Paper Title: On degenerate Whittaker space for $GL_4(\mathfrak{o}_2)$.



§ ABSTRACT 
Let _2 be a finite principal ideal local ring of length 2. 
For a representation π of _4(_2), the degenerate Whittaker space π_N, ψ is a representation of _2(_2).
We describe π_N, ψ explicitly for an irreducible strongly cuspidal representation π of _4(_2). 
This description verifies a special case of a conjecture of Prasad. We also prove that π_N, ψ is a multiplicity free representation.



The Trojan-like Colors of Low-Perihelion Kuiper Belt Objects
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    August 5, 2024
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§ INTRODUCTION
 
Let F be a finite unramified extension of _p or _p((t)). 
Let 𝔬 be the ring of integers of F with a uniformizer ϖ. 
For any positive integer l, let 𝔬_l:=𝔬/(ϖ^l).
Then _l is a finite principal ideal local ring of length l.
Note that 𝔬/(ϖ) ≅𝔽_q is a finite field of order q=p^f of characteristic p>0. 
If the characteristic of F is zero then _l is the the ring of Witt vectors over _q of length l and, if the characteristic of F is p then _l is isomorphic to _q[t]/(t^l).
We will be working with both the rings simultaneously and our proofs are uniform for both the cases.
We assume that p ≠ 2.

Let _2n(𝔬_l) be the group of invertible 2n × 2n matrices over the ring _l. 
Let 

    P_n,n = {(  g     X 
     0     g' ) : g, g' ∈_n(_l), X ∈ M_n(_l) } and 
       
     N = {(  I     X 
     0     I ) : X ∈ M_n(_l)  }≅ M_n(_l).

Let ψ_0 : 𝔬_l →ℂ^× be a non-trivial additive character which is also non-trivial when restricted to the subgroup ϖ^l-1𝔬_l for l > 1.
Let ψ : M_n(_l) →ℂ^× be the additive character defined by ψ(X) = ψ_0((X)) for all X ∈ M_n(_l), where (X) denotes the trace of X. 
Since M_n(_l) ≅ N, we treat ψ as a character of N as well.
Let (π, V) be an irreducible representation of _2n(_l). 
Define V_N,ψ := { v ∈ V : π(X)v=ψ(X)v   ∀ X ∈ N } which is the subspace of V on which N operates by the character ψ. 
The space V_N, ψ is called the (N, ψ)-Whittaker space of π, which we call degenerate Whittaker space of π. 
Since (gXg^-1) = (X) for all g ∈_n(𝔬_l) and X ∈ M_n(_l), it follows that _n(_l) sitting inside the subgroup P_n,n as g ↦(  g     0 
 0     g ) stabilizes  the character ψ of N. 
Therefore, _n(_l) operates on the space V_N,ψ giving rise to a representation of _n(_l) which we denote by π_N, ψ.
For a representation (π, V), we often omit V and write only π.
In this paper, we will be concerned with the following question.
 
Given an irreducible representation π of _2n(𝔬_l), what is the degenerate Whittaker space π_N, ψ as a representation of _n(𝔬_l)?

The above question is a variant of the question considered by Prasad <cit.> for  l=1. 
Prasad’s work has inspired  several related works, including ours.
For example, when l=1 Balasubramanian-Khurana <cit.> consider a variant by taking  different characters ψ for n=2, 3  and Balasubramanian-Dangodara-Khurana <cit.> consider analogous variant for general n. 
Another variant, when the ring _l is replaced by a non-Archimedean local field, this question has been considered by Prasad <cit.> for _4 and more recently Pandey-Venketasubramanian <cit.> for Sp_4.


Now we briefly describe the work of Prasad <cit.> for l=1. 
Note that 𝔬_1 ≅𝔽_q. 
In this case, (N, ψ)-degenerate Whittaker space π_N,ψ is a representation of _n(𝔽_q). 
Recall that for any positive integer n, we have 𝔽_q^n ^×↪_n(𝔽_q) and the embedding is unique up to conjugation.
By the work of Green <cit.>, it is well known that every irreducible cuspidal representation of _n(_q) is associated with a primitive character  of _q^n^×.
Here is the theorem of Prasad <cit.>. 
 
Let π be an irreducible cuspidal representation of _2n (𝔽_q) obtained from a primitive character θ: 𝔽_q^2n^×→ℂ^×. 
Then, as a representation of _n(𝔽_q),   

    π_N, ψ≅_𝔽_q^n^×^_n(𝔽_q) ( θ |_𝔽_q^n^× ).
     


For n=1 and π a regular representation of _2(_2), the Question <ref> is answered in a work of the second author with P. Singla <cit.>. 
In this case, π_N, ψ is a representation of _1(_l) which is isomorphic to the center of _2(_l).
More precisely, we have the following result <cit.>.

Let π be an irreducible  regular representation of _2(𝔬_l) with central character ω_π, then π_N, ψ is one dimensional and π_N, ψ≅ω_π.


A more general version of Theorem <ref> has been conjectured by Prasad which we state below.
Let L (respectively, E) be an unramified extensions of F of degree n (respectively, 2n) and 𝔒 (respectively, 𝒪) the ring of integers of L  (respectively, E). 
Write 𝔒_l = 𝔒/(ϖ^l) and 𝒪_l = 𝒪/ (ϖ^l) 
then 𝔒_l^×↪_n(_l) and 𝒪_l^×↪_2n(_l).
We also have _l⊂𝔒_l⊂𝒪_l.
It is known from Lusztig <cit.> and Aubert-Onn-Prasad <cit.> that a strongly cuspidal representation of _2n(_l) is obtained from a strongly primitive character θ : 𝒪_l^×→^× (see Section <ref> for more details).   
 
Let π be an irreducible strongly cuspidal representation of _2n(_l) associated to a strongly primitive character θ : 𝒪_l^×→^×. Then as a representation of _n(_l), we have 

    π_N, ψ≅_𝔒_l^×^_n(_l) (θ|_𝔒_l^×).


We prove the above conjecture of Prasad for n=2, l=2, which is one of the main results of this paper.
 
The Conjecture <ref> is true for n=2 and l=2.

We would like to mention that a few steps, but not all, in the proof of this theorem also work for general n and l=2.
Since, we could not achieve all the steps to prove the above conjecture of Prasad for general n (even for l=2), we restrict ourselves to the case of n=2 and l=2.


In addition, we prove the following multiplicity one result.
 
 Let π be a strongly cuspidal  representation of _4(_2). 
Then  π_N, ψ   is multiplicity free representation of _2(_2), i.e. for any irreducible representation σ of _2(_2) we have
 
    ( __2(_2)( π_N, ψ, σ)  )≤ 1.



The above theorem can be considered as a variant of a theorem of Rallis for p-adic groups <cit.>.
The theorem of Rallis is indeed a Bessel model case of Gan-Gross-Prasad (GGP) conjecture <cit.>.  
More precisely, it is the case of the GGP conjectures when one considers the restriction of representations of the group SO(6), which is closely related to _4 to the subgroup SO(3) which is closely related to _2.
The analogous multiplicity one theorem is also true in Prasad's case for n=2, which is a consequence of Theorem <ref>. 
Our proof of Theorem <ref> uses the above multiplicity one result.

We now give an outline of the paper.
Our proofs depend on the construction of the regular representations of _n() as given by Hill <cit.> and Krakovski-Onn-Singla <cit.> which we recall in Section <ref>. 
In particular, an  irreducible strongly cuspidal representation of _n(_2) is induced from certain subgroup, say T. 
Our strategy here is to restrict the representation π to the P_2,2 subgroup which we can describe by Mackey theory. 
We write an exhaustive set Ω of representatives for the double cosets in T \_4(_2)/P_2,2 in Section <ref>.
We use these representatives for many calculations.
In fact, π|_P_2,2≅δ⊕π^δ where δ varies over a set of representatives of the double cosets mentioned above. 
Then, for every δ∈Ω we describe the representation π^δ_N, ψ as a representation of _2(_2). 
In Section <ref>, we compute the dimension of π_N, ψ by computing the dimension of every π^δ_N, ψ.
In Section <ref>, we write down the character of π^δ_N, ψ for every δ.

In Section <ref>, we describe all the characters of I+ ϖ M_2(_2) appearing in π^δ_N, ψ when restricted to I+ ϖ M_2(_2).

There are three types of regular representations of _2(_2) as mentioned in Table <ref> which helps us to analyse their occurrence in π^δ_N, ψ and therefore in π_N, ψ one by one.
In fact, we describe all the irreducible representations of _2(_2) which appear in π_N, ψ without appealing to the induced representation Π := _𝔒_2^×^_2(_2)  ( θ|_𝔒_2^× ). 
In this section, we also prove the multiplicity one result mentioned above in Theorem <ref>.
In Section <ref>, we describe all the irreducible constituents of Π  and verify that all these constituents are the same as those which appear in π_N, ψ with the same multiplicity (in fact, multiplicity one), which in turn proves Theorem <ref>. 


It should be noted that the proof of Prasad's theorem in <cit.> is indirect in the sense that there is no explicit isomorphism between π_N, ψ and _𝔽_q^n^×^_n(𝔽_q) ( θ |_𝔽_q^n^× ), instead it is proved that the characters of these two representations coincide.
We also prove Theorem <ref> indirectly in the similar sense that we do not give any explicit isomorphism between π_N, ψ and Π. 

The proof in <cit.> uses character theory, while our proof is a consequence of Mackey theory and Clifford theory since the construction of π is totally explicit.





§ PRELIMINARIES 


Let F be a finite unramified extension of _p or _p((t)), and let 𝔬 be its ring of integers with a uniformizer ϖ.
For any positive integer l, let 𝔬_l=𝔬/(ϖ^l). 
Note that 𝔬/(ϖ) ≅𝔽_q is a finite field of order q=p^f of characteristic p>0. 
We assume p ≠ 2.
For any integer m with 1≤ m<l, we have the natural quotient map of rings 𝔬_l→𝔬_m. 
This quotient map induces a surjective group homomorphism _n(𝔬_l)→_n(𝔬_m) for any positive integer n. 
For any g ∈_n(_l), h ∈_n(_m) we write g̅∈_n(_m) for the image of g under the natural map and h̃∈_n(_l) for any element in the inverse image of h under the natural map.
Define K^m_l to be the kernel of the natural map _n(𝔬_l)→_n(𝔬_m) which is also called the m-th principal congruence subgroup of _n(𝔬_l). 
Note that K_l^m are normal subgroups and, for m ≥ l/2 the subgroups K_l^m are abelian.
Thus we have the following natural filtration of subgroups

    { I }⊂ K_l^l-1⊂⋯⊂ K_l^m-1⊂ K_l^m⊂⋯⊂ K_l^1⊂_n(_l).

For 1 ≤ m <l, we have K^m_l/K^m+1_l≅ (M_n(𝔽_q),+).
Let M_n(𝔬_l) denote the Pontryagin dual group of the group (M_n(𝔬_l),+).
Fix a non-trivial additive character ψ_0:𝔬_l→ℂ^× such that ψ_0|_ϖ^l-1𝔬_l≠ 1.  
For every A ∈ M_n(𝔬_l), define ϕ_A∈M_n(𝔬_l) by ϕ_A(B) = ψ_0( (AB)), where  denote the trace.
The map A ↦ϕ_A defines a group isomorphism between M_n(𝔬_l) and M_n(𝔬_l). 
Note that this isomorphism depends on the choice of the character ψ_0. 

For any 1≤ i<l, _l-i≅ϖ^i𝔬_l⊂_l and there is a natural surjection _l→_l-i written as x↦x̅. 
 Given a character ψ_0:_l→^×, its restriction to the subgroup ϖ^i_l can be seen as character of _l-i again denoted by ψ_0 satisfying the following property
 ψ_0(ϖ ^ia)=ψ_0(a̅) for all a∈_l.
 


 §.§ Regular representations of _n(_l)





  * Let x ∈ M_n(_q).
Then x is called regular if its characteristic polynomial is the same as its minimal polynomial. 
Moreover, x is called regular elliptic if its characteristic polynomial is irreducible.

  * For x ∈ M_n(_q), the character ϕ_x : M_n(𝔽_q) →ℂ^× is called regular (respectively, regular elliptic) if x is a regular (respectively, regular elliptic) element.

  * An irreducible representation of _n(𝔬_l) is called regular (respectively, strongly cuspidal) if its restriction to K^l-1_l ≅ M_n(𝔽_q) contains a character which is regular (respectively, regular elliptic).




In the study of representations of _n() the regular representations turn out to be basic objects. 
The regular representations of _n(_l) can be constructed using Clifford theory, see Gérardin <cit.>, Hill <cit.>, Krakovski-Onn-Singla <cit.>, Stasinski <cit.>, Stasinski-Stevens <cit.>. This construction is more intricate when l is odd, see <cit.>.
Now, we briefly describe the construction of regular representations of _n(_l) for even l. 

Consider the group _n(_l) for an even integer l > 1 and set
m = l/2. 
Let ρ : K^m_l →^× be a character such that ρ|_K^l-1_l is a regular character and, let the inertia group of ρ be I(ρ) : = { g∈_n(𝔬_l):ρ(g^-1xg)=ρ(x) ∀ x∈ K^m_l}.
Then the following hold.


