# Harmonic analysis on the fourfold cover of the space of ordered triangles I:
the invariant differentials

Hanlong Fang, Xiaocheng Li, and Yunfeng Zhang

###### Abstract.

Denote by $SL_{n}(\mathbb{R})$ the group of $n\times n$ real matrices with
determinant one, $A$ the subgroup consisting of the diagonal matrices with
positive entries, and $SL_{n}(\mathbb{R})/A$ the manifold of left cosets $gA$,
$g\in SL_{n}(\mathbb{R})$. In this paper, we will be concerned with the
harmonic analysis on the homogeneous space $SL_{n}(\mathbb{R})/A$ when $n=3$.
In particular, we provide explicit generators and their relations for the
algebra of the invariant differential operators. Then we prove that some of
the non-central generators are essentially self-adjoint.

###### Contents

  1. 1 Introduction
  2. 2 Structure of the Algebra of the Invariant Differentials
    1. 2.1 Generators of the invariant differentials on $SL_{n}(\mathbb{R})/A$
    2. 2.2 Relations among the generators
    3. 2.3 The center of the algebra of the invariant differentials
  3. 3 Essential Self-Adjointness
    1. 3.1 Essential self-adjointness of the central elements
    2. 3.2 Reduction to the density of $C^{\infty}_{c}(X)$ in $\text{Dom}(\Delta)$
    3. 3.3 Coordinate charts induced by the Euler angles
    4. 3.4 Presence of left derivatives in $D_{12}$
    5. 3.5 Proof of the density and Theorem 1.3
  4. A Computations in the Euler-Iwasawa Coordinates
    1. A.1 Explicit formulas for the generators of the left derivatives on $SL_{3}(\mathbb{R})/A$
    2. A.2 Explicit formulas for the generators of the left-invariant differentials on $SL_{3}(\mathbb{R})$

##  1\. Introduction

Denote by $SL_{n}(\mathbb{R})$ the group of $n\times n$ real matrices with
determinant one, $A$ the subgroup consisting of the diagonal matrices with
positive entries, and $SL_{n}(\mathbb{R})/A$ the manifold of left cosets $gA$,
$g\in SL_{n}(\mathbb{R})$. In this paper, we will be concerned with the
harmonic analysis on the homogeneous space $SL_{n}(\mathbb{R})/A$ when $n=3$.
In particular, we will restrict attention to the commutation relations and
essential self-adjointness of the invariant differential operators on
$SL_{3}(\mathbb{R})/A$; spectral decomposition of the invariant differential
operators and its interaction with Plancherel theorems are left for future
research.

The space $SL_{3}(\mathbb{R})/A$ distinguishes itself among homogeneous spaces
in various ways. 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]). 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). Secondly, $SL_{3}(\mathbb{R})/A$ is one of the simplest
examples of non-spherical homogeneous spaces. These spaces exhibit a very
different nature compared with spherical ones, and very little of their
harmonic theory is known. For instance, the finite multiplicity theorem for
induction does not hold anymore ([KO]); the algebra of invariant differential
operators is a noncommutative algebra instead of a polynomial ring ([Kn]); the
representation theory of the ring of bi-invariant functions is mysterious as
well. 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 $SL_{n}(\mathbb{R})$ in
$L^{2}\left(SL_{n}(\mathbb{R})/A\right)$ is tempered.

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.
To remedy the situation, the analytic methods are always fused with the
peculiar geometry of the underlying manifold related to the representation
theory. 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]). In
the present situation, $SL_{3}(\mathbb{R})/A$ has an inherent approachable
geometry in spite of the rather involved analysis due to its non-sphericity.
For instance, the underlying manifold is diffeomorphic to
$SO_{3}(\mathbb{R})\times\mathbb{R}^{3}$, so that an extensive calculation is
possible, just as the hyperboloid case, where the usage of the spherical
coordinates of the underlying product space $S^{p-1}\times
S^{q-1}\times\mathbb{R}$ is crucial.

Before describing our results in more detail, we first set certain notations.
To treat in a unified fashion, let $G$ denote $SL_{3}(\mathbb{R})$,
$\mathfrak{g}$ the Lie algebra of $G$, and $U(\mathfrak{g})$ the universal
enveloping algebra of the complexification of $\mathfrak{g}$. Denote by
$C^{\infty}(G)$ the space of complex-valued smooth functions on $G$. Then, the
infinitesimal action $R$ on $C^{\infty}(G)$ induced by the right regular
representation of $G$, maps $U(\mathfrak{g})$ into the algebra of algebraic
differentials on $G$. More precisely, $R$ acts on $u=X_{1}X_{2}\cdots X_{k}\in
U(\mathfrak{g})$ by

$\left(R_{u}f\right)(g):=\left(R\left(X_{1}\cdots
X_{k}\right)f\right)(g):=\left.\frac{\partial}{\partial
t_{1}}\right|_{t_{1}=0}\cdots\left.\frac{\partial}{\partial
t_{k}}\right|_{t_{k}=0}f(g\exp\left(t_{1}X_{1}\right)\cdots\exp\left(t_{k}X_{k}\right)).$
(1)

Here $X_{1},X_{2},\cdots,X_{k}\in\mathfrak{g}$, and $f\in C^{\infty}(G)$; the
exponential map is given by $\exp(X):=\gamma(1)$, where
$\gamma\colon\mathbb{R}\to G$ is the one-parameter subgroup of $G$ whose
tangent vector at the identity is equal to $X$. It is easy to verify that
$R_{u}$, $u\in U(\mathfrak{g})$, is a left $G$-invariant differential
operators on $G$. Denote by $\mathbb{D}(G)$ the algebra of the left
$G$-invariant differential operators on $G$.

For a closed subgroup $H\subset G$, denote by $\mathbb{D}(G/H)$ the algebra of
$G$-invariant differential operators on the homogeneous space $G/H$. Denote by
$\pi:G\rightarrow G/H$ the natural projection, and $\mathfrak{h}$ the Lie
algebra of $H$. Define

$\mathbb{D}^{H}(G):=\\{D\in\mathbb{D}(G)\,|\,D(f\circ R_{h})\circ
R_{h}^{-1}=Df,\,\forall h\in H\,\,{\rm and}\,\,f\in C^{\infty}(G)\\},$ (2)

where $R_{h}:g\mapsto gh$ is the right translation of $G$ for $h\in H$.
Assuming $G$ and $H$ are reductive, we have the standard isomorphism (Theorem
4.6 in Chapter 2 of [He])

$\mathbb{D}^{H}(G)/\left(\mathbb{D}^{H}(G)\cap\mathbb{D}(G)\mathfrak{h}\right)\cong\mathbb{D}(G/H).$
(3)

It is induced by the map $\mu:\mathbb{D}^{H}(G)\to\mathbb{D}(G/H)$, such that
for each $D\in\mathbb{D}^{H}(G)$, $\mu(D)$ is the element of $\mathbb{D}(G/H)$
such that

$(\mu(D)f)\circ\pi=D(f\circ\pi)\,\,\,{\rm
for\,\,all\,\,smooth\,\,functions\,\,}f\,\,{\rm on\,\,}G/H.$ (4)

Denote by $E_{ij}$ the $3\times 3$ matrix unit with a $1$ in the $i^{\rm th}$
row and $j^{\rm th}$ column. For distinct $i,j,k\in\\{1,2,3\\}$, define
differential operators on $SL_{3}(\mathbb{R})/A$

$D_{ij}=\mu\left(\frac{1}{2}\sum_{\sigma\in
S_{2}}R\left(E_{\sigma(i)\sigma(j)}E_{\sigma(j)\sigma(i)}\right)\right),$ (5)

and

$D_{ijk}=\mu\left(\frac{1}{6}\sum_{\sigma\in
S_{3}}R\left(E_{\sigma(i)\sigma(j)}E_{\sigma(j)\sigma(k)}E_{\sigma(k)\sigma(i)}\right)\right).$
(6)

Note that $D_{ij}=D_{ji}$ and $D_{ijk}=D_{jki}=D_{kij}$.

In the first part of the paper, we prove

######  Theorem 1.1.

$\mathbb{D}\left(SL_{3}(\mathbb{R})/A\right)$ is the noncommutative
associative algebra generated over $\mathbb{C}$ by $\\{D_{12}$,
$D_{13},D_{23},D_{123},D_{213}\\}$ with relations

$\left\\{\begin{aligned} &[D_{123},D_{213}]=0,\\\
&[D_{ij},D_{ik}]=D_{ijk}-D_{ikj},\,\,\,\,\,\,i,j,k\in\\{1,2,3\\}{\rm\,\,are\,\,distinct},\\\
&[D_{ijk},D_{ij}]=D_{jk}D_{ij}-D_{ij}D_{ik},\,\,\,\,\,\,i,j,k\in\\{1,2,3\\}{\rm\,\,are\,\,distinct},\\\
&2\left(D_{123}D_{213}+D_{213}D_{123}-D_{12}D_{23}D_{31}-D_{13}D_{32}D_{21}\right)=\left(D_{23}-D_{13}-D_{12}\right)^{2}.\\\
\end{aligned}\right.$ (7)

The center of $\mathbb{D}\left(SL_{3}(\mathbb{R})/A\right)$ is a polynomial
ring in $D_{123}+D_{213}$ and $D_{12}+D_{23}+D_{13}$.

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.g. see [BS1], [BS2] for the case of reductive
symmetric spaces). There is considerable literature on essential self-
adjointness for natural operators on a complete Riemannian or Hermitian
manifold. See for instance [Ga1], [Ga2], [Ga3], [Roe], [Co], [Ri] for the
Hodge-Laplace-Beltrami operator, [AV] for the $\bar{\partial}$-Laplacian, [P],
[W] for the Dirac operator, and [Ch] for certain first order differential
operators. For differential operators on $\mathbb{R}^{n}$, 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.

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]). As it can be shown that the
center of $\mathbb{D}\left(SL_{3}(\mathbb{R})/A\right)$ equals the image of
the center of $U(\mathfrak{sl}_{3}(\mathbb{R}))$, it follows

######  Proposition 1.2.

Every symmetric differential operator in the center of
$\mathbb{D}\left(SL_{3}(\mathbb{R})/A\right)$ is essentially self-adjoint.

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. 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]).

The major part of the paper is to devoted to

######  Theorem 1.3.

The differential operators $D_{12},D_{13},D_{23}$ on $SL_{3}(\mathbb{R})/A$
are essentially self-adjoint.

We now briefly describe the basic ideas for the proofs. We exploit the
universal enveloping algebra to extract the algebraic structure of
$\mathbb{D}(SL_{3}(\mathbb{R})/A)$, and the normal form theory in [FH] to
determine the center of $\mathbb{D}(SL_{3}(\mathbb{R})/A)$. To study the
essential self-adjointness of symmetric operators, we modify the scheme of
[Ba]. 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. Unfortunately, in this non-spherical case, the
left derivatives are too degenerate to span the whole space of invariant
differentials in a mild way. We make the observation that by choosing the
cutoff functions and the mollifiers in a compatible way instead of separating
the Gårding type space as the operator core, one may gain extra decays to
balance the extraordinary growth of the coefficients. In fact, the chosen
cutoff functions are annihilated by the wildest terms and contribute desired
decays thereafter.

The organization of the paper is as follows. §2 is devoted to the algebraic
structure of $\mathbb{D}(SL_{3}(\mathbb{R})/A)$. We study the presence of left
derivatives in the coordinate form of $D_{12}$ in §3.4. In §3.5, we establish
the density of $C^{\infty}_{c}\left(SL_{3}(\mathbb{R})/A\right)$ in ${\rm
Dom}(D_{12})$, and, as a consequence, prove Theorem 1.3. The explicit formulas
for the generators of the left derivatives and of the left invariant
differentials are given in Appendices A.1 and A.2, respectively.

Acknowledgement. The authors appreciate greatly Professors N. Li, W. Li, Y. Xu
and J. Yu for many helpful discussion. This research is supported by National
Key R& D Program of China (No. 2022YFA1006700). The first author is partially
supported by NSFC grant (No. 12201012).

##  2\. Structure of the Algebra of the Invariant Differentials

###  2.1. Generators of the invariant differentials on $SL_{n}(\mathbb{R})/A$

Let $S(\mathfrak{g})$ be the symmetric algebra over $\mathfrak{g}$. Then for a
basis $\\{X_{1},\cdots,X_{n}\\}$ of $\mathfrak{g}$, $S(\mathfrak{g})$ can be
identified with the algebra of polynomials

$\sum\limits_{(k_{1},\cdots,k_{n})\in\mathbb{N}^{n}}a_{k_{1}\cdots
k_{n}}X_{1}^{k_{1}}\cdots X_{n}^{k_{n}},\,\,a_{k_{1}\cdots
k_{n}}\in\mathbb{C}.$ (8)

We have the following symmetrizer map
$\lambda:S(\mathfrak{g})\to\mathbb{D}(G)$.

######  Theorem 2.1 (Theorem 4.3 in Chapter 2 of [He]).

There is a unique linear bijection $\lambda$ from $S(\mathfrak{g})$ to
$\mathbb{D}(G)$ such that $\lambda(X^{m})=R(X^{m})$ for $X\in\mathfrak{g}$ and
$m\in\mathbb{Z}^{+}$. More precisely,

$(\lambda(P)f)(g):=\left.P\left(\frac{\partial}{\partial
t_{1}},\cdots,\frac{\partial}{\partial
t_{n}}\right)f\left(g\exp(t_{1}X_{1}+\cdots+t_{n}X_{n})\right)\right|_{t_{1}=\cdots=t_{n}=0},$
(9)

for $P\in S(\mathfrak{g})$ and $f\in C^{\infty}(G)$. In particular (see Page
282 of [He]),

$\lambda(Y_{1}\cdots Y_{k})=\frac{1}{k!}\sum_{\sigma\in
S_{k}}R\left(Y_{\sigma(1)}\cdots Y_{\sigma(k)}\right),$ (10)

where $Y_{1},\cdots,Y_{k}\in\mathfrak{g}$, and $S_{k}$ is the symmetric group
of degree $n$.

Denote by $E_{ij}$ the $n\times n$ matrix unit with a $1$ in the $i^{\rm th}$
row and $j^{\rm th}$ column. Define

$\begin{split}&X_{ij}:=E_{ij},\,\,1\leq i\neq j\leq
n,\,\,\,\,\,X_{ll}:=E_{ll}-E_{nn},\,\,1\leq l\leq n-1,\end{split}$ (11)

which constitute a basis of $\mathfrak{sl}_{n}(\mathbb{R})$.

######  Proposition 2.2.

The algebra $\mathbb{D}(SL_{n}(\mathbb{R})/A)$ is generated by

$\left\\{(\mu\circ\lambda)\left(E_{i_{1}i_{2}}E_{i_{2}i_{3}}\cdots
E_{i_{k-1}i_{k}}E_{i_{k}i_{1}}\right)\left|\,\begin{matrix}2\leq k\leq
n,\,1\leq i_{1},i_{2},\cdots,i_{k}\leq n\\\ i_{1},i_{2},\cdots,i_{k}\,\,{\rm
are\,\,distinct}\end{matrix}\right.\right\\}.$ (12)

Proof of Proposition 2.2. By (3), the invariant differential operators on
$SL_{n}(\mathbb{R})/A$ are induced from the left $SL_{n}(\mathbb{R})$ and
right $A$ invariant differential operators on $SL_{n}(\mathbb{R})$. Then by
(3), it suffices to prove that $\mathbb{D}^{A}(SL_{n}(\mathbb{R}))$ is
generated by the elements in (12) and
$\mathbb{D}(SL_{n}(\mathbb{R}))\mathfrak{a}$, where $\mathfrak{a}$ is the Lie
algebra of $A$.

Take $D\in\mathbb{D}^{A}(SL_{n}(\mathbb{R}))$. There is a polynomial $P_{D}$
such that $\lambda(P_{D})=D$ by Theorem 2.1. Arrange $P_{D}$ in the following
lexicographic order,

$\small P_{D}=\sum_{k=1}^{\infty}\sum_{\begin{subarray}{c}\,\,1\leq i_{1}\leq
i_{2}\leq\cdots\leq i_{t}\leq\cdots\leq i_{k}\leq n;\\\ 1\leq j_{t}\leq
n\,\,{\rm and\,\,}(i_{t},j_{t})\neq(n,n){\rm\,\,for\,\,}1\leq t\leq k;\\\ {\rm
if\,\,}1\leq u<v\leq k{\rm\,\,and\,\,}i_{u}=i_{v},{\rm\,\,then\,\,}j_{u}\leq
j_{v}\end{subarray}}p^{i_{1}i_{2}\cdots i_{k}}_{j_{1}j_{2}\cdots
j_{k}}X_{i_{1}j_{1}}X_{i_{2}j_{2}}\cdots X_{i_{t}j_{t}}\cdots X_{i_{k}j_{k}}.$
(13)

By definition,

$(Df)(g)=\left.P_{D}\left(\frac{\partial}{\partial
t_{11}},\cdots,\frac{\partial}{\partial
t_{n(n-1)}}\right)f\left(g\exp\left(\sum_{\begin{subarray}{c}1\leq i,j\leq
n\\\ (i,j)\neq(n,n)\end{subarray}}t_{ij}\cdot
X_{ij}\right)\right)\right|_{t=0}.$ (14)

For each $a=(a_{ij})\in A$, we have

$\small\begin{split}\left(D\right.&(f\circ R_{a})\circ
R_{a}^{-1}\left.\right)(g)=\left.P_{D}\left(\frac{\partial}{\partial
t_{11}},\cdots,\frac{\partial}{\partial
t_{n(n-1)}}\right)f\left(ga^{-1}\exp\left(\sum_{\begin{subarray}{c}1\leq
i,j\leq n\\\ (i,j)\neq(n,n)\end{subarray}}t_{ij}\cdot
X_{ij}\right)a\right)\right|_{t=0}\\\
&=\left.P_{D}\left(\frac{\partial}{\partial
t_{11}},\cdots,\frac{\partial}{\partial
t_{n(n-1)}}\right)f\left(g\exp\left(\sum_{\begin{subarray}{c}1\leq i,j\leq
n\\\ i\neq j\end{subarray}}a_{ii}^{-1}t_{ij}a_{jj}\cdot
X_{ij}\right)\right)\right|_{t=0}\\\
&=\left.P_{D}\left(\cdots,a_{ii}^{-1}a_{jj}\frac{\partial}{\partial
t_{ij}},\cdots\right)f\left(g\exp\left(\sum_{\begin{subarray}{c}1\leq i,j\leq
n\\\ (i,j)\neq(n,n)\end{subarray}}t_{ij}\cdot
X_{ij}\right)\right)\right|_{t=0}.\end{split}$ (15)

Since $D$ is right $A$-invariant, $P\left(\cdots,\frac{\partial}{\partial
t_{ij}},\cdots\right)=P\left(\cdots,a_{ii}^{-1}a_{jj}\frac{\partial}{\partial
t_{ij}},\cdots\right)$; more precisely,

$\begin{split}&\sum_{k=1}^{\infty}\sum_{\begin{subarray}{c}\,\,1\leq i_{1}\leq
i_{2}\leq\cdots\leq i_{t}\leq\cdots\leq i_{k}\leq n;\\\ 1\leq j_{t}\leq
n\,\,{\rm and\,\,}(i_{t},j_{t})\neq(n,n){\rm\,\,for\,\,}1\leq t\leq k;\\\ {\rm
if\,\,}1\leq u<v\leq k{\rm\,\,and\,\,}i_{u}=i_{v},{\rm\,\,then\,\,}j_{u}\leq
j_{v}\end{subarray}}p^{i_{1}i_{2}\cdots i_{k}}_{j_{1}j_{2}\cdots
j_{k}}X_{i_{1}j_{1}}X_{i_{2}j_{2}}\cdots X_{i_{t}j_{t}}\cdots
X_{i_{k}j_{k}}\\\ =\sum_{k=1}^{\infty}&\sum_{\begin{subarray}{c}\,\,1\leq
i_{1}\leq i_{2}\leq\cdots\leq i_{t}\leq\cdots\leq i_{k}\leq n;\\\ 1\leq
j_{t}\leq n\,\,{\rm and\,\,}(i_{t},j_{t})\neq(n,n){\rm\,\,for\,\,}1\leq t\leq
k;\\\ {\rm if\,\,}1\leq u<v\leq
k{\rm\,\,and\,\,}i_{u}=i_{v},{\rm\,\,then\,\,}j_{u}\leq
j_{v}\end{subarray}}\left(p^{i_{1}i_{2}\cdots i_{k}}_{j_{1}j_{2}\cdots
j_{k}}\prod\limits_{t=1}^{k}a_{i_{t}i_{t}}^{-1}a_{j_{t}j_{t}}\right)X_{i_{1}j_{1}}X_{i_{2}j_{2}}\cdots
X_{i_{t}j_{t}}\cdots X_{i_{k}j_{k}},\end{split}$ (16)

where $\prod_{l=1}^{n}a_{ll}=1$. Then, for $1\leq t\leq k$, $1\leq
i_{t},j_{t}\leq n$, and $(i_{t},j_{t})\neq(n,n)$,

