  �  σ∈S3  R  �  Eσ(i)σ(j)Eσ(j)σ(k)Eσ(k)σ(i)  �  �  , (i, j, k) = (1, 2, 3), (2, 1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (25)  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We have the following commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D12, D13] = D123 − D132,  (26)  [D21, D23] = D213 − D231,  (27)  [D31, D32] = D312 − D321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (28)  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It suffices to prove (26);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' (27) and (28) follow from (26) by suitable  permutations of the indices therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  On the level of universal enveloping algebra, we have  E12E21E13E31 = E12E13E21E31 + E12E23E31 = E13E12E31E21 + E12E23E31  = E13E31E12E21 + E12E23E31 − E13E32E21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (29)  Similarly, we can derive  E21E12E13E31 = E13E31E21E12 + E23E12E31 − E13E21E32,  E12E21E31E13 = E31E13E12E21 + E12E31E23 − E32E13E21,  E21E12E31E13 = E31E13E21E12 + E31E23E12 − E21E32E13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (30)  8  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Also,  E12E31E23 = E12E23E31 − E12E21,  E23E12E31 = E12E23E31 − E13E31,  E23E31E12 = E12E23E31 + E23E32 − E13E31,  E31E12E23 = E12E23E31 + E32E23 − E12E21,  E31E23E12 = E12E23E31 − E12E21 + E32E23 − E31E13,  (31)  and  E32E13E21 = E13E32E21 − E12E21,  E13E21E32 = E13E32E21 − E13E31,  E21E13E32 = E13E32E21 + E23E32 − E13E31,  E32E21E13 = E13E32E21 + E32E23 − E12E21,  E21E32E13 = E13E32E21 − E12E21 + E32E23 − E31E13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (32)  By the definition,  D123 − D213 = 1  6µ (R (E12E23E31 + E12E31E23 + E23E12E31 + E23E31E12 + E31E12E23  +E31E23E12)) − 1  6µ (R (E21E13E32 + E21E32E13 + E13E21E32 + E13E32E21  +E32E13E21 + E32E21E13))  = µ (R (E12E23E31)) − µ (R (E13E32E21)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (33)  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' (29) and (30) yield  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='D12D13 − D13D12 = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ ((R (E12E21) + R (E21E12)) (R (E13E31) + R (E31E13)))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ ((R (E13E31) + R (E31E13)) (R (E12E21) + R (E21E12)))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ (R (E12E21E13E31 + E21E12E13E31 + E12E21E31E13 + E21E12E31E13))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ (R (E13E31E12E21 + E13E31E21E12 + E31E13E12E21 + E31E13E21E12))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ (R (E12E21E13E31 − E13E31E12E21)) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ (R (E21E12E13E31 − E13E31E21E12))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ (R (E12E21E31E13 − E31E13E12E21)) + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ (R (E21E12E31E13 − E31E13E21E12))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ (R (E12E23E31 + E23E12E31 + E12E31E23 + E31E23E12))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4µ (R (E13E32E21 + E13E21E32 + E32E13E21 + E21E32E13))  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='= µ (R (E12E23E31)) − µ (R (E13E32E21)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (34)  We can thus complete the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  9  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We have the following commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D123, D12] = −[D213, D12] = D23D12 − D12D13,  (35)  [D312, D31] = −[D132, D31] = D13D12 − D23D13,  (36)  [D231, D23] = −[D321, D23] = D23D31 − D12D23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (37)  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It suffices to compute [D123, D12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Computation yields that  D123D12 = µ  �  R  �  E12E23E31 + 1  6 (2E32E23 + E23E32 − 2E13E31 − E31E13 − 3E12E21)  ��  D12  = µ (R (E12E23E31E21E12)) + 1  2D23D12 − 1  2D13D12 − 1  2D12D12  = µ (R (E12E21E12E23E31)) + µ (R (E12E21E23E32)) − µ (R (E12E21E13E31))  + 1  2D23D12 − 1  2D13D12 − 1  2D12D12  = µ (R (E12E21E12E23E31)) + D12D23 − D12D13 + 1  2D23D12 − 1  2D13D12 − 1  2D12D12  (38)  Similarly,  D12D123 = µ (R (E12E21E12E23E31)) + 1  2D12D23 − 1  2D12D13 − 1  2D12D12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (39)  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We have the following equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  D123D213 + D213D123 − D12D23D31 − D13D32D21 = 1  2 (D23 − D13 − D12)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (40)  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Computation yields that  D13D32D21 = 1  8µ (R (E13E31 + E31E13) R (E23E32 + E32E23) R (E12E21 + E21E12))  = µ (R (E13E31E32E23E21E12)) + 1  8µ (R (E33 − E11) (E23E32 + E32E23) (E12E21 + E21E12))  + 1  4µ (R (E13E31 (E22 − E33) (E12E21 + E21E12))) + 1  2µ (R (E13E31E32E23 (E11 − E22)))  = µ (R (E13E31E32E23E21E12)) ,  (41)  10  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  where the last step exploits the fact that Eii −Ejj commutes with Dkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Similarly, we can derive  D213D123 = µ  �  R  �  E13E32E21 + 1  6 (2E32E23 + E23E32 − 2E13E31 − E31E13 − 3E12E21)  ��  D123  = µ (R (E13E32E21)) D123 + 1  2D32D123 − 1  2D13D123 − 1  2D12D123  = µ (R (E13E32E21E31E23E12)) + 1  2µ (R (E13E32E21E12E21)) + 1  2µ (R (E13E32E21E13E31))  − 1  2µ (R (E13E32E21E23E32)) + 1  2D32D123 − 1  2D13D123 − 1  2D12D123  = µ (R (E13E31E32E23E21E12)) − 1  2D213 (D23 − D12 − D13) + 1  2 (D23 − D12 − D13) D123  + 1  4 (D23 − D13 − D12)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (42)  Let A2 be the group consisting of the identity and the permutation of 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then,  �  σ∈A2  �  Dσ(2)σ(1)σ(3)Dσ(1)σ(2)σ(3) − Dσ(1)σ(3)Dσ(3)σ(2)Dσ(2)σ(1)  �  =  �  σ∈A2  1  2  �  Dσ(2)σ(3) − Dσ(1)σ(2) − Dσ(1)σ(3)  � �  Dσ(1)σ(2)σ(3) − Dσ(2)σ(1)σ(3)  �  +  �  σ∈A2  1  2  �  Dσ(2)σ(3)Dσ(3)σ(1) − Dσ(1)σ(2)Dσ(2)σ(3) − Dσ(1)σ(3)Dσ(1)σ(2) − Dσ(2)σ(3)Dσ(1)σ(2)  +Dσ(2)σ(3)Dσ(1)σ(3) + Dσ(1)σ(2)Dσ(1)σ(3)  �  +  �  σ∈A2  1  4  �  Dσ(2)σ(3) − Dσ(1)σ(2) − Dσ(1)σ(3)  �2  = 1  2 (D23 − D13 − D12)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (43)  Now Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We have the following commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D123, D213] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (44)  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By (41) and (42), we have  D213D123 = D13D32D21 − 1  2D213 (D23 − D12 − D13) + 1  2 (D23 − D12 − D13) D123  + 1  4 (D23 − D13 − D12)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (45)  Interchanging the indices 2 and 3 in (45), we get  D123D213 = D12D23D13 − 1  2D123 (D23 − D12 − D13) + 1  2 (D23 − D12 − D13) D213  + 1  4 (D23 − D13 − D12)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (46)  Making a subtraction, we can derive that [D123, D213] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  11  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Let D = �  i Xi,1Xi,2 · · · Xi,ki be an element of U(g), understood as a left-invariant  differential operator on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then the formal adjoint D∗ is given by  D∗ =  �  i  (−1)kiXi,kiXi,ki−1 · · · Xi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (47)  One can thus easily verify that D12, D13, D23, √−1 · D123, √−1 · D213 are formally self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  This is another reason for taking Dij, Dijk as the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The center of the algebra of the invariant differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The following elements constitute a basis of the linear space D(SL3(R)/A)  over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  Dk  12Dj  23Dl  123Dm  213, Dk  12Di  31Dl  123Dm  213, Dj  23Di  31Dl  123Dm  213,  i, j, k ∈ Z+, l, m ∈ Z≥0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Dk  12Dl  123Dm  213, Dj  23Dl  123Dm  213, Di  31Dl  123Dm  213,  i, j, k ∈ Z+, l, m ∈ Z≥0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Dl  123Dm  213, l, m ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (48)  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2, D(SL3(R)/A) is a linear span  of Dk  12Dj  23Di  31Dl  123Dm  213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  By (40) along with the commutation relations in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6, we can replace D12D23D31 by D123D213 with certain lower order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Hence,  Dk  12Dj  23Di  31Dl  123Dm  213, k, j, i ≥ 1, is generated by the elements in (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Next, we shall show that the elements in (48) are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Suppose  0 =  �  l,m∈Z≥0  r000lmDl  123Dm  213 +  �  k∈Z+  l,m∈Z≥0  rk00lmDk  12Dl  123Dm  213 +  �  j∈Z+  l,m∈Z≥0  r0j0lmDj  23Dl  123Dm  213  +  �  i∈Z+  l,m∈Z≥0  r00ilmDi  31Dl  123Dm  213 +  �  k,j∈Z+  l,m∈Z≥0  rkj0lmDk  12Dj  23Dl  123Dm  213  +  �  k,i∈Z+  l,m∈Z≥0  rk0ilmDk  12Di  31Dl  123Dm  213 +  �  j,i∈Z+  l,m∈Z≥0  r0jilmDj  23Di  31Dl  123Dm  213,  (49)  where rkjilm ̸= 0 for only finitely many nonnegative integers k, j, i, l, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Let p be the subspace of sl3(R) spanned by {Xij}1≤i,j≤3, i̸=j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Equip p with natural coordinates  x = (x12, x13, x23, x21, x31, x32) such that η = �  1≤i,j≤n, i̸=j xijXij for each η ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Denote by  j : p ֒→ g the natural injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Define a mapping P : p → SL3(R)/A by P := π ◦ exp ◦j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Then, the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  g  SL3(R)  p  SL3(R)/A  exp  π  j  P  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (50)  Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' P is a local homeomorphism near 0 ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  12  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Since exp : sl3(R) → SL3(R) is a local homeomorphism and j is an injection,  (exp ◦j)(p) is a submanifold of G locally near the identity e ∈ SL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Since p ⊕ a = sl3(R),  where a is the Lie algebra of A, we can conclude that (exp ◦j)(p) and A intersects transversally  at e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Since π : SL3(R) → SL3(R)/A is a fiber bundle which locally has a product structure, P  is a local homeomorphism between p and SL3(R)/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Let f be an arbitrary smooth function on G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1, we have  (Dijf)(eA) =  ∂2  ∂xij∂xji  f (π (exp(xijXij + xjiXji)))  ����  xij=xji=0  ,  (Dijkf)(eA) =  ∂3  ∂xij∂xjk∂xki  f (π (exp(xijXij + xjkXjk + xkiXki)))  ����  xij=xjk=xki=0  ,  (51)  for distinct i, j, k ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then, in terms of the local coordinates (x12, x13, x23, x21, x31, x32),  we can conclude that the differential operators Dk  12Dj  23Di  13Dl  123Dm  213, has leading terms at the  origin eA  � ∂  ∂x12  �k+l � ∂  ∂x23  �j+l � ∂  ∂x31  �l+i � ∂  ∂x21  �k+m � ∂  ∂x13  �m+i � ∂  ∂x32  �j+m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (52)  It is easy to verify that the differential operators appearing in (49) all have distinct leading  terms at eA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then (49) holds if and only if all the coefficients rkjilm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We complete the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We define the degree deg (D) for an invariant differential operator D as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' deg (D) := 2  for D = D12, D23, and D31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' deg (D) := 3 for D = D123 and D213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In general, for  D :=  �  k  �  Dτ  i ∈{D12,D23,D31,D123,D213}  aτ · Dτ  1Dτ  2 · · · Dτ  i · · ·Dτ  k,  (53)  define  deg (D) := max  aτ ̸=0  � k  �  i=1  degree of Dτ  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (54)  We write D = R(M) if and only if deg (D) ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The following equalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D123, Dk  12Dl  123Dm  213] = kDk  12D23Dl  123Dm  213 − kDk  12D31Dl  123Dm  213 + R(2k + 3l + 3m + 1),  [D123, Dj  23Dl  123Dm  213] = −jD12Dj  23Dl  123Dm  213 + jDj  23D31Dl  123Dm  213 + R(2j + 3l + 3m + 1),  [D123, Di  31Dl  123Dm  213] = iD12Di  31Dl  123Dm  213 − iD23Di  31Dl  123Dm  213 + R(2i + 3l + 3m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (55)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  13  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It suffices to prove the first one for the other two can be obtained by  the same method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4 repeatedly, we have  D123Dk  12Dl  123Dm  213 = D12D123Dk−1  12 Dl  123Dm  213 + (D23D12 − D12D31)Dk−1  12 Dl  123Dm  213  = D2  12D123Dk−2  12 Dl  123Dm  213 +  1  �  p=0  Dp  12(D23D12 − D12D31)Dk−1−p  12  Dl  123Dm  213  = · · ·· · ·  = Dk  12Dl  123Dm  213D123 +  k−1  �  p=0  Dp  12(D23D12 − D12D31)Dk−1−p  12  Dl  123Dm  213,  (56)  where in the last step we use [D123, D213] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Noticing that the degree of the commutator  [D1, D2] is strictly less than the degree of D1D2 , we can conclude that  k−1  �  p=0  Dp  12(D23D12 − D12D31)Dk−1−p  12  Dl  123Dm  213 − (kDk  12D23Dl  123Dm  213 − kDk  12D31Dl  123Dm  213)  (57)  is of degree less than or equal to 2k + 3l + 3m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Therefore, we complete the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The following equalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Dk  12Dj  23Dl  123Dm  213] = kDk  12Dj+1  23 Dl  123Dm  213 + (j − k)Dk−1  12 Dj−1  23 Dl+1  123Dm+1  213  − jDk+1  12 Dj  23Dl  123Dm  213 + R(2k + 2j + 3l + 3m + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Dk  12Di  31Dl  123Dm  213] = (k − i)Dk−1  12 Di−1  31 Dl+1  123Dm+1  213 − kDk  12Di+1  31 Dl  123Dm  213  + iDk+1  12 Di  31Dl  123Dm  213 + R(2k + 2i + 3l + 3m + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Dj  23Di  31Dl  123Dm  213] = jDj  23Di+1  31 Dl  123Dm  213 + (i − j)Dj−1  23 Di−1  31 Dl+1  123Dm+1  213  − iDj+1  23 Di  31Dl  123Dm  213 + R(2j + 2i + 3l + 3m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (58)  14  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  It suffices to prove the first one for the others can be obtained  similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4 repeatedly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' we have  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='D123Dk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12Dj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213 = Dk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12Dj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213D123 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='p=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='Dp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12(D23D12 − D12D23)Dk−1−p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='Dj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='31Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='p=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='Dk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12Dp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23(D23D31 − D12D23)Dj−1−p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='= Dk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12Dj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213D123 + kDk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12Dj+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23 Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213 − kDk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12Dj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23D31Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ jDk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12Dj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23D31Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213 − jDk+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 Dj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213 + R(2k + 2j + 3l + 3m + 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='= Dk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12Dj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213D123 + kDk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12Dj+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23 Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213 − kDk−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 Dj−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23 Dl+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ jDk−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 Dj−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23 Dl+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213 − jDk+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 Dj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23Dl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='123Dm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='213 + R(2k + 2j + 3l + 3m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (59)  We complete the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Similarly, we have  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For l, m ≥ 0, the following equalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D12, Dl  123Dm  213] = −lD12D23Dl−1  123Dm  213 + lD12D31Dl−1  123Dm  213  + mD12D23Dl  123Dm−1  213 − mD12D31Dl  123Dm−1  213 + R(3l + 3m),  [D23, Dl  123Dm  213] = lD12D23Dl−1  123Dm  213 − lD23D31Dl−1  123Dm  213  − mD12D23Dl  123Dm−1  213 + mD23D31Dl  123Dm−1  213 + R(3l + 3m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (60)  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The proof is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For simplicity, we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For k, j, i ≥ 1 and l, m ≥ 0, the following equalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Dj  23Dl  123Dm  213] = −lD12Dj+1  23 Dl−1  123Dm  213 + mD12Dj+1  23 Dl  123Dm−1  213  + (j + l)Dj−1  23 Dl  123Dm+1  213 − (j + m)Dj−1  23 Dl+1  123Dm  213 + R(2j + 3l + 3m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Di  31Dl  123Dm  213] = lD12Di+1  31 Dl−1  123Dm  213 − mD12Di+1  31 Dl  123Dm−1  213  + (i + m)Di−1  31 Dl+1  123Dm  213 − (i + l)Di−1  31 Dl  123Dm+1  213 + R(2i + 3l + 3m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Dk  12Dl  123Dm  213] = lDk+1  12 D23Dl−1  123Dm  213 − mDk+1  12 D23Dl  123Dm−1  213  − (k + l)Dk−1  12 Dl  123Dm+1  213 + (k + m)Dk−1  12 Dl+1  123Dm  213 + R(2k + 3l + 3m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Di  31Dl  123Dm  213] = −lD23Di+1  31 Dl−1  123Dm  213 + mD23Di+1  31 Dl  123Dm−1  213  + (i + l)Di−1  31 Dl  123Dm+1  213 − (i + m)Di−1  31 Dl+1  123Dm  213 + R(2i + 3l + 3m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (61)  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The proof is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For simplicity, we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  15  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For k, j, i ≥ 1 and l, m ≥ 0, the following equalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [D12, Dj  23Di  31Dl  123Dm  213] = −(i + l)Dj  23Di−1  31 Dl  123Dm+1  213 + (i + m)Dj  23Di−1  31 Dl+1  123Dm  213  + (j + l)Dj−1  23 Di  31Dl  123Dm+1  213 − (j + m)Dj−1  23 Di  31Dl+1  123Dm  213 + R(2j + 2i + 3l + 3m),  [D23, Dk  12Di  31Dl  123Dm  213] = −(k + l)Dk−1  12 Di  31Dl  123Dm+1  213 + (k + m)Dk−1  12 Di  31Dl+1  123Dm  213  + (i + l)Dk  12Di−1  31 Dl  123Dm+1  213 − (i + m)Dk  12Di−1  31 Dl+1  123Dm  213 + R(2k + 2i + 3l + 3m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (62)  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The proof is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For simplicity, we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The center of D(SL3(R)/A) is a polynomial ring in D123 +D213 and D12 +  D23 + D31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6, it is easy to verify that D123+D213,  D12 + D23 + D31 are in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We claim that the algebra generated by D123 + D213 and D12 + D23 + D31 is a polynomial  ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Otherwise, there is a nontrivial relation  0 =  �  K,M∈Z≥0  CKM · (D12 + D23 + D31)K(D123 + D213)M =: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (63)  Define A := max{ K | CKM ̸= 0 } and B := max{ M | CAM ̸= 0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Expand R in terms of the  basis in (48)  R =  �  l,m∈Z≥0  r000lmDl  123Dm  213 +  �  k∈Z+  l,m∈Z≥0  rk00lmDk  12Dl  123Dm  213 +  �  j∈Z+  l,m∈Z≥0  r0j0lmDj  23Dl  123Dm  213  +  �  i∈Z+  l,m∈Z≥0  r00ilmDi  31Dl  123Dm  213 +  �  k,j∈Z+  l,m∈Z≥0  rkj0lmDk  12Dj  23Dl  123Dm  213  +  �  k,i∈Z+  l,m∈Z≥0  rk0ilmDk  12Di  31Dl  123Dm  213 +  �  j,i∈Z+  l,m∈Z≥0  r0jilmDj  23Di  31Dl  123Dm  213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (64)  It is easy to verify that  rA000B = CAB ̸= 0,  (65)  which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  16  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  In what follows, we will prove that the center of D(SL3(R)/A) is generated by D123 + D213  and D12 + D23 + D31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For each z ∈ D(SL3(R)/A), write  z =  �  l,m∈Z≥0  a000lmDl  123Dm  213 +  �  k∈Z+  l,m∈Z≥0  ak00lmDk  12Dl  123Dm  213 +  �  j∈Z+  l,m∈Z≥0  a0j0lmDj  23Dl  123Dm  213  +  �  i∈Z+  l,m∈Z≥0  a00ilmDi  31Dl  123Dm  213 +  �  k,j∈Z+  l,m∈Z≥0  akj0lmDk  12Dj  23Dl  123Dm  213  +  �  k,i∈Z+  l,m∈Z≥0  ak0ilmDk  12Di  31Dl  123Dm  213 +  �  j,i∈Z+  l,m∈Z≥0  a0jilmDj  23Di  31Dl  123Dm  213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (66)  For convenience, we will use the convention that aKJILM = 0 when K, J, I, L, or M is strictly  negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Applying Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='9, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' we have  0 =[D123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z] =  �  k∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  ak00lm  �  kDk  12D23Dl  123Dm  213 − kDk  12D31Dl  123Dm  213  �  +  �  j∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a0j0lm  �  −jD12Dj  23Dl  123Dm  213 + jDj  23D31Dl  123Dm  213  �  +  �  i∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a00ilm  �  iD12Di  31Dl  123Dm  213 − iD23Di  31Dl  123Dm  213  �  (67)  +  �  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  akj0lm  �  kDk  12Dj+1  23 Dl  123Dm  213 + (j − k)Dk−1  12 Dj−1  23 Dl+1  123Dm+1  213  − jDk+1  12 Dj  23Dl  123Dm  213  �  +  �  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='i∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  ak0ilm  �  (k − i)Dk−1  12 Di−1  31 Dl+1  123Dm+1  213  − kDk  12Di+1  31 Dl  123Dm  213 + iDk+1  12 Di  31Dl  123Dm  213  �  +  �  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='i∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a0jilm  �  jDj  23Di+1  31 Dl  123Dm  213 + (i − j)Dj−1  23 Di−1  31 Dl+1  123 Dm+1  213  − iDj+1  23 Di  31Dl  123Dm  213  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  By the linear independence, we can conclude that, for L, M ≥ 0 and K, J ≥ 1,  KaK(J−1)0LM + (J − K)a(K+1)(J+1)0(L−1)(M−1) − Ja(K−1)J0LM = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (68)  for L, M ≥ 0 and K, I ≥ 1,  (K − I)a(K+1)0(I+1)(L−1)(M−1) − KaK0(I−1)LM + Ia(K−1)0ILM = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (69)  for L, M ≥ 0 and J, I ≥ 1,  Ja0J(I−1)LM + (I − J)a0(J+1)(I+1)(L−1)(M−1) − Ia0(J−1)ILM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (70)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  17  We have by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13 that  0 = [D12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z] =  �  l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a000lm  �  −lD12D23Dl−1  123 Dm  213 + lD12D31Dl−1  123 Dm  213 + mD12D23Dl  123Dm−1  213  − m D12D31Dl  123Dm−1  213  �  +  �  k∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  ak00lm  �  −lDk+1  12 D23Dl−1  123 Dm  213 + lDk+1  12 D31Dl−1  123 Dm  213  + mDk+1  12 D23Dl  123Dm−1  213  −mDk+1  12 D31Dl  123Dm−1  213  �  +  �  j∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a0j0lm  �  −lD12Dj+1  23 Dl−1  123 Dm  213  + mD12Dj+1  23 Dl  123Dm−1  213  +(j + l)Dj−1  23 Dl  123Dm+1  213  − (j + m)Dj−1  23 Dl+1  123 Dm  213  �  +  �  i∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a00ilm  �  lD12Di+1  31 Dl−1  123 Dm  213 − mD12Di+1  31 Dl  123Dm−1  213  − (i + l)Di−1  31 Dl  123Dm+1  213  (71)  +(i + m)Di−1  31 Dl+1  123Dm  213  �  +  �  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  akj0lm  �  −lDk+1  12 Dj+1  23 Dl−1  123 Dm  213  +mDk+1  12 Dj+1  23 Dl  123Dm−1  213  + (j + l)Dk  12Dj−1  23 Dl  123Dm+1  213  − (j + m)Dk  12Dj−1  23 Dl+1  123Dm  213  �  +  �  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='i∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  ak0ilm  �  lDk+1  12 Di+1  31 Dl−1  123 Dm  213 − mDk+1  12 Di+1  31 Dl  123Dm−1  213  − (i + l)Dk  12Di−1  31 Dl  123Dm+1  213  +(i + m)Dk  12Di−1  31 Dl+1  123 Dm  213  �  +  �  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='i∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a0jilm  �  −(i + l)Dj  23Di−1  31 Dl  123Dm+1  213  +(i + m)Dj  23Di−1  31 Dl+1  123 Dm  213 + (j + l)Dj−1  23 Di  31Dl  123Dm+1  213  − (j + m)Dj−1  23 Di  31Dl+1  123 Dm  213  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Then for L, M ≥ 0 and K, I ≥ 1,  (L + 1)a(K−1)0(I−1)(L+1)M − (M + 1)a(K−1)0(I−1)L(M+1) − (I + L + 1)aK0(I+1)L(M−1)  + (I + M + 1)aK0(I+1)(L−1)M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (72)  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  0 = [D23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z] =  �  l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a000lm  �  lD12D23Dl−1  123 Dm  213 − lD23D31Dl−1  123 Dm  213 − mD12D23Dl  123Dm−1  213  + m D23D31Dl  123Dm−1  213  �  +  �  k∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  ak00lm  �  lDk+1  12 D23Dl−1  123 Dm  213 − mDk+1  12 D23Dl  123Dm−1  213  − (k + l)Dk−1  12 Dl  123Dm+1  213  +(k + m)Dk−1  12 Dl+1  123 Dm  213  �  +  �  j∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a0j0lm  �  lD12Dj+1  23 Dl−1  123 Dm  213  − lDj+1  23 D31Dl−1  123 Dm  213 −mD12Dj+1  23 Dl  123Dm−1  213  + mDj+1  23 D31Dl  123Dm−1  213  �  +  �  i∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a00ilm  �  −lD23Di+1  31 Dl−1  123 Dm  213 + mD23Di+1  31 Dl  123Dm−1  213  + (i + l)Di−1  31 Dl  123Dm+1  213  − (i + m) Di−1  31 Dl+1  123 Dm  213  �  +  �  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  akj0lm  �  lDk+1  12 Dj+1  23 Dl−1  123 Dm  213 − mDk+1  12 Dj+1  23 Dl  123Dm−1  213  (73)  18  HANLONG FANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' XIAOCHENG LI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' AND YUNFENG ZHANG  − (k + l)Dk−1  12 Dj  23Dl  123Dm+1  213  +(k + m)Dk−1  12 Dj  23Dl+1  123Dm  213  �  +  �  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='i∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  ak0ilm  �  −(k + l)Dk−1  12 Di  31Dl  123Dm+1  213  + (k + m)Dk−1  12 Di  31Dl+1  123 Dm  213  + (i + l)Dk  12Di−1  31 Dl  123Dm+1  213  −(i + m)Dk  12Di−1  31 Dl+1  123Dm  213  �  +  �  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='i∈Z+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='m∈Z≥0  a0jilm  �  −lDj+1  23 Di+1  31 Dl−1  123 Dm  213  + mDj+1  23 Di+1  31 Dl  123Dm−1  213  +(i + l)Dj  23Di−1  31 Dl  123Dm+1  213  − (i + m)Dj  23Di−1  31 Dl+1  123 Dm  213  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Then for L, M ≥ 0 and K, J ≥ 1,  (L + 1)a(K−1)(J−1)0(L+1)M − (M + 1)a(K−1)(J−1)0L(M+1) − (K + L + 1)a(K+1)J0L(M−1)  + (K + M + 1)a(K+1)J0(L−1)M = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (74)  for L, M ≥ 0 and J, I ≥ 1,  − (L + 1)a0(J−1)(I−1)(L+1)M + (M + 1)a0(J−1)(I−1)L(M+1) + (I + L + 1)a0J(I+1)L(M−1)  − (I + M + 1)a0J(I+1)(L−1)M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (75)  Claim I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' If aK000M = 0 for K, M ≥ 0, then aKJ0LM ≡ 0 for K, J, L, M ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Claim I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We prove by induction on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Setting L = 0 in (68), we have for M ≥ 0  and K, J ≥ 1 that  KaK(J−1)00M = Ja(K−1)J00M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (76)  Hence, when M ≥ 0 and K, J ≥ 1,  a(K−1)J00M = (K + J − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  J!