(1) The character ρ extends to  I(ρ) and let ρ̃ be an extension of ρ to I(ρ). Then _I(ρ)^_n(𝔬_l)ρ̃ is an irreducible regular representation of _n(𝔬_l).

(2) Suppose π is a regular representation of _n(𝔬_l) such that ρ appears in π|_K^m_l. Then, there exists a character ρ̃ of I(ρ) which extends ρ such that π=_I(ρ)^_n(𝔬_l)ρ̃.



Let l be even and l=2m. 
Then, K^m_l≅ M_n(_m) and any character of M_n(_m) is ϕ_B for some B ∈ M_n(_m).
Let π be an irreducible regular representation _n(_l) such that ϕ_B appears in  π|_K^m_l. 
It can be seen by Clifford theory that ϕ_B' appears in π|_K^m_l if and only if B' = gBg^-1 for some g ∈_n(_m).




 §.§ Representations of _2(_2)


We summarise all the irreducible representations of _2(_2) below depending on their restriction to the first congruence subgroup K^1_2 which is isomorphic to M_2(_q). 
Note that a regular matrix in M_2(_q) is either non-split semisimple or split non-semisimple or split semisimple. 
It follows that the irreducible regular representation of _2(_2) can be classified into three classes described below.
If an irreducible regular representation σ is such that σ|_K^1_2 contains ϕ_B with B non-split semisimple (respectively, split non-semisimple, split semisimple) then we say the type of σ is non-split semisimple (respectively, split non-semisimple, split semisimple). 
In M_2(_q), for a given trace, there is a unique class of scalar matrix, a unique class of split non-semisimple matrices, q-1/2 distinct classes of split semisimple matrices and q-1/2 distinct classes of non-split semisimple matrices.


Fix a regular character ϕ_B of K^1_2≅ M_2(_q) and a character ω of the center Z of _2(_2).
The character ϕ_B can be extended to the inertia group I(ϕ_B). 
Observe that Z· K^1_2⊂ I(ϕ_B).
The following table describes the number of irreducible regular representation of different types when its restriction to Z · K^1_2 contains the character ω·ϕ_B.


The irreducible representations of _2(_2), which are not regular, are of the type σ⊗ (χ∘), where σ is an irreducible representation of _2(_q) considered as a representation of _2(_2) using the natural map _2(_2) →_2(_q) and χ : _2^×→^× is a character.





 §.§ Strongly primitive characters

Let E be an unramified extension of F of degree n. 
Let 𝒪 be the integral closure of 𝔬 in E with maximal ideal 𝒫 =ϖ𝒪. 
Let 𝒪_l =𝒪/𝒫^l, then 𝒪_l is isomorphic to a free 𝔬_l-module of rank n. 
Therefore, _n(𝔬_l) can be identified with the group of automorphisms of _l-module 𝒪_l. 
This identification is determined up to an inner automorphism of _n(𝔬_l).

Clearly, for a∈𝒪_l, x↦ ax is an 𝔬_l-module endomorphism of 𝒪_l and we get an embedding of the ring 𝒪_l to the ring of endomorphisms of 𝔬_l-module 𝒪_l.
Therefore, 𝒪_l^× becomes a subgroup of _n(𝔬_l) under the same embedding.




  * A character ϕ : 𝒪_1≅_q^n→ℂ^× is called primitive if there does not exist a proper subfield 𝔽_q^d of _q^n and a character ϕ_0 of _q^d such that  ϕ(x)=ϕ_0(__q^n|_q^d(x)) for all x∈_q^n, where __q^n|_q^d denote the trace map for the field extension _q^n/_q^d. 

  * For l >1, a character θ : 𝒪_l^×→ℂ^× is called strongly primitive if its restriction to the kernel of the map 𝒪_l^×→𝒪_l-1^×, which is isomorphic to 𝒪_1, is a primitive character.


Strongly cuspidal representations of _n(𝔬_l) are associated to strongly primitive characters of 𝒪_l^×, which we explain for even l=2m.
Let x ∈ M_n(_q) be a regular elliptic element and x̃∈ M_n(_m) be such that it maps to x under the natural quotient M_n(_m) → M_n(_q).
Since K^m_l≅ M_n(_m), ϕ_x̃ is a character of K^m_l. 
Then the inertia group I(ϕ_x̃) ≅𝒪_l^× K^m_l.
Let ϕ̃_x̃ be an extension of ϕ_x̃ to I(ϕ_x̃). 
Write θ = ϕ̃_x̃|_𝒪_l^×.
Then θ is a strongly primitive character of 𝒪_l^× associated to the strongly cuspidal representation π = _I(ϕ̃_x̃)^_n(_l)ϕ̃_x̃, see Aubert-Onn-Prasad <cit.>, Lusztig <cit.>.
More precisely, we have the following result <cit.>.

There is a canonical bijective correspondence between strongly cuspidal representations of _n(𝔬_l) and (E/F) orbits of strongly primitive characters of 𝒪_l^×.




  * Let L be a degree 2 unramified extension of F and 𝔒 the integral closure of 𝔬 in L.
Let E be a degree 4 unramified extension of F containing L with 𝒪 the integral closure of 𝔬 in E.
Throughout the paper, we denote 𝔬_2, 𝔒_2 and 𝒪_2 for 𝔬/(ϖ^2),  𝔒/ (ϖ^2) and 𝒪/(ϖ^2) respectively.

  * We denote the subgroup  I+ϖ M_2(𝔬_2) of _2(𝔬_2) by J^1_2 and the subgroup I+ϖ M_4(𝔬_2) of _4(_2) by K^1_2.





§ DOUBLE COSETS T\ G/P

For a representation π of a finite group H and an automorphism τ of H,  we define a representation π^τ given by π^τ := π∘τ.
In particular, if H ⊂H̃ is a subgroup and x ∈H̃ normalises H, then we get an automorphism τ_x of H given by τ_x(y)= x^-1yx. In this case, we write π^x instead of π^τ_x. 

Throughout this section,  G=_4(𝔬_2) and P=P_2,2⊂ G.
We fix an irreducible strongly cuspidal representation π = _I(ϕ_x)^_4 (_2) (ϕ̃_x) where x is a regular elliptic element of M_4(_q).
We will choose an embedding _q^4^×↪_4(_q) and take x ∈_q^4^×.
We will also fix an embedding of 𝒪_2^×↪_4(_2) such that 𝒪_2^× maps onto _q^4^× under the quotient  map _4(_2) →_4(_q).
Since x is a regular elliptic element, we have I(ϕ_x) = 𝒪_2^×· K^1_2 and I(ϕ_x) / K^1_2≅_q^4^×. 
For convenience, we write T=I(ϕ_x) and then π = _T^G ϕ̃_x.
Therefore, π_N, ψ = (^G_Pπ)_N, ψ = (_P^G _T^G ϕ̃_x)_N,ψ.
By Mackey theory 


_P^G_T^Gϕ̃_x≅δ∈ T \ G/P⊕_δ^-1T δ∩ P^Pϕ̃_x^δ^-1.

 
For δ∈ T \ G/P, we will write π^δ = _δ^-1T δ∩ P^Pϕ̃_x^δ^-1.
Now we describe the double cosets T \ G/P. 
We write P̅ for the image of P in _4(𝔽_q) and T̅ = T/K^1_2≅_q^4^×.
Since K^1_2 is the kernel of the quotient map _4(_2) →_4(_q) and K^1_2⊂ T, the quotient map induces a bijection between set of double cosets
T\_4(𝔬_2)/P and T̅\_4(𝔽_q)/P̅. 
 
The number of distinct cosets T̅\_4(_q)/P̅ is q+1.


We know that _4(𝔽_q)/P̅ = Gr(4,2) the Grassmannian, i.e. the set of all 2-dimensional subspaces in a 4-dimensional vector space over _q. 
For a group G acting on a set X we write, ↷ G  X. Naturally, ↷
_4(𝔽_q) Gr(4,2) transitively and distinct orbits of the restricted action to the subgroup T̅↪_4(𝔽_q) gives the distinct double cosets T̅\G̅/P̅.

Now, consider _q^4 as a 4-dimensional vector space over _q. 
A general element of Gr(4,2) can be written as {x,y} the span of vectors x, y ∈_q^4 which are linearly independent  over _q. 
Further, a representative of any orbit under the action ↷ _q^4^× Gr(4,2) can be taken to be {1,y} where y∈𝔽_q^4\𝔽_q. 
It can be easily verified that {1,y} and  {1,y'} are in the same orbit if and only if
y^'=py+qry+s for some p,q,r,s∈𝔽_q where { py+q,ry+s } is a linearly independent set of vectors of 𝔽_q^4, which is equivalent to say that ∃[ p q; r s ]∈_2(_q) such that y^'=py+qry+s. 
 Now consider the action ↷ _2(_q) _q^4\_q given by 
    [ p q; r s ]· y=py+qry+s,    where [ p q; r s ]∈_2(_q) and  y ∈_q^4\𝔽_q.

Thus, it is clear that there is a bijection between the orbits in the action↷ _q^4^× Gr(2,4) and the action ↷ _2(_q) _q^4\_q.
We use the action ↷ _2(_q) _q^4\_q to compute the number of distinct orbits.


  * If y∈_q^2\_q, the stabiliser of {1,y} is _q^2^× and therefore the orbit size of {1,y} is q^2-q.

  * If y∈_q^4\_q^2, the stabiliser of {1,y} is _q^× and therefore the orbit size of {1,y} is  q(q^2-1).

Thus, there is a single orbit with stabiliser _q^2^× and the number of orbits with stabiliser _q^× is |_q^4\_q^2|q(q^2-1)=q. 
Hence, the number of distinct double cosets is q+1.


Now we write down an exhaustive, although not exclusive, set of representatives for these double cosets for which we need to fix an embedding of _q^4^× inside _4(_q). 
We fix an embedding of _q^4 in M_4(_q) as described below.

Let α∈_q^×\_q^× 2 and then _q^2=_q(√(α)). 
Let a, b ∈_q be such that a+b√(α)∈_q^2^×\_q^2^× 2 and 

    _𝔽_q^2|_q(a+b√(α)) = a^2 -b^2 α  is a square free element in _q,
 
where _𝔽_q^2|_q denotes the norm map for the field extension _q^2/_q. 
Write β = √(a+b √(α)). 
Then _q^4=_q(β).
Note that _q(β^2)=_q^2. For x ∈_q^4, the map _q^4→_q^4 given by y ↦ xy is _q-linear and thus gives rise to an embedding _q^4↪ M_4(_q).
Similarly, there is an embedding _q^2↪ M_2(_q).
By fixing the ordered basis  {1,β^2,β,β^3} of _q^4 as a _q-vector space, we write an explicit embedding which is given by 

    x= a_0+a_1β^2+a_2β+a_3β^3 ↦[ X_1 X_2; X_3 X_1 ],
 
where X_1, X_2, X_3 ∈ M_2(_q) are as follows:


X_1=[            a_0 -a_1(a^2-b^2α);            a_1      a_0+2aa_1 ], X_2=[         -a_3(a^2-b^2α) -(a_2+2aa_3)(a^2-b^2α);              a_2+2aa_3   a_3(3a^2+b^2α)+2aa_2 ], X_3=[            a_2 -a_3(a^2-b^2α);            a_3      a_2+2aa_3 ]. 
Note that X_1 and X_3 are the images of a_0 + a_1 β^2 and a_2 + a_3 β^2 under the fixed embedding _q^2↪ M_2(_q) obtained by the ordered basis {1, β^2} of _q^2 as _q-vector space. More explicitly,

    a_0+a_1β^2↦[            a_0 -a_1(a^2-b^2α);            a_1      a_0+2aa_1 ].

From now onwards, we identify _q^4^× as a subgroup of _4(_q) and _q^2^× as a subgroup of _2(_q) via the above mentioned embedding.


For u,v,w ∈_q, define A_u,v =
[ 1 0 0 0; 0 1 0 0; 0 u 1 0; 0 v 0 1 ] and 
A_w=
[ 1 0 0 0; 0 0 1 0; 0 1 0 0; 0 w 0 1 ]. 


 
By taking the embedding _q^4^×↪_4(_q) obtained from Equation (<ref>), an exhaustive, but not exclusive, set of coset representatives of T̅\G̅/P̅ is given by


{ A_u,v, A_w : u,v,w ∈𝔽_q }.




Recall that _4(𝔽_q)/P̅≅ Gr(4,2) where P̅ is the stabiliser of the subspace W_0={ 1,β^2 }≅_q^2, therefore transitivity of the action ↷
_4(𝔽_q) Gr(4,2) 
implies given any W∈ Gr(4,2), we get an element g∈_4(𝔽_q) such that g· W_0=W. 
Note that a representative of an orbit for the action  ↷ _q^4^× Gr(4,2) can be taken to be W={1, y} for y ∈_q^4\_q. 
Write y = b_0+b_1β^2+b_2β+b_3β^3  for some b_0, b_1, b_2, b_3 ∈_q then,


     W  = { 1,b_1β^2+b_2β+b_3β^3 }.