$p^{i_{1}i_{2}\cdots i_{k}}_{j_{1}j_{2}\cdots j_{k}}=p^{i_{1}i_{2}\cdots
i_{k}}_{j_{1}j_{2}\cdots
j_{k}}\prod\limits_{t=1}^{k}a_{i_{t}i_{t}}^{-1}a_{j_{t}j_{t}}.$ (17)

Without loss of generality, we can assume that

$P_{D}=p^{i_{1}i_{2}\cdots i_{k}}_{j_{1}j_{2}\cdots
j_{k}}\prod\limits_{t=1}^{k}a_{i_{t}i_{t}}^{-1}a_{j_{t}j_{t}}X_{i_{1}j_{1}}X_{i_{2}j_{2}}\cdots
X_{i_{t}j_{t}}\cdots X_{i_{k}j_{k}}.$ (18)

Write
$\mathcal{I}_{k}:=\left((i_{1},j_{1}),(i_{2},j_{2}),\cdots,(i_{k},j_{k})\right)$,
and

$\prod\limits_{t=1}^{k}a_{i_{t}i_{t}}^{-1}a_{j_{t}j_{t}}=\prod\limits_{l=1}^{n}a_{ll}^{\sigma_{l}(\mathcal{I}_{k})},$
(19)

where for $1\leq l\leq n$,

$\sigma_{l}(\mathcal{I}_{k}):={\rm cardinality\,\,of\,\,}\\{1\leq t\leq
n\,|\,j_{t}=l\,\\}\,\,-{\rm cardinality\,\,of\,\,}\\{1\leq t\leq
n\,|\,i_{t}=l\,\\}.$ (20)

If $p^{i_{1}i_{2}\cdots i_{k}}_{j_{1}j_{2}\cdots j_{k}}\neq 0$, then
$\sigma_{1}(\mathcal{I}_{k})=\sigma_{2}(\mathcal{I}_{k})=\cdots=\sigma_{n}(\mathcal{I}_{k})$.
Moreover, since $\sum_{l=1}^{n}\sigma_{l}(\mathcal{I}_{k})=0$,

$\sigma_{1}(\mathcal{I}_{k})=\sigma_{2}(\mathcal{I}_{k})=\cdots=\sigma_{n}(\mathcal{I}_{k})=0.$
(21)

Construct a directed graph $(V,A)$ as follows. Let $V:=\\{1,2,\cdots,n\\}$ be
the set of vertices. The set of directed edges $A$ consists of directed edges
$(i_{t},j_{t})$ from $i_{t}$ to $j_{t}$, $1\leq t\leq k$, for $(i_{t},j_{t})$
appearing in $\mathcal{I}_{k}$. It is clear that the indegree equals the
outdegree for each vertex in $V$ by (21).

Now, if the directed graph $(V,A)$ contains a self-loop (a directed edge
connects a vertex to itself), then
$P_{D}\in\mathbb{D}(SL_{n}(\mathbb{R}))\mathfrak{a}$, that is, $P_{D}$ is the
composition of a differential operator in $\mathbb{D}(SL_{n}(\mathbb{R}))$ and
a vector field generated by $\mathfrak{a}$. Otherwise, we can partition $A$
into simple cycles, that is,

$\\{(i_{t},j_{t})|1\leq t\leq k\\}=\bigsqcup_{l_{s}\geq
2}\left\\{(i^{s}_{1},i^{s}_{2}),(i^{s}_{2},i^{s}_{3}),\cdots,(i^{s}_{l_{s}-1},i^{s}_{l_{s}}),(i^{s}_{l_{s}},i^{s}_{1})\left|\begin{matrix}i^{s}_{u}\neq
i^{s}_{v}\,\,{\rm for\,\,}\\\ 1\leq u<v\leq
l_{s}\end{matrix}\right.\right\\}.$ (22)

Noticing that each simple cycle in (22) corresponds to an element in (12), we
can verify that

$P_{D}-\prod_{l_{s}\geq
2}(\mu\circ\lambda)\left(X_{i^{s}_{1}i^{s}_{2}}X_{i^{s}_{2}i^{s}_{3}}\cdots
X_{i^{s}_{l_{s}-1}i^{s}_{l_{s}}}X_{i^{s}_{l_{s}}i^{s}_{1}}\right)$ (23)

is a $SL_{n}(\mathbb{R})$-invariant differential operator with degree strictly
less than that of $P_{D}$. By induction, we can conclude that $P_{D}$ is
generated by the elements in (12).

We complete the proof of Proposition 2.2.

###  2.2. Relations among the generators

By Proposition 2.2 and Theorem 2.1, the algebra of the
$SL_{3}(\mathbb{R})$-invariant differential operators on
$SL_{3}(\mathbb{R})/A$ are generated by

$D_{ij}=\mu\left(\frac{1}{2}\sum_{\sigma\in
S_{2}}R\left(E_{\sigma(i)\sigma(j)}E_{\sigma(j)\sigma(i)}\right)\right),\,\,(i,j)=(1,2),(1,3),(2,3),$
(24)

and

$D_{ijk}=\mu\left(\frac{1}{6}\sum_{\sigma\in
S_{3}}R\left(E_{\sigma(i)\sigma(j)}E_{\sigma(j)\sigma(k)}E_{\sigma(k)\sigma(i)}\right)\right),\,\,(i,j,k)=(1,2,3),(2,1,3).$
(25)

######  Proposition 2.3.

We have the following commutation relations.

$[D_{12},D_{13}]=D_{123}-D_{132},$ (26) $[D_{21},D_{23}]=D_{213}-D_{231},$
(27) $[D_{31},D_{32}]=D_{312}-D_{321}.$ (28)

Proof of Proposition 2.3. It suffices to prove (26); (27) and (28) follow from
(26) by suitable permutations of the indices therein.

On the level of universal enveloping algebra, we have

$\begin{split}&E_{12}E_{21}E_{13}E_{31}=E_{12}E_{13}E_{21}E_{31}+E_{12}E_{23}E_{31}=E_{13}E_{12}E_{31}E_{21}+E_{12}E_{23}E_{31}\\\
&\,\,\,\,\,\,\,\,\,\,=E_{13}E_{31}E_{12}E_{21}+E_{12}E_{23}E_{31}-E_{13}E_{32}E_{21}.\\\
\end{split}$ (29)

Similarly, we can derive

$\begin{split}&E_{21}E_{12}E_{13}E_{31}=E_{13}E_{31}E_{21}E_{12}+E_{23}E_{12}E_{31}-E_{13}E_{21}E_{32},\\\
&E_{12}E_{21}E_{31}E_{13}=E_{31}E_{13}E_{12}E_{21}+E_{12}E_{31}E_{23}-E_{32}E_{13}E_{21},\\\
&E_{21}E_{12}E_{31}E_{13}=E_{31}E_{13}E_{21}E_{12}+E_{31}E_{23}E_{12}-E_{21}E_{32}E_{13}.\\\
\end{split}$ (30)

Also,

$\begin{split}&E_{12}E_{31}E_{23}=E_{12}E_{23}E_{31}-E_{12}E_{21},\\\
&E_{23}E_{12}E_{31}=E_{12}E_{23}E_{31}-E_{13}E_{31},\\\
&E_{23}E_{31}E_{12}=E_{12}E_{23}E_{31}+E_{23}E_{32}-E_{13}E_{31},\\\
&E_{31}E_{12}E_{23}=E_{12}E_{23}E_{31}+E_{32}E_{23}-E_{12}E_{21},\\\
&E_{31}E_{23}E_{12}=E_{12}E_{23}E_{31}-E_{12}E_{21}+E_{32}E_{23}-E_{31}E_{13},\\\
\end{split}$ (31)

and

$\begin{split}&E_{32}E_{13}E_{21}=E_{13}E_{32}E_{21}-E_{12}E_{21},\\\
&E_{13}E_{21}E_{32}=E_{13}E_{32}E_{21}-E_{13}E_{31},\\\
&E_{21}E_{13}E_{32}=E_{13}E_{32}E_{21}+E_{23}E_{32}-E_{13}E_{31},\\\
&E_{32}E_{21}E_{13}=E_{13}E_{32}E_{21}+E_{32}E_{23}-E_{12}E_{21},\\\
&E_{21}E_{32}E_{13}=E_{13}E_{32}E_{21}-E_{12}E_{21}+E_{32}E_{23}-E_{31}E_{13}.\\\
\end{split}$ (32)

By the definition,

$\begin{split}&D_{123}-D_{213}=\frac{1}{6}\mu\left(R\left(E_{12}E_{23}E_{31}+E_{12}E_{31}E_{23}+E_{23}E_{12}E_{31}+E_{23}E_{31}E_{12}+E_{31}E_{12}E_{23}\right.\right.\\\
&\,\,\,\,\,\,\,\,\,\,\left.\left.+E_{31}E_{23}E_{12}\right)\right)-\frac{1}{6}\mu\left(R\left(E_{21}E_{13}E_{32}+E_{21}E_{32}E_{13}+E_{13}E_{21}E_{32}+E_{13}E_{32}E_{21}\right.\right.\\\
&\,\,\,\,\,\,\,\,\,\,\left.\left.+E_{32}E_{13}E_{21}+E_{32}E_{21}E_{13}\right)\right)\\\
&=\mu\left(R\left(E_{12}E_{23}E_{31}\right)\right)-\mu\left(R\left(E_{13}E_{32}E_{21}\right)\right).\\\
\end{split}$ (33)

Similarly, (29) and (30) yield

$\begin{split}&D_{12}D_{13}-D_{13}D_{12}=\frac{1}{4}\mu\left(\left(R\left(E_{12}E_{21}\right)+R\left(E_{21}E_{12}\right)\right)\left(R\left(E_{13}E_{31}\right)+R\left(E_{31}E_{13}\right)\right)\right)\\\
&\,\,\,\,\,\,\,\,\,-\frac{1}{4}\mu\left(\left(R\left(E_{13}E_{31}\right)+R\left(E_{31}E_{13}\right)\right)\left(R\left(E_{12}E_{21}\right)+R\left(E_{21}E_{12}\right)\right)\right)\\\
&=\frac{1}{4}\mu\left(R\left(E_{12}E_{21}E_{13}E_{31}+E_{21}E_{12}E_{13}E_{31}+E_{12}E_{21}E_{31}E_{13}+E_{21}E_{12}E_{31}E_{13}\right)\right)\\\
&\,\,\,\,\,\,\,\,\,-\frac{1}{4}\mu\left(R\left(E_{13}E_{31}E_{12}E_{21}+E_{13}E_{31}E_{21}E_{12}+E_{31}E_{13}E_{12}E_{21}+E_{31}E_{13}E_{21}E_{12}\right)\right)\\\
&=\frac{1}{4}\mu\left(R\left(E_{12}E_{21}E_{13}E_{31}-E_{13}E_{31}E_{12}E_{21}\right)\right)+\frac{1}{4}\mu\left(R\left(E_{21}E_{12}E_{13}E_{31}-E_{13}E_{31}E_{21}E_{12}\right)\right)\\\
&\,\,\,\,\,+\frac{1}{4}\mu\left(R\left(E_{12}E_{21}E_{31}E_{13}-E_{31}E_{13}E_{12}E_{21}\right)\right)+\frac{1}{4}\mu\left(R\left(E_{21}E_{12}E_{31}E_{13}-E_{31}E_{13}E_{21}E_{12}\right)\right)\\\
&=\frac{1}{4}\mu\left(R\left(E_{12}E_{23}E_{31}+E_{23}E_{12}E_{31}+E_{12}E_{31}E_{23}+E_{31}E_{23}E_{12}\right)\right)\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\frac{1}{4}\mu\left(R\left(E_{13}E_{32}E_{21}+E_{13}E_{21}E_{32}+E_{32}E_{13}E_{21}+E_{21}E_{32}E_{13}\right)\right)\\\
&=\mu\left(R\left(E_{12}E_{23}E_{31}\right)\right)-\mu\left(R\left(E_{13}E_{32}E_{21}\right)\right).\\\
\end{split}$ (34)

We can thus complete the proof of Proposition 2.3.

######  Proposition 2.4.

We have the following commutation relations.

$\begin{split}[D_{123},D_{12}]=-[D_{213},D_{12}]&=D_{23}D_{12}-D_{12}D_{13},\end{split}$
(35)
$\begin{split}[D_{312},D_{31}]=-[D_{132},D_{31}]&=D_{13}D_{12}-D_{23}D_{13},\end{split}$
(36)
$\begin{split}[D_{231},D_{23}]=-[D_{321},D_{23}]&=D_{23}D_{31}-D_{12}D_{23}.\end{split}$
(37)

Proof of Proposition 2.4. It suffices to compute $[D_{123},D_{12}]$.

Computation yields that

$\small\begin{split}&D_{123}D_{12}=\mu\left(R\left(E_{12}E_{23}E_{31}+\frac{1}{6}\left(2E_{32}E_{23}+E_{23}E_{32}-2E_{13}E_{31}-E_{31}E_{13}-3E_{12}E_{21}\right)\right)\right)D_{12}\\\
&=\mu\left(R\left(E_{12}E_{23}E_{31}E_{21}E_{12}\right)\right)+\frac{1}{2}D_{23}D_{12}-\frac{1}{2}D_{13}D_{12}-\frac{1}{2}D_{12}D_{12}\\\
&=\mu\left(R\left(E_{12}E_{21}E_{12}E_{23}E_{31}\right)\right)+\mu\left(R\left(E_{12}E_{21}E_{23}E_{32}\right)\right)-\mu\left(R\left(E_{12}E_{21}E_{13}E_{31}\right)\right)\\\
&\,\,\,\,\,\,\,\,\,\,\,+\frac{1}{2}D_{23}D_{12}-\frac{1}{2}D_{13}D_{12}-\frac{1}{2}D_{12}D_{12}\\\
&=\mu\left(R\left(E_{12}E_{21}E_{12}E_{23}E_{31}\right)\right)+D_{12}D_{23}-D_{12}D_{13}+\frac{1}{2}D_{23}D_{12}-\frac{1}{2}D_{13}D_{12}-\frac{1}{2}D_{12}D_{12}\\\
\end{split}$ (38)

Similarly,

$\begin{split}&D_{12}D_{123}=\mu\left(R\left(E_{12}E_{21}E_{12}E_{23}E_{31}\right)\right)+\frac{1}{2}D_{12}D_{23}-\frac{1}{2}D_{12}D_{13}-\frac{1}{2}D_{12}D_{12}.\\\
\end{split}$ (39)

Proposition 2.3 follows.

######  Proposition 2.5.

We have the following equality.

$\begin{split}&D_{123}D_{213}+D_{213}D_{123}-D_{12}D_{23}D_{31}-D_{13}D_{32}D_{21}=\frac{1}{2}\left(D_{23}-D_{13}-D_{12}\right)^{2}.\\\
\end{split}$ (40)

Proof of Proposition 2.5. Computation yields that

$\small\begin{split}&D_{13}D_{32}D_{21}=\frac{1}{8}\mu\left(R\left(E_{13}E_{31}+E_{31}E_{13}\right)R\left(E_{23}E_{32}+E_{32}E_{23}\right)R\left(E_{12}E_{21}+E_{21}E_{12}\right)\right)\\\
&=\mu\left(R\left(E_{13}E_{31}E_{32}E_{23}E_{21}E_{12}\right)\right)+\frac{1}{8}\mu\left(R\left(E_{33}-E_{11}\right)\left(E_{23}E_{32}+E_{32}E_{23}\right)\left(E_{12}E_{21}+E_{21}E_{12}\right)\right)\\\
&\,\,\,\,\,\,+\frac{1}{4}\mu\left(R\left(E_{13}E_{31}\left(E_{22}-E_{33}\right)\left(E_{12}E_{21}+E_{21}E_{12}\right)\right)\right)+\frac{1}{2}\mu\left(R\left(E_{13}E_{31}E_{32}E_{23}\left(E_{11}-E_{22}\right)\right)\right)\\\
&=\mu\left(R\left(E_{13}E_{31}E_{32}E_{23}E_{21}E_{12}\right)\right),\\\
\end{split}$ (41)

where the last step exploits the fact that $E_{ii}-E_{jj}$ commutes with
$D_{kl}$. Similarly, we can derive

$\footnotesize\begin{split}&D_{213}D_{123}=\mu\left(R\left(E_{13}E_{32}E_{21}+\frac{1}{6}\left(2E_{32}E_{23}+E_{23}E_{32}-2E_{13}E_{31}-E_{31}E_{13}-3E_{12}E_{21}\right)\right)\right)D_{123}\\\
&=\mu\left(R\left(E_{13}E_{32}E_{21}\right)\right)D_{123}+\frac{1}{2}D_{32}D_{123}-\frac{1}{2}D_{13}D_{123}-\frac{1}{2}D_{12}D_{123}\\\
&=\mu\left(R\left(E_{13}E_{32}E_{21}E_{31}E_{23}E_{12}\right)\right)+\frac{1}{2}\mu\left(R\left(E_{13}E_{32}E_{21}E_{12}E_{21}\right)\right)+\frac{1}{2}\mu\left(R\left(E_{13}E_{32}E_{21}E_{13}E_{31}\right)\right)\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\frac{1}{2}\mu\left(R\left(E_{13}E_{32}E_{21}E_{23}E_{32}\right)\right)+\frac{1}{2}D_{32}D_{123}-\frac{1}{2}D_{13}D_{123}-\frac{1}{2}D_{12}D_{123}\\\
&=\mu\left(R\left(E_{13}E_{31}E_{32}E_{23}E_{21}E_{12}\right)\right)-\frac{1}{2}D_{213}\left(D_{23}-D_{12}-D_{13}\right)+\frac{1}{2}\left(D_{23}-D_{12}-D_{13}\right)D_{123}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{1}{4}\left(D_{23}-D_{13}-D_{12}\right)^{2}.\\\
\end{split}$ (42)

Let $A_{2}$ be the group consisting of the identity and the permutation of $2$
and $3$. Then,

$\begin{split}&\sum_{\sigma\in
A_{2}}\left(D_{\sigma(2)\sigma(1)\sigma(3)}D_{\sigma(1)\sigma(2)\sigma(3)}-D_{\sigma(1)\sigma(3)}D_{\sigma(3)\sigma(2)}D_{\sigma(2)\sigma(1)}\right)\\\
&=\sum_{\sigma\in
A_{2}}\frac{1}{2}\left(D_{\sigma(2)\sigma(3)}-D_{\sigma(1)\sigma(2)}-D_{\sigma(1)\sigma(3)}\right)\left(D_{\sigma(1)\sigma(2)\sigma(3)}-D_{\sigma(2)\sigma(1)\sigma(3)}\right)\\\
&+\sum_{\sigma\in
A_{2}}\frac{1}{2}\left(D_{\sigma(2)\sigma(3)}D_{\sigma(3)\sigma(1)}-D_{\sigma(1)\sigma(2)}D_{\sigma(2)\sigma(3)}-D_{\sigma(1)\sigma(3)}D_{\sigma(1)\sigma(2)}-D_{\sigma(2)\sigma(3)}D_{\sigma(1)\sigma(2)}\right.\\\
&\,\,\,\,\left.+D_{\sigma(2)\sigma(3)}D_{\sigma(1)\sigma(3)}+D_{\sigma(1)\sigma(2)}D_{\sigma(1)\sigma(3)}\right)+\sum_{\sigma\in
A_{2}}\frac{1}{4}\left(D_{\sigma(2)\sigma(3)}-D_{\sigma(1)\sigma(2)}-D_{\sigma(1)\sigma(3)}\right)^{2}\\\
&=\frac{1}{2}\left(D_{23}-D_{13}-D_{12}\right)^{2}.\\\ \end{split}$ (43)

Now Proposition 2.5 follows.

######  Proposition 2.6.

We have the following commutation relations.