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' · (K − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' · a(K+J−1)000M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (77)  We thus prove Claim I when L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Suppose Claim I holds for 0 ≤ L ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By setting L = l in (74), we can conclude that for  M ≥ 0 and K, J ≥ 1,  a(K−1)(J−1)0(l+1)M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (78)  Then, Claim I holds for L = l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Therefore, we complete the proof of Claim I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Claim II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' If aK000M = 0 for K, M ≥ 0, then aK0ILM ≡ 0 for K, I, L, M ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Claim II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Similarly, we prove by induction on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Setting L = 0 in (69),  KaK0(I−1)0M = Ia(K−1)0I0M, M ≥ 0, K ≥ 1, I ≥ 1,  (79)  and hence  a(K−1)0I0M = (K + I − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  I!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' · (K − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' · a(K+I−1)000M = 0, M ≥ 0, K ≥ 1, I ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (80)  Suppose Claim II holds for 0 ≤ L ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Setting L = l in (72), we can conclude that  a(K−1)0(I−1)(l+1)M = 0, M ≥ 0, K ≥ 1, I ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (81)  We complete the proof of Claim II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Claim III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' If aK000M = 0 for K, M ≥ 0, then a0JILM ≡ 0 for K, J, L, M ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  19  Proof of Claim III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By Claim II, it is clear that for I, M ≥ 0  a00I0M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (82)  Similarly to the proofs of Claims I and II, we can then complete the proof of Claim III by (70)  and (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Define  z′ := z −  �  K,M∈Z≥0  aK000M(D12 + D23 + D31)K(D123 + D213)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (83)  Then z′ belongs to the center and can be written as  z′ =  �  l,m∈Z≥0  a′  000lmDl  123Dm  213 +  �  k∈Z+  l,m∈Z≥0  a′  k00lmDk  12Dl  123Dm  213 +  �  j∈Z+  l,m∈Z≥0  a′  0j0lmDj  23Dl  123Dm  213  +  �  i∈Z+  l,m∈Z≥0  a′  00ilmDi  31Dl  123Dm  213 +  �  k,j∈Z+  l,m∈Z≥0  a′  kj0lmDk  12Dj  23Dl  123Dm  213  +  �  k,i∈Z+  l,m∈Z≥0  a′  k0ilmDk  12Di  31Dl  123Dm  213 +  �  j,i∈Z+  l,m∈Z≥0  a′  0jilmDj  23Di  31Dl  123Dm  213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (84)  such that for K, M ≥ 0  a′  K000M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (85)  Now by Claims I, II, III, we can conclude that  z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (86)  Therefore, we complete the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It follows from Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Essential Self-Adjointness  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Essential self-adjointness of the central elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' A classical result due to [Seg],  [NS] says that if τ is the quasi-regular representation of G in L2(G/H), D is a symmetric  element in the center of the universal enveloping algebra, and the operator τ(D) is an invariant  differential operator on X, then τ(D) is essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It leads to a proof of Proposition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Recall that the center of U(sl3(R)) is a polynomial ring Z = R[h, k]  (see Page 984 of [Ca]), where  h = −(X11 − X22)2 + (X11 − X22)X11 − X2  11 − 3E12E21 − 3E13E31 − 3E23E32 + 3X11,  (87)  k = 2(X11 − X22)3 − 3(X11 − X22)2X11 − 3(X11 − X22)X2  11 + 2X3  11  + 9E12E21(X11 − X22) − 18E12E21X11 − 18E13E31(X11 − X22) + 9E13E31X11  + 9E23E32(X11 − X22) + 9E23E32X11 − 27E12E23E31 − 27E21E13E32  + 18(X11 − X22)X11 − 9X2  11 − 18(X11 − X22) + 9X11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (88)  20  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  It is clear that h, k ∈ DA(SL3(R)), and by (31) and (32)  µ(h) = −3µ(E12E21) − 3µ(E13E31) − 3µ(E23E32) = −3(D12 + D13 + D23),  µ(k) = −27µ(E12E31E23) − 27µ(E21E13E32) = −27(D123 + D213).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (89)  By the above result due to [Seg], [NS], we complete the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Reduction to the density of C∞  c (X) in Dom(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Assume that X := G/H is a homo-  geneous space with a G-invariant measure dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Denote by C∞(X) the space of smooth functions  on X, and by C∞  c (X) the space of functions in C∞(X) with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Assume ∆ ∈ D(X) is formally self-adjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', such that the following equality holds  (∆f, g) = (f, ∆g)  (90)  for all f, g ∈ C∞  c (X), which also implies that the above equality also holds for all f ∈ C∞  c (X)  and g in the space C∞  c (X)∗ of distributions (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3 Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' II of [He]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It is clear that ∆  is naturally defined on  Dom(∆) :=  �  f ∈ L2(X) : ∆f ∈ L2(X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (91)  It is also clear that the graph of the operator ∆ : Dom(∆) → L2(X) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Suppose that ∆ is a formally self-adjoint differential operator on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' If the graph  of ∆ on the domain C∞  c (X) is dense in the graph of ∆ on the domain Dom(∆) with respect to  the L2 × L2 norm, ∆ is symmetric on Dom(∆), that is,  (∆f, g) = (f, ∆g) for all f, g ∈ Dom(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (92)  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Since C∞  c (X) is dense in Dom(∆) with respect to the graph L2 × L2  norm, for f, g ∈ Dom(∆) there exist sequences {fn}∞  n=1, {gn}∞  n=1 ⊂ C∞  c (X), such that  lim  n→∞ ∥fn − f∥L2 = lim  n→∞ ∥∆fn − ∆f∥L2 = lim  n→∞ ∥gn − g∥L2 = lim  n→∞ ∥∆gn − ∆g∥L2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (93)  Then,  |(∆f, g) − (f, ∆g)| ≤ |(∆f, g) − (∆f, gn)| + |(∆f, gn) − (∆fn, gn)|  + |(∆fn, gn) − (fn, ∆gn)| + |(fn, ∆gn) − (fn, ∆g)| + |(fn, ∆g) − (f, ∆g)|  = |(∆f, g − gn)| + |(∆f − ∆fn, gn)| + |(fn, ∆gn − ∆g)| + |(fn − f, ∆g)|  ≤ ∥∆f∥L2 · ∥g − gn∥L2 + ∥∆f − ∆fn∥L2 · ∥gn∥L2 + ∥fn∥L2 · ∥∆gn − ∆g∥L2  + ∥fn − f∥L2 · ∥∆g∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (94)  Taking the limit of (94) as n approaches infinity, we can conclude Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Assume that the operator ∆ is symmetric on Dom(∆), that is,  (∆f, g) = (f, ∆g) for all f, g ∈ Dom(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (95)  Then ∆ is self-adjoint on Dom(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  21  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It suffices to prove that Dom(∆∗) = Dom(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Firstly, the symmetric  property (95) implies the domain Dom(∆∗) of the adjoint ∆∗ of ∆ contains Dom(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Next, we shall prove that Dom(∆∗) ⊂ Dom(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By definition, g ∈ Dom(∆∗) means g ∈ L2  and there exist g∗ ∈ L2 such that (∆f, g) = (f, g∗) holds for all f ∈ Dom(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In particular,  (∆f, g) = (f, g∗) holds for all f ∈ C∞  c (X), so that g∗ = ∆g as distribution thus also in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Therefore, g ∈ Dom(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We complete the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Coordinate charts induced by the Euler angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In this subsection, we describe  certain geometric properties of X = SL3(R)/A for explicit computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For convenience, in  the following, we will use X and SL3(R)/A interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  According to the Iwasawa decomposition SL3(R) = KNA, each x ∈ SL3(R) can be uniquely  written as x = kxnxax, where kx is an unimodular orthogonal matrix, nx is an upper unitrian-  gular matrix, and ax is a unimodular diagonal matrix with positive entries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' more explicitly,  x =  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  λ1  0  0  0  λ2  0  0  0  λ−1  1 λ−1  2  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (96)  Since SL3(R)/A is diffeomorphic to SO3(R) × R3, without loss of generality, we can identify  each x ∈ SL3(R)/A with (kx, z12, z13, z23) ∈ SO3(R) × R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We parametrize SO3(R) by the following variant of Euler angles (Tait-Bryan angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Define  Γ : [−π, π] × [−π  2, π  2] × [−π, π] → SO3(R) by (α, β, γ) �→ kx, where  kx :=  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8 :=  \uf8eb  \uf8ed  cos β cos γ  − sin β  cos β sin γ  sin α sin γ + cos α cos γ sin β  cos α cos β  cos α sin β sin γ − cos γ sin α  cos γ sin α sin β − cos α sin γ  cos β sin α  cos α cos γ + sin α sin β sin γ  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (97)  It is easy to verify that when −π  2 < α < π  2, −π  3 < β < π  3, −π  2 < γ < π  2, Γ is bijective onto the  image, and the inverse map is given by  α = arctan  �k32  k22  �  , β = arctan  �  −k12  �  1 − k2  12  �  , γ = arctan  �k13  k11  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (98)  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We denote by UEI the image under Γ of −π  2 < α <  π  2, −π  3 < β <  π  3,  −π  2 < γ < π  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It is clear that U EI is a neighborhood of the identity element in SO3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Denote by (k0 · U EI) the left translation of U EI by k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then, we can construct an atlas  {((k0 · UEI) × R3, Λk0)}k0∈SO3(R) of SL3(R)/A as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For each k0 ∈ SO3(R), define a map  Λk0 : (k0 · UEI) × R3 → (−π  2, π  2) × (−π  3, π  3) × (−π  2, π  2) × R3 by  \uf8eb  \uf8edk0 ��  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8 , z12, z13, z23  \uf8f6  \uf8f8 �→ (α, β, γ, z12, z13, z23),  (99)  where α, β, γ are related to (kij) by (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  22  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We call the coordinates (α, β, γ, z12, z13, z23), which is associated with the  coordinate chart ((k0 · UEI) × R3, Λk0) as above, the Euler-Iwasawa coordinates of SL3(R)/A  attached to k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Notice that the Euler-Iwasawa coordinates attached to k0, combined with the global coordi-  nates (λ1, λ2) of A (see (96)), parametrize the open neighborhood (k0 · U EI)×R3×R2 ⊂ SL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We denote the corresponding atlas of SL3(R) by  ��  (k0 · U EI) × R3 × R2, �Λk0  ��  k0∈SO3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We call the coordinates (α, β, γ, z12, z13, z23, λ1, λ2), which is associated with  the coordinate chart  �  (k0 · UEI) × R3 × R2, �Λk0  �  as above, the Euler-Iwasawa coordinates of  SL3(R) attached to k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  One can verify that an invariant measure on X in the coordinate chart ((k0 · UEI) × R3, Λk0)  takes the form  dµ = dk dn = cos β dα dβ dγ dz12 dz13 dz23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (100)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Presence of left derivatives in D12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We first recall the notion of left derivatives as  in [Ba].