  Case 1: If b_1 ≠ 0, we can take b_1=1 and  get W={1,β^2+b_2β+b_3β^3} 
 where b_2,b_3∈𝔽_q. 
 
 We define a linear map on _q^4 which maps W_0 to W in such a way that 1 ↦ 1, β^2 ↦β^2+ b_2 β +b_3 β^3, β↦β, β^3 ↦β^3.
 Then a representative for the double coset corresponding to W can be taken to be A_b_2, b_3.

  Case 2: If b_1=0, b_2 ≠ 0 we can take b_2=1 and get W={1,β+b_3β^3} where b_3∈𝔽_q. Similar to the Case 1, a representative for the double coset corresponding to W can be taken to be A_b_3.

  Case 3: If b_1=b_2=0, b_3≠ 0 we can take b_3=1 and get W={1,β^3}.
For this subspace, we claim that its representative can be taken to be either A_u,v  for some u,v ∈_q or A_w for some w ∈_q.
Note that β^-3 action takes { 1,β^3} to {1,β^-3}.
Write β^-3=b_0+b_1β^2+b_2β+b_3β^3 for some b_0,b_1,b_2,b_3.
We claim that either b_1 ≠ 0 or b_2 ≠ 0 and then the proof follows from the earlier two cases. 
If b_1=b_2=0, then, β^-3 = b_0+b_3β^3 which gives 1=b_0β^3+b_3β^6
= b_0β^3-(2ab_3)(a^2-b^2α)+(3a^2+b^2α)b_3β^2. 
Comparing the coefficients both sides, we get, b_0=0 and -2ab_3(a^2-b^2α)=1 and (3a^2+b^2α)b_3=0 which implies 3a^2+b^2α=0__q^2|_q(a+b√(α))=4a^2, which contradicts our assumption (<ref>).


 
For every w ∈_q there exists u,v ∈_q such that T̅ A_wP̅ = T̅ A_u,vP̅.    


It can be verified that T̅ A_wP̅ = T̅ A_u,vP̅ if and only if 

    u^2[w(1+2aw)]-uv[1-w^2(3a^2+b^2α)]-v^2[2a+w(3a^2+b^2α)]
    
        +(1+2aw+(a^2-b^2α)w^2)=0.

Therefore, it is enough to prove that for any w ∈_q, there always exists  u,v ∈_q such that Equation (<ref>) holds. 
For a given w ∈_q, we consider the above equation as a quadratic equation in u. 
If the coefficient of u^2 is 0, i.e. w(1+2aw)=0, then the Equation (<ref>) is linear and one can easily verify that there exists u, v ∈_q satisfying the equation.
Now we assume that the coefficient of u^2 is not zero. 
Then Equation (<ref>) has a solution u ∈_q if and only if the discriminant is a square element in _q. 
The discriminant D(v) is given by 


    D(v) = v^2((1+2aw)^2 -(a^2-b^2α)w^2) -4w(1+2aw)((1+aw)^2-(bw)^2α).

 
Clearly, the coefficient of v^2 is not zero, otherwise a^2 - b^2 α∈_q^× 2. 
Then, 
    |{ D(v) : v ∈_q}| = q+1/2.

Since the number of non-square elements in _q is q-1/2, there exists v ∈_q such that D(v) ∈_q^× 2. Hence we are through.

By using Lemma <ref> and Lemma <ref> we conclude the following.

The set Ω := { A_u,v : u,v ∈_q} is an exhaustive, but not exclusive, set of representatives for T̅\G̅ / P̅. 

We state the following lemma without proof which provides an equivalent condition for two elements from Ω representing the same double coset for T̅\G̅/P̅.

For δ = A_u,v, δ'=A_u', v'∈Ω, let 

    C(δ, δ') =      u^2-u^'^2 +uvu^'^2-u^2u^' v^' + 2a(v^2u^'^2-u^2v^'^2) + 2a(uv-u^' v^') 
        +(3a^2+b^2α)(u^' v^' v^2-uvv^'^2)+(v^2-v^'^2)(a^2-b^2α).

Then,
T δ P = T δ' P if and only if 
 C(δ, δ')=0.
 In particular,  T δ P = TIP  if and only if δ =I, where I is the 4× 4 identity matrix.


Here is a list of notations which we will use throughout this paper.



  * We denote the identity matrix of a suitable order by I, which will be clear from the context.

  * Let Ω_0⊂Ω be such that every double coset in T̅\G̅/P̅ has a representative in Ω_0 and distinct elements in Ω_0 represent distinct double cosets in T̅\G̅/P̅.

  * Since the quotient map G →G̅ induces a bijection between T \ G /P and T̅\G̅ /P̅, for  a representative δ∈G̅ of a double coset in T̅\G̅/P̅ we write its lift to G by the same letter δ for the corresponding representative of the double coset in T\ G/P.

  * For δ=A_u,v =[ I 0; U I ] where U=[ 0 u; 0 v ], write 


L_δ=X_1U-UX_1+X_3-UX_2U. In particular, L_I=X_3.


  * If L_δ is invertible, then write
ℬ_δ=L_δ(X_1+X_2U)L_δ^-1+X_1-UX_2.

In particular, ℬ_I=2X_1 and (ℬ_δ)= (2X_1). 


  * Write X=a_1^2-a_2a_3-2aa_3^2 and Y=a_2^2+a_3^2(a^2-b^2α)+2aa_2a_3.



Observe that Y≠ 0 otherwise a_2=a_3=0 and then x will not be a regular elliptic element of M_4(_q). 
It can be checked that 

    (L_δ)   = -u^2X-uv(2aX+Y)-v^2(2aY+X(a^2-b^2α))+Y.





§ THE DIMENSION OF Π_N,Ψ

For a finite dimensional representation ρ of a finite group H, we write Θ_ρ for the character of ρ.
Suppose ρ' be an irreducible representation of the group H. 
The multiplicity of ρ' in ρ is defined to be the dimension of _H(ρ, ρ'). 
We say that ρ is multiplicity free if the multiplicity of any irreducible representation in ρ is ≤ 1.
If we define ⟨Θ_ρ,Θ_ρ'⟩_H := 1|H|g∈ H∑Θ_ρ(g) Θ̅_ρ'(g), then by <cit.> we have

    (_H(V,V'))= ⟨Θ_ρ,Θ_ρ'⟩_H.

We fix x ∈_q^4^× as in Equation (<ref>) and
consider a strongly cuspidal representation π=_T^Gϕ̃_x of _4(_2) (see Theorem <ref>).
If we write π^δ =_δ^-1T δ∩ P^Pϕ̃_x^δ^-1, then by Mackey theory, 


π_N,ψ =δ∈Ω_0⊕π^δ_N,ψ. 

Therefore, (π_N, ψ) = δ∈Ω_0∑(π^δ_N, ψ).
In what follows we compute the dimension of π^δ_N, ψ for all δ∈Ω with the help of following series of lemmas.

Write ϖ N={[   I ϖ X;   0   I ]:X∈ M_2(_2)}.
Then, for every δ∈Ω, we have 

    π^δ_N, ψ≅γ∈ (δ^-1 Tδ∩ P) \ P/N⊕_ϖ N(ϕ̃_δ^-1xδ^γ^-1,ψ).
  


By Mackey theory,
π^δ_N,ψ≅γ∈ (δ^-1 Tδ∩ P) \ P/N⊕_γ^-1(δ^-1 Tδ∩ P)γ∩ N(ϕ̃_δ^-1xδ^γ^-1,ψ). 

The Lemma follows from the fact that the subgroup γ^-1(δ^-1 Tδ∩ P)γ∩ N=ϖ N for all γ∈ (δ^-1Tδ∩ P) \ P /N, which we prove now. 
For any γ∈ P, we have 

    γ^-1(δ^-1 Tδ∩ P)γ∩ N=γ^-1(δ^-1 Tδ∩ P∩ N)γ=γ^-1(δ^-1 Tδ∩ N)γ.
 
Since δ^-1 Tδ ( K^1_2)≅𝔽_q^4^×
we get, (δ^-1 Tδ∩ N ) ( K^1_2) = { I}. 
Then, δ^-1Tδ∩ N= ϖ N. 
Also, ϖ N is a normal subgroup of P, therefore γ^-1(δ^-1 Tδ∩ N)γ=ϖ N.

Note that (δ^-1Tδ∩ P) \ P /N is in bijection with (δ^-1𝔽_q^4^×δ∩P̅) \P̅/N̅.

A set of representatives of the double coset (δ^-1𝔽_q^4^×δ∩P̅) \P̅/N̅
 is given as 


  * For δ=I, {[ g_1   0;   0 g_2 ]:g_1∈_q^2^×\_2(_q), g_2∈_2(_q)} is a set of distinct representatives for 
(_q^4^×∩P̅)\P̅/N̅.

  * For δ≠ I, {[ g_1   0;   0 g_2 ]:g_1∈_q^×\_2(_q), g_2∈_2(_q)} is a set of distinct representatives for (δ^-1_q^4^×δ∩P̅)\P̅/N̅.



Clearly, P̅/N̅≅_2(_q)×_2(_q).
Then (a) follows by noting that _q^4^×∩P̅≅_q^2^× and, (b) follows by noting that δ^-1_q^4^×δ∩P̅≅_q^× for δ≠ I.




    
  * (π^I_N,ψ)=q(q-1).
    
  * Let δ≠ I. If π^δ_N,ψ≠ 0 then, (π^δ_N,ψ)=q(q^2-1).



For γ∈ P we have 

    ⟨ϕ̃_δ^-1xδ^γ^-1,ψ⟩_ϖ N=1  if ϕ̃_δ^-1xδ^γ^-1|_ϖ N =ψ|_ϖ N and  0 otherwise.

Write γ = [ g̃_1    0;    0 g̃_2 ] and
using the definition of ϕ_x on ϖ N we get,
ϕ̃_δ^-1xδ^γ^-1|_ϖ N =ψ|_ϖ N if and only if the character X ↦ψ_0(((g̃_2^-1L̃_δg̃_1-I)X)) of ϖ N is trivial (recall L_δ as defined in Notation <ref><ref>), which is equivalent to
L_δg_1= g_2. 


  * Let δ=I.
Note that L_I=X_3. By Lemma <ref> (a), we get 

    (π^I_N,ψ)     = |{(g_1,g_2)∈(_q^2^×\_2(_q))×_2(_q):g_2=X_3g_1}|
        = q(q-1).


  * Let δ≠ I.  
By Lemma <ref> (b), we get

    (π^δ_N,ψ)
        = 1|ϖ M_2(𝔬_2)|∑_γ∈(δ^-1 Tδ∩ P) \ P/N
    
    X∈ϖ M_2(𝔬_2)ψ_0(((g̃_2^-1L̃_δg̃_1-I)X))
    
        =|{(g_1,g_2)∈(_q^×\_2(_q))×_2(_q):g_2=L_δg_1}|
        =
     
            q(q^2-1)     if L_δ is invertible,
    
            0     otherwise.




There exists exactly one δ∈Ω_0 such that π^δ_N,ψ=0.


First, we prove that there exists a δ =A_u,v∈Ω for some u,v ∈_q such that π^δ_N, ψ =0. 
From the proof of Lemma <ref> it follows that π^δ_N, ψ =0 if and only if L_δ is not invertible, i.e. (L_δ)=0.

Recall that (L_δ)=
-(u^2X+uv(2aX+Y)+v^2(2aY+X(a^2-b^2α))-Y) where Y ≠ 0, see (<ref>).
We claim that there exists u, v ∈_q such that (L_δ)=0.

If X=0, then (L_δ)=0 ⇒ (1- uv - 2av^2)Y =0 and it can be easily verified that there exist u,v ∈ F_q satisfying this condition.
Now we assume X ≠ 0 and consider the expression for (L_δ) as a quadratic polynomial in variable u, whose discriminant is
(Y^2-4aXY+4X^2b^2α)v^2+4XY.
Note that if for some v ∈_q, the discriminant is a square in _q then there exists a u ∈_q such that (L_δ) =0.
Write the discriminant as C_1v^2+C_2 for C_1=Y^2-4aXY+4X^2b^2α and C_2=4XY. 
Note that C_1 ≠ 0 otherwise,


 Y^2-4aXY+4X^2b^2α = (Y-2aX)^2 - 4X^2 (a^2 -b^2 α) =0 


i.e. a^2 - b^2 α  is a square element in _q,
which contradicts our assumption in (<ref>).
Since C_1 ≠ 0, | { C_1 v^2 + C_2 : v ∈ F_q}| = q+1/2 but the number of non-square elements in _q is q-1/2.
Therefore, there exists a v ∈_q such that C_1 v^2 + C_2 is a square in _q.
This completes the proof of the existence of δ∈Ω such that (L_δ)= 0.

 

Now we prove the uniqueness, i.e. if π^δ_N, ψ=0 and π^δ'_N, ψ=0 then T δ P = T δ' P.
Write δ =A_u,v and δ' =A_u',v' then 

    (L_δ)= 0   and   (L_δ')=0.