$\begin{split}[D_{123},D_{213}]&=0.\\\ \end{split}$ (44)

Proof of Proposition 2.6. By (41) and (42), we have

$\begin{split}&D_{213}D_{123}=D_{13}D_{32}D_{21}-\frac{1}{2}D_{213}\left(D_{23}-D_{12}-D_{13}\right)+\frac{1}{2}\left(D_{23}-D_{12}-D_{13}\right)D_{123}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{1}{4}\left(D_{23}-D_{13}-D_{12}\right)^{2}.\\\
\end{split}$ (45)

Interchanging the indices $2$ and $3$ in (45), we get

$\begin{split}&D_{123}D_{213}=D_{12}D_{23}D_{13}-\frac{1}{2}D_{123}\left(D_{23}-D_{12}-D_{13}\right)+\frac{1}{2}\left(D_{23}-D_{12}-D_{13}\right)D_{213}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{1}{4}\left(D_{23}-D_{13}-D_{12}\right)^{2}.\\\
\end{split}$ (46)

Making a subtraction, we can derive that $[D_{123},D_{213}]=0$.

######  Remark 2.7.

Let $D=\sum_{i}X_{i,1}X_{i,2}\cdots X_{i,k_{i}}$ be an element of
$U(\mathfrak{g})$, understood as a left-invariant differential operator on
$G$. Then the formal adjoint $D^{*}$ is given by

$D^{*}=\sum_{i}(-1)^{k_{i}}X_{i,k_{i}}X_{i,k_{i}-1}\cdots X_{i,1}.$ (47)

One can thus easily verify that $D_{12},D_{13},D_{23},\sqrt{-1}\cdot
D_{123},\sqrt{-1}\cdot D_{213}$ are formally self-adjoint. This is another
reason for taking $D_{ij}$, $D_{ijk}$ as the generators.

###  2.3. The center of the algebra of the invariant differentials

######  Proposition 2.8.

The following elements constitute a basis of the linear space
$\mathbb{D}(SL_{3}(\mathbb{R})/A)$ over $\mathbb{C}$.

$\left\\{\begin{aligned}
&D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m},\,\,D_{12}^{k}D_{31}^{i}D_{123}^{l}D_{213}^{m},\,\,D_{23}^{j}D_{31}^{i}D_{123}^{l}D_{213}^{m},\,\,\,\,i,j,k\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}};\\\
&D_{12}^{k}D_{123}^{l}D_{213}^{m},\,\,D_{23}^{j}D_{123}^{l}D_{213}^{m},\,\,D_{31}^{i}D_{123}^{l}D_{213}^{m},\,\,\,\,i,j,k\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}};\\\ &D_{123}^{l}D_{213}^{m},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}.\end{aligned}\right.$ (48)

Proof of Proposition 2.8. According to Proposition 2.2,
$\mathbb{D}(SL_{3}(\mathbb{R})/A)$ is a linear span of
$D_{12}^{k}D_{23}^{j}D_{31}^{i}D_{123}^{l}D_{213}^{m}$. By (40) along with the
commutation relations in Proposition 2.3, 2.4, and 2.6, we can replace
$D_{12}D_{23}D_{31}$ by $D_{123}D_{213}$ with certain lower order terms.
Hence, $D_{12}^{k}D_{23}^{j}D_{31}^{i}D_{123}^{l}D_{213}^{m}$, $k,j,i\geq 1$,
is generated by the elements in (48).

Next, we shall show that the elements in (48) are linearly independent.
Suppose

$\begin{split}0=&\sum_{\begin{subarray}{c}l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{000lm}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{k00lm}D_{12}^{k}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{0j0lm}D_{23}^{j}D_{123}^{l}D_{213}^{m}\\\
&+\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{+}\\\ l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{00ilm}D_{31}^{i}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}k,j\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{kj0lm}D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m}\\\
&+\sum_{\begin{subarray}{c}k,i\in\mathbb{Z}^{+}\\\ l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{k0ilm}D_{12}^{k}D_{31}^{i}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}j,i\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{0jilm}D_{23}^{j}D_{31}^{i}D_{123}^{l}D_{213}^{m},\\\
\end{split}$ (49)

where $r_{kjilm}\neq 0$ for only finitely many nonnegative integers
$k,j,i,l,m$.

Let $\mathfrak{p}$ be the subspace of $\mathfrak{sl}_{3}(\mathbb{R})$ spanned
by $\\{X_{ij}\\}_{1\leq i,j\leq 3,\,i\neq j}$. Equip $\mathfrak{p}$ with
natural coordinates $x=(x_{12},x_{13},x_{23},x_{21},x_{31},x_{32})$ such that
$\eta=\sum_{1\leq i,j\leq n,\,i\neq j}x_{ij}X_{ij}$ for each
$\eta\in\mathfrak{p}$. Denote by $j:\mathfrak{p}\hookrightarrow\mathfrak{g}$
the natural injection. Define a mapping $\mathcal{P}:\mathfrak{p}\rightarrow
SL_{3}(\mathbb{R})/A$ by $\mathcal{P}:=\pi\circ\exp\circ j$. Then, the
following diagram commutes.

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(50)

Claim. $\mathcal{P}$ is a local homeomorphism near $0\in\mathfrak{p}$.

Proof of Claim. Since $\exp:\mathfrak{sl}_{3}(\mathbb{R})\rightarrow
SL_{3}(\mathbb{R})$ is a local homeomorphism and $j$ is an injection,
$(\exp\circ j)(\mathfrak{p})$ is a submanifold of $G$ locally near the
identity ${e}\in SL_{3}(\mathbb{R})$. Since
$\mathfrak{p}\oplus\mathfrak{a}=\mathfrak{sl}_{3}(\mathbb{R})$, where
$\mathfrak{a}$ is the Lie algebra of $A$, we can conclude that $(\exp\circ
j)(\mathfrak{p})$ and $A$ intersects transversally at ${e}$. Since
$\pi:SL_{3}(\mathbb{R})\rightarrow SL_{3}(\mathbb{R})/A$ is a fiber bundle
which locally has a product structure, $\mathcal{P}$ is a local homeomorphism
between $\mathfrak{p}$ and $SL_{3}(\mathbb{R})/A$.

Let $f$ be an arbitrary smooth function on $G/H$. By Theorem 2.1, we have

$\begin{split}&(D_{ij}f)(eA)=\left.\frac{\partial^{2}}{\partial
x_{ij}\partial{x_{ji}}}f\left(\pi\left(\exp(x_{ij}X_{ij}+x_{ji}X_{ji})\right)\right)\right|_{x_{ij}=x_{ji}=0},\\\
&(D_{ijk}f)(eA)=\left.\frac{\partial^{3}}{\partial
x_{ij}\partial{x_{jk}}\partial
x_{ki}}f\left(\pi\left(\exp(x_{ij}X_{ij}+x_{jk}X_{jk}+x_{ki}X_{ki})\right)\right)\right|_{x_{ij}=x_{jk}=x_{ki}=0},\\\
\end{split}$ (51)

for distinct $i,j,k\in\\{1,2,3\\}$. Then, in terms of the local coordinates
$(x_{12},x_{13},x_{23},x_{21},x_{31},x_{32})$, we can conclude that the
differential operators $D_{12}^{k}D_{23}^{j}D_{13}^{i}D_{123}^{l}D_{213}^{m}$,
has leading terms at the origin $eA$

$\left(\frac{\partial}{\partial
x_{12}}\right)^{k+l}\left(\frac{\partial}{\partial
x_{23}}\right)^{j+l}\left(\frac{\partial}{\partial
x_{31}}\right)^{l+i}\left(\frac{\partial}{\partial
x_{21}}\right)^{k+m}\left(\frac{\partial}{\partial
x_{13}}\right)^{m+i}\left(\frac{\partial}{\partial x_{32}}\right)^{j+m}.$ (52)

It is easy to verify that the differential operators appearing in (49) all
have distinct leading terms at $eA$ . Then (49) holds if and only if all the
coefficients $r_{kjilm}=0$.

We complete the proof of Proposition 2.8.

We define the degree ${\rm deg\,}(D)$ for an invariant differential operator
$D$ as follows. ${\rm deg\,}(D):=2$ for $D=D_{12},D_{23}$, and $D_{31}$; ${\rm
deg\,}(D):=3$ for $D=D_{123}$ and $D_{213}$. In general, for

$D:=\sum_{k}\sum_{D^{\tau}_{i}\in\\{D_{12},D_{23},D_{31},D_{123},D_{213}\\}}a_{\tau}\cdot
D^{\tau}_{1}D^{\tau}_{2}\cdots D^{\tau}_{i}\cdots D^{\tau}_{k},$ (53)

define

${\rm deg\,}(D):=\max\limits_{a_{\tau}\neq
0}\left\\{\sum_{i=1}^{k}{\rm\,degree\,\,of\,\,}D^{\tau}_{i}\right\\}.$ (54)

We write $D=R(M)$ if and only if ${\rm deg\,}(D)\leq M$.

######  Lemma 2.9.

The following equalities hold.

$\begin{split}&[D_{123},D_{12}^{k}D_{123}^{l}D_{213}^{m}]=kD_{12}^{k}D_{23}D_{123}^{l}D_{213}^{m}-kD_{12}^{k}D_{31}D_{123}^{l}D_{213}^{m}+R(2k+3l+3m+1),\\\
&[D_{123},D_{23}^{j}D_{123}^{l}D_{213}^{m}]=-jD_{12}D_{23}^{j}D_{123}^{l}D_{213}^{m}+jD_{23}^{j}D_{31}D_{123}^{l}D_{213}^{m}+R(2j+3l+3m+1),\\\
&[D_{123},D_{31}^{i}D_{123}^{l}D_{213}^{m}]=iD_{12}D_{31}^{i}D_{123}^{l}D_{213}^{m}-iD_{23}D_{31}^{i}D_{123}^{l}D_{213}^{m}+R(2i+3l+3m+1).\\\
\end{split}$ (55)

Proof of Lemma 2.9. It suffices to prove the first one for the other two can
be obtained by the same method. Applying Proposition 2.4 repeatedly, we have

$\begin{split}D_{123}D_{12}^{k}&D_{123}^{l}D_{213}^{m}=D_{12}D_{123}D_{12}^{k-1}D_{123}^{l}D_{213}^{m}+(D_{23}D_{12}-D_{12}D_{31})D_{12}^{k-1}D_{123}^{l}D_{213}^{m}\\\
&=D_{12}^{2}D_{123}D_{12}^{k-2}D_{123}^{l}D_{213}^{m}+\sum_{p=0}^{1}D_{12}^{p}(D_{23}D_{12}-D_{12}D_{31})D_{12}^{k-1-p}D_{123}^{l}D_{213}^{m}\\\
&=\cdots\cdots\\\
&=D_{12}^{k}D_{123}^{l}D_{213}^{m}D_{123}+\sum_{p=0}^{k-1}D_{12}^{p}(D_{23}D_{12}-D_{12}D_{31})D_{12}^{k-1-p}D_{123}^{l}D_{213}^{m},\\\
\end{split}$ (56)

where in the last step we use $[D_{123},D_{213}]=0$. Noticing that the degree
of the commutator $[D_{1},D_{2}]$ is strictly less than the degree of
$D_{1}D_{2}$ , we can conclude that

$\begin{split}&\sum_{p=0}^{k-1}D_{12}^{p}(D_{23}D_{12}-D_{12}D_{31})D_{12}^{k-1-p}D_{123}^{l}D_{213}^{m}-(kD_{12}^{k}D_{23}D_{123}^{l}D_{213}^{m}-kD_{12}^{k}D_{31}D_{123}^{l}D_{213}^{m})\\\
\end{split}$ (57)

is of degree less than or equal to $2k+3l+3m+1$.

Therefore, we complete the proof of Lemma 2.9.

######  Lemma 2.10.

The following equalities hold.

$\begin{split}&[D_{123},D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m}]=kD_{12}^{k}D_{23}^{j+1}D_{123}^{l}D_{213}^{m}+(j-k)D_{12}^{k-1}D_{23}^{j-1}D_{123}^{l+1}D_{213}^{m+1}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-jD_{12}^{k+1}D_{23}^{j}D_{123}^{l}D_{213}^{m}+R(2k+2j+3l+3m+1),\\\
&[D_{123},D_{12}^{k}D_{31}^{i}D_{123}^{l}D_{213}^{m}]=(k-i)D_{12}^{k-1}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m+1}-kD_{12}^{k}D_{31}^{i+1}D_{123}^{l}D_{213}^{m}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+iD_{12}^{k+1}D_{31}^{i}D_{123}^{l}D_{213}^{m}+R(2k+2i+3l+3m+1),\\\
&[D_{123},D_{23}^{j}D_{31}^{i}D_{123}^{l}D_{213}^{m}]=jD_{23}^{j}D_{31}^{i+1}D_{123}^{l}D_{213}^{m}+(i-j)D_{23}^{j-1}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m+1}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-iD_{23}^{j+1}D_{31}^{i}D_{123}^{l}D_{213}^{m}+R(2j+2i+3l+3m+1).\\\
\end{split}$ (58)

Proof of Lemma 2.10. It suffices to prove the first one for the others can be
obtained similarly. Applying Proposition 2.4 repeatedly, we have

$\begin{split}&D_{123}D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m}=D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m}D_{123}+\sum_{p=0}^{k-1}D_{12}^{p}(D_{23}D_{12}-D_{12}D_{23})D_{12}^{k-1-p}D_{31}^{j}D_{123}^{l}D_{213}^{m}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\sum_{p=0}^{j-1}D_{12}^{k}D_{23}^{p}(D_{23}D_{31}-D_{12}D_{23})D_{23}^{j-1-p}D_{123}^{l}D_{213}^{m}\\\
&=D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m}D_{123}+kD_{12}^{k}D_{23}^{j+1}D_{123}^{l}D_{213}^{m}-kD_{12}^{k}D_{23}^{j}D_{31}D_{123}^{l}D_{213}^{m}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+jD_{12}^{k}D_{23}^{j}D_{31}D_{123}^{l}D_{213}^{m}-jD_{12}^{k+1}D_{23}^{j}D_{123}^{l}D_{213}^{m}+R(2k+2j+3l+3m+1)\\\
&=D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m}D_{123}+kD_{12}^{k}D_{23}^{j+1}D_{123}^{l}D_{213}^{m}-kD_{12}^{k-1}D_{23}^{j-1}D_{123}^{l+1}D_{213}^{m+1}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+jD_{12}^{k-1}D_{23}^{j-1}D_{123}^{l+1}D_{213}^{m+1}-jD_{12}^{k+1}D_{23}^{j}D_{123}^{l}D_{213}^{m}+R(2k+2j+3l+3m+1).\\\
\end{split}$ (59)

We complete the proof of Lemma 2.10.

Similarly, we have

######  Lemma 2.11.

For $l,m\geq 0$, the following equalities hold.

$\begin{split}&[D_{12},D_{123}^{l}D_{213}^{m}]=-lD_{12}D_{23}D_{123}^{l-1}D_{213}^{m}+lD_{12}D_{31}D_{123}^{l-1}D_{213}^{m}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+mD_{12}D_{23}D_{123}^{l}D_{213}^{m-1}-mD_{12}D_{31}D_{123}^{l}D_{213}^{m-1}+R(3l+3m),\\\
&[D_{23},D_{123}^{l}D_{213}^{m}]=lD_{12}D_{23}D_{123}^{l-1}D_{213}^{m}-lD_{23}D_{31}D_{123}^{l-1}D_{213}^{m}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-mD_{12}D_{23}D_{123}^{l}D_{213}^{m-1}+mD_{23}D_{31}D_{123}^{l}D_{213}^{m-1}+R(3l+3m).\\\
\end{split}$ (60)

Proof of Lemma 2.11. The proof is similar. For simplicity, we omit it here.

######  Lemma 2.12.

For $k,j,i\geq 1$ and $l,m\geq 0$, the following equalities hold.

$\begin{split}&[D_{12},D_{23}^{j}D_{123}^{l}D_{213}^{m}]=-lD_{12}D_{23}^{j+1}D_{123}^{l-1}D_{213}^{m}+mD_{12}D_{23}^{j+1}D_{123}^{l}D_{213}^{m-1}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+(j+l)D_{23}^{j-1}D_{123}^{l}D_{213}^{m+1}-(j+m)D_{23}^{j-1}D_{123}^{l+1}D_{213}^{m}+R(2j+3l+3m),\\\
&[D_{12},D_{31}^{i}D_{123}^{l}D_{213}^{m}]=lD_{12}D_{31}^{i+1}D_{123}^{l-1}D_{213}^{m}-mD_{12}D_{31}^{i+1}D_{123}^{l}D_{213}^{m-1}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+(i+m)D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}-(i+l)D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}+R(2i+3l+3m),\\\
&[D_{23},D_{12}^{k}D_{123}^{l}D_{213}^{m}]=lD_{12}^{k+1}D_{23}D_{123}^{l-1}D_{213}^{m}-mD_{12}^{k+1}D_{23}D_{123}^{l}D_{213}^{m-1}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-(k+l)D_{12}^{k-1}D_{123}^{l}D_{213}^{m+1}+(k+m)D_{12}^{k-1}D_{123}^{l+1}D_{213}^{m}+R(2k+3l+3m),\\\
&[D_{23},D_{31}^{i}D_{123}^{l}D_{213}^{m}]=-lD_{23}D_{31}^{i+1}D_{123}^{l-1}D_{213}^{m}+mD_{23}D_{31}^{i+1}D_{123}^{l}D_{213}^{m-1}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+(i+l)D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}-(i+m)D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}+R(2i+3l+3m).\\\
\end{split}$ (61)

Proof of Lemma 2.12. The proof is similar. For simplicity, we omit it here.

######  Lemma 2.13.

For $k,j,i\geq 1$ and $l,m\geq 0$, the following equalities hold.

$\begin{split}&[D_{12},D_{23}^{j}D_{31}^{i}D_{123}^{l}D_{213}^{m}]=-(i+l)D_{23}^{j}D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}+(i+m)D_{23}^{j}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}\\\
&\,\,\,\,\,\,\,\,+(j+l)D_{23}^{j-1}D_{31}^{i}D_{123}^{l}D_{213}^{m+1}-(j+m)D_{23}^{j-1}D_{31}^{i}D_{123}^{l+1}D_{213}^{m}+R(2j+2i+3l+3m),\\\
&[D_{23},D_{12}^{k}D_{31}^{i}D_{123}^{l}D_{213}^{m}]=-(k+l)D_{12}^{k-1}D_{31}^{i}D_{123}^{l}D_{213}^{m+1}+(k+m)D_{12}^{k-1}D_{31}^{i}D_{123}^{l+1}D_{213}^{m}\\\
&\,\,\,\,\,\,\,\,+(i+l)D_{12}^{k}D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}-(i+m)D_{12}^{k}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}+R(2k+2i+3l+3m).\\\
\end{split}$ (62)

Proof of Lemma 2.13. The proof is similar. For simplicity, we omit it here.

######  Proposition 2.14.

The center of $\mathbb{D}(SL_{3}(\mathbb{R})/A)$ is a polynomial ring in
$D_{123}+D_{213}$ and $D_{12}+D_{23}+D_{31}$.

Proof of Proposition 2.14. By Propositions 2.3, 2.4, 2.6, it is easy to verify
that $D_{123}+D_{213}$, $D_{12}+D_{23}+D_{31}$ are in the center.

We claim that the algebra generated by $D_{123}+D_{213}$ and
$D_{12}+D_{23}+D_{31}$ is a polynomial ring. Otherwise, there is a nontrivial
relation

$0=\sum_{\begin{subarray}{c}K,M\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}C_{KM}\cdot(D_{12}+D_{23}+D_{31})^{K}(D_{123}+D_{213})^{M}=:R.$
(63)

Define $A:=\max\\{\,K\,|\,C_{KM}\neq 0\,\\}$ and $B:=\max\\{\,M\,|\,C_{AM}\neq
0\,\\}$. Expand $R$ in terms of the basis in (48)

$\begin{split}R=&\sum_{\begin{subarray}{c}l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{000lm}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{k00lm}D_{12}^{k}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{0j0lm}D_{23}^{j}D_{123}^{l}D_{213}^{m}\\\
&+\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{+}\\\ l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{00ilm}D_{31}^{i}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}k,j\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{kj0lm}D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m}\\\
&+\sum_{\begin{subarray}{c}k,i\in\mathbb{Z}^{+}\\\ l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{k0ilm}D_{12}^{k}D_{31}^{i}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}j,i\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}r_{0jilm}D_{23}^{j}D_{31}^{i}D_{123}^{l}D_{213}^{m}.\\\
\end{split}$ (64)

It is easy to verify that

$r_{A000B}=C_{AB}\neq 0,$ (65)

which is a contradiction.