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Notice that the infinitesimal action L on C∞(SL3(R)) induced by the left regular  representation, maps U(sl3(R)) into the algebra of differentials on SL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' More precisely, for  u = X1X2 · · · Xk, X1, X2, · · · , Xk ∈ sl3(R), and f ∈ C∞(SL3(R)),  (Luf)(g) :=  ∂  ∂t1  ����  t1=0  · · ∂  ∂tk  ����  tk=0  f(exp (−tkXk) · · · exp (−t1X1) g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (101)  Then, L also induces differential operators on SL3(R)/A, that is, for X1, X2, · · · , Xk ∈ sl3(R),  and f ∈ C∞(SL3(R)/A),  (L (X1 · · · Xk) · f) (xA) :=  ∂  ∂t1  ����  t1=0  · · ∂  ∂tk  ����  tk=0  f(exp (−tkXk) · · · exp (−t1X1) xA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (102)  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We call a differential operator on SL3(R)/A a left derivative if it is induced  by an element u ∈ U(sl3(R)) through the above infinitesimal action L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For convenience, denote  left derivatives by Lu, u ∈ U(sl3(R)), as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In each coordinate chart ((k0 · U EI) × R3, Λk0), D12 takes the form  D12 = (sec β sin γ)  ∂2  ∂α∂z12  + (cos γ)  ∂2  ∂β∂z12  + (tan β sin γ)  ∂2  ∂γ∂z12  +  �  z2  12 + 1  �  ∂2  ∂z12∂z12  + (z23)  ∂2  ∂z13∂z12  + (−z13)  ∂2  ∂z23∂z12  + 2z12  ∂  ∂z12  ,  (103)  where (α, β, γ, z12, z13, z23) are the Euler-Iwasawa coordinates attached to k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' (103) is independent of k0, which is also a consequence of D12 being G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' According to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1, we first compute the left-invariant differ-  ential �D12 on SL3(R) associated with D12, where, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2,  �D12 = 1  2 (R (E12) R (E21) + R (E21) R (E12)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (104)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  23  Consider the Euler-Iwasawa coordinates (α, β, γ, z12, z13, z23, λ1, λ2) attached to k0 (see Def-  inition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then, by Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�D12 = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='sec β sin γ ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂α + cos γ ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂β + tan β sin γ ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂γ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+λ2z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� �λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� �λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='sec β sin γ ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂α + cos γ ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ tan β sin γ ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂γ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ λ2z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='= (sec β sin γ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂α∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (cos γ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂β∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (tan β sin γ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂γ∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (z23)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (−z13)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ 2z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (λ1z12)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12∂λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− (λ2z12)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12∂λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(105)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='Let f be an arbitary smooth function on SL3(R)/A and �f its lift to SL3(R) via the natural  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='projection π : SL3(R) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then,  (D12f)(α, β, γ, z12, z13, z23) =  �  �D12 �f  �  (α, β, γ, z12, z13, z23, λ1, λ2)  = (sec β sin γ)  ∂2  ∂α∂z12  + (cos γ)  ∂2  ∂β∂z12  + (tan β sin γ)  ∂2  ∂γ∂z12  +  �  z2  12 + 1  �  ∂2  ∂z12∂z12  + (z23)  ∂2  ∂z13∂z12  + (−z13)  ∂2  ∂z23∂z12  + 2z12  ∂  ∂z12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (106)  We complete the proof of Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Next, we introduce a special class P of linear differential operators defined on U EI × R3, in  terms of the Euler-Iwasawa coordinates (α, β, γ, z12, z13, z23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The class P consists of  �  ui∈U(sl3(R))  are monomials  Pi (sin α, cos α, sin β, cos β, sin γ, cos γ)  cosli β  Lui,  (107)  where li ≥ 0, Pi are polynomials in trigonometric functions, Lui are left derivatives on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' If D1, D2 ∈ P, then D1 ◦ D2 ∈ P, where  (D1 ◦ D2)f := D1(D2f), f ∈ C∞(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (108)  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Notice that LX12, LX13, LX23, LX21, LX31, LX32 constitute a set of  generators of all left derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By explicit formulas of LXij in Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1, we can verify Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Each differential operator D ∈ P can be written as  D =  �  ui∈U(sl3(R))  are monomials  riLui,  (109)  24  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  where ri are bounded functions on U EI × R3 ⊂ SL3(R)/A, and Lui are left derivatives on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It follows easily from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The following differential operators are in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  ∂  ∂α,  ∂  ∂β ,  ∂  ∂γ ,  ∂  ∂z13  ,  ∂  ∂z23  ,  z12  ∂  ∂z12  − z23  ∂  ∂z23  ,  ∂  ∂z12  + z23  ∂  ∂z13  , z13  ∂  ∂z13  + z23  ∂  ∂z23  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (110)  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1, we have  ∂  ∂α = (LX23 − LX32) ,  ∂  ∂β = cos α (LX12 − LX21) − sin α (LX31 − LX13) ,  ∂  ∂γ = cos α cos β (LX31 − LX13) + sin α cos β (LX12 − LX21) − sin β (LX23 − LX32) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (111)  Computation yields that  1  4  � ∂  ∂γ ,  � ∂  ∂γ , LX12  ��  + LX12 −  �1  2 sin α tan β  � ∂  ∂α −  �1  2 cos α  � ∂  ∂β −  �1  2 sin α sec β  � ∂  ∂γ  = −  ��3  4 cos α sin 2β  �  z12 +  �3  4 cos α cos 2β cos γ − 3  4 sin α sin β sin γ  ��  ∂  ∂z12  −  ��  −3  4 sin α sin β sin γ + 3  4 cos α cos 2β cos γ  �  z23  �  ∂  ∂z13  −  ��  −3  4 cos α sin 2β  �  z23 +  �3  4 sin α sin β cos γ + 3  4 cos α cos 2β sin γ  ��  ∂  ∂z23  =: Lγγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (112)  Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10 Lγγ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Similarly,  Lγγγγ :=  � ∂  ∂γ ,  � ∂  ∂γ , Lγγ  ��  + Lγγ = −  �3  4 cos α sin 2β  � �  z12  ∂  ∂z12  − z23  ∂  ∂z23  �  ∈ P,  (113)  and hence  cos 2β  � ∂  ∂β , sin α  � ∂  ∂α, Lγγγγ  �  − cos α · Lγγγγ  �  + 2 sin 2β  �  sin α  � ∂  ∂α, Lγγγγ  �  − cos α · Lγγγγ  �  = 3  2  �  z12  ∂  ∂z12  − z23  ∂  ∂z23  �  ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (114)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  25  Compuation yields  �L := Lγγ +  �3  4 cos α sin 2β  � �  z12  ∂  ∂z12  − z23  ∂  ∂z23  �  = −  �3  4 cos α cos 2β cos γ − 3  4 sin α sin β sin γ  � � ∂  ∂z12  + z23  ∂  ∂z13  �  −  �3  4 sin α sin β cos γ + 3  4 cos α cos 2β sin γ  �  ∂  ∂z23  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �Lα := sin α  � ∂  ∂α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �L  �  − cos α · �L  =  �3  4 cos 2β cos γ  � � ∂  ∂z12  + z23  ∂  ∂z13  �  +  �3  4 cos 2β sin γ  �  ∂  ∂z23  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �Lαγ  1  := − sin γ  � ∂  ∂γ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �Lα  �  + cos γ · �Lα =  �3  4 cos 2β  � � ∂  ∂z12  + z23  ∂  ∂z13  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �Lαγ  2  := cos γ  � ∂  ∂γ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �Lα  �  + sin γ · �Lα =  �3  4 cos 2β  �  ∂  ∂z23  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (115)  and hence  �Lαγβ  1  := − sin 2β  � ∂  ∂β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �Lαγ  1  �  + 2 cos 2β · �Lαγ  1  = 3  2  � ∂  ∂z12  + z23  ∂  ∂z13  �  ∈ P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �Lαγβ  2  := − sin 2β  � ∂  ∂β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �Lαγ  2  �  + 2 cos 2β · �Lαγ  2  = 3  2  ∂  ∂z23  ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(116)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='Consider  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='LX12 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α cos β sin 2γ − sin α tan β sin2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin β tan β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α sin β sin 2γ − cos α sin2 β cos2 γ − cos α cos2 β sin2 γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− sin α sec β sin2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α tan β sin 2γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4 cos α sin 2β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α cos β sin 2γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin 2β cos2 γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin 2β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (− sin α sin β sin γ + cos α cos 2β cos γ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (sin α sin β cos γ + cos α cos 2β sin γ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(117)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='HANLONG FANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' XIAOCHENG LI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' AND YUNFENG ZHANG  = −  �  sin α cos β sin 2γ + 1  2 cos α sin 2β cos 2γ  � �  z13  ∂  ∂z13  + z23  ∂  ∂z23  �  −  �  − sin α cos β cos 2γ + 1  2 cos α sin 2β sin 2γ  �  ∂  ∂z13  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  By a similar argument, one can show  z13  ∂  ∂z13  + z23  ∂  ∂z23  ,  ∂  ∂z13  ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (118)  We complete the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Denote the differentials  ∂  ∂z13,  ∂  ∂z23, z12  ∂  ∂z12 −z23  ∂  ∂z23,  ∂  ∂z12 +z23  ∂  ∂z13, z13  ∂  ∂z13 +z23  ∂  ∂z23  by δ1, δ2, δ3, δ4, δ5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then, we can represent  δi =  �  vi,j∈U(sl3(R))  are monomials  pi,jLvi,j, 1 ≤ i ≤ 5,  (119)  where pi,j are bounded functions on X, and Lvi,j are left derivatives on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We first note that the differential δi is globally defined on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By  Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11, δi can be written on the open set U EI × R3 ⊂ SL3(R)/A as  δi =  �  vi,j∈U(sl3(R))  are monomials  pi,jLvi,j,  (120)  where pi,j are bounded functions on U EI × R3, and Lvi,j are left derivatives on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For k0 ∈ SO3(R), δi can be written on the open set (k0 · U EI) × R3 ⊂ SL3(R)/A as  δi =  �  vk0  i,j∈U(sl3(R))  are monomials  pk0  i,jLvk0  i,j,  (121)  where pk0  i,j are bounded functions on (k0 · U EI) × R3, and Lvk0  i,j are left derivatives on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Take the Euler-Iwasawa coordinates (α, β, γ, z12, z13, z23) attached to k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It  is clear that the explicit formula of the differential δ in terms of the Euler-Iwasawa coordinates  attached to k0, coincides with that in terms of the Euler-Iwasawa coordinates attached to the  identity of SO3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Let Ad be the adjoint action of SL3(R) on its lie algebra defined by  Ad(g)X = d  dt  ����  t=0  g · exp (tX) · g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (122)  Then the adjoint action Ad acts on the universal enveloping algebra U(sl3(R)) by  Ad(g)(X1 · X2 · · · · · Xr) = Ad(g)X1 · Ad(g)X2 · · · · · Ad(g)Xr,  (123)  where X1, X2, · · · , Xr ∈ sl3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Since for u ∈ U(sl3(R)), the left derivatives LAd(k0)u and Lu  are intertwined by the left k0-translation, one sees that the explicit formula of LAd(k0)u in terms  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  27  of the Euler-Iwasawa coordinates attached to k0, coincides with the explicit formula of Lu in  terms of the Euler-Iwasawa coordinates attached to the identity of SO3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Now by (120), in terms of the Euler-Iwasawa coordinates attached to k0, we can derive  δi =  �  vi,j∈U(sl3(R))  are monomials  pi,jLAd(k0)vi,j,  (124)  where pi,j and vi,j are exactly the same as in (120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We complete the proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Take points k1, k2, · · · , kM ∈ SO3(R) such that {(kl · UEI)}M  l=1 is an open cover of SO3(R),  and a partition of unity {ρl}M  l=1 subordinate to {(kl · U EI)}M  l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By Claim, we have  δi =  M  �  l=1  �  v  kl  i,j∈U(sl3(R))  are monomials  �  ρl · pkl  i,j  �  Lv  kl  i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (125)  Noticing that the above functions ρl · pkl  i,j are bounded on X, we conclude Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We can write  D12 −  �  z2  12 + 1  �  ∂2  ∂z12∂z12  − (z23)  ∂2  ∂z13∂z12  − (−z13)  ∂2  ∂z23∂z12  − 2z12  ∂  ∂z12  =  \uf8eb  \uf8ed  �  wj∈so3(R)  qjLwj  \uf8f6  \uf8f8  ∂  ∂z12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (126)  Here qj are bounded functions on X which are independent of the variables z12, z23, z13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Lwj are  left derivatives on X induced by the elements in so3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The proof is similar to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7, in each  coordinate chart ((k0 · UEI) × R3, Λk0),  D12 = (sec β sin γ)  ∂2  ∂α∂z12  + (cos γ)  ∂2  ∂β∂z12  + (tan β sin γ)  ∂2  ∂γ∂z12  +  �  z2  12 + 1  �  ∂2  ∂z12∂z12  + (z23)  ∂2  ∂z13∂z12  + (−z13)  ∂2  ∂z23∂z12  + 2z12  ∂  ∂z12  ,  (127)  where (α, β, γ, z12, z13, z23) are the Euler-Iwasawa coordinates attached to k0 ∈ SO3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11 and (111) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 , we can write on the open set U EI × R3 ⊂ X  D12 −  �  z2  12 + 1  �  ∂2  ∂z12∂z12  − (z23)  ∂2  ∂z13∂z12  − (−z13)  ∂2  ∂z23∂z12  − 2z12  ∂  ∂z12  =  \uf8eb  \uf8ed  �  wj∈so3(R)  qjLwj  \uf8f6  \uf8f8  ∂  ∂z12  ,  (128)  such that qj are bounded functions on U EI × R3 which are independent of the variables  z12, z23, z13, and Lwj are left derivatives on X induced by the elements in so3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  28  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For k0 ∈ SO3(R), we can write on the open set (k0 · UEI) × R3 ⊂ SL3(R)/A  D12 −  �  z2  12 + 1  �  ∂2  ∂z12∂z12  − (z23)  ∂2  ∂z13∂z12  − (−z13)  ∂2  ∂z23∂z12  − 2z12  ∂  ∂z12  =  \uf8eb  \uf8ec  \uf8ed  �  wk0  j ∈so3(R)  qk0  j Lwk0  j  \uf8f6  \uf8f7  \uf8f8  ∂  ∂z12  ,  (129)  such that qk0  j  are bounded functions on (k0 · U EI) × R3, which are independent of the variables  z12, z23, z13, and Lwk0  j  are left derivatives on X induced by the elements in so3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Take the Euler-Iwasawa coordinates attached to k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' One verifies (see Remark  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='8) that the explicit formula of the left hand side of (129) in terms of the Euler-Iwasawa  coordinates attached to k0, coincides with that in terms of the Euler-Iwasawa coordinates  attached to the identity of SO3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Now by (128), in terms of the Euler-Iwasawa coordinates attached to k0,  D12 −  �  z2  12 + 1  �  ∂2  ∂z12∂z12  − (z23)  ∂2  ∂z13∂z12  − (−z13)  ∂2  ∂z23∂z12  − 2z12  ∂  ∂z12  =  \uf8eb  \uf8ed  �  wj∈so3(R)  qjLAd(k0)wj  \uf8f6  \uf8f8  ∂  ∂z12  ,  (130)  where qj and wj are exactly the same as in (128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Noticing that Ad(k0)wj ∈ so3(R) for all  k0 ∈ SO3(R) and wj ∈ so3(R), we complete the proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Take the partition of unity {ρl}M  l=1 subordinate to {(kl · UEI)}M  l=1 as in the proof of Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We have by Claim  D12 −  �  z2  12 + 1  �  ∂2  ∂z12∂z12  − (z23)  ∂2  ∂z13∂z12  − (−z13)  ∂2  ∂z23∂z12  − 2z12  ∂  ∂z12  =  M  �  l=1  \uf8eb  \uf8ec  \uf8ed  �  w  kl  j ∈so3(R)  �  ρl · qkl  j  �  Lw  kl  j  \uf8f6  \uf8f7  \uf8f8  ∂  ∂z12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (131)  Noticing that the functions ρl · qkl  j  are bounded on X and are independent of the variables  z12, z23, z13, we conclude Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Proof of the density and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In the following, we shall establish the density  of C∞  c (SL3(R)/A) in Dom(D12) for the invariant differential operator D12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Firstly, we construct a sequence of cutoff functions {χn}∞  n=1 on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Take ξ(r) to be a smooth  function on [0, +∞) such that  (1) 0 ≤ ξ(r) ≤ 1 for r ∈ [0, +∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (2) ξ(r) ≡ 1 for 0 ≤ r ≤ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (3) ξ(r) ≡ 0 for r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  29  For n = 1, 2, · · ·, define  χn(x) := χn (kx, z12, z13, z23) := ξ  �|z13|2 + |z23|2  n  �  ξ  �ln (|z12|2 + 1)  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (132)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The following properties hold for the cutoff functions {χn}∞  n=1 defined by (132).