If X= 0, then (<ref>) gives 1-uv-2av^2=0 and 1-u'v'-2av'^2=0.
It can be easily verified that u,v and u', v' satisfy the condition in Lemma <ref> giving T δ P = T δ' P. 

Now assume that X ≠ 0.
By eliminating Y from the two equations in (<ref>) 
we get

    X · C(δ, δ') = 0.

Since X ≠ 0, using Lemma <ref>, we get T δ P = T δ' P.

 
Let π be a strongly cuspidal representation of _4(𝔬_2) then  
    (π_N,ψ)=q^3(q-1).



Note that (π_N,ψ) =δ∈Ω_0∑(π^δ_N,ψ).
Recall that | Ω_0|= q+1 and by Lemma <ref>, there exists a unique δ∈Ω_0 such that δ≠ I and π^δ_N, ψ = 0.
Using Lemma <ref>,  we get

    (π_N,ψ) = q(q-1)+(q-1) · q(q^2-1)
    = q^3(q-1).
 

One of the main theorems in this paper is that π_N, ψ is isomorphic to _𝔒_2^×^_2(𝔬_2) ( θ|_𝔒_2^×) where π is a strongly cuspidal representation of _4(_2) associated to a strongly primitive character θ of 𝒪_2^×. 
Proposition <ref> proves that these two representations have the same dimension.



§ CHARACTERS OF Π^Δ_N,Ψ



For a finite dimensional representation ρ of a group we write ω_ρ for its central character, if it exists.


Let ℳ, 𝒩 be subgroups of a finite group 𝒫 such that 𝒩 is a normal subgroup and 𝒫 = ℳ⋉𝒩. 
Let ψ : 𝒩→^× be a character of 𝒩 and ℳ_ψ = { m ∈ℳ : ψ(mnm^-1) = ψ(n)   for all   n ∈𝒩}.
Let (ρ, V) be a finite dimensional representation of 𝒫 and V_𝒩, ψ = { v ∈ V : ρ(n)v = ψ(n) v  for all  n ∈𝒩} which is a representation of ℳ_ψ denoted by (ρ_𝒩, ψ, V_𝒩, ψ).
The character of ρ_𝒩, ψ is given by


Θ_ρ_𝒩, ψ (m) = 1|𝒩|n ∈𝒩∑Θ_ρ (mn) ψ(n).




The projection map : V → V_𝒩, ψ  is given by (v) = 1/|𝒩|n ∈𝒩∑ρ(n) ψ (n) v. 
Then, 

    Θ_ρ_𝒩, ψ (m) 
        =  (ρ_𝒩, ψ (m)) 
    = (ρ(m)|_V_𝒩, ψ) 
    = (ρ∘)
        = 1/|𝒩|∑_n ∈𝒩(ρ(mn)) ψ(n) 
    = 1/|𝒩|∑_n ∈𝒩Θ_ρ(mn) ψ(n).


The main application of the above lemma is the following.
 
Let ρ be a finite dimensional representation of P_n,n = M ⋉ N where ℳ≅_n(_l) ×_n(_l) and N ≅ M_n(_l). 
Then, g ∈_n(_l) we have

    Θ_ρ_N, ψ(g) = 1|N|∑_X ∈ NΘ_ρ[ g X; 0 g ]ψ(Xg^-1).



We first calculate Θ_π^δ_N, ψ on J^1_2 =I + ϖ M_2(_2), for which we need Θ_π^δ.
By using the formula for the character of an induced representation,

the value Θ_π^δ[ I+ϖ A     X;     0 I+ϖ A ] equals

    1|δ^-1 Tδ∩ P|∑_S,R∈_2(𝔬_2) 
     Q∈ M_2(𝔬_2) ϕ̃_x^δ^-1([ S Q; 0 R ]^-1[ I+ϖ A     X;     0 I+ϖ A ][ S Q; 0 R ]).

It is clear that this value is zero if X∉ϖ M_2(_2). 
Then by using Corollary <ref>, 
Θ_π^δ_N,ψ(I+ϖ A) is given by

    1.0!1/q^8|δ^-1Tδ∩ P|∑_
     S,R∈_2(𝔬_2) 
     X∈ϖ M_2(𝔬_2) 
     Q∈ M_2(𝔬_2)ϕ̃_x^δ^-1[ I+ϖ S^-1AS          0;          0 I+ϖ R^-1AR ]ϕ̃_x^δ^-1[                     I ϖ(S^-1AQ-S^-1QR^-1AR);                     0                     I ]
    
    ϕ̃_x^δ^-1[      I S^-1XR;      0      I ]ψ(X)


We compute this by first summing over X, then over Q, and then over (S,R).


For δ∈Ω, let π^δ_N,ψ≠ 0 and recall L_δ and ℬ_δ from Notation <ref> <ref>, <ref>.


  * For any S, R ∈_2(_2) we have


X∈ϖ M_2(𝔬_2)∑ϕ̃_x^δ^-1[      I S^-1XR;      0      I ]ψ(X)=
    q^4    if  S=RL̃_δ, 

    0    otherwise.



  * For any δ and S,R∈_2(_2) with S=R L̃_δ, 
we have 

 Q∈ M_2(𝔬_2)∑ϕ̃_x^δ^-1[                     I ϖ(S^-1AQ-S^-1QR^-1AR);                     0                     I ]=q^8.
 

 
  * For S=RL̃_δ, we have

    ϕ̃_x^δ^-1[ I+ϖ S^-1AS          0;          0 I+ϖ R^-1AR ]=ψ_0(ϖ (Rℬ̃_̃δ̃R^-1A)).








  * Recall that ϕ_x(I+ϖ A)=ψ_0(ϖ (x̃A)). Since X∈ϖ M_2(_2), 
for any δ∈Ω, we have

    X∈ϖ M_2(𝔬_2)∑ϕ̃_x^δ^-1[      I S^-1XR;      0      I ]ψ(X)     =X∈ϖ M_2(𝔬_2)∑ψ_0(ϖ (    RL̃_δS^-1-I)X)
    
        =
    
         |ϖ M_2(𝔬_2)|     if  S=RL̃_δ, 
    
        0    otherwise.


  * Assume S=RL̃_δ. The statement follows if we prove that the character 

    Q↦ϕ̃_x^δ^-1[                     I ϖ(S^-1AQ-S^-1QR^-1AR);                     0                     I ]
 
is the trivial character of M_2(𝔬_2).
Since ϕ̃_x^δ^-1|_J^1_2 = ϕ_δ^-1 x δ|_J^1_2, we get 

    ϕ̃_x^δ^-1[                      I ϖ (S^-1AQ-S^-1QR^-1AR);                      0                      I ]     = ψ_0(ϖ (L̃_δ(S^-1AQ-S^-1QR^-1AR)))
        
        =ψ_0(ϖ (R^-1AQ-R^-1QR^-1AR))
        
        = 1.



  * Since S=RL̃_δ, we get,

    ϕ̃_x^δ^-1[ I+ϖ S^-1AS          0;          0 I+ϖ R^-1AR ]    = ϕ̃_x^δ^-1[ I+ϖL̃_δ^-1R^-1ARL̃_δ                    0;                    0           I+ϖ R^-1AR ]
        = ψ_0(ϖ  (Rℬ̃_̃δ̃R^-1A)).
 
The last equality follows from the definition of ϕ_x.  



For δ∈Ω with π^δ_N,ψ≠ 0 and A∈ M_2(_2), we have

    Θ_π^δ_N,ψ(I+ϖ A)=q^12|δ^-1Tδ∩ P|R∈_2(_q)∑ψ_0((Rℬ_δR^-1A̅)).




Using the sum over X, over Q and over (S,R) from Lemma <ref> in Equation (<ref>), we get



    Θ_π^δ_N,ψ(I+ϖ A)=q^12/q^8|δ^-1Tδ∩ P|∑_S,R∈_2(_2)
    
    S=RL̃_δψ_0(ϖ(Rℬ̃_̃δ̃R^-1A)).

The above sum does not change if we replace R by R' and S by S' such that R=R' ( ϖ) and S=S' ( ϖ). Therefore, we get

    Θ_π^δ_N,ψ(I+ϖ A)=q^12· q^8/q^8|δ^-1Tδ∩ P|∑_R∈_2(_q)ψ_0((Rℬ_δR^-1A̅)).
 



Note that |δ^-1 T δ∩ P| =  (q^2-1)q^12    if δ=I, 
 (q-1)q^12    if δ≠ I.


We denote A∼ B if the two matrices A and B are conjugate. 
Recall that  any character of M_2(_q) is of the form ϕ_B for B∈ M_2(_q). 
Since M_2(_q) ≅ J^1_2 we write character of J^1_2 also as ϕ_B.
  
Consider δ∈Ω such that π^δ_N, ψ≠ 0. 


  * ϕ_B appears in π^δ_N, ψ|_J^1_2 if and only if B ∼ℬ_δ.

  * The multiplicity of ϕ_ℬ_δ in π^δ_N, ψ is given by


⟨Θ_π^δ_N,ψ,ϕ_ℬ_δ⟩_J^1_2=
1/q^2-1· |(2X_1)|     if δ = I,

1/q-1· |(ℬ_δ)|     if δ≠ I


where (B) denotes the stabiliser of B under the conjugation action of _2(_q) on M_2(_q).



Using Proposition <ref>,  we get

    ⟨Θ_π^δ_N,ψ,ϕ_B⟩_J^1_2 
    = q^12|δ^-1Tδ∩ P|·1q^4∑_R∈_2(_q) 
    
    A∈ M_2(_q)ψ_0((Rℬ_δR^-1-B)A).

Part (a) follows by observing that 

    1/q^4·∑_A∈ M_2(_q)ψ_0((Rℬ_δR^-1-B)A) = {[                 0  if  Rℬ_δR^-1≠ B,;                 1 if  Rℬ_δR^-1 = B. ].

Part (b) follows from Remark <ref> and Equation (<ref>).




  * Under the considered embedding _2(_2) ↪_4(_2) the image of the center of _2(_2) is equal to the center of _4(_2) which are isomorphic to _2^×. We write Z for both the center of _2(_2) as well as that of _4(_2).

  * Let the central character of representation π be ω_π i.e. Θ_π(z)=ω_π(z)(π)=ϕ̃_x(z)(π) for z ∈ Z. 
It can easily be seen that for δ∈Ω, if π^δ_N,ψ≠ 0 then it has a central character which is equal to ω_π.


Proposition <ref> describes the character of π^δ_N,ψ at elements of J^1_2 and now we describe the character of π^δ_N,ψ on remaining elements of _2(_2).

Let δ∈Ω be such that π^δ_N,ψ≠ 0. Then,


  * For z∈ Z and k∈ J^1_2, we have Θ_π^δ_N,ψ(z· k)=ω_π(z)Θ_π^δ_N,ψ(k) . 
In particular,
Θ_π^δ_N,ψ(z)=ω_π(z)·(π^δ_N,ψ).

  * Let δ≠ I. 
For g∉ Z· J^1_2, we have  Θ_π^δ_N,ψ(g)=0.

  * If g is not conjugate to any element in 𝔒_2^×· J^1_2, then Θ_π^I_N,ψ(g)=0 .

  * For g∈(𝔒_2^×· J^1_2)\(Z· J^1_2), we have


Θ_π^I_N,ψ(g)=1q^4(q^2-1)[R∈𝖭(𝔒_2^×· J^1_2)∑ϕ̃_x[ R^-1gR      0;      0 R^-1gR ]] 


where 𝖭(𝔒_2^×· J^1_2) is the normaliser of 𝔒_2^×· J^1_2 in _2(_2).




Since π^δ = _δ^-1 Tδ∩ P^Pϕ̃_x^δ^-1, we get 



    Θ_π^δ[ g X; 0 g ]
    = 1|δ^-1 Tδ∩ P|∑_γ∈ Pϕ̃_x^δ^-1(γ^-1[ g X; 0 g ]γ)

where the summation is over γ=[ S Q; 0 R ]∈ P is such that 

    γ^-1[ g X; 0 g ]γ=[                    S^-1gS S^-1gQ+S^-1XR-S^-1QR^-1gR;                         0                    R^-1gR ]∈δ^-1 T δ∩ P.




  * It follows from the Remark <ref> (b).


  * For δ≠ I, 
δ^-1Tδ∩ P= Z· (K^1_2∩ P).
By Equation (<ref>), both S^-1gS and R^-1gR ∈ Z· J^1_2.
Therefore, if g is not conjugate to Z · J^1_2 (equivalently, g ∉ Z · J^1_2) then there does not exist any γ∈ P such that γ^-1[ g X; 0 g ]γ∈ (δ^-1 Tδ∩ P) and hence Θ_π^δ[ g X; 0 g ] =0.
By Corollary <ref>, we conclude Θ_π^δ_N, ψ(g)=0.

  * For δ=I, T∩ P = Δ(𝔒_2^×)· (K^1_2∩ P) where Δ(𝔒_2^×) is the diagonal embedding of 𝔒_2^× in _4(_2). 
By Equation (<ref>), both S^-1gS and R^-1gR ∈𝔒_2^×· J^1_2. 
Therefore, if g is not conjugate to any element in 𝔒_2^×· J^1_2, then Θ_π^I[ g X; 0 g ] =0. 
By Corollary <ref>, we conclude Θ_π^I_N, ψ(g)=0.