In what follows, we will prove that the center of
$\mathbb{D}(SL_{3}(\mathbb{R})/A)$ is generated by $D_{123}+D_{213}$ and
$D_{12}+D_{23}+D_{31}$. For each $z\in\mathbb{D}(SL_{3}(\mathbb{R})/A)$, write

$\begin{split}z=&\sum_{\begin{subarray}{c}l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{000lm}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{k00lm}D_{12}^{k}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0j0lm}D_{23}^{j}D_{123}^{l}D_{213}^{m}\\\
&+\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{+}\\\ l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{00ilm}D_{31}^{i}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}k,j\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{kj0lm}D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m}\\\
&+\sum_{\begin{subarray}{c}k,i\in\mathbb{Z}^{+}\\\ l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{k0ilm}D_{12}^{k}D_{31}^{i}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}j,i\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0jilm}D_{23}^{j}D_{31}^{i}D_{123}^{l}D_{213}^{m}.\\\
\end{split}$ (66)

For convenience, we will use the convention that $a_{KJILM}=0$ when $K,J,I,L$,
or $M$ is strictly negative.

Applying Lemmas 2.9, 2.10, we have

$\footnotesize\begin{split}0=&[D_{123},z]=\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{k00lm}\left(kD_{12}^{k}D_{23}D_{123}^{l}D_{213}^{m}-kD_{12}^{k}D_{31}D_{123}^{l}D_{213}^{m}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\
&+\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0j0lm}\left(-jD_{12}D_{23}^{j}D_{123}^{l}D_{213}^{m}+jD_{23}^{j}D_{31}D_{123}^{l}D_{213}^{m}\right)\\\
&+\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{00ilm}\left(iD_{12}D_{31}^{i}D_{123}^{l}D_{213}^{m}-iD_{23}D_{31}^{i}D_{123}^{l}D_{213}^{m}\right)\\\
\end{split}$ (67)
$\footnotesize\begin{split}\,\,&+\sum_{\begin{subarray}{c}k,j\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{kj0lm}\left(kD_{12}^{k}D_{23}^{j+1}D_{123}^{l}D_{213}^{m}+(j-k)D_{12}^{k-1}D_{23}^{j-1}D_{123}^{l+1}D_{213}^{m+1}-jD_{12}^{k+1}D_{23}^{j}D_{123}^{l}D_{213}^{m}\right)\\\
&+\sum_{\begin{subarray}{c}k,i\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{k0ilm}\left((k-i)D_{12}^{k-1}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m+1}-kD_{12}^{k}D_{31}^{i+1}D_{123}^{l}D_{213}^{m}+iD_{12}^{k+1}D_{31}^{i}D_{123}^{l}D_{213}^{m}\right)\\\
&+\sum_{\begin{subarray}{c}j,i\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0jilm}\left(jD_{23}^{j}D_{31}^{i+1}D_{123}^{l}D_{213}^{m}+(i-j)D_{23}^{j-1}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m+1}-iD_{23}^{j+1}D_{31}^{i}D_{123}^{l}D_{213}^{m}\right).\\\
\end{split}$

By the linear independence, we can conclude that, for $L,M\geq 0$ and $K,J\geq
1$,

$Ka_{K(J-1)0LM}+(J-K)a_{(K+1)(J+1)0(L-1)(M-1)}-Ja_{(K-1)J0LM}=0;$ (68)

for $L,M\geq 0$ and $K,I\geq 1$,

$(K-I)a_{(K+1)0(I+1)(L-1)(M-1)}-Ka_{K0(I-1)LM}+Ia_{(K-1)0ILM}=0;$ (69)

for $L,M\geq 0$ and $J,I\geq 1$,

$Ja_{0J(I-1)LM}+(I-J)a_{0(J+1)(I+1)(L-1)(M-1)}-Ia_{0(J-1)ILM}=0.$ (70)

We have by Lemmas 2.12, 2.13 that

$\footnotesize\begin{split}0&=[D_{12},z]=\sum_{\begin{subarray}{c}l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{000lm}\left(-lD_{12}D_{23}D_{123}^{l-1}D_{213}^{m}+lD_{12}D_{31}D_{123}^{l-1}D_{213}^{m}+mD_{12}D_{23}D_{123}^{l}D_{213}^{m-1}\right.\\\
&-m\left.D_{12}D_{31}D_{123}^{l}D_{213}^{m-1}\right)+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{k00lm}\left(-lD_{12}^{k+1}D_{23}D_{123}^{l-1}D_{213}^{m}+lD_{12}^{k+1}D_{31}D_{123}^{l-1}D_{213}^{m}\right.\\\
&+mD_{12}^{k+1}D_{23}D_{123}^{l}D_{213}^{m-1}\left.-mD_{12}^{k+1}D_{31}D_{123}^{l}D_{213}^{m-1}\right)+\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0j0lm}\left(-lD_{12}D_{23}^{j+1}D_{123}^{l-1}D_{213}^{m}\right.\,\,\,\\\
&+mD_{12}D_{23}^{j+1}D_{123}^{l}D_{213}^{m-1}\left.+(j+l)D_{23}^{j-1}D_{123}^{l}D_{213}^{m+1}-(j+m)D_{23}^{j-1}D_{123}^{l+1}D_{213}^{m}\right)\\\
&+\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{00ilm}\left(lD_{12}D_{31}^{i+1}D_{123}^{l-1}D_{213}^{m}-mD_{12}D_{31}^{i+1}D_{123}^{l}D_{213}^{m-1}-(i+l)D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}\right.\\\
\end{split}$ (71)
$\footnotesize\begin{split}\,\,&\left.+(i+m)D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}\right)+\sum_{\begin{subarray}{c}k,j\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{kj0lm}\left(-lD_{12}^{k+1}D_{23}^{j+1}D_{123}^{l-1}D_{213}^{m}\right.\\\
&\left.+mD_{12}^{k+1}D_{23}^{j+1}D_{123}^{l}D_{213}^{m-1}+(j+l)D_{12}^{k}D_{23}^{j-1}D_{123}^{l}D_{213}^{m+1}-(j+m)D_{12}^{k}D_{23}^{j-1}D_{123}^{l+1}D_{213}^{m}\right)\\\
&+\sum_{k,i\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}}a_{k0ilm}\left(lD_{12}^{k+1}D_{31}^{i+1}D_{123}^{l-1}D_{213}^{m}-mD_{12}^{k+1}D_{31}^{i+1}D_{123}^{l}D_{213}^{m-1}-(i+l)D_{12}^{k}D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}\right.\\\
&\left.+(i+m)D_{12}^{k}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}\right)+\sum_{\begin{subarray}{c}j,i\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0jilm}\left(-(i+l)D_{23}^{j}D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}\right.\\\
&\left.+(i+m)D_{23}^{j}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}+(j+l)D_{23}^{j-1}D_{31}^{i}D_{123}^{l}D_{213}^{m+1}-(j+m)D_{23}^{j-1}D_{31}^{i}D_{123}^{l+1}D_{213}^{m}\right).\\\
\end{split}$

Then for $L,M\geq 0$ and $K,I\geq 1$,

$\begin{split}&(L+1)a_{(K-1)0(I-1)(L+1)M}-(M+1)a_{(K-1)0(I-1)L(M+1)}-(I+L+1)a_{K0(I+1)L(M-1)}\,\,\,\,\,\,\\\
&\,\,\,\,\,\,\,\,+(I+M+1)a_{K0(I+1)(L-1)M}=0.\end{split}$ (72)

Similarly,

$\footnotesize\begin{split}&0=[D_{23},z]=\sum_{\begin{subarray}{c}l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{000lm}\left(lD_{12}D_{23}D_{123}^{l-1}D_{213}^{m}-lD_{23}D_{31}D_{123}^{l-1}D_{213}^{m}-mD_{12}D_{23}D_{123}^{l}D_{213}^{m-1}\right.\\\
&+m\left.D_{23}D_{31}D_{123}^{l}D_{213}^{m-1}\right)+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{k00lm}\left(lD_{12}^{k+1}D_{23}D_{123}^{l-1}D_{213}^{m}-mD_{12}^{k+1}D_{23}D_{123}^{l}D_{213}^{m-1}\right.\\\
&-(k+l)D_{12}^{k-1}D_{123}^{l}D_{213}^{m+1}\left.+(k+m)D_{12}^{k-1}D_{123}^{l+1}D_{213}^{m}\right)+\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0j0lm}\left(lD_{12}D_{23}^{j+1}D_{123}^{l-1}D_{213}^{m}\right.\\\
&-lD_{23}^{j+1}D_{31}D_{123}^{l-1}D_{213}^{m}\left.-mD_{12}D_{23}^{j+1}D_{123}^{l}D_{213}^{m-1}+mD_{23}^{j+1}D_{31}D_{123}^{l}D_{213}^{m-1}\right)\\\
&+\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{00ilm}\left(-lD_{23}D_{31}^{i+1}D_{123}^{l-1}D_{213}^{m}+mD_{23}D_{31}^{i+1}D_{123}^{l}D_{213}^{m-1}+(i+l)D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}\right.\\\
&-(i+m)\left.D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}\right)+\sum_{\begin{subarray}{c}k,j\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{kj0lm}\left(lD_{12}^{k+1}D_{23}^{j+1}D_{123}^{l-1}D_{213}^{m}-mD_{12}^{k+1}D_{23}^{j+1}D_{123}^{l}D_{213}^{m-1}\right.\,\,\,\,\,\,\,\\\
\end{split}$ (73)
$\footnotesize\begin{split}&-(k+l)D_{12}^{k-1}D_{23}^{j}D_{123}^{l}D_{213}^{m+1}\left.+(k+m)D_{12}^{k-1}D_{23}^{j}D_{123}^{l+1}D_{213}^{m}\right)\\\
&+\sum_{\begin{subarray}{c}k,i\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{k0ilm}\left(-(k+l)D_{12}^{k-1}D_{31}^{i}D_{123}^{l}D_{213}^{m+1}+(k+m)D_{12}^{k-1}D_{31}^{i}D_{123}^{l+1}D_{213}^{m}\right.\\\
&+(i+l)D_{12}^{k}D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}\left.-(i+m)D_{12}^{k}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}\right)+\sum_{\begin{subarray}{c}j,i\in\mathbb{Z}^{+},\,\,l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0jilm}\left(-lD_{23}^{j+1}D_{31}^{i+1}D_{123}^{l-1}D_{213}^{m}\right.\,\,\\\
&+mD_{23}^{j+1}D_{31}^{i+1}D_{123}^{l}D_{213}^{m-1}\left.+(i+l)D_{23}^{j}D_{31}^{i-1}D_{123}^{l}D_{213}^{m+1}-(i+m)D_{23}^{j}D_{31}^{i-1}D_{123}^{l+1}D_{213}^{m}\right).\\\
\end{split}$

Then for $L,M\geq 0$ and $K,J\geq 1$,

$\begin{split}&(L+1)a_{(K-1)(J-1)0(L+1)M}-(M+1)a_{(K-1)(J-1)0L(M+1)}-(K+L+1)a_{(K+1)J0L(M-1)}\\\
&\,\,\,\,\,\,\,\,+(K+M+1)a_{(K+1)J0(L-1)M}=0;\end{split}$ (74)

for $L,M\geq 0$ and $J,I\geq 1$,

$\begin{split}&-(L+1)a_{0(J-1)(I-1)(L+1)M}+(M+1)a_{0(J-1)(I-1)L(M+1)}+(I+L+1)a_{0J(I+1)L(M-1)}\,\,\,\,\,\,\\\
&\,\,\,\,\,\,\,\,-(I+M+1)a_{0J(I+1)(L-1)M}=0.\end{split}$ (75)

Claim I. If $a_{K000M}=0$ for $K,M\geq 0$, then $a_{KJ0LM}\equiv 0$ for
$K,J,L,M\geq 0$.

Proof of Claim I. We prove by induction on $L$. Setting $L=0$ in (68), we have
for $M\geq 0$ and $K,J\geq 1$ that

$Ka_{K(J-1)00M}=Ja_{(K-1)J00M}.$ (76)

Hence, when $M\geq 0$ and $K,J\geq 1$,

$a_{(K-1)J00M}=\frac{(K+J-1)!}{J!\cdot(K-1)!}\cdot a_{(K+J-1)000M}=0.$ (77)

We thus prove Claim I when $L=0$.

Suppose Claim I holds for $0\leq L\leq l$. By setting $L=l$ in (74), we can
conclude that for $M\geq 0$ and $K,J\geq 1$,

$a_{(K-1)(J-1)0(l+1)M}=0.$ (78)

Then, Claim I holds for $L=l+1$. Therefore, we complete the proof of Claim I.

Claim II. If $a_{K000M}=0$ for $K,M\geq 0$, then $a_{K0ILM}\equiv 0$ for
$K,I,L,M\geq 0$.

Proof of Claim II. Similarly, we prove by induction on $L$. Setting $L=0$ in
(69),

$Ka_{K0(I-1)0M}=Ia_{(K-1)0I0M},\,\,M\geq 0,K\geq 1,I\geq 1,$ (79)

and hence

$a_{(K-1)0I0M}=\frac{(K+I-1)!}{I!\cdot(K-1)!}\cdot a_{(K+I-1)000M}=0,\,\,M\geq
0,K\geq 1,I\geq 1.$ (80)

Suppose Claim II holds for $0\leq L\leq l$. Setting $L=l$ in (72), we can
conclude that

$a_{(K-1)0(I-1)(l+1)M}=0,\,\,M\geq 0,K\geq 1,I\geq 1.$ (81)

We complete the proof of Claim II.

Claim III. If $a_{K000M}=0$ for $K,M\geq 0$, then $a_{0JILM}\equiv 0$ for
$K,J,L,M\geq 0$.

Proof of Claim III. By Claim II, it is clear that for $I,M\geq 0$

$a_{00I0M}=0.$ (82)

Similarly to the proofs of Claims I and II, we can then complete the proof of
Claim III by (70) and (75).

Define

$z^{\prime}:=z-\sum_{\begin{subarray}{c}K,M\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{K000M}(D_{12}+D_{23}+D_{31})^{K}(D_{123}+D_{213})^{M}.$
(83)

Then $z^{\prime}$ belongs to the center and can be written as

$\begin{split}z^{\prime}=&\sum_{\begin{subarray}{c}l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{000lm}^{\prime}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}k\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{k00lm}^{\prime}D_{12}^{k}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0j0lm}^{\prime}D_{23}^{j}D_{123}^{l}D_{213}^{m}\\\
&+\sum_{\begin{subarray}{c}i\in\mathbb{Z}^{+}\\\ l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{00ilm}^{\prime}D_{31}^{i}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}k,j\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{kj0lm}^{\prime}D_{12}^{k}D_{23}^{j}D_{123}^{l}D_{213}^{m}\\\
&+\sum_{\begin{subarray}{c}k,i\in\mathbb{Z}^{+}\\\ l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{k0ilm}^{\prime}D_{12}^{k}D_{31}^{i}D_{123}^{l}D_{213}^{m}+\sum_{\begin{subarray}{c}j,i\in\mathbb{Z}^{+}\\\
l,m\in\mathbb{Z}{{}^{\geq
0}}\end{subarray}}a_{0jilm}^{\prime}D_{23}^{j}D_{31}^{i}D_{123}^{l}D_{213}^{m}.\\\
\end{split}$ (84)

such that for $K,M\geq 0$

$a^{\prime}_{K000M}=0.$ (85)

Now by Claims I, II, III, we can conclude that

$z^{\prime}=0.$ (86)

Therefore, we complete the proof of Theorem 2.14.

Proof of Theorem 1.1. It follows from Propositions 2.2, 2.3, 2.4, 2.5, 2.6,
2.8, 2.14.

##  3\. Essential Self-Adjointness

###  3.1. Essential self-adjointness of the central elements

A classical result due to [Seg], [NS] says that if $\tau$ is the quasi-regular
representation of $G$ in $L^{2}(G/H)$, $D$ is a symmetric element in the
center of the universal enveloping algebra, and the operator $\tau(D)$ is an
invariant differential operator on $X$, then $\tau(D)$ is essentially self-
adjoint. It leads to a proof of Proposition 1.2 as follows.

Proof of Proposition 1.2. Recall that the center of
$U(\mathfrak{sl}_{3}(\mathbb{R}))$ is a polynomial ring $Z=\mathbb{R}[h,k]$
(see Page 984 of [Ca]), where

$h=-(X_{11}-X_{22})^{2}+(X_{11}-X_{22})X_{11}-X_{11}^{2}-3E_{12}E_{21}-3E_{13}E_{31}-3E_{23}E_{32}+3X_{11},$
(87)
$\begin{split}&k=2(X_{11}-X_{22})^{3}-3(X_{11}-X_{22})^{2}X_{11}-3(X_{11}-X_{22})X_{11}^{2}+2X_{11}^{3}\\\
&\,\,\,\,\,\,\,\,\,+9E_{12}E_{21}(X_{11}-X_{22})-18E_{12}E_{21}X_{11}-18E_{13}E_{31}(X_{11}-X_{22})+9E_{13}E_{31}X_{11}\,\,\,\,\,\,\,\,\,\,\,\,\\\
&\,\,\,\,\,\,\,\,\,+9E_{23}E_{32}(X_{11}-X_{22})+9E_{23}E_{32}X_{11}-27E_{12}E_{23}E_{31}-27E_{21}E_{13}E_{32}\\\
&\,\,\,\,\,\,\,\,\,+18(X_{11}-X_{22})X_{11}-9X_{11}^{2}-18(X_{11}-X_{22})+9X_{11}.\end{split}$
(88)

It is clear that $h,k\in\mathbb{D}^{A}(SL_{3}(\mathbb{R}))$, and by (31) and
(32)

$\begin{split}&\mu(h)=-3\mu(E_{12}E_{21})-3\mu(E_{13}E_{31})-3\mu(E_{23}E_{32})=-3(D_{12}+D_{13}+D_{23}),\\\
&\mu(k)=-27\mu(E_{12}E_{31}E_{23})-27\mu(E_{21}E_{13}E_{32})=-27(D_{123}+D_{213}).\\\
\end{split}$ (89)

By the above result due to [Seg], [NS], we complete the proof of Proposition
1.2.

###  3.2. Reduction to the density of $C^{\infty}_{c}(X)$ in
$\text{Dom}(\Delta)$

Assume that $X:=G/H$ is a homogeneous space with a $G$-invariant measure ${\rm
d}\mu$. Denote by $C^{\infty}(X)$ the space of smooth functions on $X$, and by
$C^{\infty}_{c}(X)$ the space of functions in $C^{\infty}(X)$ with compact
support.

Assume $\Delta\in\mathbb{D}(X)$ is formally self-adjoint, i.e., such that the
following equality holds

$(\Delta f,g)=(f,\Delta g)$ (90)

for all $f,g\in C_{c}^{\infty}(X)$, which also implies that the above equality
also holds for all $f\in C_{c}^{\infty}(X)$ and $g$ in the space
$C_{c}^{\infty}(X)^{*}$ of distributions (see Section 2.3 Ch. II of [He]). It
is clear that $\Delta$ is naturally defined on

$\text{Dom}(\Delta):=\left\\{f\in L^{2}(X):\Delta f\in L^{2}(X)\right\\}.$
(91)

It is also clear that the graph of the operator $\Delta:\text{Dom}(\Delta)\to
L^{2}(X)$ is closed.