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (a) {χn}∞  n=1 ⊂ C∞  c (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (b) 0 ≤ χn(x) ≤ 1, and limn→∞ χn(x) = 1, for each x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (c) For non-negative integers i1, i2, i3 such that i1 + i2 + i3 ≥ 1,  lim  n→∞  ∂i1+i2+i3  ∂zi1  12∂zi2  13∂zi3  23  χn(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (133)  (d) For n = 1, 2, · · ·, we have  ∂χn  ∂α = ∂χn  ∂β = ∂χn  ∂γ ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (134)  (e) For n = 1, 2, · · ·, we have  �  z23  ∂  ∂z13  − z13  ∂  ∂z23  �  χn ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (135)  (f) There exists M > 0 such that for all n = 1, 2, · · ·,  sup  x∈X  ����(1 + |z12|)(1 + |z13| + |z23|) · ∂χn  ∂z12  (x)  ���� + sup  x∈X  ����  �  |z12|2 + 1  � ∂2χn  ∂z12∂z12  (x)  ���� ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (136)  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It is clear that (a), (b), (c), (d) follow from the definition in (132).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (e) follows from the computation that  z23  ∂χn  ∂z13  − z13  ∂χn  ∂z23  = ξ  �ln (|z12|2 + 1)  n  �  ξ′  �|z13|2 + |z23|2  n  � �z23z13  n  − z13z23  n  �  ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (137)  Computation yields  ∂χn  ∂z12  (kx, z12, z13, z23) = ξ  �|z13|2 + |z23|2  n  �  ξ′  �ln (|z12|2 + 1)  n  � 2z12  |z12|2 + 1 · 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (138)  Then,  sup  x∈X  ����(1 + |z12|)(1 + |z13| + |z23|) · ∂χn  ∂z12  (x)  ����  = sup  x∈X  �����ξ  �|z13|2 + |z23|2  n  �  ξ′  �ln (|z12|2 + 1)  n  ����� · 2|z12|(1 + |z12|)  |z12|2 + 1  1 + |z13| + |z23|  n  �  ≤ 4  sup  r∈[0,+∞)  |ξ′(r)| · sup  x∈X  �����ξ  �|z13|2 + |z23|2  n  ����� · 1 + |z13| + |z23|  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (139)  Notice that when 1 + |z13| + |z23| ≥ 4n,  |z13|2 + |z23|2  n  ≥ (|z13| + |z23|)2  2n  ≥ (4n − 1)2  2n  > 4,  (140)  30  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  and hence ξ  �  |z13|2+|z23|2  n  �  = 0 by the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' As a conclusion, we derive  sup  x∈X  ����(1 + |z12|)(1 + |z13| + |z23|) · ∂χn  ∂z12  (x)  ���� ≤ 16  sup  r∈[0,+∞)  |ξ′(r)| ,  (141)  which is a universal constant independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Similarly, from the computation  ∂2χn  ∂z12∂z12  (kx, z12, z13, z23) = ξ  �|z13|2 + |z23|2  n  � �  ξ′′  �ln (|z12|2 + 1)  n  � 4z2  12  (|z12|2 + 1)2 · 1  n2 + ξ′  �ln (|z12|2 + 1)  n  �  2(1 − z2  12)  (|z12|2 + 1)2 · 1  n  �  ,  (142)  we can conclude  sup  x∈X  ����  �  |z12|2 + 1  �  ∂2χn  ∂z2  12  (x)  ���� ≤ 4  sup  r∈[0,+∞)  (|ξ′(r)| + |ξ′′(r)|) ,  (143)  which is a universal constant independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We thus establish (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  As a conclusion, we complete the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Let {ψm}∞  m=1 be a smooth approximate identity at the identity element Id of SL3(R) satis-  fying the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (1) ψm are smooth non-negative functions on SL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (2) ψm, are compactly supported in the open set Um ⊂ SL3(R), where ∩∞  m=1Um = {Id}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (3)  �  SL3(R) ψm(g) dg = 1, where dg is a fixed Haar measure on SL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  For f ∈ L2(X), define convolutions ψm ∗ f by  (ψm ∗ f)(x) :=  �  SL3(R)  ψm(g) · f  �  g−1x  �  dg,  m = 1, 2, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (144)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The following holds for each f ∈ L2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (a) ψm ∗ f are smooth functions on X for m = 1, 2, · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (b) ψm ∗ f ∈ L2(X) for m = 1, 2, · · ·, and ψm ∗ f → f in L2(X) as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (c) For each u ∈ U (sl3(R)), Lu(ψm ∗ f) ∈ L2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Assume that ∆ is an SL3(R)-invariant differential operator on X, and that f, ∆f ∈ L2(X),  then  (d) ∆(ψm ∗ f) ∈ L2(X) for m = 1, 2, · · ·, and ∆(ψm ∗ f) → ∆f in L2(X) as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The arguments are standard and are given here for the sake of  completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For each u ∈ U(sl3(R)), it holds by definition that Lu(ψm ∗ f) = Lu(ψm) ∗ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  This implies (a) since {Lu}u∈U(sl3(R)) provide all partial derivatives with respect to any local  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  31  coordinates at each point of the manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By the Cauchy-Schwarz inequality,  �  X  |(ψm ∗ f)(x)|2dµ =  �  X  ����  �  SL3(R)  ψm(g) · f  �  g−1x  �  dg  ����  2  dµ  ≤  �  X  \uf8eb  \uf8ec  \uf8ed  �  SL3(R)  ψm (�g) d�g  \uf8f6  \uf8f7  \uf8f8  \uf8eb  \uf8ec  \uf8ed  �  SL3(R)  ψm(g) ·  ��f  �  g−1x  ���2 dg  \uf8f6  \uf8f7  \uf8f8 dµ =  �  X  |f (x)|2 dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (145)  For each ǫ > 0, we can find continuous function fǫ with compact support such that  �  X  |fǫ (x) − f(x)|2 dµ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (146)  By the Cauchy-Schwarz inequality again,  �  X  |(ψm ∗ fǫ)(x) − fǫ(x)|2dµ =  �  X  ����  �  SL3(R)  ψm(g)  �  fǫ  �  g−1x  �  − fǫ(x)  �  dg  ����  2  dµ  ≤  �  X  ��  SL3(R)  ψm (�g) d�g  � ��  SL3(R)  ψm(g) ·  ��fǫ  �  g−1x  �  − fǫ(x)  ��2 dg  �  dµ  =  �  SL3(R)  ψm(g)  ��  X  ��fǫ  �  g−1x  �  − fǫ(x)  ��2 dµ  �  dg ≤ sup  g∈Um  �  X  ��fǫ  �  g−1x  �  − fǫ(x)  ��2 dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (147)  Since fǫ is uniform continuous with compact support, we conclude  lim  m→∞  �  X  |(ψm ∗ fǫ)(x) − fǫ(x)|2dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (148)  Similarly to (145), we can derive by (146)  �  X  |(ψm ∗ f)(x) − f(x)|2dµ ≤  �  X  |(ψm ∗ fǫ)(x) − fǫ(x)|2dµ +  �  X  |f (x) − fǫ(x)|2 dµ  +  �  X  |(ψm ∗ (f − fǫ))(x)|2dµ ≤  �  X  |(ψm ∗ fǫ)(x) − fǫ(x)|2dµ + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (149)  Then, by(148) and (149), we have  lim  m→∞  �  X  |(ψm ∗ f)(x) − f(x)|2dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (150)  (b) is now proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Thanks to the fact that Lu(ψm ∗ f) = (Luψm) ∗ f, the same argument as in  (145) gives (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Next, assume f, ∆f ∈ L2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We first show that ∆(ψm ∗ f) = ψm ∗ (∆f) in the sense of  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Recall the notion that {Xij}1≤i,j≤3, (i,j)̸=(3,3)is a basis of sl3(R) (see (11)) and the  lift �w of w ∈ C∞ (SL3(R)/A) is defined by �w := w ◦ π (see (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For clarity, in the following,  we denote the points of X by xA and the points of SL3(R) by x, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  32  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  For each invariant differential ∆, there exists a polynomial P∆ by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 such that  (∆w)(xA) = P∆  �  · · , ∂  ∂tij  , · · ·  �  �w  \uf8eb  \uf8ec  \uf8ec  \uf8edx exp  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  1≤i,j≤3  (i,j)̸=(3,3)  tijXij  \uf8f6  \uf8f7  \uf8f7  \uf8f8  \uf8f6  \uf8f7  \uf8f7  \uf8f8  ��������  t11=t12=···=t32=0  =: P∆  � ∂  ∂tij  �  �w  �  x exp  ��  tijXij  ������  tij=0  ,  w ∈ C∞ (SL3(R)/A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (151)  Let ∆∗ be the formal adjoint of ∆ (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then ∆∗ is still an invariant differential  operator on X ([Ba]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For each h ∈ C∞  c (X),  (∆(ψm ∗ f), h) = (ψm ∗ f, ∆∗h)  =  �  X  ��  SL3(R)  ψm(g) · f  �  g−1xA  �  dg  �  P∆∗  � ∂  ∂tij  �  �h  �  x exp  ��  tijXij  ������  tij=0  dµ  =  �  SL3(R)  ψm(g)  ��  X  f (xA) P∆∗  � ∂  ∂tij  �  �h  �  gx exp  ��  tijXij  ������  tij=0  dµ  �  dg  =  �  X  f (xA) P∆∗  � ∂  ∂tij  � ��  SL3(R)  ψ∗m(g)�h  �  g−1x exp  ��  tijXij  ��  dg  �����  tij=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (152)  where ψ∗  m(g) := ψm(g−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Noticing that ψ∗  m ∗ �h = �  ψ∗  m ∗ h, we have  �  X  f (xA) P∆  � ∂  ∂tij  � �  �  ψ∗  m ∗ h  � �  x exp  ��  tijXij  �����  tij=0 = (f, ∆∗(ψ∗  m ∗ h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (153)  Substituting into (152), we have  (∆(ψm ∗ f), h) = (∆f, ψ∗  m ∗ h) =  �  X  (∆f)(xA) ·  ��  SL3(R)  ψm  �  g−1�  h (g−1xA) dg  �  dµ  =  �  X  ��  SL3(R)  ψm (�g) · (∆f)  �  �g−1�xA  �  d�g  �  h (�xA) dµ = (ψm ∗ (∆f), h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (154)  Similarly to (145), we have  |(∆(ψm ∗ f), h)| = |(ψm ∗ (∆f), h)| ≤  �  X  |(∆f)(x)|2 dµ ·  �  X  |h(x)|2 dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (155)  Now ψm ∗ ∆f lies in L2(X) thanks to (b), we also have ∆(ψm ∗ f) = ψm ∗ ∆f in L2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Using  (b) again, ∆(ψm ∗f) = ψm ∗∆f → ∆f in L2(X) as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' We complete the proof of Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Now, we proceed to prove  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' D12 is an SL3(R)-invariant, formally self-adjoint differential operator on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  The graph of D12 on the domain C∞  c (X) is dense in the graph of D12 on the domain Dom(D12),  with respect to the graph L2 × L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  33  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Firstly, we show that D12 is formally self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It suffices to  prove in each coordinate chart that D12 coincides with D∗  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' This is can be done by a direct  computation, thanks to the invariant measure given by (100) and the explicit formula for D12  given in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7 (or see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7 for a coordinate-free approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  For each f ∈ Dom(D12), consider functions χn · (ψm ∗ f), m, n = 1, 2, · · ·, where χn and  (ψm ∗ f) are defined by (132) and (144), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It is clear that χn · (ψm ∗ f) ∈ C∞  c (X) by  (a) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15 and (a) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Further, by (b) and (d) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='16, to prove  the density of C∞  c (X) in Dom(D12), it suffices to prove that for fixed m = 1, 2, · · ·,  lim  n→∞ ∥χn · (ψm ∗ f) − ψm ∗ f∥L2(X) = 0,  (156)  and  lim  n→∞ ∥D12 (χn · (ψm ∗ f)) − D12(ψm ∗ f)∥L2(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (157)  (156) follows easily from (b) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15, and Lebesgue’s dominated convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  In the following, we shall establish (157) for m = 1, 2, · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' computation yields that  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='D12 (χn · (ψm ∗ f)) = χn · D12 (ψm ∗ f) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='wj∈so3(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='qjLwj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂χn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8 · (ψm ∗ f)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='wj∈so3(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='qjLwjχn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8 · ∂ (ψm ∗ f)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ ∂χn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12 \uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='wj∈so3(R)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='qjLwj (ψm ∗ f)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂2χn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(ψm ∗ f) + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂χn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂ (ψm ∗ f)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� � ∂χn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(ψm ∗ f) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂χn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂χn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂ (ψm ∗ f)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ ∂χn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(ψm ∗ f) + 2z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂χn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(ψm ∗ f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (158)  Since Lwj, wj ∈ so3(R), are linear combinations of  ∂  ∂α,  ∂  ∂β,  ∂  ∂γ, by (d) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15 we have  \uf8eb  \uf8ed  �  wj∈so3(R)  qjLwj  � ∂χn  ∂z12  �\uf8f6  \uf8f8 · (ψm ∗ f) =  \uf8eb  \uf8ed  �  wj∈so3(R)  qjLwjχn  \uf8f6  \uf8f8 · ∂ (ψm ∗ f)  ∂z12  ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (159)  Similarly, by (e) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15,  �  z23  ∂  ∂z13  − z13  ∂  ∂z23  � � ∂χn  ∂z12  �  (ψm ∗ f) =  �  z23  ∂χn  ∂z13  − z13  ∂χn  ∂z23  �  ∂ (ψm ∗ f)  ∂z12  ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (160)  34  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Therefore,  D12 (χn (ψm ∗ f)) = χnD12 (ψm ∗ f) + ∂χn  ∂z12  \uf8eb  \uf8ed  �  wj∈so3(R)  qjLwj (ψm ∗ f)  \uf8f6  \uf8f8  +  �  z2  12 + 1  �  ∂2χn  ∂z12∂z12  (ψm ∗ f) + 2  �  z2  12 + 1  � ∂χn  ∂z12  ∂ (ψm ∗ f)  ∂z12  + ∂χn  ∂z12 �  z23  ∂  ∂z13  − z13  ∂  ∂z23  �  (ψm ∗ f) + 2z12  ∂χn  ∂z12  (ψm ∗ f)  =: An + Bn + Cn + Dn + En + Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (161)  In what follows, we shall prove that the functions Bn, Cn, Dn, En, Fn converge to 0 and An  converges to D12(ψm ∗ f) in L2(X), when n approaches ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Thanks to (f) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15 and  the fact that qj are bounded functions on X, we have the pointwise estimate  |Bn(x)| ≤  \uf8eb  \uf8edM  sup  wj∈so3(R)  x∈X  |qj(x)|  \uf8f6  \uf8f8  �  wj∈so3(R)  ��Lwj (ψm ∗ f) (x)  �� ,  (162)  where the right hand side is an L2 function by (c) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then by (c) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15,  we conclude by Lebesgue’s dominated convergence theorem that  lim  n→∞ ∥Bn∥L2(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (163)  Similarly, by (f) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15, we have pointwisely  |Cn(x)| + |Fn(x)| ≤ 2M · |(ψm ∗ f) (x)| ,  (164)  where the right hand side is an L2 function by (b) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then by (c) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15,  we conclude by Lebesgue’s dominated convergence theorem that  lim  n→∞ ∥Cn∥L2(X) + lim  n→∞ ∥Fn∥L2(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (165)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  35  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' we have  �  z2  12 + 1  � ∂χn  ∂z12  ∂ (ψm ∗ f)  ∂z12  = ∂χn  ∂z12 � ∂  ∂z12  + z23  ∂  ∂z13  �  (ψm ∗ f) − z23  ∂χn  ∂z12  ∂ (ψm ∗ f)  ∂z13  + z12  ∂χn  ∂z12 �  z12  ∂  ∂z12  − z23  ∂  ∂z23  �  (ψm ∗ f) + z12z23  ∂χn  ∂z12  ∂ (ψm ∗ f)  ∂z23  = ∂χn  ∂z12  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  v4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j∈U(sl3(R))  are monomials  p4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='jLv4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j (ψm ∗ f)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 − z23  ∂χn  ∂z12  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j∈U(sl3(R))  are monomials  p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='jLv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j (ψm ∗ f)  \uf8f6  \uf8f7  \uf8f7  \uf8f8  + z12  ∂χn  ∂z12  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j∈U(sl3(R))  are monomials  p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='jLv3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j (ψm ∗ f)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 + z12z23  ∂χn  ∂z12  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j∈U(sl3(R))  are monomials  p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='jLv2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='j (ψm ∗ f)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (166)  Thanks to (f) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15 and the fact that p1,j, p2,j, p3,j, p4,j are bounded functions on X,  |Dn(x)| ≤  \uf8eb  \uf8edM sup  1≤i≤4  x∈X  |pi,j(x)|  \uf8f6  \uf8f8  4  �  i=1  �  vi,j∈U(sl3(R))  are monomials  |Lvi,j (ψm ∗ f) (x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (167)  Then by (c) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15, we conclude by Lebesgue’s dominated convergence theorem that  lim  n→∞ ∥Dn∥L2(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (168)  Similarly, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13,  ∂χn  ∂z12 �  z23  ∂  ∂z13  − z13  ∂  ∂z23  �  (ψm ∗ f) = z23  ∂χn  ∂z12 \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  v1,j∈U(sl3(R))  are monomials  p4,jLv4,j (ψm ∗ f)  \uf8f6  \uf8f7  \uf8f7  \uf8f8  − z13  ∂χn  ∂z12 \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  v2,j∈U(sl3(R))  are monomials  p2,jLv2,j (ψm ∗ f)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (169)  By (c), (f) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15, and (c) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='16, we can conclude  lim  n→∞ ∥En∥L2(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (170)  Finally, we consider An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' It is clear that  |An(x) − (D12 (ψm ∗ f)) (x)| = | (χn − 1) · (D12 (ψm ∗ f)) (x)| ≤ | (D12 (ψm ∗ f)) (x)|,  (171)  where the right hand side is an L2 function by (d) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then by (b) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='15,  lim  n→∞ ∥An − D12 (ψm ∗ f) ∥L2(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (172)  36  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  As a conclusion, we complete the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='17, the graph of D12 on the domain C∞  c (X) is dense  in the graph of D12 on the domain Dom(D12), with respect to the graph L2 × L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then  D12 is essentially self-adjoint by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Notice that the conjugations of the normalizer of A in SL3(R) (or equivalently the permuta-  tions of the indices 1, 2, 3) are isometries of L2(X), and such isometries transform D12 to D13  and D23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then, it is clear that D13 and D23 are essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  We complete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='17, one can see that our cutoff functions are specifi-  cally designed to treat the operator D12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Firstly, the first order derivative of our cutoff functions  with respect to the variable z12 contributes a decay of the order (z12 log z12)−1, which is better  than the more standard z−1  12 decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By stretching the cutoff functions in an inhomogeneous  way, one can balance a linear growth of the other variables z13, z23 exactly by the remaining  factor (log z12)−1, so that certain terms in D12 can be dealt with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Secondly, some other terms  in D12 force us to arrange the cutoff functions so that they are totally annihilated by those  differentials such as z23  ∂  ∂z13 −z13  ∂  ∂z23, in order to kill the wild terms such as  ∂  ∂z12(ψm ∗f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Recall  that the singled out differential z23  ∂  ∂z13 − z13  ∂  ∂z23 has an intrinsic geometric explanation as the  rotation caused by an element in SO3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Because of these restrictions, the cutoff functions are  left with little wiggle room, and in particular when we perturb the cutoff functions for instance  by convolutions, they are no longer suitable for our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Therefore, to deal with the cases D123  and D213, one may want to decompose the operators in new geometric coordinates so that the  separation of terms is more transparent and suitable cutoff functions can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Computations in the Euler-Iwasawa Coordinates  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Explicit formulas for the generators of the left derivatives on SL3(R)/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  37  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In the coordinate chart ((k0 · U EI) × R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Λk0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' where k0 is the identity of SO3(R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− LX12 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α cos β sin 2γ − sin α tan β sin2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin β tan β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α sin β sin 2γ − cos α sin2 β cos2 γ − cos α cos2 β sin2 γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− sin α sec β sin2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α tan β sin 2γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4 cos α sin 2β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α cos β sin 2γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin 2β cos2 γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin 2β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (− sin α sin β sin γ + cos α cos 2β cos γ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='sin α cos β sin 2γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin 2β cos 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z13 + (− sin α sin β sin γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ cos α cos 2β cos γ) z23 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− sin α cos β cos 2γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin 2β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α cos β sin 2γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin 2β − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin 2β sin2 γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (sin α sin β cos γ + cos α cos 2β sin γ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (173)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' According to the definition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (−LX12f)(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z23) = d  dt  �  f  �  �α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �β(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �γ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z12(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z13(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z23(t)  ������  t=0  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (174)  where  �  �α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �β(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �γ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z12(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z13(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z23(t)  �  are the Euler-Iwasawa coordinates of  \uf8eb  \uf8ed  1  t  0  0  1  0  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  cos β cos γ  − sin β  cos β sin γ  sin α sin γ + cos α cos γ sin β  cos α cos β  cos α sin β sin γ − cos γ sin α  cos γ sin α sin β − cos α sin γ  cos β sin α  cos α cos γ + sin α sin β sin γ  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8  =:  \uf8eb  \uf8ed  1  t  0  0  1  0  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (175)  Applying the Gram-Schmidt process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' we can obtain that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' up to order 1 in t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  \uf8eb  \uf8ed  1  t  0  0  1  0  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  �k11  �k12  �k13  �k21  �k22  �k23  �k31  �k32  �k33  \uf8f6  \uf8f7  \uf8f8  \uf8eb  \uf8ed  1  �z12  �z13  0  1  �z23  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  ∆1  0  0  0  ∆2  0  0  0  ∆3  \uf8f6  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' (176)  38  HANLONG FANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' XIAOCHENG LI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' AND YUNFENG ZHANG  where  ∆1 = 1 + t (k11k21) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' ∆2 = 1 + t (k12k22) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' ∆3 = 1 + t (k13k23) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k11 = k11 + t  �  k21 − k2  11k21  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k21 = k21 + t  �  −k11k2  21  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k31 = k31 + t (−k11k21k31) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k12 = k12 + t  �  k2  13k22 − k11k12k21  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k22 = k22 + t  �  −k12k2  21 + k13k23k22  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k32 = k32 + t (k12k23k33 − k11k22k31) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k13 = k13 + t  �  k2  13k23  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k23 = k23 + t  �  −k13 + k13k2  23  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k33 = k33 + t (k13k23k33) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �z12 = (1 + t (k11k21 − k12k22)) z12 + t (k11k22 + k12k21) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �z13 = (1 + t (k11k21 − k13k23)) z13 + t (k11k22 + k12k21) z23 + t (k11k23 + k13k21) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �z23 = (1 + t (k12k22 − k13k23)) z23 + t (k12k23 + k13k22) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (177)  Then for smooth functions f on X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' we have by (98) that  (−LX12f)(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z23) = ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='d�α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='dt (0) + ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='d�β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='dt (0) + ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='d�γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='dt (0) + ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='d�z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='dt (0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='d�z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='dt (0) + ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='d�z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='dt (0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�(k21k32 − k22k31)(k12k21 + k11k22)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='22 + k2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂α +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k11k12k21 − k2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13k22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 − k2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�−k13k21 − k11k12k13k22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11 + k2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ ((k11k21 − k12k22) z12 + (k11k22 + k12k21)) ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ ((k11k21 − k13k23) z13 + (k11k22 + k12k21) z23 + (k11k23 + k13k21)) ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ ((k12k22 − k13k23) z23 + (k12k23 + k13k22)) ∂f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (178)  Substituting (97) into the above formula, we can derive (173).