  * Using Equation (<ref>), if γ^-1[ g X; 0 g ]γ∈δ^-1Tδ∩ P then, 

    (S^-1gQ+S^-1XR-S^-1QR^-1gR)∈ϖ M_2(_2).



For δ=I and g∈ (𝔒_2^×· J^1_2)\ (Z· J^1_2), by Corollary <ref>, Θ_π^I_N,ψ(g) equals

    1|N|·|T∩ P|(S,R,Q,X)∈𝒥(g)∑ϕ̃_x[                    S^-1gS S^-1gQ+S^-1XR-S^-1QR^-1gR;                         0                    R^-1gR ]ψ(Xg^-1)
 
where 


𝒥(g)={ (S,R,Q,X) : 
[                      S, R ∈_2(_2)  and  Q, X ∈ M_2(_2),; S^-1gS, R^-1gR∈𝔒_2^×· J^1_2 
 with  S^-1gS=R^-1gR ( ϖ),;                    S^-1gQ+S^-1XR-S^-1QR^-1gR∈ϖ M_2(𝔬_2) ]}.




The set 𝒥(g) describes the conditions on γ and X such that the matrix in (<ref>) belongs to T ∩ P.
 
Note that

    S^-1gQ+S^-1XR-S^-1QR^-1gR=0 ( ϖ) 
        gQR^-1+X-QR^-1g=0 ( ϖ) 
        X=Q'g - g Q' ( ϖ),

where Q'=QR^-1.
If Q'_1 and Q'_2 are two solutions of  X=Q'g-gQ' ( ϖ), then (Q'_1-Q'_2) ( ϖ) belongs to the centraliser of g ( ϖ)∈_q^2\_q.
This gives 𝒥(g) consists of tuples (S,R,Q,X) such that


  * S,R ∈𝖭(𝔒_2^×· J^1_2) with RS^-1∈𝔒_2^×· J^1_2.

  * Q' belongs to the homogeneous space of centraliser of g ( ϖ).

  * X ( ϖ) belongs to [  1  0;  0 -1 ]·_q^2.

Hence, 

    |{Q∈ M_2(_2):(S,R,Q,X)∈𝒥(g)}|=q^6=|{X∈ M_2(_2):(S,R,Q,X)∈𝒥(g)}|.

From Equation (<ref>), Θ_π^I_N,ψ(g) is given by

    1|N||T∩ P|(S,R,Q,X)∈𝒥(g)∑   ϕ̃_x[ S^-1gS      0;      0 R^-1gR ]ϕ̃_x[                                I S^-1Q+S^-1g^-1XR-S^-1g^-1QR^-1gR;                                0                                I ]ψ(Xg^-1).


The theorem follows easily from the following two lemmas, which can be verified by using the non-degeneracy of the trace pairing and using R^-1 gR ( ϖ), RS^-1 ( ϖ) and X_3 ∈_q^2^×. 
For more details, look at the PhD Thesis of first author <cit.>.


For fixed R,S ∈_2(_2) such that RS^-1∈𝔒_2^×· J^1_2, we have

    ∑_X,Q ∈ M_2(_2)  s.t.
    (S,R,Q,X) ∈𝒥(g) ϕ̃_x[                                I S^-1Q+S^-1g^-1XR-S^-1g^-1QR^-1gR;                                0                                I ]ψ(Xg^-1)= q^12.



Let S,R∈𝖭(𝔒_2^×· J^1_2) such that SR^-1∈𝔒_2^×· J^1_2 then, for any g ∈𝔒_2^×· J^1_2 we have

    ϕ̃_x[ S^-1gS      0;      0 R^-1gR ]=ϕ̃_x[ R^-1gR      0;      0 R^-1gR ].





§ A DESCRIPTION OF Π_N, Ψ

Since π_N, ψ = δ∈Ω_0⊕π^δ_N, ψ, we describe all the irreducible representations of _2(_2) appearing in π^δ_N, ψ for all δ∈Ω. 


Let π be a strongly cuspidal representation of _4(_2).
For δ∈Ω with δ≠ I and an irreducible representation σ of _2(_2), if σ appears in π^δ_N,ψ then σ  is a regular representation.
 

Let B ∈ M_2(_q) be such that ⟨Θ_σ,ϕ_B⟩_J^1_2≠ 0.
Since σ appears in π^δ_N,ψ then ϕ_B appears in π^δ_N, ψ|_J^1_2.
By Corollary <ref>, B∼ℬ_δ and,
therefore it is enough to prove that ℬ_δ is not a scalar matrix. 

 Now we assume that ℬ_δ is a scalar matrix and deduce a contradiction.
Recall that (ℬ_δ) = (2X_1), then 
ℬ_δ=[ 2(a_0+aa_1)           0;           0 2(a_0+aa_1) ]. 
Write

    ℬ_δ = L_δ(X_1+X_2Y)L_δ^-1+X_1-YX_2 =L_δ(X_1+X_2Y+L_δ^-1(X_1-YX_2)L_δ)L_δ^-1.
 
This gives X_1+X_2Y+L_δ^-1(X_1-YX_2)L_δ is diagonalizable and both the columns of L_δ are its eigenvectors with eigenvalue 2(a_0+aa_1). Write 

    L_δ 
    =[ L_δ^1 L_δ^2;       ],
 
where L_δ^1 and L_δ^1 are the columns of the matrix L_δ. 
Therefore, 

    (X_1+X_2Y+L_δ^-1(X_1-YX_2)L_δ)· L_δ^1=2(a_0+aa_1)· L_δ^1. 
    (X_1+X_2Y+L_δ^-1(X_1-YX_2)L_δ)· L_δ^2=2(a_0+aa_1)· L_δ^2.

Equation (<ref>) gives

    u=-2a_1(a_3(a^2-b^2α)+aa_2)a_2^2+(3a^2+b^2α)a_3^2+4aa_2a_3 and v=2a_1(a_2+aa_3)a_2^2+(3a^2+b^2α)a_3^2+4aa_2a_3.

Equation (<ref>) gives two quadratic equations 

    Q_1(u,v)=0   and   Q_2(u,v)=0.


If a_1=0, then Equation (<ref>) gives u=v=0 and hence δ =I, which contradicts our assumption δ≠ I.



Now we assume that a_1≠ 0. 
Substituting the values of u,v from Equation (<ref>) in Equations (<ref>), one arrives at a contradiction to our assumption x ∈_q^4\_q^2, for which we skip the details.




Let π be a strongly cuspidal representation of _4(_2). For δ∈Ω and δ≠ I, the representation π^δ_N, ψ is multiplicity free. 
In other words, if σ is a regular representation of _2(𝔬_2) such that ω_π=ω_σ and
ϕ_ℬ_δ appears in σ|_J^1_2, then σ appears in π^δ_N, ψ with multiplicity one.


For δ∈Ω and δ≠ I, by 
Theorem <ref> (b), we have
Θ_π^δ_N,ψ(g)=0 if g∉ Z· J^1_2. 
Using ω_π = ω_σ we get,

    ⟨Θ_π^δ_N,ψ, Θ_σ⟩__2(𝔬_2)    = 1|_2(_2)|g=z· k∈ Z· J^1_2∑Θ_π^δ_N,ψ(z· k)Θ_σ(z· k) 
        = |Z/(J^1_2∩ Z)||_2(_2)|k∈ J^1_2∑Θ_π^δ_N,ψ(k)Θ_σ(k) 
        = |Z/(J^1_2∩ Z)|· |J^1_2||_2(_2)|⟨Θ_π^δ_N,ψ,Θ_σ⟩_J^1_2.

By Corollary <ref>, we have
π^δ_N,ψ|_J^1_2 
= B ∼ℬ_δ⊕( |(B)|q-1) ϕ_B 
and by Remark <ref>, it follows that
σ|_J^1_2 = B ∼ℬ_δ⊕ϕ_B.
Therefore, 

    ⟨Θ_π^δ_N,ψ,Θ_σ⟩_J^1_2
    = ( |(ℬ_δ)|q-1) ·|Conjugacy class of ℬ_δ|,
 which gives
 ⟨Θ_π^δ_N,ψ, Θ_σ⟩__2(𝔬_2)=1.


Note that for δ≠ I, the number of σ satisfying the property in 
Proposition <ref> is |(ℬ_δ)|/q-1 and each having dimension |_2(_q)|/|(ℬ_δ)|.
This gives  (π^δ_N,ψ)=q(q^2-1) = ( |(ℬ_δ)|/q-1) ·( |_2(_q)|/|(ℬ_δ)|). 
Therefore, an irreducible representation σ appears in π^δ_N,ψ if and only if σ satisfies the properties mentioned in Proposition <ref>.

 
The representation π^I_N,ψ≅_𝔒_2^×· J^1_2^_2(𝔬_2)( ϕ̃_x|_𝔒_2^×· J^1_2). 


It is enough to verify that the characters of the two representations are the same.
The character of π^I_N, ψ is given in Theorem <ref>. 
The character of  _𝔒_2^×· J^1_2^_2(𝔬_2)( ϕ̃_x|_𝔒_2^×· J^1_2) can be calculated using the formula for the character of an induced representation.
We leave the verification to the reader that these two characters are the same.

 
The representation π^I_N,ψ  is multiplicity free.  


First consider X_1∈_q^2\_q. 
Then π^I_N,ψ is an irreducible regular representation of non-split semisimple type and hence nothing to prove.


Now consider X_1∈_q. 
By Clifford theory, there exists a character ζ:_2(_2) →^× such that ζ|_J^1_2 =ϕ_-(2X_1).
Then θ' := (θ·ϕ_2X_1) ·ζ|_𝔒_2^×· J^1_2 is trivial on J^1_2 and therefore θ' factors through (𝔒_2^×· J^1_2) / J^1_2≅_q^2^×. 
Since π^I_N,ψ≅_𝔒_2^×· J^1_2^_2(_2)( θ·ϕ_2X_1), we get π^I_N,ψ⊗ ζ≅_𝔒_2^×· J^1_2^_2(_2)( θ' ) (see <cit.>). 
It follows that π^I_N,ψ⊗ ζ is trivial on J^1_2 and hence factors through _2(_2)/ J^1_2 ≅_2(_q).
It is enough to prove that π^I_N, ψ⊗ζ≅__q^2^×^_2(_q) (θ') is multiplicity free representation as a representation of _2(_q).
It follows from the fact that (_2(_q), _q^2^×) is a strong Gelfand pair which can be easily verified by looking at the character table of _2(_q).   



For δ, δ' ∈Ω, ℬ_δ∼ℬ_δ^' if and only if Tδ P=Tδ^' P.


First, assume that Tδ P=Tδ^'P. 
Then π^δ_N, ψ = π^δ'_N, ψ and by Corollary <ref>, we get ℬ_δ∼ℬ_δ^'. 

For the converse, assume that ℬ_δ∼ℬ_δ^'. 
Then, (ℬ_δ)=( ℬ_δ^'). 
It can be checked that for δ =A_u,v, the determinant of ℬ_δ is given by

    (ℬ_δ)=4(a_0+aa_1)^2-[4a_1^2(A)
    -((u+av)^2-(bv)^2α)·(a_2^2+a_3^2(3a^2+b^2α)+4aa_2a_3)^2 (L_δ)]

where A=A(u,v) is a quadratic polynomial in u and v. 

Moreover, for δ=A_u,v and δ' = A_u', v' it can be deduced that the conditions for (ℬ_δ) = ( ℬ_δ') comes out to be the following for which we refer the reader to <cit.> and omit the details

    M · (a^2a_3^2+2aa_2a_3+a_2^2-a_3^2b^2α)· C(δ, δ') =0,

where 

    M = 4a_1^4b^2α  - 4a_1^2(a^3a_3^2+2a^2a_2a_3+aa_2^2+3aa_3^2b^2α+2a_2a_3b^2α)
     +(a_2^2 +a_3^2(3a^2+ b^2α)+4aa_2a_3)^2.

Observe that the second factor in (<ref>) is not zero, since 
    a^2a_3^2+2aa_2a_3+a_2^2-a_3^2b^2α = (a_2+aa_3)^2-(a_3b)^2α = 0  a_2=0, a_3=0
 
and then x ∈_q^2, a contradiction.

Now we claim that M≠ 0.
Note that M is a quadratic polynomial in a_1^2.
If M=0, then the discriminant of this quadratic polynomial is a square in _q.
But, the discriminant is given by

    16(a^2-b^2α)((a_2+aa_3)^2-(a_3b)^2α)^2

which is not a square in _q (see the assumption (<ref>)).
This proves M ≠ 0.

Therefore, by Equation (<ref>) we get C(δ, δ')=0 and Lemma <ref> gives  Tδ P = Tδ'P.

We have described each π^δ_N, ψ and proved that for distinct δ, δ' ∈Ω_0 the restriction of π^δ_N, ψ and π^δ'_N, ψ to J^1_2 are distinct, which gives the following result. 
 