######  Lemma 3.1.

Suppose that $\Delta$ is a formally self-adjoint differential operator on $X$.
If the graph of $\Delta$ on the domain $C^{\infty}_{c}(X)$ is dense in the
graph of $\Delta$ on the domain ${\rm Dom}(\Delta)$ with respect to the
$L^{2}\times L^{2}$ norm, $\Delta$ is symmetric on ${\rm Dom}(\Delta)$, that
is,

$(\Delta f,g)=(f,\Delta g)\,\,\,{\rm for\,\,all\,\,}f,g\in{\rm Dom}(\Delta).$
(92)

Proof of Lemma 3.1. Since $C^{\infty}_{c}(X)$ is dense in ${\rm Dom}(\Delta)$
with respect to the graph $L^{2}\times L^{2}$ norm, for $f,g\in{\rm
Dom}(\Delta)$ there exist sequences $\\{f_{n}\\}_{n=1}^{\infty}$,
$\\{g_{n}\\}_{n=1}^{\infty}\subset C^{\infty}_{c}(X)$, such that

$\lim_{n\rightarrow\infty}\|f_{n}-f\|_{L^{2}}=\lim_{n\rightarrow\infty}\|\Delta
f_{n}-\Delta
f\|_{L^{2}}=\lim_{n\rightarrow\infty}\|g_{n}-g\|_{L^{2}}=\lim_{n\rightarrow\infty}\|\Delta
g_{n}-\Delta g\|_{L^{2}}=0.$ (93)

Then,

$\begin{split}&|(\Delta f,g)-(f,\Delta g)|\leq|(\Delta f,g)-(\Delta
f,g_{n})|+|(\Delta f,g_{n})-(\Delta f_{n},g_{n})|\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+|(\Delta f_{n},g_{n})-(f_{n},\Delta
g_{n})|+|(f_{n},\Delta g_{n})-(f_{n},\Delta g)|+|(f_{n},\Delta g)-(f,\Delta
g)|\\\ &=|(\Delta f,g-g_{n})|+|(\Delta f-\Delta f_{n},g_{n})|+|(f_{n},\Delta
g_{n}-\Delta g)|+|(f_{n}-f,\Delta g)|\\\ &\leq\|\Delta
f\|_{L^{2}}\cdot\|g-g_{n}\|_{L^{2}}+\|\Delta f-\Delta
f_{n}\|_{L^{2}}\cdot\|g_{n}\|_{L^{2}}+\|f_{n}\|_{L^{2}}\cdot\|\Delta
g_{n}-\Delta g\|_{L^{2}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\|f_{n}-f\|_{L^{2}}\cdot\|\Delta
g\|_{L^{2}}.\\\ \end{split}$ (94)

Taking the limit of (94) as $n$ approaches infinity, we can conclude Lemma
3.1.

######  Lemma 3.2.

Assume that the operator $\Delta$ is symmetric on ${\rm Dom}(\Delta)$, that
is,

$(\Delta f,g)=(f,\Delta g)\,\,\,{\rm for\,\,all\,\,}f,g\in{\rm Dom}(\Delta).$
(95)

Then $\Delta$ is self-adjoint on ${\rm Dom}(\Delta)$.

Proof of Lemma 3.2. It suffices to prove that
$\text{Dom}(\Delta^{*})=\text{Dom}(\Delta)$. Firstly, the symmetric property
(95) implies the domain $\text{Dom}(\Delta^{*})$ of the adjoint $\Delta^{*}$
of $\Delta$ contains $\text{Dom}(\Delta)$.

Next, we shall prove that $\text{Dom}(\Delta^{*})\subset\text{Dom}(\Delta)$.
By definition, $g\in\text{Dom}(\Delta^{*})$ means $g\in L^{2}$ and there exist
$g^{*}\in L^{2}$ such that $(\Delta f,g)=(f,g^{*})$ holds for all
$f\in\text{Dom}(\Delta)$. In particular, $(\Delta f,g)=(f,g^{*})$ holds for
all $f\in C_{c}^{\infty}(X)$, so that $g^{*}=\Delta g$ as distribution thus
also in $L^{2}$. Therefore, $g\in{\rm Dom}(\Delta)$.

We complete the proof of Lemma 3.2.

###  3.3. Coordinate charts induced by the Euler angles

In this subsection, we describe certain geometric properties of
$X=SL_{3}(\mathbb{R})/A$ for explicit computation. For convenience, in the
following, we will use $X$ and $SL_{3}(\mathbb{R})/A$ interchangeably.

According to the Iwasawa decomposition $SL_{3}(\mathbb{R})=KNA$, each $x\in
SL_{3}(\mathbb{R})$ can be uniquely written as $x=k_{x}n_{x}a_{x}$, where
$k_{x}$ is an unimodular orthogonal matrix, $n_{x}$ is an upper unitriangular
matrix, and $a_{x}$ is a unimodular diagonal matrix with positive entries;
more explicitly,

$x=\left(\begin{matrix}k_{11}&k_{12}&k_{13}\\\ k_{21}&k_{22}&k_{23}\\\
k_{31}&k_{32}&k_{33}\end{matrix}\right)\left(\begin{matrix}1&z_{12}&z_{13}\\\
0&1&z_{23}\\\ 0&0&1\\\
\end{matrix}\right)\left(\begin{matrix}\lambda_{1}&0&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{1}^{-1}\lambda_{2}^{-1}\end{matrix}\right).$ (96)

Since $SL_{3}(\mathbb{R})/A$ is diffeomorphic to
$SO_{3}(\mathbb{R})\times\mathbb{R}^{3}$, without loss of generality, we can
identify each $x\in SL_{3}(\mathbb{R})/A$ with
$(k_{x},z_{12},z_{13},z_{23})\in SO_{3}(\mathbb{R})\times\mathbb{R}^{3}$.

We parametrize $SO_{3}(\mathbb{R})$ by the following variant of Euler angles
(Tait-Bryan angles). Define
$\Gamma:[-\pi,\pi]\times[-\frac{\pi}{2},\frac{\pi}{2}]\times[-\pi,\pi]\rightarrow
SO_{3}(\mathbb{R})$ by $(\alpha,\beta,\gamma)\mapsto k_{x}$, where

$\footnotesize k_{x}:=\left(\begin{matrix}k_{11}&k_{12}&k_{13}\\\
k_{21}&k_{22}&k_{23}\\\
k_{31}&k_{32}&k_{33}\end{matrix}\right):=\left(\begin{matrix}\cos{\beta}\cos{\gamma}&-\sin{\beta}&\cos{\beta}\sin{\gamma}\\\
\sin{\alpha}\sin{\gamma}+\cos{\alpha}\cos{\gamma}\sin{\beta}&\cos{\alpha}\cos{\beta}&\cos{\alpha}\sin{\beta}\sin{\gamma}-\cos{\gamma}\sin{\alpha}\\\
\cos{\gamma}\sin{\alpha}\sin{\beta}-\cos{\alpha}\sin{\gamma}&\cos{\beta}\sin{\alpha}&\cos{\alpha}\cos{\gamma}+\sin{\alpha}\sin{\beta}\sin{\gamma}\end{matrix}\right).$
(97)

It is easy to verify that when $-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$,
$-\frac{\pi}{3}<\beta<\frac{\pi}{3}$, $-\frac{\pi}{2}<\gamma<\frac{\pi}{2}$,
$\Gamma$ is bijective onto the image, and the inverse map is given by

$\alpha=\arctan\left({\frac{k_{32}}{k_{22}}}\right),\,\,\beta=\arctan\left(\frac{-k_{12}}{\sqrt{1-k_{12}^{2}}}\right),\,\,\gamma=\arctan\left({\frac{k_{13}}{k_{11}}}\right).$
(98)

######  Definition 3.3.

We denote by $\underline{U}_{EI}$ the image under $\Gamma$ of
$-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$, $-\frac{\pi}{3}<\beta<\frac{\pi}{3}$,
$-\frac{\pi}{2}<\gamma<\frac{\pi}{2}$. It is clear that $\underline{U}_{EI}$
is a neighborhood of the identity element in $SO_{3}(\mathbb{R})$.

Denote by $(k_{0}\cdot\underline{U}_{EI})$ the left translation of
$\underline{U}_{EI}$ by $k_{0}$. Then, we can construct an atlas
$\left\\{\left(\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3},\Lambda_{k_{0}}\right)\right\\}_{k_{0}\in
SO_{3}(\mathbb{R})}$ of $SL_{3}(\mathbb{R})/A$ as follows. For each $k_{0}\in
SO_{3}(\mathbb{R})$, define a map
$\Lambda_{k_{0}}:\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3}\rightarrow(-\frac{\pi}{2},\frac{\pi}{2})\times(-\frac{\pi}{3},\frac{\pi}{3})\times(-\frac{\pi}{2},\frac{\pi}{2})\times\mathbb{R}^{3}$
by

$\left(k_{0}\cdot\left(\begin{matrix}k_{11}&k_{12}&k_{13}\\\
k_{21}&k_{22}&k_{23}\\\
k_{31}&k_{32}&k_{33}\end{matrix}\right),z_{12},z_{13},z_{23}\right)\mapsto(\alpha,\beta,\gamma,z_{12},z_{13},z_{23}),\\\
$ (99)

where $\alpha,\beta,\gamma$ are related to $(k_{ij})$ by (98).

######  Definition 3.4.

We call the coordinates $(\alpha,\beta,\gamma,z_{12},z_{13},z_{23})$, which is
associated with the coordinate chart
$\left(\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3},\Lambda_{k_{0}}\right)$
as above, the Euler-Iwasawa coordinates of $SL_{3}(\mathbb{R})/A$ attached to
$k_{0}$.

Notice that the Euler-Iwasawa coordinates attached to $k_{0}$, combined with
the global coordinates $(\lambda_{1},\lambda_{2})$ of $A$ (see (96)),
parametrize the open neighborhood
$\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3}\times\mathbb{R}^{2}\subset
SL_{3}(\mathbb{R})$. We denote the corresponding atlas of $SL_{3}(\mathbb{R})$
by
$\left\\{\left(\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3}\times\mathbb{R}^{2},\widehat{\Lambda}_{k_{0}}\right)\right\\}_{k_{0}\in
SO_{3}(\mathbb{R})}$.

######  Definition 3.5.

We call the coordinates $(\alpha,\beta,\gamma,z_{12}$,
$z_{13},z_{23},\lambda_{1},\lambda_{2})$, which is associated with the
coordinate chart
$\left(\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3}\times\mathbb{R}^{2},\widehat{\Lambda}_{k_{0}}\right)$
as above, the Euler-Iwasawa coordinates of $SL_{3}(\mathbb{R})$ attached to
$k_{0}$.

One can verify that an invariant measure on $X$ in the coordinate chart
$\left(\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3},\Lambda_{k_{0}}\right)$
takes the form

${\rm d}\mu={\rm d}k\ {\rm d}n=\cos\beta\,{\rm d}\alpha\ {\rm d}\beta\ {\rm
d}\gamma\ {\rm d}z_{12}\ {\rm d}z_{13}\ {\rm d}z_{23}.$ (100)

###  3.4. Presence of left derivatives in $D_{12}$

We first recall the notion of left derivatives as in [Ba]. Notice that the
infinitesimal action $L$ on $C^{\infty}(SL_{3}(\mathbb{R}))$ induced by the
left regular representation, maps $U(\mathfrak{sl}_{3}(\mathbb{R}))$ into the
algebra of differentials on $SL_{3}(\mathbb{R})$. More precisely, for
$u=X_{1}X_{2}\cdots X_{k}$,
$X_{1},X_{2},\cdots,X_{k}\in\mathfrak{sl}_{3}(\mathbb{R})$, and $f\in
C^{\infty}(SL_{3}(\mathbb{R}))$,

$(L_{u}f)(g):=\left.\frac{\partial}{\partial
t_{1}}\right|_{t_{1}=0}\cdots\left.\frac{\partial}{\partial
t_{k}}\right|_{t_{k}=0}f(\exp\left(-t_{k}X_{k}\right)\cdots\exp\left(-t_{1}X_{1}\right)g).$
(101)

Then, $L$ also induces differential operators on $SL_{3}(\mathbb{R})/A$, that
is, for $X_{1},X_{2},\cdots,X_{k}\in\mathfrak{sl}_{3}(\mathbb{R})$, and $f\in
C^{\infty}(SL_{3}(\mathbb{R})/A)$,

$\left(L\left(X_{1}\cdots X_{k}\right)\cdot
f\right)(xA):=\left.\frac{\partial}{\partial
t_{1}}\right|_{t_{1}=0}\cdots\left.\frac{\partial}{\partial
t_{k}}\right|_{t_{k}=0}f(\exp\left(-t_{k}X_{k}\right)\cdots\exp\left(-t_{1}X_{1}\right)xA).$
(102)

######  Definition 3.6.

We call a differential operator on $SL_{3}(\mathbb{R})/A$ a left derivative if
it is induced by an element $u\in U(\mathfrak{sl}_{3}(\mathbb{R}))$ through
the above infinitesimal action $L$. For convenience, denote left derivatives
by $L_{u}$, $u\in U(\mathfrak{sl}_{3}(\mathbb{R}))$, as well.

######  Lemma 3.7.

In each coordinate chart
$\left(\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3},\Lambda_{k_{0}}\right)$,
$D_{12}$ takes the form

$\begin{split}&D_{12}=\left(\sec\beta\sin\gamma\right)\frac{\partial^{2}}{\partial\alpha\partial
z_{12}}+\left(\cos\gamma\right)\frac{\partial^{2}}{\partial\beta\partial
z_{12}}+\left(\tan\beta\sin\gamma\right)\frac{\partial^{2}}{\partial\gamma\partial
z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,+\left(z_{12}^{2}+1\right)\frac{\partial^{2}}{\partial
z_{12}\partial z_{12}}+\left(z_{23}\right)\frac{\partial^{2}}{\partial
z_{13}\partial z_{12}}+\left(-z_{13}\right)\frac{\partial^{2}}{\partial
z_{23}\partial z_{12}}+2z_{12}\frac{\partial}{\partial z_{12}},\\\
\end{split}$ (103)

where $(\alpha,\beta,\gamma,z_{12},z_{13},z_{23})$ are the Euler-Iwasawa
coordinates attached to $k_{0}$.

######  Remark 3.8.

(103) is independent of $k_{0}$, which is also a consequence of $D_{12}$ being
$G$-invariant.

Proof of Lemma 3.7. According to Theorem 2.1, we first compute the left-
invariant differential $\widehat{D}_{12}$ on $SL_{3}(\mathbb{R})$ associated
with $D_{12}$, where, by Proposition 2.2,

$\widehat{D}_{12}=\frac{1}{2}\left(R\left(E_{12}\right)R\left(E_{21}\right)+R\left(E_{21}\right)R\left(E_{12}\right)\right).$
(104)

Consider the Euler-Iwasawa coordinates $(\alpha,\beta,\gamma$,
$z_{12},z_{13},z_{23},\lambda_{1},\lambda_{2})$ attached to $k_{0}$ (see
Definition 3.5). Then, by Lemmas A.7, A.10 in Appendix A.2,

$\footnotesize\begin{split}&\widehat{D}_{12}=\frac{1}{2}\left(\frac{\lambda_{2}}{\lambda_{1}}\left\\{\sec\beta\sin\gamma\frac{\partial}{\partial\alpha}+\cos\gamma\frac{\partial}{\partial\beta}+\tan\beta\sin\gamma\frac{\partial}{\partial\gamma}+\left(z_{12}^{2}+1\right)\frac{\partial}{\partial
z_{12}}+z_{23}\frac{\partial}{\partial z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}\right\\}\right.\\\
&\,\,\,\,\,\,\,\,\,\,\left.+\lambda_{2}z_{12}\frac{\partial}{\partial\lambda_{1}}-\frac{\lambda_{2}^{2}}{\lambda_{1}}z_{12}\frac{\partial}{\partial\lambda_{2}}\right)\left(\frac{\lambda_{1}}{\lambda_{2}}\frac{\partial}{\partial
z_{12}}\right)+\frac{1}{2}\left(\frac{\lambda_{1}}{\lambda_{2}}\frac{\partial}{\partial
z_{12}}\right)\left(\frac{\lambda_{2}}{\lambda_{1}}\left\\{\sec\beta\sin\gamma\frac{\partial}{\partial\alpha}+\cos\gamma\frac{\partial}{\partial\beta}\right.\right.\\\
&\,\,\,\,\,\,\,\,\,\,\left.\left.+\tan\beta\sin\gamma\frac{\partial}{\partial\gamma}+\left(z_{12}^{2}+1\right)\frac{\partial}{\partial
z_{12}}+z_{23}\frac{\partial}{\partial z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}\right\\}+\lambda_{2}z_{12}\frac{\partial}{\partial\lambda_{1}}-\frac{\lambda_{2}^{2}}{\lambda_{1}}z_{12}\frac{\partial}{\partial\lambda_{2}}\right)\\\
&=\left(\sec\beta\sin\gamma\right)\frac{\partial^{2}}{\partial\alpha\partial
z_{12}}+\left(\cos\gamma\right)\frac{\partial^{2}}{\partial\beta\partial
z_{12}}+\left(\tan\beta\sin\gamma\right)\frac{\partial^{2}}{\partial\gamma\partial
z_{12}}+\left(z_{12}^{2}+1\right)\frac{\partial^{2}}{\partial z_{12}\partial
z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,+\left(z_{23}\right)\frac{\partial^{2}}{\partial
z_{13}\partial z_{12}}+\left(-z_{13}\right)\frac{\partial^{2}}{\partial
z_{23}\partial z_{12}}+2z_{12}\frac{\partial}{\partial z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,+\left(\lambda_{1}z_{12}\right)\frac{\partial^{2}}{\partial
z_{12}\partial\lambda_{1}}-\left(\lambda_{2}z_{12}\right)\frac{\partial^{2}}{\partial
z_{12}\partial\lambda_{2}}+\left(\frac{\lambda_{1}}{2}\right)\frac{\partial}{\partial\lambda_{1}}-\left(\frac{\lambda_{2}}{2}\right)\frac{\partial}{\partial\lambda_{2}}\\\
\end{split}$ (105)

Let $f$ be an arbitary smooth function on $SL_{3}(\mathbb{R})/A$ and
$\widetilde{f}$ its lift to $SL_{3}(\mathbb{R})$ via the natural projection
$\pi:SL_{3}(\mathbb{R})\rightarrow X$. Then,

$\begin{split}&(D_{12}f)(\alpha,\beta,\gamma,z_{12},z_{13},z_{23})=\left(\widehat{D}_{12}\widetilde{f}\right)(\alpha,\beta,\gamma,z_{12},z_{13},z_{23},\lambda_{1},\lambda_{2})\\\
&=\left(\sec\beta\sin\gamma\right)\frac{\partial^{2}}{\partial\alpha\partial
z_{12}}+\left(\cos\gamma\right)\frac{\partial^{2}}{\partial\beta\partial
z_{12}}+\left(\tan\beta\sin\gamma\right)\frac{\partial^{2}}{\partial\gamma\partial
z_{12}}+\left(z_{12}^{2}+1\right)\frac{\partial^{2}}{\partial z_{12}\partial
z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,+\left(z_{23}\right)\frac{\partial^{2}}{\partial
z_{13}\partial z_{12}}+\left(-z_{13}\right)\frac{\partial^{2}}{\partial
z_{23}\partial z_{12}}+2z_{12}\frac{\partial}{\partial z_{12}}.\\\
\end{split}$ (106)

We complete the proof of Lemma (3.7).

Next, we introduce a special class $\mathcal{P}$ of linear differential
operators defined on $\underline{U}_{EI}\times\mathbb{R}^{3}$, in terms of the
Euler-Iwasawa coordinates $(\alpha,\beta,\gamma,z_{12},z_{13}$, $z_{23})$.

######  Definition 3.9.

The class $\mathcal{P}$ consists of

$\sum_{\begin{subarray}{c}u_{i}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}\frac{P_{i}\left(\sin\alpha,\cos\alpha,\sin\beta,\cos\beta,\sin\gamma,\cos\gamma\right)}{\cos^{l_{i}}\beta}L_{u_{i}},$
(107)

where $l_{i}\geq 0$, $P_{i}$ are polynomials in trigonometric functions,
$L_{u_{i}}$ are left derivatives on $X$.

######  Lemma 3.10.

If $D_{1},D_{2}\in\mathcal{P}$, then $D_{1}\circ D_{2}\in\mathcal{P}$, where

$(D_{1}\circ D_{2})f:=D_{1}(D_{2}f),\,\,f\in C^{\infty}(X).$ (108)

Proof of Lemma 3.10. Notice that $L_{X_{12}}$, $L_{X_{13}}$, $L_{X_{23}}$,
$L_{X_{21}}$, $L_{X_{31}}$, $L_{X_{32}}$ constitute a set of generators of all
left derivatives. By explicit formulas of $L_{X_{ij}}$ in Lemmas A.1, A.2,
A.3, A.4, A.5, A.6 in Appendix A.1, we can verify Lemma 3.10.

######  Lemma 3.11.

Each differential operator $D\in\mathcal{P}$ can be written as

$D=\sum_{\begin{subarray}{c}u_{i}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}r_{i}L_{u_{i}},$ (109)

where $r_{i}$ are bounded functions on
$\underline{U}_{EI}\times\mathbb{R}^{3}\subset SL_{3}(\mathbb{R})/A$, and
$L_{u_{i}}$ are left derivatives on $X$.

Proof of Lemma 3.11. It follows easily from Definition 3.9.

######  Lemma 3.12.

The following differential operators are in $\mathcal{P}$.