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  39  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In the coordinate chart ((k0 · U EI) × R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Λk0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' where k0 is the identity of SO3(R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− LX13 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α cos β sin 2γ + cos α tan β sin2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α sin β tan β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α sin β sin 2γ − sin α sin2 β cos2 γ − sin α cos2 β sin2 γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4 sin α sin 2β sin 2γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α tan β sin 2γ + cos α sec β sin2 γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α sin 2β cos2 γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α sin 2β − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α cos β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (sin α cos 2β cos γ + cos α sin β sin γ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α sin 2β cos 2γ − cos α cos β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z13 + (sin α cos 2β cos γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ cos α sin β sin γ) z23 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α sin 2β sin 2γ + cos α cos β cos 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α sin 2β − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin α sin 2β sin2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos α cos β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ (sin α cos 2β sin γ − cos α sin β cos γ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (179)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Similarly, consider the Iwasawa decomposition  \uf8eb  \uf8ed  1  0  t  0  1  0  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  �k11  �k12  �k13  �k21  �k22  �k23  �k31  �k32  �k33  \uf8f6  \uf8f7  \uf8f8  \uf8eb  \uf8ed  1  �z12  �z13  0  1  �z23  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  ∆1  0  0  0  ∆2  0  0  0  ∆3  \uf8f6  \uf8f8 ,  (180)  where (kij) ∈ SO3(R) is parametrized by (97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then up to order 1 in t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' we have  ∆1 = 1 + t (k11k31) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' ∆2 = 1 + t (k12k32) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' ∆3 = 1 + t (k13k33) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k11 = k11 + t  �  k31 − k2  11k31  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k21 = k21 + t (−k11k31k21) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k31 = k31 + t  �  −k11k2  31  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k12 = k12 + t  �  k32k2  13 − k11k12k31  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k22 = k22 + t (−k12k31k21 + k13k32k23) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k32 = k32 + t  �  −k12k2  31 + k13k33k32  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k13 = k13 + t  �  k2  13k33  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k23 = k23 + t (k13k23k33) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k33 = k33 + t  �  −k13 + k13k2  33  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �z12 = (1 + t (k11k31 − k12k32)) z12 + t (k11k32 + k12k31) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �z13 = (1 + t (k11k31 − k13k33)) z13 + t (k11k32 + k12k31) z23 + t (k11k33 + k13k31) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �z23 = (1 + t (k12k32 − k13k33)) z23 + t (k12k33 + k13k32) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (181)  40  HANLONG FANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' XIAOCHENG LI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' AND YUNFENG ZHANG  Computation yields that  (−LX13f)(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z23) = d  dt  �  f  �  �α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �β(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �γ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z12(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z13(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z23(t)  ������  t=0  =  �(k21k32 − k22k31)(k12k31 + k11k32)  k2  22 + k2  32  � ∂f  ∂α +  �  k11k12k31 − k2  13k32  �  1 − k2  12  �  ∂f  ∂β  +  �−k13k31 − k11k12k13k32  k2  11 + k2  13  � ∂f  ∂γ + ((k11k31 − k12k32) z12 + (k11k32 + k12k31)) ∂f  ∂z12  + ((k11k31 − k13k33) z13 + (k11k32 + k12k31) z23 + (k11k33 + k13k31)) ∂f  ∂z13  + ((k12k32 − k13k33) z23 + (k12k33 + k13k32)) ∂f  ∂z23  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (182)  Substituting (97) into the above formula, we can derive (179).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In the coordinate chart ((k0 · U EI) × R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Λk0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' where k0 is the identity of SO3(R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− LX23 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='cos 2α cos2 γ − cos2 α + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4 sin 2α sin 2β cos 2γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos 2α cos β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂β +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4 sin 2α cos2 β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='� ∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin2 β cos2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α cos2 β − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos 2α sin β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin 2β cos γ − cos 2α cos β sin γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α cos 2γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin2 β cos 2γ − cos 2α sin β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin 2β cos γ − cos 2α cos β sin γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin 2γ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin2 β sin 2γ + cos 2α sin β cos 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α cos2 β + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α cos2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin2 β sin2 γ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 cos 2α sin β sin 2γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 sin 2α sin 2β sin γ + cos 2α cos β cos γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='∂z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (183)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For (kij) ∈ SO3(R), consider the Iwasawa decomposition  \uf8eb  \uf8ed  1  0  0  0  1  t  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  �k11  �k12  �k13  �k21  �k22  �k23  �k31  �k32  �k33  \uf8f6  \uf8f7  \uf8f8  \uf8eb  \uf8ed  1  �z12  �z13  0  1  �z23  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  ∆1  0  0  0  ∆2  0  0  0  ∆3  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (184)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  41  Up to order 1 in t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  ∆1 = 1 + t (k21k31) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' ∆2 = 1 + t (k22k32) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' ∆3 = 1 + t (k23k33) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k11 = k11 + t (−k21k31k11) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k21 = k21 + t  �  k31 − k2  21k31  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k31 = k31 + t  �  −k21k2  31  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k12 = k12 + t (k22k33k13 − k21k32k11) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k22 = k22 + t  �  k32k2  23 − k21k22k31  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k32 = k32 + t  �  −k22k2  31 + k23k33k32  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k13 = k13 + t (k23k33k13) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k23 = k23 + t  �  k2  23k33  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �k33 = k33 + t  �  −k23 + k23k2  33  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �z12 = (1 + t (k21k31 − k22k32)) z12 + t (k21k32 + k22k31) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �z13 = (1 + t (k21k31 − k23k33)) z13 + t (k21k32 + k22k31) z23 + t (k21k33 + k23k31) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �z23 = (1 + t (k22k32 − k23k33)) z23 + t (k22k33 + k23k32) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (185)  Then by (98),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (−LX23f)(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' z23) = d  dt  �  f  �  �α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �β(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �γ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z12(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z13(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' �z23(t)  ������  t=0  =  �−k2  22k2  31 − k2  22k2  32 − k2  23k2  32  k2  22 + k2  32  � ∂f  ∂α +  �  k21k32k11 − k22k33k13  �  1 − k2  12  �  ∂f  ∂β  +  �  −k11k13k22k32  k2  11 + k2  13  � ∂f  ∂γ + ((k21k31 − k22k32) z12 + (k21k32 + k22k31)) ∂f  ∂z12  + ((k21k31 − k23k33) z13 + (k21k32 + k22k31) z23 + (k21k33 + k23k31)) ∂f  ∂z13  + ((k22k32 − k23k33) z23 + (k22k33 + k23k32)) ∂f  ∂z23  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (186)  Substituting (97) into the above formula, we can derive (183).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In the coordinate chart ((k0 · U EI) × R3, Λk0), where k0 is the identity of SO3(R),  LX12 − LX21 = (sin α tan β) ∂  ∂α + (cos α) ∂  ∂β + (sin α sec β) ∂  ∂γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (187)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For (kij) ∈ SO3(R), we have up to order 1 in t  \uf8eb  \uf8ec  \uf8ed  �k11  �k12  �k13  �k21  �k22  �k23  �k31  �k32  �k33  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ed  1  −t  0  t  1  0  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8 =  \uf8eb  \uf8ed  k11 − tk21  k12 − tk22  k13 − tk23  k21 + tk11  k22 + tk12  k23 + tk13  k31  k32  k33  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' (188)  Then by (98),  (LX12 − LX21)f =  � −k12k32  k2  22 + k2  32  � ∂f  ∂α +  �  k22  �  1 − k2  12  �  ∂f  ∂β +  �k13k21 − k11k23  k2  11 + k2  13  � ∂f  ∂γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (189)  Substituting (97) into the above formula, we can derive (187).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  42  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In the coordinate chart ((k0 · U EI) × R3, Λk0), where k0 is the identity of SO3(R),  LX13 − LX31 = − (cos α tan β) ∂  ∂α + (sin α) ∂  ∂β − (cos α sec β) ∂  ∂γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (190)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For (kij) ∈ SO3(R), we have up to order 1 in t  \uf8eb  \uf8ec  \uf8ed  �k11  �k12  �k13  �k21  �k22  �k23  �k31  �k32  �k33  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ed  1  0  −t  0  1  0  t  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8 =  \uf8eb  \uf8ed  k11 − tk31  k12 − tk32  k13 − tk33  k21  k22  k23  k31 + tk11  k32 + tk12  k33 + tk13  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' (191)  Then by (98),  (LX13 − LX31)f =  � k12k22  k2  22 + k2  32  � ∂f  ∂α +  �  k32  �  1 − k2  12  �  ∂f  ∂β +  �k13k31 − k11k33  k2  11 + k2  13  � ∂f  ∂γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (192)  Substituting (97) into the above formula, we can derive (190).