For δ, δ' ∈Ω_0 with δ≠δ' the representations π^δ_N, ψ and π^δ'_N, ψ have no irreducible representation in common.



Let π be a strongly cuspidal representation of _4(_2). 
Then π_N,ψ is multiplicity free.


For δ, δ' ∈Ω_0 such that δ≠δ', by Corollary <ref>, π^δ_N,ψ and π^δ'_N,ψ have no irreducible representation in common.
By Proposition <ref> and Corollary <ref> π^δ_N, ψ is multiplicity free for all δ∈Ω_0 and hence π_N, ψ is multiplicity free.



Recall that | Ω_0| =q+1 (Lemma <ref>) and |{δ∈Ω_0 : π^δ_N, ψ≠ 0 }| =q (Lemma <ref>). 
Therefore, there are exactly q conjugacy classes represented by ℬ_δ such that ϕ_ℬ_δ appears in π_N, ψ restricted to K^1_2. 
Moreover, all the ℬ_δ's  have distinct determinant (Theorem <ref>) and then, 

    {(ℬ_δ) : δ∈Ω_0} = _q.


Note that the trace of each ℬ_δ is the same as (2X_1). 
We conclude the following :


(1) all the q-1 semisimple (both split and non-split) conjugacy classes with trace (2X_1) are represented by ℬ_δ for some δ∈Ω_0. 

(2) The determinant of the scalar matrix and the split non-semisimple matrices with trace (2X_1) are the same and exactly one of these two conjugacy classes will be represented by ℬ_δ for some δ∈Ω_0.

We summarise the results discussed in Section <ref> in the theorem below.

 
Let π=_I(ϕ_x)^_4(_2)ϕ̃_x be a strongly cuspidal representation of _4(_2). 


  * Let X_1∈_q, then π_N,ψ consists of the following representations of _2(_2).


  * The representation π^I_N,ψ = _𝔒_2^×· J^1_2^_2(_2)( ϕ̃_2X_1), which is not a regular representation and it is also not irreducible.

  * An irreducible regular representation σ of _2(_2) with multiplicity one such that ω_σ=ω_π and σ|_J^1_2 contains the character ϕ_B with B a semisimple matrix of trace (2X_1). 



  * Let X_1∈_q^2\_q, then π_N,ψ consists of the following representations of _2(_2).


  * The irreducible regular representation π^I_N,ψ=_𝔒_2^×· J^1_2^_2(_2)( ϕ̃_2X_1) with multiplicity one.

  * An irreducible regular representation σ of _2(_2) with multiplicity one such that ω_σ=ω_π and σ|_J^1_2 contains the character ϕ_B with B a regular matrix of trace (2X_1) and B  2X_1. 







§ A DESCRIPTION OF Π= _𝔒_2^×^_2(𝔬_2)(Θ|_𝔒_2^×)



Recall from Section <ref> that we have fixed a, b, α∈_q such that α is a square free element in _q and a+b √(α) is a square free element in _q^2= _q(√(α)). We have also fixed an embedding _q^2↪ M_2(_q) as given in (<ref>).
We state the following easy lemma without proof. 
It provides a set of representatives for various double cosets, which will be useful later in this section.
 
For y ∈_q, z ∈_q^× write γ(y,z) 
=[ 1 y; 0 z ] and 

    Γ = {γ(y,z) : y ∈_q, z ∈_q^×}.




  * The set Γ is a set of distinct representatives of _q^2^×\_2(_q).

  * For γ(y,z) and γ(y',z') ∈Γ ,

𝔽_q^2^× γ(y,z) 𝔽_q^2^×=𝔽_q^2^× γ(y',z') 𝔽_q^2^×  if and only if  



    y^2+2ay(z-1)+(1+z^2)(a^2-b^2α)z=y'^2+2ay'(z'-1)+(1+z'^2)(a^2-b^2α)z'.
 



We fix a subset Γ_0⊂Γ representing distinct  double cosets for _q^2^×\_2(_q)/_q^2^×.

  * Let T_1 = {[  m  0;  0 m' ] : m, m' ∈_q^×}. 
For _q^2^×\_2(_q)/T_1, a set of distinct representatives  is given by
Γ_1 := {γ(y,1): y∈_q}.

  * Let T_2 = {[ m n; 0 m ] :m∈_q^×,  n∈_q  }. 
For _q^2^×\_2(_q)/T_2, a set of distinct representatives  is given by
Γ_2 : ={γ(0,z) : z∈_q^×}.




Let H,K be subgroups of G=_2(_2). 
Assume that the quotient map G →G̅= _2(_q) induces a bijection between the double cosets K\ G/H and K̅\G̅/H̅, where H̅, K̅ denote the images of H, K in G̅.
Then, for a representative γ∈G̅ of a double coset in K̅\G̅/H̅, we write its lift to G by the same letter γ as a representative of the corresponding double coset in K\ G/H.

The following proposition describes the characters ϕ_B of J^1_2≅ M_2(_q) which appear in the restriction of Π to J^1_2.

For B∈ M_2(_q), ϕ_B appears in Π|_J^1_2 if and only if ∃ m, n ∈_q such that

    B∼ 2X_1-[               m n(a^2-b^2α)+2am;               n              -m ].
 


Recall that the character ϕ_B:M_2(_q)→^× is given by ϕ_B(A)=ψ_0((BA)). 
Since Π is an induced representation, by Mackey theory we have 
    Π|_J^1_2=γ∈𝔒_2^×\_2(_2)/J^1_2⊕_γ^-1𝔒_2^×γ∩ J^1_2^J^1_2(θ^γ^-1).
 
Using Frobenius reciprocity, we get 

    ⟨Θ_Π,ϕ_B⟩_J^1_2
    = γ∈𝔒_2^×\_2(_2)/J^1_2∑⟨θ^γ^-1,ϕ_B⟩_γ^-1(𝔒_2^×)γ∩ J^1_2.

Since J^1_2 is a normal subgroup, γ^-1(𝔒_2^×)γ∩ J^1_2 = γ^-1 (𝔒_2^×∩ J^1_2) γ = γ^-1 K γ where K=𝔒_2^×∩ J^1_2. 
Note that there is a bijection between 𝔒_2^×\_2(_2)/J^1_2 and _q^2^×\_2(_q). Using  the Lemma <ref> (a) we get, 

    ⟨Θ_Π,ϕ_B⟩_J^1_2 
    =γ∈Γ∑⟨θ^γ^-1,ϕ_B⟩_γ^-1(K)γ =γ∈Γ∑⟨θ,ϕ_B^γ⟩_K.

Note that θ|_K=ϕ_2X_1|_K, which gives

    ⟨θ,ϕ_B^γ⟩_K
    =⟨ϕ_2X_1,ϕ_B^γ⟩_K    =1|K|A∈ K∑ψ_0(  (2X_1A))ψ_0((γ Bγ^-1 A)) 
        =1|K|A∈ K∑ψ_0( (2X_1-γ Bγ^-1)A).

We know that :M_2(_q)× M_2(_q)→_q is a non-degenerate bilinear map. 
Using the isomorphism J^1_2≅ M_2(_q), K is identified with {[            b_0 -b_1(a^2-b^2α);            b_1      b_0+2ab_1 ] : b_0,b_1∈_q}.
It can be checked that


{ C∈ M_2(_q):(CD)=0 ∀ D∈ K}={[               m n(a^2-b^2α)+2am;               n              -m ]:m,n∈_q}.


Note that ⟨θ,ϕ_B^γ⟩_K is either 0 or 1. 
Moreover, ⟨θ,ϕ_B^γ⟩_K =1 if and only if 

    ∃ m,n ∈_q such that  2X_1-γ Bγ^-1 = [               m n(a^2-b^2α)+2am;               n              -m ].

Therefore, ϕ_B appears in Π|_J^1_2 if and only if  B satisfies (<ref>) for some γ∈Γ and m,n ∈_q.




If X_1∈_q^2\_q, then ϕ_2X_1 appears in Π|_J^1_2 with multiplicity one.


Using Equation (<ref>) , (<ref>) and (<ref>) for B=2X_1, we have

    ⟨Θ_Π,ϕ_2X_1⟩_J^1_2    
    =1|K|γ∈Γ, A∈ K∑ψ_0( (2X_1- γ 2X_1γ^-1)A)
    
        = |{γ∈Γ: 2X_1-γ 2X_1γ^-1∈{[               m n(a^2-b^2α)+2am;               n              -m ]:m,n∈_q}}|
    
        =|{γ(y,z) ∈Γ :y^2+2ay(z-1)+(1+z^2)(a^2-b^2α)/z=2(a^2-b^2α) }|
    
    
    
    
    
        =1.



Let B = 2X_1-[               m n(a^2-b^2α)+2am;               n              -m ] for some m,n ∈_q.


  * If X_1∈_q, then either B=2X_1 or B is a semisimple matrix with trace (2X_1).

  * If X_1 ∈_q^2\_q, then B is a regular matrix with trace (2X_1).



It is clear that (B)= (2X_1).
The characteristic polynomial of B is given by

    (λ-2(a_0+aa_1))^2 -( 4a_1^2b^2α + (m+an)^2 -b^2n^2 α).
 
Clearly, if D := 4a_1^2b^2α + (m+an)^2 -b^2n^2 α≠ 0, then B will be semisimple. 
If D = 0, then B is either scalar or split non-semisimple with trace (2X_1).


  * Let X_1 ∈_q i.e. a_1=0. 
Then, D=0   m=0, n=0  B=2X_1 which is scalar. 

  * Let X_1 ∈_q^2\_q, i.e. a_1 ≠ 0.
If D=0 and B is the scalar matrix with trace (2X_1) then by comparing the entries we get m=-2aa_1, n=2a_1 and 4a_1 b^2α =0.
Since a_1 ≠ 0, 4a_1 b^2 α =0 is not possible, a contradiction.
In fact, the choice m=-2aa_1, n=2a_1 gives a split non-semisimple regular matrix. 

Let ϕ_B be a character of J^1_2 and σ = _I(ϕ_B)^_2(𝔬_2)ϕ̃_B.
By Mackey theory, it follows that 

    ⟨Θ_Π,Θ_σ⟩__2(_2) 
    =⟨θ, Θ_σ⟩_𝔒_2^×    =γ∈𝔒_2^×\_2(𝔬_2)/I(ϕ_B)∑⟨θ, ϕ̃_B^γ⟩_γ(I(ϕ_B)) γ^-1∩𝔒_2^×.




Let B ∈ M_2(_q) be a non-split semisimple matrix with trace (2X_1). 
Let ϕ̃_B : I(ϕ_B) →^× be an extension of ϕ_B and σ = _I(ϕ_B)^_2(𝔬_2)ϕ̃_B.
Then,

    ⟨Θ_Π,Θ_σ⟩__2(_2)=⟨θ,ϕ̃_B⟩_𝔒_2^×+⟨θ,ϕ̃_B^γ_0⟩_𝔒_2^×+∑_γ∈Γ_0 such that 
    γ∉{I,γ_0}⟨ϕ̃_x,ϕ̃_B^γ⟩_Z· K.

where γ_0∈Γ_0 is a non-trivial element in 𝖭__2(_q)(_q^2^×)  the normalizer of _q^2^×, and K=I+ϖ𝔒_2≅_q^2.


We use Equation (<ref>) to prove the lemma.
Note that I(ϕ_B)=𝔒_2^×· J^1_2 and if we write K=I+ϖ𝔒_2, then


γ(𝔒_2^×· J^1_2)γ^-1∩𝔒_2^×
=𝔒_2^×    if γ∈𝖭__2(_2)(𝔒_2^×· J^1_2)

Z· K     if γ∉𝖭__2(_2)(𝔒_2^×· J^1_2).


Note that there is a bijection between 𝔒_2^×\_2(𝔬_2)/𝔒_2^×· J^1_2 and _q^2^×\_2(_q)/_q^2^× for which a set of representatives Γ_0 are given in Lemma <ref>.
Since the index of _q^2^× in 
𝖭__2(_q)(_q^2^×) is 2, we get the index of 𝔒_2^× in  𝖭__2(_2)(𝔒_2^×) is 2. 
Now, the lemma follows by observing that θ|_Z · K = ϕ̃_x|_Z · K.



 
Let ϕ̃_2X_1 : I(ϕ_2X_1) →^× be given by ϕ̃_2X_1=θ·ϕ_2X_1 and σ=_𝔒_2^×· J^1_2^_2(𝔬_2)(θ·ϕ_2X_1), then σ is a subrepresentation of Π.
Moreover, if X_1∈_q^2\_q, then σ is an irreducible representation of _2(_2) and σ appears in Π with multiplicity one.



First part follows easily from the transitivity of induction.
For X_1 ∈_q^2\_q, B= 2X_1 is non-split semisimple, we use Equation (<ref>) to compute ⟨Θ_Π,Θ_σ⟩__2(_2).
Since ϕ̃_2X_1=θ·ϕ_2X_1and θ is a strongly primitive character of 𝔒_2^×, we get 

    ⟨θ,ϕ̃_2X_1⟩_𝔒_2^×=1
     and ⟨θ,ϕ̃_2X_1^γ_0^-1⟩_𝔒_2^×=⟨θ^γ_0,ϕ̃_2X_1⟩_𝔒_2^×= ⟨θ^γ_0, θ⟩_𝔒_2^× =0 .