$\left\\{\begin{aligned}
&\frac{\partial}{\partial\alpha},\,\,\frac{\partial}{\partial\beta},\,\,\frac{\partial}{\partial\gamma},\,\,\frac{\partial}{\partial
z_{13}},\,\,\frac{\partial}{\partial z_{23}},\\\
&z_{12}\frac{\partial}{\partial z_{12}}-z_{23}\frac{\partial}{\partial
z_{23}},\,\,\frac{\partial}{\partial z_{12}}+z_{23}\frac{\partial}{\partial
z_{13}},\,\,z_{13}\frac{\partial}{\partial
z_{13}}+z_{23}\frac{\partial}{\partial z_{23}}.\\\ \end{aligned}\right.$ (110)

Proof of Lemma 3.12. By Lemmas A.4, A.5, A.6 in Appendix A.1, we have

$\begin{split}&\frac{\partial}{\partial\alpha}=\left(L_{X_{23}}-L_{X_{32}}\right),\\\
&\frac{\partial}{\partial\beta}=\cos\alpha\left(L_{X_{12}}-L_{X_{21}}\right)-\sin\alpha\left(L_{X_{31}}-L_{X_{13}}\right),\\\
&\frac{\partial}{\partial\gamma}=\cos\alpha\cos\beta\left(L_{X_{31}}-L_{X_{13}}\right)+\sin\alpha\cos\beta\left(L_{X_{12}}-L_{X_{21}}\right)-\sin\beta\left(L_{X_{23}}-L_{X_{32}}\right).\end{split}$
(111)

Computation yields that

$\small\begin{split}&\frac{1}{4}\left[\frac{\partial}{\partial\gamma},\left[\frac{\partial}{\partial\gamma},L_{X_{12}}\right]\right]+L_{X_{12}}-\left(\frac{1}{2}\sin\alpha\tan\beta\right)\frac{\partial}{\partial\alpha}-\left(\frac{1}{2}\cos\alpha\right)\frac{\partial}{\partial\beta}-\left(\frac{1}{2}\sin\alpha\sec\beta\right)\frac{\partial}{\partial\gamma}\\\
&=-\left(\left(\frac{3}{4}\cos\alpha\sin
2\beta\right)z_{12}+\left(\frac{3}{4}\cos\alpha\cos
2\beta\cos\gamma-\frac{3}{4}\sin\alpha\sin\beta\sin\gamma\right)\right)\frac{\partial}{\partial
z_{12}}\\\
&\,\,\,\,\,\,\,-\left(\left(-\frac{3}{4}\sin\alpha\sin\beta\sin\gamma+\frac{3}{4}\cos\alpha\cos
2\beta\cos\gamma\right)z_{23}\right)\frac{\partial}{\partial
z_{13}}\,\,\,\,\,\,\,\,\,\,\,\,\\\
&\,\,\,\,\,\,\,-\left(\left(-\frac{3}{4}\cos\alpha\sin
2\beta\right)z_{23}+\left(\frac{3}{4}\sin\alpha\sin\beta\cos\gamma+\frac{3}{4}\cos\alpha\cos
2\beta\sin\gamma\right)\right)\frac{\partial}{\partial
z_{23}}=:L^{\gamma\gamma}.\\\ \end{split}$ (112)

Then, by Lemma 3.10 $L^{\gamma\gamma}\in\mathcal{P}$. Similarly,

$\begin{split}&L^{\gamma\gamma\gamma\gamma}:=\left[\frac{\partial}{\partial\gamma},\left[\frac{\partial}{\partial\gamma},L^{\gamma\gamma}\right]\right]+L^{\gamma\gamma}=-\left(\frac{3}{4}\cos\alpha\sin
2\beta\right)\left(z_{12}\frac{\partial}{\partial
z_{12}}-z_{23}\frac{\partial}{\partial z_{23}}\right)\in\mathcal{P},\\\
\end{split}$ (113)

and hence

$\small\begin{split}&\cos
2\beta\left[\frac{\partial}{\partial\beta},\sin\alpha\left[\frac{\partial}{\partial\alpha},L^{\gamma\gamma\gamma\gamma}\right]-\cos\alpha\cdot
L^{\gamma\gamma\gamma\gamma}\right]+2\sin
2\beta\left(\sin\alpha\left[\frac{\partial}{\partial\alpha},L^{\gamma\gamma\gamma\gamma}\right]-\cos\alpha\cdot
L^{\gamma\gamma\gamma\gamma}\right)\\\
&\,\,\,\,\,\,\,\,\,\,\,=\frac{3}{2}\left(z_{12}\frac{\partial}{\partial
z_{12}}-z_{23}\frac{\partial}{\partial z_{23}}\right)\in\mathcal{P}.\\\
\end{split}$ (114)

Compuation yields

$\begin{split}&\widehat{L}:=L^{\gamma\gamma}+\left(\frac{3}{4}\cos\alpha\sin
2\beta\right)\left(z_{12}\frac{\partial}{\partial
z_{12}}-z_{23}\frac{\partial}{\partial z_{23}}\right)\\\
&\,\,\,\,\,\,=-\left(\frac{3}{4}\cos\alpha\cos
2\beta\cos\gamma-\frac{3}{4}\sin\alpha\sin\beta\sin\gamma\right)\left(\frac{\partial}{\partial
z_{12}}+z_{23}\frac{\partial}{\partial z_{13}}\right)\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,-\left(\frac{3}{4}\sin\alpha\sin\beta\cos\gamma+\frac{3}{4}\cos\alpha\cos
2\beta\sin\gamma\right)\frac{\partial}{\partial z_{23}},\\\
&\widehat{L}^{\alpha}:=\sin\alpha\left[\frac{\partial}{\partial\alpha},\widehat{L}\right]-\cos\alpha\cdot\widehat{L}\\\
&\,\,\,\,\,\,\,\,\,=\left(\frac{3}{4}\cos
2\beta\cos\gamma\right)\left(\frac{\partial}{\partial
z_{12}}+z_{23}\frac{\partial}{\partial z_{13}}\right)+\left(\frac{3}{4}\cos
2\beta\sin\gamma\right)\frac{\partial}{\partial z_{23}},\\\
&\widehat{L}_{1}^{\alpha\gamma}:=-\sin\gamma\left[\frac{\partial}{\partial\gamma},\widehat{L}^{\alpha}\right]+\cos\gamma\cdot\widehat{L}^{\alpha}=\left(\frac{3}{4}\cos
2\beta\right)\left(\frac{\partial}{\partial
z_{12}}+z_{23}\frac{\partial}{\partial z_{13}}\right),\\\
&\widehat{L}_{2}^{\alpha\gamma}:=\cos\gamma\left[\frac{\partial}{\partial\gamma},\widehat{L}^{\alpha}\right]+\sin\gamma\cdot\widehat{L}^{\alpha}=\left(\frac{3}{4}\cos
2\beta\right)\frac{\partial}{\partial z_{23}},\\\ \end{split}$ (115)

and hence

$\begin{split}&\widehat{L}_{1}^{\alpha\gamma\beta}:=-\sin
2\beta\left[\frac{\partial}{\partial\beta},\widehat{L}_{1}^{\alpha\gamma}\right]+2\cos
2\beta\cdot\widehat{L}_{1}^{\alpha\gamma}=\frac{3}{2}\left(\frac{\partial}{\partial
z_{12}}+z_{23}\frac{\partial}{\partial z_{13}}\right)\in\mathcal{P},\\\
&\widehat{L}_{2}^{\alpha\gamma\beta}:=-\sin
2\beta\left[\frac{\partial}{\partial\beta},\widehat{L}_{2}^{\alpha\gamma}\right]+2\cos
2\beta\cdot\widehat{L}_{2}^{\alpha\gamma}=\frac{3}{2}\frac{\partial}{\partial
z_{23}}\in\mathcal{P}.\\\ \end{split}$ (116)

Consider

$\begin{split}&L_{X_{12}}+\left(\frac{1}{2}\cos\alpha\cos\beta\sin
2\gamma-\sin\alpha\tan\beta\sin^{2}\gamma-\frac{1}{2}\cos\alpha\sin\beta\tan\beta\sin
2\gamma\right)\frac{\partial}{\partial\alpha}\\\
&\,\,\,\,\,\,+\left(-\frac{1}{2}\sin\alpha\sin\beta\sin
2\gamma-\cos\alpha\sin^{2}\beta\cos^{2}\gamma-\cos\alpha\cos^{2}\beta\sin^{2}\gamma\right)\frac{\partial}{\partial\beta}\\\
&\,\,\,\,\,\,+\left(-\sin\alpha\sec\beta\sin^{2}\gamma-\frac{1}{2}\cos\alpha\tan\beta\sin
2\gamma+\frac{1}{4}\cos\alpha\sin 2\beta\sin
2\gamma\right)\frac{\partial}{\partial\gamma}\\\
&\,\,\,\,\,\,+\left(\frac{1}{2}\sin\alpha\cos\beta\sin
2\gamma+\frac{1}{2}\cos\alpha\sin
2\beta\cos^{2}\gamma+\frac{1}{2}\cos\alpha\sin
2\beta\right)\left(z_{12}\frac{\partial}{\partial
z_{12}}-z_{23}\frac{\partial}{\partial z_{23}}\right)\\\
&\,\,\,\,\,\,+\left(-\sin\alpha\sin\beta\sin\gamma+\cos\alpha\cos
2\beta\cos\gamma\right)\left(\frac{\partial}{\partial
z_{12}}+z_{23}\frac{\partial}{\partial z_{13}}\right)\\\
&\,\,\,\,\,\,+\left(\sin\alpha\sin\beta\cos\gamma+\cos\alpha\cos
2\beta\sin\gamma\right)\frac{\partial}{\partial z_{23}}\ \end{split}$ (117)
$\begin{split}&=-\left(\sin\alpha\cos\beta\sin
2\gamma+\frac{1}{2}\cos\alpha\sin 2\beta\cos
2\gamma\right)\left(z_{13}\frac{\partial}{\partial
z_{13}}+z_{23}\frac{\partial}{\partial
z_{23}}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\
&\,\,\,\,\,\,\,-\left(-\sin\alpha\cos\beta\cos
2\gamma+\frac{1}{2}\cos\alpha\sin 2\beta\sin
2\gamma\right)\frac{\partial}{\partial z_{13}}.\\\ \end{split}$

By a similar argument, one can show

$z_{13}\frac{\partial}{\partial z_{13}}+z_{23}\frac{\partial}{\partial
z_{23}},\,\,\frac{\partial}{\partial z_{13}}\in\mathcal{P}.$ (118)

We complete the proof of Lemma 3.12.

######  Lemma 3.13.

Denote the differentials $\frac{\partial}{\partial
z_{13}},\frac{\partial}{\partial z_{23}},z_{12}\frac{\partial}{\partial
z_{12}}-z_{23}\frac{\partial}{\partial z_{23}}$, $\frac{\partial}{\partial
z_{12}}+z_{23}\frac{\partial}{\partial z_{13}}$,
$z_{13}\frac{\partial}{\partial z_{13}}+z_{23}\frac{\partial}{\partial
z_{23}}$ by $\delta_{1},\delta_{2},\delta_{3},\delta_{4},\delta_{5}$,
respectively. Then, we can represent

$\delta_{i}=\sum_{\begin{subarray}{c}v_{i,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{i,j}L_{v_{i,j}},\,\,\,1\leq i\leq 5,$ (119)

where $p_{i,j}$ are bounded functions on $X$, and $L_{v_{i,j}}$ are left
derivatives on $X$.

Proof of Lemma 3.13. We first note that the differential $\delta_{i}$ is
globally defined on $X$. By Lemmas 3.12, 3.11, $\delta_{i}$ can be written on
the open set $\underline{U}_{EI}\times\mathbb{R}^{3}\subset
SL_{3}(\mathbb{R})/A$ as

$\delta_{i}=\sum_{\begin{subarray}{c}v_{i,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{i,j}L_{v_{i,j}},$ (120)

where $p_{i,j}$ are bounded functions on
$\underline{U}_{EI}\times\mathbb{R}^{3}$, and $L_{v_{i,j}}$ are left
derivatives on $X$.

Claim. For $k_{0}\in SO_{3}(\mathbb{R})$, $\delta_{i}$ can be written on the
open set $\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3}\subset
SL_{3}(\mathbb{R})/A$ as

$\delta_{i}=\sum_{\begin{subarray}{c}v_{i,j}^{k_{0}}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{i,j}^{k_{0}}L_{v_{i,j}^{k_{0}}},$ (121)

where $p_{i,j}^{k_{0}}$ are bounded functions on
$\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3}$, and
$L_{v_{i,j}^{k_{0}}}$ are left derivatives on $X$.

Proof of Claim. Take the Euler-Iwasawa coordinates
$(\alpha,\beta,\gamma,z_{12},z_{13},z_{23})$ attached to $k_{0}$. It is clear
that the explicit formula of the differential $\delta$ in terms of the Euler-
Iwasawa coordinates attached to $k_{0}$, coincides with that in terms of the
Euler-Iwasawa coordinates attached to the identity of $SO_{3}(\mathbb{R})$.

Let Ad be the adjoint action of $SL_{3}(\mathbb{R})$ on its lie algebra
defined by

${\rm Ad}(g)X=\left.\frac{d}{dt}\right|_{t=0}g\cdot\exp\left(tX\right)\cdot
g^{-1}.$ (122)

Then the adjoint action Ad acts on the universal enveloping algebra
$U(\mathfrak{sl}_{3}(\mathbb{R}))$ by

${\rm Ad}(g)(X_{1}\cdot X_{2}\cdot\cdots\cdot X_{r})={\rm Ad}(g)X_{1}\cdot{\rm
Ad}(g)X_{2}\cdot\cdots\cdot{\rm Ad}(g)X_{r},$ (123)

where $X_{1},X_{2},\cdots,X_{r}\in\mathfrak{sl}_{3}(\mathbb{R})$. Since for
$u\in U(\mathfrak{sl}_{3}(\mathbb{R}))$, the left derivatives $L_{{\rm
Ad}(k_{0})u}$ and $L_{u}$ are intertwined by the left $k_{0}$-translation, one
sees that the explicit formula of $L_{{\rm Ad}(k_{0})u}$ in terms of the
Euler-Iwasawa coordinates attached to $k_{0}$, coincides with the explicit
formula of $L_{u}$ in terms of the Euler-Iwasawa coordinates attached to the
identity of $SO_{3}(\mathbb{R})$.

Now by (120), in terms of the Euler-Iwasawa coordinates attached to $k_{0}$,
we can derive

$\delta_{i}=\sum_{\begin{subarray}{c}v_{i,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{i,j}L_{{\rm Ad}(k_{0})v_{i,j}},$ (124)

where $p_{i,j}$ and $v_{i,j}$ are exactly the same as in (120).

We complete the proof of Claim.

Take points $k_{1},k_{2},\cdots,k_{M}\in SO_{3}(\mathbb{R})$ such that
$\left\\{\left(k_{l}\cdot\underline{U}_{EI}\right)\right\\}_{l=1}^{M}$ is an
open cover of $SO_{3}(\mathbb{R})$, and a partition of unity
$\\{\rho_{l}\\}_{l=1}^{M}$ subordinate to
$\left\\{\left(k_{l}\cdot\underline{U}_{EI}\right)\right\\}_{l=1}^{M}$. By
Claim, we have

$\small\begin{split}&\delta_{i}=\sum_{l=1}^{M}\sum_{\begin{subarray}{c}v_{i,j}^{k_{l}}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}\left(\rho_{l}\cdot
p_{i,j}^{k_{l}}\right)L_{v_{i,j}^{k_{l}}}.\\\ \end{split}$ (125)

Noticing that the above functions $\rho_{l}\cdot p_{i,j}^{k_{l}}$ are bounded
on $X$, we conclude Lemma 3.13.

######  Lemma 3.14.

We can write

$\begin{split}&D_{12}-\left(z_{12}^{2}+1\right)\frac{\partial^{2}}{\partial
z_{12}\partial z_{12}}-\left(z_{23}\right)\frac{\partial^{2}}{\partial
z_{13}\partial z_{12}}-\left(-z_{13}\right)\frac{\partial^{2}}{\partial
z_{23}\partial z_{12}}-2z_{12}\frac{\partial}{\partial z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left(\sum_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}L_{w_{j}}\right)\frac{\partial}{\partial
z_{12}}.\\\ \end{split}$ (126)

Here $q_{j}$ are bounded functions on $X$ which are independent of the
variables $z_{12},z_{23},z_{13}$; $L_{w_{j}}$ are left derivatives on $X$
induced by the elements in $\mathfrak{so}_{3}(\mathbb{R})$.

Proof of Lemma 3.14. The proof is similar to that of Lemma 3.13. By Lemma 3.7,
in each coordinate chart
$\left(\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3},\Lambda_{k_{0}}\right)$,

$\begin{split}&D_{12}=\left(\sec\beta\sin\gamma\right)\frac{\partial^{2}}{\partial\alpha\partial
z_{12}}+\left(\cos\gamma\right)\frac{\partial^{2}}{\partial\beta\partial
z_{12}}+\left(\tan\beta\sin\gamma\right)\frac{\partial^{2}}{\partial\gamma\partial
z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,+\left(z_{12}^{2}+1\right)\frac{\partial^{2}}{\partial
z_{12}\partial z_{12}}+\left(z_{23}\right)\frac{\partial^{2}}{\partial
z_{13}\partial z_{12}}+\left(-z_{13}\right)\frac{\partial^{2}}{\partial
z_{23}\partial z_{12}}+2z_{12}\frac{\partial}{\partial z_{12}},\\\
\end{split}$ (127)

where $(\alpha,\beta,\gamma,z_{12},z_{13},z_{23})$ are the Euler-Iwasawa
coordinates attached to $k_{0}\in SO_{3}(\mathbb{R})$.

By Lemma 3.11 and (111) in Lemma 3.12 , we can write on the open set
$\underline{U}_{EI}\times\mathbb{R}^{3}\subset X$

$\begin{split}&D_{12}-\left(z_{12}^{2}+1\right)\frac{\partial^{2}}{\partial
z_{12}\partial z_{12}}-\left(z_{23}\right)\frac{\partial^{2}}{\partial
z_{13}\partial z_{12}}-\left(-z_{13}\right)\frac{\partial^{2}}{\partial
z_{23}\partial z_{12}}-2z_{12}\frac{\partial}{\partial z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left(\sum_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}L_{w_{j}}\right)\frac{\partial}{\partial
z_{12}},\\\ \end{split}$ (128)

such that $q_{j}$ are bounded functions on
$\underline{U}_{EI}\times\mathbb{R}^{3}$ which are independent of the
variables $z_{12},z_{23},z_{13}$, and $L_{w_{j}}$ are left derivatives on $X$
induced by the elements in $\mathfrak{so}_{3}(\mathbb{R})$.

Claim. For $k_{0}\in SO_{3}(\mathbb{R})$, we can write on the open set
$\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3}\subset
SL_{3}(\mathbb{R})/A$

$\begin{split}&D_{12}-\left(z_{12}^{2}+1\right)\frac{\partial^{2}}{\partial
z_{12}\partial z_{12}}-\left(z_{23}\right)\frac{\partial^{2}}{\partial
z_{13}\partial z_{12}}-\left(-z_{13}\right)\frac{\partial^{2}}{\partial
z_{23}\partial z_{12}}-2z_{12}\frac{\partial}{\partial z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left(\sum_{\begin{subarray}{c}w_{j}^{k_{0}}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}^{k_{0}}L_{w_{j}^{k_{0}}}\right)\frac{\partial}{\partial
z_{12}},\\\ \end{split}$ (129)

such that $q_{j}^{k_{0}}$ are bounded functions on
$\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3}$, which are
independent of the variables $z_{12},z_{23},z_{13}$, and $L_{w_{j}^{k_{0}}}$
are left derivatives on $X$ induced by the elements in
$\mathfrak{so}_{3}(\mathbb{R})$.

Proof of Claim. Take the Euler-Iwasawa coordinates attached to $k_{0}$. One
verifies (see Remark 3.8) that the explicit formula of the left hand side of
(129) in terms of the Euler-Iwasawa coordinates attached to $k_{0}$, coincides
with that in terms of the Euler-Iwasawa coordinates attached to the identity
of $SO_{3}(\mathbb{R})$.