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In the coordinate chart ((k0 · U EI) × R3, Λk0), where k0 is the identity of SO3(R),  LX23 − LX32 = ∂  ∂α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (193)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For (kij) ∈ SO3(R), we have up to order 1 in t  \uf8eb  \uf8ec  \uf8ed  �k11  �k12  �k13  �k21  �k22  �k23  �k31  �k32  �k33  \uf8f6  \uf8f7  \uf8f8 =  \uf8eb  \uf8ed  1  0  0  0  1  −t  0  t  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8 =  \uf8eb  \uf8ed  k11  k12  k13  k21 − tk31  k22 − tk32  k23 − tk33  k31 + tk21  k32 + tk22  k33 + tk23  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' (194)  We derive (193) by (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Explicit formulas for the generators of the left-invariant differentials on SL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  In this subsection, we compute the left-invariant differential operators R (Eij), 1 ≤ i ̸= j ≤ 3  on SL3(R), where Eij is the 3 × 3 matrix unit with a 1 in the ith row and jth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Recall the Euler-Iwasawa coordinates (α, β, γ, z12, z13, z23, λ1, λ2) of SL3(R) attached to k0  (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Then,  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In each coordinate chart  �  (k0 · U EI) × R3 × R2, �Λk0  �  ,  R (E12) (α, β, γ, z12, z13, z23, λ1, λ2) = λ1  λ2  ∂  ∂z12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (195)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For matrix-valued functions g := �  1≤i,j≤3 gij(t)Eij, write g = O(2) if  dgij  dt (t)  ����  t=0  = 0, 1 ≤ i, j ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (196)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  43  Computation yields that for each k0 ∈ SO3(R),  k0 ·  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  λ1  0  0  0  λ2  0  0  0  λ−1  1 λ−1  2  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  t  0  0  1  0  0  0  1  \uf8f6  \uf8f8  = k0 ·  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12 + λ1  λ2t  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  λ1  0  0  0  λ2  0  0  0  λ−1  1 λ−1  2  \uf8f6  \uf8f8 + O(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (197)  For smooth functions f on SL3(R), we can derive that  (R (E12) f) (α, β, γ, z12, z13, z23, λ1, λ2)  = d  dt  ���  f  �  α, β, γ, z12 + λ1  λ2  t, z13, z23, λ1, λ2  ������  t=0  = λ1  λ2  ∂  ∂z12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (198)  We complete the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In each coordinate chart  �  (k0 · U EI) × R3 × R2, �Λk0  �  ,  R (E13) (α, β, γ, z12, z13, z23, λ1, λ2) = λ2  1λ2  ∂  ∂z13  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (199)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The proof is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For each k0 ∈ SO3(R), computation yields  k0 ·  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  λ1  0  0  0  λ2  0  0  0  λ−1  1 λ−1  2  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  0  t  0  1  0  0  0  1  \uf8f6  \uf8f8  = k0 ·  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13 + λ2  1λ2t  0  1  z23  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  λ1  0  0  0  λ2  0  0  0  λ−1  1 λ−1  2  \uf8f6  \uf8f8 + O(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (200)  Then,  R (E13) f = d  dt  �  f  �  α, β, γ, z12, z13 + λ2  1λ2t, z23, λ1, λ2  ������  t=0  = λ2  1λ2  ∂f  ∂z13  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (201)  We complete the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In each coordinate chart  �  (k0 · U EI) × R3 × R2, �Λk0  �  ,  R (E23) (α, β, γ, z12, z13, z23, λ1, λ2) = λ1λ2  2  � ∂  ∂z23  + z12  ∂  ∂z13  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (202)  44  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The proof is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For each k0 ∈ SO3(R), computation yields  k0 ·  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13  0  1  z23  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  λ1  0  0  0  λ2  0  0  0  λ−1  1 λ−1  2  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  0  0  0  1  t  0  0  1  \uf8f6  \uf8f8  = k0 ·  \uf8eb  \uf8ed  k11  k12  k13  k21  k22  k23  k31  k32  k33  \uf8f6  \uf8f8  \uf8eb  \uf8ed  1  z12  z13 + λ1λ2  2z12t  0  1  z23 + λ1λ2  2t  0  0  1  \uf8f6  \uf8f8  \uf8eb  \uf8ed  λ1  0  0  0  λ2  0  0  0  λ−1  1 λ−1  2  \uf8f6  \uf8f8 + O(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (203)  Then,  R (E13) f = d  dt  �  f  �  α, β, γ, z12, z13 + λ1λ2  2z12t, z23 + λ1λ2  2t, λ1, λ2  ������  t=0  = λ1λ2  2  � ∂  ∂z23  + z12  ∂  ∂z13  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (204)  We complete the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In each coordinate chart  �  (k0 · U EI) × R3 × R2, �Λk0  �  ,  R (E21) (α, β, γ, z12, z13, z23, λ1, λ2) = λ2  λ1  �  sec β sin γ ∂  ∂α + cos γ ∂  ∂β + tan β sin γ ∂  ∂γ  +  �  z2  12 + 1  �  ∂  ∂z12  + z23  ∂  ∂z13  − z13  ∂  ∂z23  �  + λ2z12  ∂  ∂λ1  − λ2  2  λ1  z12  ∂  ∂λ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (205)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' The proof is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' For each k0 ∈ SO3(R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' computation yields  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k0 · K · Z · Λ(t) :=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8 = O(2)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='+ k0 ·  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�k11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�k12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z13 + λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1 z23t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23 − λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1 z13t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ec  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1 + λ2z12t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ2 − λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1 z12t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (206)  where  �k11 = k11 + λ1  λ2  k12t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k21 = k21 + λ1  λ2  k22t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k31 = k31 + λ1  λ2  k32t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k12 = k12 − λ1  λ2  k11t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k22 = k22 − λ1  λ2  k21t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  �k32 = k32 − λ1  λ2  k31t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (207)  Denote by  �  �α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t), �λ1(t), �λ2(t)  �  the corresponding Euler-Iwasawa  coordinates of k0 · K · Z · Λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By (98) we have up to order 1 in t,  �α = arctan  �  k32 − λ1  λ2 k31t  k22 − λ1  λ2 k31t  �  , �β = arctan  \uf8eb  \uf8ec  \uf8ec  \uf8ed  −  �  k12 − λ1  λ2 k11t  �  �  1 −  �  k12 − λ1  λ2 k11t  �2  \uf8f6  \uf8f7  \uf8f7  \uf8f8 , �γ = arctan  �  k13  k11 + λ1  λ2 k12t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (208)  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  45  We can thus complete the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='10 by direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In each coordinate chart  �  (k0 · U EI) × R3 × R2, �Λk0  �  ,  R (E31) (α, β, γ, z12, z13, z23, λ1, λ2) = λ−2  1 λ−1  2  �  ((sec β sin γ) z23 − (sec β cos γ) z12) ∂  ∂α  + ((cos γ) z23 + (sin γ) z12) ∂  ∂β + ((tan β sin γ) z23 − (tan β cos γ) z12 − 1) ∂  ∂γ  +  �  z23 + z2  12z23  �  ∂  ∂z12  +  �  1 + z2  13 + z2  23 − z12z13z23  �  ∂  ∂z13  +  �  −z12 − z12z2  23  �  ∂  ∂z23  �  + λ−1  1 λ−1  2 z13  ∂  ∂λ1  − λ−2  1 z12z23  ∂  ∂λ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (209)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Computation yields  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='K · Z · Λ(t) :=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='k23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k11 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(k12z23 + k13) t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k12 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(k11z23 + k13z12) t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k13 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(k12z12 − k11) t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k21 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(k22z23 + k23) t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k22 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(k21z23 + k23z12) t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k23 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(k22z12 − k21) t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k31 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(k32z23 + k33) t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k32 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(k31z23 + k33z12) t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k33 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='(k32z12 − k31) t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8 \uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z12 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23 + z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z13 + λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 + z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='13 + z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23 − z12z13z23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z23 − λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='z12 + z12z2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8 \uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ1 + λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 z13t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ2 − λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 z12z23t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='− λ−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='1 λ−2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='2 (z13 − z12z23)t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (210)  Denote by  �  �α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t), �λ1(t), �λ2(t)  �  the corresponding Euler-Iwasawa  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By (98) we have up to order 1 in t,  �α(t) = arctan  �k32 + λ−2  1 λ−1  2 (k31z23 + k33z12) t  k22 + λ−2  1 λ−1  2 (k21z23 + k23z12) t  �  ,  �β(t) = arctan  \uf8eb  \uf8ed  −  �  k12 + λ−2  1 λ−1  2 (k11z23 + k13z12) t  �  �  1 −  �  k12 + λ−2  1 λ−1  2 (k11z23 + k13z12) t  �2  \uf8f6  \uf8f8 ,  �γ(t) = arctan  �k13 + λ−2  1 λ−1  2 (k12z12 − k11) t  k11 + λ−2  1 λ−1  2 (k12z23 + k13) t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (211)  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='11 follows from a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  46  HANLONG FANG, XIAOCHENG LI, AND YUNFENG ZHANG  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' In each coordinate chart  �  (k0 · U EI) × R3 × R2, �Λk0  �  ,  R (E32) (α, β, γ, z12, z13, z23, λ1, λ2) = λ−1  1 λ−2  2  �  sec β cos γ ∂  ∂α − sin γ ∂  ∂β + tan β cos γ ∂  ∂γ  + (z13 − z12z23)  ∂  ∂z12  + z13z23  ∂  ∂z13  +  �  z2  23 + 1  �  ∂  ∂z23  �  + λ−1  1 λ−1  2 z23  ∂  ∂λ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (212)  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Computation yields  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='K · Z · Λ(t) :=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8ed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='k12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='1 λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='2 z23t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='  (213)  Denote by  �  �α(t), �β(t), �γ(t), �z12(t), �z13(t), �z23(t), �λ1(t), �λ2(t)  �  the corresponding Euler-Iwasawa  coordinates of K · Z · Λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' By (98) we have up to order 1 in t,  �α = arctan  �k32 + λ−1  1 λ−2  2 k33t  k22 + λ−1  1 λ−2  2 k23t  �  , �β = arctan  \uf8eb  \uf8ed  −  �  k12 + λ−1  1 λ−2  2 k13t  �  �  1 −  �  k12 + λ−1  1 λ−2  2 k13t  �2  \uf8f6  \uf8f8 ,  �γ = arctan  �k13 − λ−1  1 λ−2  2 k12t  k11  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (214)  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='12 follows from a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  References  [AV] Andreotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', and Vesentini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', Carleman estimates for the Laplace-Beltrami equation on complex man-  ifolds, Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content=', Invariant differential operators on a semisimple symmetric space and finite multiplic-  ities in a Plancherel formula, Ark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content=' 2, 175–187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [BH] Bopp, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', and Harinck, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', Formule de Plancherel pour GL(n, C)/U(p, q), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='  [BIK] Benoist, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', Inoue, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', and Kobayashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', Temperedness criterion of the tensor product of parabolic  induction for GLn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Algebra 617 (2023), 1-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [BK1] Benoist, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', and Kobayashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', Tempered reductive homogeneous spaces, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' 17 (2015),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' 12, 3015-3036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [BK2] Benoist, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', and Kobayashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', Tempered homogeneous spaces II, in: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Fisher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Kleinbock, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Soifer  (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' ), Dynamics, Geometry, Number Theory: The Impact of Margulis on Modern Mathematics, The  University of Chicago Press, 2022, 213-245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  [BK3] Benoist, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', and Kobayashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', Tempered homogeneous spaces III, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content=' 3,  833-869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  HARMONIC ANALYSIS ON THE SPACE OF ORDERED TRIANGLE  47  [BK4] Benoist, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', and Kobayashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', Tempered homogeneous spaces IV, in: Journal of the Institute of Math-  ematics of Jussieu, Cambridge University Press, 7 June, 2022, 1-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content=', and Schlichtkrull, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=', The Plancherel decomposition for a reductive symmetric space I:  Spherical functions, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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+page_content='cn)  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content=' Zhang, School of Mathematical Sciences, Peking University, Beijing, Beijing, 100871, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
+page_content='  (yunfengzhang108@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfpvgE/content/2301.00529v1.pdf'}
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