It remains to prove that ⟨θ,ϕ̃_2X_1^γ^-1⟩_Z· K=0 for γ∉{ I, γ_0}, which can be easily verified using Equation (<ref>) and we skip the details. 



Let B ∈ M_2(_q) be a non-split semisimple matrix with trace (2X_1) and B 2X_1. 
Let ϕ̃_B be an extension of ϕ_B to its inertia group I(ϕ_B)  such that ϕ̃_B|_Z = ω_Π and σ=_I(ϕ_B)^_2(𝔬_2)ϕ̃_B.
Then, σ appears in Π with multiplicity one.


Since B is non-split semisimple matrix, we can take that B ∈_q^2\_q.
We use Equation (<ref>) to prove 
⟨Θ_Π,Θ_σ⟩__2(_2)=1.
We first claim that 
⟨θ,ϕ̃_B ⟩_𝔒_2^×=0, 
⟨θ,ϕ̃_B ^γ_0^-1⟩_𝔒_2^×=0. 
Since 1+ ϖ𝔒_2 ⊂𝔒_2^×, it is enough to prove that ⟨θ,ϕ_B ⟩_1+ϖ𝔒_2=0 and ⟨θ,ϕ_B ^γ_0^-1⟩_1+ϖ𝔒_2=0.
We identify 1+ϖ𝔒_2 with _q^2 under the isomorphism J^1_2≅ M_2(_q). 
Recall that : M_2(_q)× M_2(_q)→_q is a non-degenerate bilinear map and so is its restriction to _q^2. 
Therefore, B_1, B_2 ∈_q^2 satisfy the following


ϕ_B_1|__q^2 = ϕ_B_2|__q^2 if and only if B_1=B_2. 


Note that θ|_1+ϖ𝔒_2=ϕ_2X_1|_1+ϖ𝔒_2. 
Since B  2X_1, we get ϕ_B|_1+ϖ𝔒_2≠ϕ_2X_1|_1+ϖ𝔒_2 and ϕ_B^γ_0^-1|_1+ϖ𝔒_2≠ϕ_2X_1|_1+ϖ𝔒_2.
Hence the claim follows.

One can easily verify that there exists unique γ∈Γ_0\{ I, γ_0 } with ⟨ϕ̃_x,ϕ̃_B^γ^-1⟩_Z· K=1, for which we skip the details.



Let B = [             m_1               0;               0 4(a_0+aa_1)-m_1 ] be a split semisimple matrix such that m_1≠ 2(a_0+aa_1). 
Let ϕ̃_B be an extension of ϕ_B to its inertia group I(ϕ_B)  such that ϕ̃_B|_Z = ω_Π and σ=_I(ϕ_B)^_2(𝔬_2)ϕ̃_B. 
Then, σ appears in Π with multiplicity one.


We use Equation (<ref>) by noting that for the given B, the inertia group I(ϕ_B) is such that I(ϕ_B)/J^1_2 is the diagonal torus T_1 in _2(_q).
Moreover, there is a bijection between 𝔒_2^×\_2(_2)/I(ϕ_B) and _q^2^×\_2(_q)/T_1 for which a set of representatives Γ_1 is known from Lemma <ref> (c).
One can easily verify that that there exists a unique γ∈Γ_1 satisfying 
⟨θ,ϕ̃_B^γ⟩_γ I(ϕ_B)γ^-1∩𝔒_2^×=1.




Let X_1∈_q^2\_q and B=[ 2(a_0+aa_1)           1;           0 2(a_0+aa_1) ]. Let ϕ̃_B be an extension of ϕ_B to its inertia group I(ϕ_B) such that ϕ̃_B|_Z = ω_Π and σ = _I(ϕ_B)^_2(𝔬_2)ϕ̃_B. 
Then, σ appears in Π with multiplicity one.


For the given B the inertia group I(ϕ_B) is such that I(ϕ_B)/J^1_2 = T_2 as defined in Lemma <ref>. The rest of the proof is similar to that of Lemma <ref>.



We summarise the results from Section <ref> in the following theorem.

Let Π = _𝔒_2^×^_2(𝔬_2)(θ|_𝔒_2^×). Recall that θ = ϕ_2X_1 when restricted to 𝔒_2^×∩ J^1_2≅ 1+ ϖ𝔒_2.


  * Let X_1∈_q, then Π consists of the following representations of _2(_2).


  * The representation _𝔒_2^×· J^1_2^_2(_2)( ϕ̃_2X_1), which is not a regular representation and it is also not irreducible.

  * An irreducible regular representation σ of _2(_2) with multiplicity one such that ω_σ=ω_Π and σ|_J^1_2 contains the character ϕ_B where B is a semisimple matrix of trace (2X_1).



  * Let X_1∈_q^2\_q, then Π consists of the following representations of _2(_2).


  * The irreducible regular representation _𝔒_2^×· J^1_2^_2(_2)( ϕ̃_2X_1) with multiplicity one.

  * An irreducible regular representation σ of _2(_2) with multiplicity one such that ω_σ=ω_Π and σ|_J^1_2 contains the character ϕ_B where B is a regular matrix of trace (2X_1) and B  2X_1. 



Using Theorem <ref> and Theorem <ref>, we obtain our main result verifying the Conjecture <ref> of Prasad for the case n=2, l=2.

Let π be an irreducible strongly cuspidal representation of _4(_2) associated to a strongly primitive character θ : 𝒪_2^×→^×. 
Then as a representation of _2(_2) we have 

    π_N, ψ≅_𝔒_2^×^_2(_2) (θ|_𝔒_2^×).




§ ACKNOWLEDGEMENT

The authors thank Dipendra Prasad for several helpful conversations and insightful comments. 
This work is a part of AP's Thesis at IIT Delhi and she acknowledges the support of PMRF fellowship (1400772). 
SPP acknowledges the support of the SERB MATRICS grant (MTR/2022/000782).



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 Paper Title: Enumerating Finite Braid Group Orbits on $SL_2(\C)$-Character Varieties.







 









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	Enumerating Finite Braid Group Orbits on SL_2()-Character Varieties
    Amal Vayalinkal
    August 5, 2024
===================================================================

§ ABSTRACT 
		We analyze finite orbits of the natural braid group action on the character variety of the n times punctured sphere. Building on recent results relating middle convolution and finite complex reflection groups, our work implements Katz's middle convolution to explicitly classify finite orbits in the SL_2()-character variety of the punctured sphere. We provide theoretical results on the existence of finite orbits arising from the imprimitive finite complex reflection groups and formulas for constructing such examples when they exist. In the primitive finite complex reflection groups, we perform an exhaustive search and provide computational results. Our contributions also include Magma computer code for middle convolution and for computing the orbit under this action when it is known to be finite. 
	

	

§ INTRODUCTION

	Dynamics of the natural braid group action on character varieties has been studied for over a century. The growing interest in classifying the finite orbits of this action has led to a wealth of new techniques and examples. Inspired by Boalch <cit.> who explored finite complex reflection groups as a source of finite orbits, Lam-Landesman-Litt <cit.> showed, up to mild conditions, all finite orbits come from finite complex reflection groups via Katz's middle convolution operation <cit.>. With the goal of completing their analysis, we search the finite complex reflection groups to enumerate all possibilities and study them. We classify all finite orbits arising via middle convolution from the finite complex reflection groups. 
	
	Let Σ_0, T+1 denote the T+1-punctured sphere. Given T and n, a rank n representation of the fundamental group of Σ_0, T+1 can be identified with a tuple of matrices (A_1, … A_T+1) ∈ GL_n()^T+1 satisfying 
    ∏_i = 1^T+1 A_i = Id

	
	The mapping class group of Σ_0, T+1, denoted  _0, T+1 has a well-defined action on such representations. It is well known that _0, T+1 is isomorphic to the spherical braid group on T+1 strands, denoted B_T+1.  This group has T generators σ_1,  …, σ_T and acts on T+1-tuples in the following way: 
	
    σ_i(A_1, A_2, …, A_T, A_T+1) = (A_1, …, A_i-1, A_iA_i+1A_i^-1, A_i, A_i+2, …, A_T+1)
	
	If we further consider such representations up to simultaneous conjugation, this action descends to an action on the character variety i.e. on the isomorphism classes of representations.
	
		Conjugacy classes of tuples [A_1, ⋯, A_T+1] with finite orbit under this action are called MCG-finite and we say a tuple is irreducible if the associated representation is irreducible.
	

		Recently, Lam-Landesman-Litt <cit.> classified finite orbits of rank two tuples with at least one matrix of infinite order.  They proved that such Zariski dense representations come from pullbacks of curves or arise via middle convolution from finite complex reflection groups.
	
	The first kind, “pullback" representations were classified by Diarra <cit.>. 
	
	Finite irreducible complex reflection groups, which can be viewed as generalizations of Coxeter groups, were classified by Shephard and Todd in 1954 as belonging to an infinite family G(m, p, n) for each rank n or being one of 34 exceptional cases denoted G_3 to G_37 [9]. In view of this classification, Lam-Landesman-Litt <cit.> prove that for T+1 ≥ 7, MCG-finite representations do not come from groups of rank five or greater. Their paper leaves open for analysis the infinite family of rank three and four and most of the exceptional groups of rank three and four, as well as the case when T + 1 < 7, which we investigate.
	
	Katz <cit.> originally introduced middle convolution in the language of perverse sheaves and one of our contributions is implementing computer code to explicitly compute the middle convolution given a tuple of matrices, following the algebraic reformulation of Dettweiler-Reiter <cit.>. We also provide code for computing the braid group orbit for MCG-finite 4-tuples and 5-tuples in the  SL_2()-character variety of the punctured sphere. This computer code is publicly accessible online at the following link:  <amalvayalin.github.io/research.html> and we hope it encourages further applications of this operation.
	
	
		A 𝐧𝐢𝐜𝐞 T-tuple is a tuple (up to simultaneous conjugation) from a finite complex reflection group such that its middle convolution produces a rank 2 MCG-finite T+1-tuple. 
	
	For the precise definition of when a tuple is nice, see Definition <ref>. An explanation for how we get MCG-finite T+1-tuples from a T-tuple is provided in Remark <ref>. Since middle convolution depends on a parameter λ (see Section <ref>), the same nice T-tuple can give rise to distinct conjugacy classes of rank 2 MCG-finite T+1-tuples, as discussed in Remark <ref>.
	
	With this definition in mind, our problem of studying rank 2 MCG-finite tuples changes to studying nice tuples in the finite complex reflection groups. 
	Using properties of complex reflection groups, as well as prior work on this topic, it remains to study nice n and n+1-tuples in the rank n = 3, 4 primitive groups, nice n+1-tuples in the imprimitive groups of rank 2 <n ≤ 5 and  nice n-tuples in imprimitive groups of rank n ≥ 3. More details on why these are the only remaining cases is provided in Remark <ref>, Remark <ref> and <cit.>.
	
	
	In the imprimitive groups, we either prove no such nice tuples exist, or provide a system of equations for an infinite family of solutions. We found nice tuples in all the primitive reflection groups of rank three. Among the rank four primitive reflection groups, all but G_29 and G_31 contain nice tuples. For each nice tuple in the primitive groups, we provide representatives (up to simultaneous conjugation and braid group orbit equivalence) before and after middle convolution. We also list the size of the subgroup generated by the rank 2 matrices, and in most cases we provide an approximate `size' for the braid group orbit. 
	 
		There are no nice n-tuples in G(m, p, n) for n ≥ 5. 
	
	
		In the rank 3 imprimitive groups, an infinite family of nice 3-tuples exist in G(m, 1, 3) and G(m, m, 3) and in each case we provide conditions on the eigenvalues of the product of the matrices in the tuple. Nice 4-tuples exist in G(m, p, 3) for m >1 if and only if one of the following holds: 
		
			
  * p = m and m ≠ 3. 
			
  * p = 1 and (m, 3) = 1.
			
  * p ≠ 1, m and p = 3. 
		
		In all three cases the parameter for middle convolution is a root of unity of order m.
	
	 
		In the rank 4 imprimitive groups, nice 4-tuples and nice 5-tuples exist in G(m, p, 4) if and only if p = m and m =2 or m = 4. 
		
	The proof of these theorems, as well as further conditions on the eigenvalues of the product of the matrices and equations for how to construct such examples are provided in Section <ref>.
	
		In our computations for the primitive groups, nice tuples within each group are partitioned into different 𝐭𝐲𝐩𝐞𝐬 based on eigenvalues (see Definition <ref> and Remark <ref>). 
	 
		For each primitive reflection group of rank three, we summarize the distinct rank 2 MCG-finite tuples (up to simultaneous conjugation and braid group equivalence) produced by taking middle convolution of the different nice tuples (up to type equivalence). We also note how many of the rank 2 tuples generate an infinite subgroup. 
		