Now by (128), in terms of the Euler-Iwasawa coordinates attached to $k_{0}$,

$\begin{split}&D_{12}-\left(z_{12}^{2}+1\right)\frac{\partial^{2}}{\partial
z_{12}\partial z_{12}}-\left(z_{23}\right)\frac{\partial^{2}}{\partial
z_{13}\partial z_{12}}-\left(-z_{13}\right)\frac{\partial^{2}}{\partial
z_{23}\partial z_{12}}-2z_{12}\frac{\partial}{\partial z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left(\sum_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}L_{{\rm
Ad}(k_{0})w_{j}}\right)\frac{\partial}{\partial z_{12}},\\\ \end{split}$ (130)

where $q_{j}$ and $w_{j}$ are exactly the same as in (128). Noticing that
${\rm Ad}(k_{0})w_{j}\in\mathfrak{so}_{3}(\mathbb{R})$ for all $k_{0}\in
SO_{3}(\mathbb{R})$ and $w_{j}\in\mathfrak{so}_{3}(\mathbb{R})$, we complete
the proof of Claim.

Take the partition of unity $\\{\rho_{l}\\}_{l=1}^{M}$ subordinate to
$\left\\{\left(k_{l}\cdot\underline{U}_{EI}\right)\right\\}_{l=1}^{M}$ as in
the proof of Lemma 3.13. We have by Claim

$\begin{split}&D_{12}-\left(z_{12}^{2}+1\right)\frac{\partial^{2}}{\partial
z_{12}\partial z_{12}}-\left(z_{23}\right)\frac{\partial^{2}}{\partial
z_{13}\partial z_{12}}-\left(-z_{13}\right)\frac{\partial^{2}}{\partial
z_{23}\partial z_{12}}-2z_{12}\frac{\partial}{\partial z_{12}}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\sum_{l=1}^{M}\left(\sum_{\begin{subarray}{c}w_{j}^{k_{l}}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}\left(\rho_{l}\cdot
q_{j}^{k_{l}}\right)L_{w_{j}^{k_{l}}}\right)\frac{\partial}{\partial
z_{12}}.\\\ \end{split}$ (131)

Noticing that the functions $\rho_{l}\cdot q_{j}^{k_{l}}$ are bounded on $X$
and are independent of the variables $z_{12},z_{23},z_{13}$, we conclude Lemma
3.14.

###  3.5. Proof of the density and Theorem 1.3

In the following, we shall establish the density of
$C^{\infty}_{c}\left(SL_{3}(\mathbb{R})/A\right)$ in ${\rm Dom}(D_{12})$ for
the invariant differential operator $D_{12}$.

Firstly, we construct a sequence of cutoff functions
$\\{\chi_{n}\\}_{n=1}^{\infty}$ on $X$. Take $\xi(r)$ to be a smooth function
on $[0,+\infty)$ such that

  1. (1)

$0\leq\xi(r)\leq 1$ for $r\in[0,+\infty)$;

  2. (2)

$\xi(r)\equiv 1$ for $0\leq r\leq 1$;

  3. (3)

$\xi(r)\equiv 0$ for $r\geq 2$.

For $n=1,2,\cdots$, define

$\chi_{n}(x):=\chi_{n}\left(k_{x},z_{12},z_{13},z_{23}\right):=\xi\left(\frac{|z_{13}|^{2}+|z_{23}|^{2}}{n}\right)\cdot\xi\left(\frac{\ln\left(|z_{12}|^{2}+1\right)}{n}\right).$
(132)

######  Lemma 3.15.

The following properties hold for the cutoff functions
$\\{\chi_{n}\\}_{n=1}^{\infty}$ defined by (132).

  1. (a)

$\\{\chi_{n}\\}_{n=1}^{\infty}\subset C^{\infty}_{c}(X)$.

  2. (b)

$0\leq\chi_{n}(x)\leq 1$, and $\lim_{n\to\infty}\chi_{n}(x)=1$, for each $x\in
X$.

  3. (c)

For non-negative integers $i_{1},i_{2},i_{3}$ such that $i_{1}+i_{2}+i_{3}\geq
1$,

$\lim_{n\to\infty}\frac{\partial^{i_{1}+i_{2}+i_{3}}}{\partial
z_{12}^{i_{1}}\partial z_{13}^{i_{2}}\partial z_{23}^{i_{3}}}\chi_{n}(x)=0.$
(133)

  4. (d)

For $n=1,2,\cdots$, we have

$\frac{\partial\chi_{n}}{\partial\alpha}=\frac{\partial\chi_{n}}{\partial\beta}=\frac{\partial\chi_{n}}{\partial\gamma}\equiv
0.$ (134)

  5. (e)

For $n=1,2,\cdots$, we have

$\left(z_{23}\frac{\partial}{\partial z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}\right)\chi_{n}\equiv 0.$ (135)

  6. (f)

There exists $\mathcal{M}>0$ such that for all $n=1,2,\cdots$,

$\sup_{x\in
X}\left|(1+|z_{12}|)(1+|z_{13}|+|z_{23}|)\cdot\frac{\partial\chi_{n}}{\partial
z_{12}}(x)\right|+\sup_{x\in
X}\left|\left(|z_{12}|^{2}+1\right)\cdot\frac{\partial^{2}\chi_{n}}{\partial
z_{12}\partial z_{12}}(x)\right|\leq\mathcal{M}.$ (136)

Proof of Lemma 3.15. It is clear that (a), (b), (c), (d) follow from the
definition in (132).

(e) follows from the computation that

$z_{23}\frac{\partial\chi_{n}}{\partial
z_{13}}-z_{13}\frac{\partial\chi_{n}}{\partial
z_{23}}=\xi\left(\frac{\ln\left(|z_{12}|^{2}+1\right)}{n}\right)\cdot\xi^{\prime}\left(\frac{|z_{13}|^{2}+|z_{23}|^{2}}{n}\right)\cdot\left(\frac{z_{23}z_{13}}{n}-\frac{z_{13}z_{23}}{n}\right)\equiv
0.$ (137)

Computation yields

$\begin{split}&\frac{\partial\chi_{n}}{\partial
z_{12}}\left(k_{x},z_{12},z_{13},z_{23}\right)=\xi\left(\frac{|z_{13}|^{2}+|z_{23}|^{2}}{n}\right)\cdot\xi^{\prime}\left(\frac{\ln\left(|z_{12}|^{2}+1\right)}{n}\right)\cdot\frac{2z_{12}}{|z_{12}|^{2}+1}\cdot\frac{1}{n}.\\\
\end{split}$ (138)

Then,

$\begin{split}&\sup_{x\in
X}\left|(1+|z_{12}|)(1+|z_{13}|+|z_{23}|)\cdot\frac{\partial\chi_{n}}{\partial
z_{12}}(x)\right|\\\ &=\sup_{x\in
X}\left(\left|\xi\left(\frac{|z_{13}|^{2}+|z_{23}|^{2}}{n}\right)\xi^{\prime}\left(\frac{\ln\left(|z_{12}|^{2}+1\right)}{n}\right)\right|\cdot\frac{2|z_{12}|(1+|z_{12}|)}{|z_{12}|^{2}+1}\cdot\frac{1+|z_{13}|+|z_{23}|}{n}\right)\\\
&\leq 4\sup_{r\in[0,+\infty)}\left|\xi^{\prime}(r)\right|\cdot\sup_{x\in
X}\left(\left|\xi\left(\frac{|z_{13}|^{2}+|z_{23}|^{2}}{n}\right)\right|\cdot\frac{1+|z_{13}|+|z_{23}|}{n}\right).\\\
\end{split}$ (139)

Notice that when $1+|z_{13}|+|z_{23}|\geq 4n$,

$\begin{split}&\frac{|z_{13}|^{2}+|z_{23}|^{2}}{n}\geq\frac{(|z_{13}|+|z_{23}|)^{2}}{2n}\geq\frac{(4n-1)^{2}}{2n}>4,\\\
\end{split}$ (140)

and hence $\xi\left(\frac{|z_{13}|^{2}+|z_{23}|^{2}}{n}\right)=0$ by the
definition. As a conclusion, we derive

$\begin{split}&\sup_{x\in
X}\left|(1+|z_{12}|)(1+|z_{13}|+|z_{23}|)\cdot\frac{\partial\chi_{n}}{\partial
z_{12}}(x)\right|\leq 16\sup_{r\in[0,+\infty)}\left|\xi^{\prime}(r)\right|,\\\
\end{split}$ (141)

which is a universal constant independent of $n$. Similarly, from the
computation

$\begin{split}&\frac{\partial^{2}\chi_{n}}{\partial z_{12}\partial
z_{12}}\left(k_{x},z_{12},z_{13},z_{23}\right)=\xi\left(\frac{|z_{13}|^{2}+|z_{23}|^{2}}{n}\right)\cdot\\\
&\cdot\left(\xi^{\prime\prime}\left(\frac{\ln\left(|z_{12}|^{2}+1\right)}{n}\right)\cdot\frac{4z_{12}^{2}}{(|z_{12}|^{2}+1)^{2}}\cdot\frac{1}{n^{2}}+\xi^{\prime}\left(\frac{\ln\left(|z_{12}|^{2}+1\right)}{n}\right)\cdot\frac{2(1-z_{12}^{2})}{(|z_{12}|^{2}+1)^{2}}\cdot\frac{1}{n}\right),\\\
\end{split}$ (142)

we can conclude

$\begin{split}&\sup_{x\in
X}\left|\left(|z_{12}|^{2}+1\right)\cdot\frac{\partial^{2}\chi_{n}}{\partial
z_{12}^{2}}(x)\right|\leq
4\sup_{r\in[0,+\infty)}\left(\left|\xi^{\prime}(r)\right|+\left|\xi^{\prime\prime}(r)\right|\right),\\\
\end{split}$ (143)

which is a universal constant independent of $n$. We thus establish (f).

As a conclusion, we complete the proof of Lemma 3.15.

Let $\\{\psi_{m}\\}_{m=1}^{\infty}$ be a smooth approximate identity at the
identity element $\rm Id$ of $SL_{3}(\mathbb{R})$ satisfying the following
properties.

  1. (1)

$\psi_{m}$ are smooth non-negative functions on $SL_{3}(\mathbb{R})$.

  2. (2)

$\psi_{m}$, are compactly supported in the open set $U_{m}\subset
SL_{3}(\mathbb{R})$, where $\cap_{m=1}^{\infty}U_{m}=\\{\rm Id\\}$.

  3. (3)

$\int_{SL_{3}(\mathbb{R})}\psi_{m}(g)\,\mathrm{d}g=1$, where $\mathrm{d}g$ is
a fixed Haar measure on $SL_{3}(\mathbb{R})$.

For $f\in L^{2}(X)$, define convolutions $\psi_{m}*f$ by

$(\psi_{m}*f)(x):=\int_{SL_{3}(\mathbb{R})}\psi_{m}(g)\cdot
f\left(g^{-1}x\right){\rm d}g,\,\,\,\,\,\,m=1,2,\cdots.$ (144)

######  Lemma 3.16.

The following holds for each $f\in L^{2}(X)$.

  1. (a)

$\psi_{m}*f$ are smooth functions on $X$ for $m=1,2,\cdots$.

  2. (b)

$\psi_{m}*f\in L^{2}(X)$ for $m=1,2,\cdots$, and $\psi_{m}*f\rightarrow f$ in
$L^{2}(X)$ as $m\rightarrow\infty$.

  3. (c)

For each $u\in U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)$,
$L_{u}(\psi_{m}*f)\in L^{2}(X)$.

Assume that $\Delta$ is an $SL_{3}(\mathbb{R})$-invariant differential
operator on $X$, and that $f,\Delta f\in L^{2}(X)$, then

  1. (d)

$\Delta(\psi_{m}*f)\in L^{2}(X)$ for $m=1,2,\cdots$, and
$\Delta(\psi_{m}*f)\rightarrow\Delta f$ in $L^{2}(X)$ as $m\rightarrow\infty$.

Proof of Lemma 3.16. The arguments are standard and are given here for the
sake of completeness. For each $u\in U(\mathfrak{sl}_{3}(\mathbb{R}))$, it
holds by definition that $L_{u}(\psi_{m}*f)=L_{u}(\psi_{m})*f$. This implies
(a) since $\\{L_{u}\\}_{u\in U(\mathfrak{sl}_{3}(\mathbb{R}))}$ provide all
partial derivatives with respect to any local coordinates at each point of the
manifold $X$. By the Cauchy-Schwarz inequality,

$\begin{split}&\int_{X}|(\psi_{m}*f)(x)|^{2}{\rm
d}\mu=\int_{X}\left|\int_{SL_{3}(\mathbb{R})}\psi_{m}(g)\cdot
f\left(g^{-1}x\right){\rm d}g\right|^{2}{\rm d}\mu\\\
&\leq\int\limits_{X}\left(\int\limits_{SL_{3}(\mathbb{R})}\psi_{m}\left(\widetilde{g}\right){\rm
d}\widetilde{g}\right)\left(\int\limits_{SL_{3}(\mathbb{R})}\psi_{m}(g)\cdot\left|f\left(g^{-1}x\right)\right|^{2}{\rm
d}g\right){\rm d}\mu=\int_{X}\left|f\left(x\right)\right|^{2}{\rm d}\mu.\\\
\end{split}$ (145)

For each $\epsilon>0$, we can find continuous function $f_{\epsilon}$ with
compact support such that

$\int_{X}\left|f_{\epsilon}\left(x\right)-f(x)\right|^{2}{\rm d}\mu<\epsilon.$
(146)

By the Cauchy-Schwarz inequality again,

$\begin{split}&\int_{X}|(\psi_{m}*f_{\epsilon})(x)-f_{\epsilon}(x)|^{2}{\rm
d}\mu=\int_{X}\left|\int_{SL_{3}(\mathbb{R})}\psi_{m}(g)\left(f_{\epsilon}\left(g^{-1}x\right)-f_{\epsilon}(x)\right){\rm
d}g\right|^{2}{\rm d}\mu\\\
&\leq\int_{X}\left(\int_{SL_{3}(\mathbb{R})}\psi_{m}\left(\widetilde{g}\right){\rm
d}\widetilde{g}\right)\left(\int_{SL_{3}(\mathbb{R})}\psi_{m}(g)\cdot\left|f_{\epsilon}\left(g^{-1}x\right)-f_{\epsilon}(x)\right|^{2}{\rm
d}g\right){\rm d}\mu\\\
&=\int_{SL_{3}(\mathbb{R})}\psi_{m}(g)\left(\int_{X}\left|f_{\epsilon}\left(g^{-1}x\right)-f_{\epsilon}(x)\right|^{2}{\rm
d}\mu\right){\rm d}g\leq\sup_{g\in
U_{m}}\int_{X}\left|f_{\epsilon}\left(g^{-1}x\right)-f_{\epsilon}(x)\right|^{2}{\rm
d}\mu.\\\ &\\\ \end{split}$ (147)

Since $f_{\epsilon}$ is uniform continuous with compact support, we conclude

$\lim_{m\rightarrow\infty}\int_{X}|(\psi_{m}*f_{\epsilon})(x)-f_{\epsilon}(x)|^{2}{\rm
d}\mu=0.$ (148)

Similarly to (145), we can derive by (146)

$\begin{split}&\int_{X}|(\psi_{m}*f)(x)-f(x)|^{2}{\rm
d}\mu\leq\int_{X}|(\psi_{m}*f_{\epsilon})(x)-f_{\epsilon}(x)|^{2}{\rm
d}\mu+\int_{X}\left|f\left(x\right)-f_{\epsilon}(x)\right|^{2}{\rm d}\mu\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\int_{X}|(\psi_{m}*(f-f_{\epsilon}))(x)|^{2}{\rm
d}\mu\leq\int_{X}|(\psi_{m}*f_{\epsilon})(x)-f_{\epsilon}(x)|^{2}{\rm
d}\mu+2\epsilon.\\\ \end{split}$ (149)

Then, by(148) and (149), we have

$\lim_{m\rightarrow\infty}\int_{X}|(\psi_{m}*f)(x)-f(x)|^{2}{\rm d}\mu=0.$
(150)

(b) is now proved. Thanks to the fact that
$L_{u}(\psi_{m}*f)=(L_{u}\psi_{m})*f$, the same argument as in (145) gives
(c).

Next, assume $f,\Delta f\in L^{2}(X)$. We first show that
$\Delta(\psi_{m}*f)=\psi_{m}*(\Delta f)$ in the sense of distribution. Recall
the notion that $\\{X_{ij}\\}_{1\leq i,j\leq 3,\,(i,j)\neq(3,3)}$is a basis of
$\mathfrak{sl}_{3}(\mathbb{R})$ (see (11)) and the lift $\widetilde{w}$ of
$w\in C^{\infty}\left(SL_{3}(\mathbb{R})/A\right)$ is defined by
$\widetilde{w}:=w\circ\pi$ (see (3)). For clarity, in the following, we denote
the points of $X$ by $xA$ and the points of $SL_{3}(\mathbb{R})$ by $x$, $g$.

For each invariant differential $\Delta$, there exists a polynomial
$P_{\Delta}$ by Theorem 2.1 such that

$\begin{split}&(\Delta
w)(xA)=\left.P_{\Delta}\left(\cdots,\frac{\partial}{\partial
t_{ij}},\cdots\right)\widetilde{w}\left(x\exp\left(\sum_{\begin{subarray}{c}1\leq
i,j\leq 3\\\
(i,j)\neq(3,3)\end{subarray}}t_{ij}X_{ij}\right)\right)\right|_{t_{11}=t_{12}=\cdots=t_{32}=0}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=:\left.P_{\Delta}\left(\frac{\partial}{\partial
t_{ij}}\right)\widetilde{w}\left(x\exp\left(\sum
t_{ij}X_{ij}\right)\right)\right|_{t_{ij}=0},\,\,\,\,\,\,\,\,w\in
C^{\infty}\left(SL_{3}(\mathbb{R})/A\right).\end{split}$ (151)

Let $\Delta^{*}$ be the formal adjoint of $\Delta$ (see Remark 2.7). Then
$\Delta^{*}$ is still an invariant differential operator on $X$ ([Ba]). For
each $h\in C^{\infty}_{c}(X)$,

$\begin{split}&(\Delta(\psi_{m}*f),h)=(\psi_{m}*f,\Delta^{*}h)\\\
&=\int_{X}\left(\int_{SL_{3}(\mathbb{R})}\psi_{m}(g)\cdot
f\left(g^{-1}xA\right){\rm
d}g\right)\overline{\left.P_{\Delta^{*}}\left(\frac{\partial}{\partial
t_{ij}}\right)\widetilde{h}\left(x\exp\left(\sum
t_{ij}X_{ij}\right)\right)\right|_{t_{ij}=0}\,}{\rm d}\mu\\\
&=\int_{SL_{3}(\mathbb{R})}\psi_{m}(g)\left(\int_{X}f\left(xA\right)\overline{\left.P_{\Delta^{*}}\left(\frac{\partial}{\partial
t_{ij}}\right)\widetilde{h}\left(gx\exp\left(\sum
t_{ij}X_{ij}\right)\right)\right|_{t_{ij}=0}}\,{\rm d}\mu\right){\rm d}g\\\
&=\int_{X}f\left(xA\right)\overline{P_{\Delta^{*}}\left(\frac{\partial}{\partial
t_{ij}}\right)\left.\left(\int_{SL_{3}(\mathbb{R})}\psi_{m}^{*}(g)\widetilde{h}\left(g^{-1}x\exp\left(\sum
t_{ij}X_{ij}\right)\right)\,{\rm d}g\right)\right|_{t_{ij}=0}}.\\\
\end{split}$ (152)

where $\psi_{m}^{*}(g):=\psi_{m}(g^{-1})$. Noticing that
$\psi^{*}_{m}*\widetilde{h}=\widetilde{\psi^{*}_{m}*h}$, we have

$\begin{split}&\int_{X}f\left(xA\right)\overline{P_{\Delta}\left(\frac{\partial}{\partial
t_{ij}}\right)\left.\left(\widetilde{\psi_{m}^{*}*h}\right)\left(x\exp\left(\sum
t_{ij}X_{ij}\right)\right)\right|_{t_{ij}=0}}=(f,\Delta^{*}(\psi_{m}^{*}*h)).\\\
\end{split}$ (153)

Substituting into (152), we have

$\begin{split}&(\Delta(\psi_{m}*f),h)=(\Delta
f,\psi_{m}^{*}*h)=\int_{X}(\Delta
f)(xA)\cdot\left(\int_{SL_{3}(\mathbb{R})}\psi_{m}\left(g^{-1}\right)\cdot\overline{h\left(g^{-1}xA\right)}\,{\rm
d}g\right)\,{\rm d}\mu\\\
&=\int_{X}\left(\int_{SL_{3}(\mathbb{R})}\psi_{m}\left(\widetilde{g}\right)\cdot(\Delta
f)\left(\widetilde{g}^{-1}\widetilde{x}A\right)\,{\rm
d}\widetilde{g}\right)\cdot\overline{h\left(\widetilde{x}A\right)}\,{\rm
d}\mu=(\psi_{m}*(\Delta f),h).\\\ \end{split}$ (154)

Similarly to (145), we have

$\begin{split}&|(\Delta(\psi_{m}*f),h)|=|(\psi_{m}*(\Delta
f),h)|\leq\int_{X}\left|(\Delta f)(x)\right|^{2}\,{\rm
d}\mu\cdot\int_{X}\left|h(x)\right|^{2}\,{\rm d}\mu.\\\ \end{split}$ (155)

Now $\psi_{m}*\Delta f$ lies in $L^{2}(X)$ thanks to (b), we also have
$\Delta(\psi_{m}*f)=\psi_{m}*\Delta f$ in $L^{2}(X)$. Using (b) again,
$\Delta(\psi_{m}*f)=\psi_{m}*\Delta f\to\Delta f$ in $L^{2}(X)$ as
$m\to\infty$. We complete the proof of Lemma 3.16.

Now, we proceed to prove

######  Theorem 3.17.

$D_{12}$ is an $SL_{3}(\mathbb{R})$-invariant, formally self-adjoint
differential operator on $X$. The graph of $D_{12}$ on the domain
$C^{\infty}_{c}(X)$ is dense in the graph of $D_{12}$ on the domain ${\rm
Dom}(D_{12})$, with respect to the graph $L^{2}\times L^{2}$ norm.

Proof of Theorem 3.17. Firstly, we show that $D_{12}$ is formally self-
adjoint. It suffices to prove in each coordinate chart that $D_{12}$ coincides
with $D_{12}^{*}$. This is can be done by a direct computation, thanks to the
invariant measure given by (100) and the explicit formula for $D_{12}$ given
in Lemma 3.7 (or see Remark 2.7 for a coordinate-free approach).

For each $f\in{\rm Dom}(D_{12})$, consider functions
$\chi_{n}\cdot(\psi_{m}*f)$, $m,n=1,2,\cdots$, where $\chi_{n}$ and
$(\psi_{m}*f)$ are defined by (132) and (144), respectively. It is clear that
$\chi_{n}\cdot(\psi_{m}*f)\in C^{\infty}_{c}(X)$ by (a) in Lemma 3.15 and (a)
in Lemma 3.16. Further, by (b) and (d) in Lemma 3.16, to prove the density of
$C^{\infty}_{c}(X)$ in ${\rm Dom}(D_{12})$, it suffices to prove that for
fixed $m=1,2,\cdots$,

$\begin{split}&\lim_{n\to\infty}\|\chi_{n}\cdot(\psi_{m}*f)-\psi_{m}*f\|_{L^{2}(X)}=0,\\\
\end{split}$ (156)

and

$\begin{split}&\lim_{n\to\infty}\|D_{12}\left(\chi_{n}\cdot(\psi_{m}*f)\right)-D_{12}(\psi_{m}*f)\|_{L^{2}(X)}=0.\\\
\end{split}$ (157)

(156) follows easily from (b) in Lemma 3.15, and Lebesgue’s dominated
convergence theorem. In the following, we shall establish (157) for
$m=1,2,\cdots$.

By Lemma 3.14, computation yields that

$\begin{split}&D_{12}\left(\chi_{n}\cdot\left(\psi_{m}*f\right)\right)=\chi_{n}\cdot
D_{12}\left(\psi_{m}*f\right)+\left(\sum_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}L_{w_{j}}\left(\frac{\partial\chi_{n}}{\partial
z_{12}}\right)\right)\cdot\left(\psi_{m}*f\right)\\\
&+\left(\sum_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}L_{w_{j}}\chi_{n}\right)\cdot\frac{\partial\left(\psi_{m}*f\right)}{\partial
z_{12}}+\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(\sum_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}L_{w_{j}}\left(\psi_{m}*f\right)\right)\\\
&+\left(z_{12}^{2}+1\right)\frac{\partial^{2}\chi_{n}}{\partial z_{12}\partial
z_{12}}\cdot\left(\psi_{m}*f\right)+2\left(z_{12}^{2}+1\right)\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\frac{\partial\left(\psi_{m}*f\right)}{\partial z_{12}}\\\
&+\left(z_{23}\frac{\partial}{\partial z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}\right)\left(\frac{\partial\chi_{n}}{\partial
z_{12}}\right)\cdot\left(\psi_{m}*f\right)+\left(z_{23}\frac{\partial\chi_{n}}{\partial
z_{13}}-z_{13}\frac{\partial\chi_{n}}{\partial
z_{23}}\right)\cdot\frac{\partial\left(\psi_{m}*f\right)}{\partial z_{12}}\\\
&+\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(z_{23}\frac{\partial}{\partial
z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}\right)\left(\psi_{m}*f\right)+2z_{12}\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(\psi_{m}*f\right).\\\ \end{split}$ (158)

Since $L_{w_{j}}$, $w_{j}\in\mathfrak{so}_{3}(\mathbb{R})$, are linear
combinations of $\frac{\partial}{\partial\alpha}$,
$\frac{\partial}{\partial\beta}$, $\frac{\partial}{\partial\gamma}$, by (d) in
Lemma 3.15 we have

$\left(\sum_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}L_{w_{j}}\left(\frac{\partial\chi_{n}}{\partial
z_{12}}\right)\right)\cdot\left(\psi_{m}*f\right)=\left(\sum_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}L_{w_{j}}\chi_{n}\right)\cdot\frac{\partial\left(\psi_{m}*f\right)}{\partial
z_{12}}\equiv 0.$ (159)

Similarly, by (e) in Lemma 3.15,

$\left(z_{23}\frac{\partial}{\partial z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}\right)\left(\frac{\partial\chi_{n}}{\partial
z_{12}}\right)\cdot\left(\psi_{m}*f\right)=\left(z_{23}\frac{\partial\chi_{n}}{\partial
z_{13}}-z_{13}\frac{\partial\chi_{n}}{\partial
z_{23}}\right)\cdot\frac{\partial\left(\psi_{m}*f\right)}{\partial
z_{12}}\equiv 0.$ (160)

Therefore,

$\begin{split}&D_{12}\left(\chi_{n}\left(\psi_{m}*f\right)\right)=\chi_{n}D_{12}\left(\psi_{m}*f\right)+\frac{\partial\chi_{n}}{\partial
z_{12}}\left(\sum_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\end{subarray}}q_{j}L_{w_{j}}\left(\psi_{m}*f\right)\right)\\\
&+\left(z_{12}^{2}+1\right)\frac{\partial^{2}\chi_{n}}{\partial z_{12}\partial
z_{12}}\left(\psi_{m}*f\right)+2\left(z_{12}^{2}+1\right)\frac{\partial\chi_{n}}{\partial
z_{12}}\frac{\partial\left(\psi_{m}*f\right)}{\partial z_{12}}\\\
&+\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(z_{23}\frac{\partial}{\partial
z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}\right)\left(\psi_{m}*f\right)+2z_{12}\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(\psi_{m}*f\right)\\\
&=:A_{n}+B_{n}+C_{n}+D_{n}+E_{n}+F_{n}.\\\ \end{split}$ (161)

In what follows, we shall prove that the functions $B_{n}$, $C_{n}$, $D_{n}$,
$E_{n}$, $F_{n}$ converge to $0$ and $A_{n}$ converges to $D_{12}(\psi_{m}*f)$
in $L^{2}(X)$, when $n$ approaches $\infty$. Thanks to (f) in Lemma 3.15 and
the fact that $q_{j}$ are bounded functions on $X$, we have the pointwise
estimate

$\begin{split}|B_{n}(x)|\leq&\left(\mathcal{M}\sup_{\begin{subarray}{c}w_{j}\in\mathfrak{so}_{3}(\mathbb{R})\\\
x\in
X\end{subarray}}\left|q_{j}(x)\right|\right)\sum_{w_{j}\in\mathfrak{so}_{3}(\mathbb{R})}\left|L_{w_{j}}\left(\psi_{m}*f\right)(x)\right|,\\\
\ \end{split}$ (162)

where the right hand side is an $L^{2}$ function by (c) in Lemma 3.16. Then by
(c) in Lemma 3.15, we conclude by Lebesgue’s dominated convergence theorem
that

$\begin{split}&\lim_{n\to\infty}\|B_{n}\|_{L^{2}(X)}=0.\\\ \end{split}$ (163)

Similarly, by (f) in Lemma 3.15, we have pointwisely

$\begin{split}|C_{n}(x)|+|F_{n}(x)|\leq
2\mathcal{M}\cdot\left|\left(\psi_{m}*f\right)(x)\right|,\\\ \end{split}$
(164)

where the right hand side is an $L^{2}$ function by (b) in Lemma 3.16. Then by
(c) in Lemma 3.15, we conclude by Lebesgue’s dominated convergence theorem
that

$\begin{split}&\lim_{n\to\infty}\|C_{n}\|_{L^{2}(X)}+\lim_{n\to\infty}\|F_{n}\|_{L^{2}(X)}=0.\\\
\end{split}$ (165)

By Lemma 3.13, we have

$\small\begin{split}&\left(z_{12}^{2}+1\right)\frac{\partial\chi_{n}}{\partial
z_{12}}\frac{\partial\left(\psi_{m}*f\right)}{\partial
z_{12}}=\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(\frac{\partial}{\partial
z_{12}}+z_{23}\frac{\partial}{\partial
z_{13}}\right)\left(\psi_{m}*f\right)-z_{23}\frac{\partial\chi_{n}}{\partial
z_{12}}\frac{\partial\left(\psi_{m}*f\right)}{\partial z_{13}}\\\
&+z_{12}\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(z_{12}\frac{\partial}{\partial
z_{12}}-z_{23}\frac{\partial}{\partial
z_{23}}\right)\left(\psi_{m}*f\right)+z_{12}z_{23}\frac{\partial\chi_{n}}{\partial
z_{12}}\frac{\partial\left(\psi_{m}*f\right)}{\partial z_{23}}\\\
&=\frac{\partial\chi_{n}}{\partial
z_{12}}\left(\sum_{\begin{subarray}{c}v_{4,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{4,j}L_{v_{4,j}}\left(\psi_{m}*f\right)\right)-z_{23}\frac{\partial\chi_{n}}{\partial
z_{12}}\left(\sum_{\begin{subarray}{c}v_{1,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{1,j}L_{v_{1,j}}\left(\psi_{m}*f\right)\right)\\\
&+z_{12}\frac{\partial\chi_{n}}{\partial
z_{12}}\left(\sum_{\begin{subarray}{c}v_{3,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{3,j}L_{v_{3,j}}\left(\psi_{m}*f\right)\right)+z_{12}z_{23}\frac{\partial\chi_{n}}{\partial
z_{12}}\left(\sum_{\begin{subarray}{c}v_{2,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{2,j}L_{v_{2,j}}\left(\psi_{m}*f\right)\right).\\\
\end{split}$ (166)

Thanks to (f) in Lemma 3.15 and the fact that $p_{1,j}$, $p_{2,j}$, $p_{3,j}$,
$p_{4,j}$ are bounded functions on $X$,

$\begin{split}|D_{n}(x)|\leq&\left(\mathcal{M}\sup_{\begin{subarray}{c}1\leq
i\leq 4\\\ x\in
X\end{subarray}}\left|p_{i,j}(x)\right|\right)\sum_{i=1}^{4}\sum_{\begin{subarray}{c}v_{i,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}|L_{v_{i,j}}\left(\psi_{m}*f\right)(x)|.\\\
\end{split}$ (167)

Then by (c) in Lemma 3.15, we conclude by Lebesgue’s dominated convergence
theorem that

$\begin{split}&\lim_{n\to\infty}\|D_{n}\|_{L^{2}(X)}=0.\\\ \end{split}$ (168)

Similarly, by Lemma 3.13,

$\begin{split}&\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(z_{23}\frac{\partial}{\partial
z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}\right)\left(\psi_{m}*f\right)=z_{23}\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(\sum_{\begin{subarray}{c}v_{1,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{4,j}L_{v_{4,j}}\left(\psi_{m}*f\right)\right)\\\
\\\ &\,\,\,\,\,\,\,\,\,\,\,-z_{13}\frac{\partial\chi_{n}}{\partial
z_{12}}\cdot\left(\sum_{\begin{subarray}{c}v_{2,j}\in
U\left(\mathfrak{sl}_{3}(\mathbb{R})\right)\\\ {\rm
are\,\,monomials}\end{subarray}}p_{2,j}L_{v_{2,j}}\left(\psi_{m}*f\right)\right).\\\
\end{split}$ (169)

By (c), (f) in Lemma 3.15, and (c) in Lemma 3.16, we can conclude

$\begin{split}&\lim_{n\to\infty}\|E_{n}\|_{L^{2}(X)}=0.\\\ \end{split}$ (170)

Finally, we consider $A_{n}$. It is clear that

$\begin{split}&|A_{n}(x)-\left(D_{12}\left(\psi_{m}*f\right)\right)(x)|=|\left(\chi_{n}-1\right)\cdot\left(D_{12}\left(\psi_{m}*f\right)\right)(x)|\leq|\left(D_{12}\left(\psi_{m}*f\right)\right)(x)|,\\\
\end{split}$ (171)

where the right hand side is an $L^{2}$ function by (d) in Lemma 3.16. Then by
(b) in Lemma 3.15,

$\begin{split}&\lim_{n\to\infty}\|A_{n}-D_{12}\left(\psi_{m}*f\right)\|_{L^{2}(X)}=0.\\\
\end{split}$ (172)

As a conclusion, we complete the proof of Theorem 3.17.

Proof of Theorem 1.3. By Theorem 3.17, the graph of $D_{12}$ on the domain
$C^{\infty}_{c}(X)$ is dense in the graph of $D_{12}$ on the domain
$\text{Dom}(D_{12})$, with respect to the graph $L^{2}\times L^{2}$ norm. Then
$D_{12}$ is essentially self-adjoint by Lemmas 3.1, 3.2.

Notice that the conjugations of the normalizer of $A$ in $SL_{3}(\mathbb{R})$
(or equivalently the permutations of the indices $1,2,3$) are isometries of
$L^{2}(X)$, and such isometries transform $D_{12}$ to $D_{13}$ and $D_{23}$.
Then, it is clear that $D_{13}$ and $D_{23}$ are essentially self-adjoint.

We complete the proof of Theorem 1.3.

######  Remark 3.18.

In the proof of Theorem 3.17, one can see that our cutoff functions are
specifically designed to treat the operator $D_{12}$. Firstly, the first order
derivative of our cutoff functions with respect to the variable $z_{12}$
contributes a decay of the order $(z_{12}\log z_{12})^{-1}$, which is better
than the more standard $z_{12}^{-1}$ decay. By stretching the cutoff functions
in an inhomogeneous way, one can balance a linear growth of the other
variables $z_{13},z_{23}$ exactly by the remaining factor $(\log
z_{12})^{-1}$, so that certain terms in $D_{12}$ can be dealt with. Secondly,
some other terms in $D_{12}$ force us to arrange the cutoff functions so that
they are totally annihilated by those differentials such as
$z_{23}\frac{\partial}{\partial z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}$, in order to kill the wild terms such as $\frac{\partial}{\partial
z_{12}}(\psi_{m}*f)$. Recall that the singled out differential
$z_{23}\frac{\partial}{\partial z_{13}}-z_{13}\frac{\partial}{\partial
z_{23}}$ has an intrinsic geometric explanation as the rotation caused by an
element in $SO_{3}(\mathbb{R})$. 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. Therefore, to deal with the cases $D_{123}$ and $D_{213}$, 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.

##  Appendix A Computations in the Euler-Iwasawa Coordinates

###  A.1. Explicit formulas for the generators of the left derivatives on
$SL_{3}(\mathbb{R})/A$

######  Lemma A.1.

In the coordinate chart
$\left(\left(k_{0}\cdot\underline{U}_{EI}\right)\times\mathbb{R}^{3},\Lambda_{k_{0}}\right)$,
where $k_{0}$ is the identity of $SO_{3}(\mathbb{R})$,

$\begin{split}&-L_{X_{12}}=\left(\frac{1}{2}\cos\alpha\cos\beta\sin
2\gamma-\sin\alpha\tan\beta\sin^{2}\gamma-\frac{1}{2}\cos\alpha\sin\beta\tan\beta\sin
2\gamma\right)\frac{\partial}{\partial\alpha}\\\
&+\left(-\frac{1}{2}\sin\alpha\sin\beta\sin
2\gamma-\cos\alpha\sin^{2}\beta\cos^{2}\gamma-\cos\alpha\cos^{2}\beta\sin^{2}\gamma\right)\frac{\partial}{\partial\beta}\\\
&+\left(-\sin\alpha\sec\beta\sin^{2}\gamma-\frac{1}{2}\cos\alpha\tan\beta\sin
2\gamma+\frac{1}{4}\cos\alpha\sin 2\beta\sin
2\gamma\right)\frac{\partial}{\partial\gamma}\\\
&+\left(\left(\frac{1}{2}\sin\alpha\cos\beta\sin
2\gamma+\frac{1}{2}\cos\alpha\sin
2\beta\cos^{2}\gamma+\frac{1}{2}\cos\alpha\sin 2\beta\right)z_{12}\right.\\\
&\,\,\,\,\,\,\,\,\,\,\,\,+\left(-\sin\alpha\sin\beta\sin\gamma+\cos\alpha\cos
2\beta\cos\gamma\right)\biggr{)}\frac{\partial}{\partial z_{12}}\\\
&+\left(\left(\sin\alpha\cos\beta\sin 2\gamma+\frac{1}{2}\cos\alpha\sin
2\beta\cos
2\gamma\right)z_{13}+\left(-\sin\alpha\sin\beta\sin\gamma\right.\right.\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\left.\left.+\cos\alpha\cos
2\beta\cos\gamma\right)z_{23}+\left(-\sin\alpha\cos\beta\cos
2\gamma+\frac{1}{2}\cos\alpha\sin 2\beta\sin
2\gamma\right)\right)\frac{\partial}{\partial
z_{13}}\,\,\,\,\,\,\,\,\,\,\,\,\\\
&+\left(\left(\frac{1}{2}\sin\alpha\cos\beta\sin
2\gamma-\frac{1}{2}\cos\alpha\sin 2\beta-\frac{1}{2}\cos\alpha\sin
2\beta\sin^{2}\gamma\right)z_{23}\right.\\\
&\,\,\,\,\,\,\,\,\,\,\,\,+\left(\sin\alpha\sin\beta\cos\gamma+\cos\alpha\cos
2\beta\sin\gamma\right)\biggr{)}\frac{\partial}{\partial z_{23}}.\\\
\end{split}$ (173)

Proof of Lemma A.1. According to the definition,

$\begin{split}&(-L_{X_{12}}f)(\alpha,\beta,\gamma,z_{12},z_{13},z_{23})=\left.\frac{d}{dt}\left(f\left(\widetilde{\alpha}(t),\widetilde{\beta}(t),\widetilde{\gamma}(t),\widetilde{z}_{12}(t),\widetilde{z}_{13}(t),\widetilde{z}_{23}(t)\right)\right)\right|_{t=0},\\\
\end{split}$ (174)

where
$\left(\widetilde{\alpha}(t),\widetilde{\beta}(t),\widetilde{\gamma}(t),\widetilde{z}_{12}(t),\widetilde{z}_{13}(t),\widetilde{z}_{23}(t)\right)$
are the Euler-Iwasawa coordinates of

$\footnotesize\begin{split}\left(\begin{matrix}1&t&0\\\ 0&1&0\\\ 0&0&1\\\
\end{matrix}\right)&\left(\begin{matrix}\cos{\beta}\cos{\gamma}&-\sin{\beta}&\cos{\beta}\sin{\gamma}\\\
\sin{\alpha}\sin{\gamma}+\cos{\alpha}\cos{\gamma}\sin{\beta}&\cos{\alpha}\cos{\beta}&\cos{\alpha}\sin{\beta}\sin{\gamma}-\cos{\gamma}\sin{\alpha}\\\
\cos{\gamma}\sin{\alpha}\sin{\beta}-\cos{\alpha}\sin{\gamma}&\cos{\beta}\sin{\alpha}&\cos{\alpha}\cos{\gamma}+\sin{\alpha}\sin{\beta}\sin{\gamma}\end{matrix}\right)\left(\begin{matrix}1&z_{12}&z_{13}\\\