			
  * In G_23, there are three nice 3-tuples, resulting in six distinct MCG-finite 4-tuples after middle convolution. Three of the six generate infinite subgroups. This group also has one nice 4-tuple, which after middle convolution produces one MCG-finite 5-tuple that generates an infinite subgroup. 
			
  * In G_24,  there is only one nice 3-tuple, which produced three distinct MCG-finite 4-tuples after middle convolution, all generating infinite subgroups. There does not exist any nice 4-tuples in this group. 
			
  * In G_25, there are four nice 3-tuples, resulting in eight distinct MCG-finite 4-tuples after middle convolution. Five of these generate infinite subgroups. This group also has three nice 4-tuples, which produced two distinct MCG-finite 4-tuples after middle convolution. Only one of these 4-tuples generates an infinite subgroup. 
			
  * In G_26, there are three nice 3-tuples, resulting in seven distinct MCG-finite 4-tuples after middle convolution. Five of these generate an infinite subgroup. In this group there are no nice 4-tuples. 
			
  * In G_27, there are three nice 3-tuples, resulting in nine distinct MCG-finite 4-tuples after middle convolution. Six of these generate an infinite subgroup.
			This group also has one nice 4-tuple, which produced one MCG-finite 5-tuple after middle convolution. This 5-tuple does not generate an infinite subgroup.
		
	
	
	 
		For each primitive reflection group of rank four, we summarize the distinct rank 2 MCG-finite tuples (up to simultaneous conjugation and braid group equivalence) produced by taking middle convolution of the different nice tuples (up to type equivalence). Only nice tuples in G_32 produces an MCG-finite tuple that generates an infinite subgroup after middle convolution. 
		
			
  * In G_28, there is only one nice 4-tuple, which produces one MCG-finite 5-tuple after middle convolution. 
			
  * In G_29 there are no nice 4-tuples nor any nice 5-tuples. 
			
  * In G_30, there are three nice 4-tuples, each producing one distinct MCG-finite 5-tuple after middle convolution. For each of these 5-tuples, after tensoring with a character so the matrices are in SL_2( ) we end up with one matrix being identity, which means these tuples reduce to MCG-finite 4-tuples instead. There are no nice 5-tuples in this group. 
			
  * In G_31 there are no nice 4-tuples nor any nice 5-tuples.
			
  * In G_32 there are three nice 4-tuples, resulting in four distinct MCG-finite 5-tuples after middle convolution. One of these 5-tuples generates an infinite subgroup. After tensoring with a character so the matrices are in SL_2(), for two of the remaining three 5-tuples, we get the identity matrix, so these reduce to MCG-finite 4-tuples instead. This group also contains at least two nice 5-tuples, producing at least one example of an MCG-finite 6-tuple.  The 6-tuple did not generate an infinite subgroup. The search for nice 5-tuples in this group was too large computationally, so this list may be incomplete. 
					
		
	

 §.§ Prior Work

	MCG-finite representations correspond to algebraic solutions to nonlinear differential equatons, such as the Painlevé VI equation and more generally the  Schlesinger equations <cit.>.
	Regarded as a nonlinear analogue of the hypergeometric equation, the Painlevé VI (PVI) equation posed over 100 years ago, has inspired much work on the study of finite orbits. Algebraic solutions to the PVI equation correspond to irreducible MCG-finite representations when n = 2 and T+1 = 4 and motivates progress towards a nonlinear Schwartz list <cit.>. Among the many advances made in this area, notable contributions on finite orbits come from Hitchin <cit.>, Dubrovin-Mazzocco <cit.>, Boalch <cit.>, Kitaev <cit.> and an eventual computer-aided classification by Lisovy-Tykhyy <cit.>. 
	
	The work of Dubrovin-Mazzocco originally studied the PVI equation in connection with reflection groups <cit.>. Expanding on this idea,  solutions constructed from finite reflection groups through special cases of middle convolution, like the “Okamoto transform,"  were explored by others in <cit.>, <cit.>, <cit.>. More generally, a computer-aided classification was provided by Tykhyy in <cit.>, with explicit orbits listed when T+1 = 4, 5 and conjectural descriptions of the different possibilities for T+1 ≥ 6.  We found exact matches for orbits given in this list, except in two cases where the closest match differs in just one choice of parameter. As both these examples almost match with the same orbit they provided, we suspect this difference might be due to a typo. Details on these two tuples are provided in Section <ref> and Section <ref> and the differing coordinates are marked with an asterisk in the tables.
	
	The connection between finite complex reflection groups and middle convolutions was further explored more recently in the work of Lam-Landesman-Litt <cit.> who proved that all MCG-finite tuples with at least one matrix of infinite order are either pullback representations (classified in <cit.>) or come from finite complex reflection groups via middle convolution. 
	
	
	Our work investigates the finite complex reflection groups for MCG-finite tuples and examines their middle convolution in relation to the braid group orbit. In the applicable cases, we also explore how the examples we find relate to the classifications provided by Lisovy-Tykhyy <cit.> and Tykhyy <cit.>. Note that the analysis in these two papers exclude the tuples where all the matrices share a common eigenvector; these correspond to non-irreducible cases and were classified by Cousin-Moussard <cit.>.
	
	

§ BACKGROUND

	

 §.§ Middle Convolution
 
	Defined by Katz <cit.>, middle convolution  is an invertible functor in the category of perverse sheaves depending on some parameter λ∈^× and is denoted MC_λ. Middle convolution can also be viewed as a functor in the category of local systems on the punctured sphere to itself, and with this interpretation, Katz proved that all irreducible rigid local systems on the punctured sphere can be constructed from rank 1 local systems via scaling or middle convolution.  
	
	Letting k be a field and F_T denote the free group on T generators, middle convolution can equivalently be understood as a functor on the category of finite dimensional k[F_T] modules to itself as described by Dettweiler-Reiter <cit.>. The relation to local systems on the T+1-punctured sphere is by taking k = and viewing a tuple of T matrices as defining the action of [F_T] on the module i.e. a representation of [F_T]. In this way irreducible tuples correspond to irreducible representations.
	While middle convolution preserves irreducibility <cit.>, it typically changes the dimension of the module (rank of the local system on T+1-punctured sphere) and does not preserve basic properties e.g. finiteness of the subgroup generated by the matrices (i.e. monodromy group). Important to the study of finite orbits, one desirable property of middle convolution is that it is equivariant with the action of the braid group B_T <cit.>. One way to construct nontrivial finite orbits is to take irreducible tuples generating a finite subgroup, hence finite braid group orbit, and taking the middle convolution to (hopefully) get a tuple generating an infinite group which will necessarily be irreducible and have finite braid group orbit. 
	
	The purely algebraic analogue by Dettweiler-Reiter <cit.> gives an explicit description of middle convolution in terms of tuples of matrices and relates some properties of middle convolution to the matrices in the tuple.
	
		Given a tuple of T matrices A = [A_1, A_2, …, A_T] with A_i ∈ GL_n(), the 𝐩𝐫𝐨𝐝𝐮𝐜𝐭 of such a tuple is the product of the matrices in the tuple, in the ordering given by the tuple. So the product of the tuple 𝐀 is A_1A_2⋯ A_T. The 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 of the tuple A is the tuple [A_T^-1, …, A_1^-1], denoted 𝐀^-1. The 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐩𝐫𝐨𝐝𝐮𝐜𝐭 of a tuple is the matrix product of the inverse tuple.
	
	
		Given a tuple of T matrices A = [A_1, A_2, …, A_T] with A_i ∈ GL_n(), we denote the middle convolution of 𝐀 with parameter λ∈^× as MC_λ(A) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃]. Note that MC_λ(A) will again be a tuple of T matrices, well-defined up to simultaneous conjugation, with Ã_̃ĩ∈ GL_n'() for some n' not necessarily equal to n. 
	
	
	
		Middle convolution does not preserve the product of the matrices as it is only well-defined up to simultaneous conjugation. 
	
	
	Letting V denote the underlying vector space associated to the [F_T] module defined by a tuple 𝐀. Then the relation between n and n' is as follows. 	
	
		Let V denote the [F_T]-module defined by the tuple A = [A_1, A_2, …, A_T] and let λ∈^×. 
		
			
  * If λ≠ 1 then 
    (MC_λ(V)) = ∑_i = 1^T(A_i -I) - ((V) - (λ· A_1A_2 ⋯ A_T -I))

		
	 
	
		Proofs of this statement, as well as further details and properties of the middle convolution can be found in <cit.> and <cit.>. 
	
	To get an MCG-finite T+1-tuple of rank 2, we want to find a T-tuple of reflections (hence (A_i -I) = 1) such that their product has a nontrivial eigenvalue of multiplicity T-2. This is why we search for nice tuples, Definition <ref> in Section <ref>.

	Another property of middle convolution that will greatly simplify our computations is the following. 
	 (Dettweiler and Reiter, 2000)
		If 𝐀 is an irreduble tuple of rank greater than 1, then the modules MC_λ^-1(𝐀^-1) and MC_λ(𝐀)^-1 are isomorphic. 
	
	
		Proved in <cit.>.
	
	
	Our code for middle convolution uses the formulation given by Dettweiler-Reiter in <cit.> as opposed to their formulation in <cit.> which only differs by changing the order of blocks in the matrices. The code requires a matrix group W containing the matrices in the tuple 𝐀 and the base ring of W should contain the parameter λ used for middle convolution. Note that the final function Compute_MC will automatically check if our intermediate constructions act invariantly on the subspaces K and L, as defined in the notation of <cit.>. The full computer code can be found online at  <amalvayalin.github.io/research.html>.
	
	

 §.§ Finite Complex Reflection Groups
 
	A real reflection can be identified as a matrix with one eigenvalue -1 and all other eigenvalues are 1, and pseudo-reflections refers to matrices with one eigenvalue a root of unity and all other eigenvalues 1.  A finite complex reflection group (FCRG) is a finite group generated by pseudo-reflections. 
	As we are only working with complex groups, the distinction between real or complex eigenvalues is not relevant and henceforth reflections can refer to real reflections or pseudo-reflections.
	
	All irreducible finite complex reflection groups  were classified by Shephard-Todd <cit.>, as belonging to an infinite family of imprimitive reflection groups or being one of 34 exceptional primitive cases. 
	 
		The rank of a reflection group is the smallest dimension of the vector space on which the group acts irreducibly.
		Every finite complex reflection group of rank n has a minimal generating set consisting of either n or n+1 reflections. The group is said to be well-generated if it can be generated by n reflections. 
	
	As we want irreducible MCG-finite T+1-tuples, we know that our nice T-tuple (see Defintion <ref>) must at least be of length n to generate an irreducible FCRG, so in each group we only need to search for T-tuples with T > n. Among the rank three and four primitive irreducible groups, only G_31 is not well-generated (see Cartan matrices in <cit.>. An imprimitive group G(m, p, n) is well-generated if p = m or p = 1. 
	
	Associated to each FCRG of rank n is a set of n invariants called the degrees of the reflection group <cit.>, <cit.>. An important property of these invariants relevant for our purposes is their relation to the eigenvalues of the group elements. Note that we already know the eigenvalues for any element of the group must be roots of unity since the group is finite. 
	
		Let W be a FCRG of rank n and let {d_i}_i = 1, …, n denote the degrees of W. Suppose λ is some primitive d-th root of unity that is an eigenvalue for some element of W. Then, d | d_i for some degree d_i. Moreover, the maximum dimension of the eigenspace corresponding to λ for any element of the group is the number of degrees d_i that is divisible by d, and for some element in W the eigenspace corresponding to λ achieves this maximal dimension. 
	
	
		Proved in <cit.>.
	
	The theorem above will be helpful in our computations for the primitive groups as we know how large our field needs to be to ensure the code finds all eigenvalues. 
	
	Further properties specific to imprimitive reflection groups are described in Section <ref>.
	
	

 §.§ Representations and Character Varieties

	
	Given a finitely generated group Γ, a representations of Γ is a homomorphism ρ: Γ⟶ G where G < GL_n() is some algebraic group. Considering the vector space V the matrices ρ(γ) act on, ρ is said to be irreducible if there is no nonzero subspace U ⊊ V that is left invariant by the matrix ρ(γ) for all γ∈Γ. Two representations ρ and ρ' are isomorphic if there exists some g ∈ G such that ρ(γ) = g^-1ρ'(γ) g for all γ∈Γ. 	The character of a representation  ρ is the map χ_ρ: Γ⟶ given by χ_ρ(γ) = ρ(γ). Isomorphic representations have the same character since trace is fixed under conjugation. The character variety 𝔛(Γ, G) parametrizes the isomorphism classes of representations ρ: Γ⟶ G in terms of suitable traces.  
	
	In the case of the punctured sphere Σ_0, T+1, the fundamental group is π_1(Σ_0, T+1) = γ_1, …, γ_T+1|γ_1*γ_2*⋯ *γ_T+1 = 1. Any representation of Γ = π_1(Σ_0, T+1) is identified by ρ(γ_i) = M_i ∈ G such that M_1 · M_2 ⋯ M_T+1 = I is the identity matrix. Two such representations  ρ and ρ' are equivalent if the tuple [ρ(γ_1), …, ρ(γ_T+1)] is related to [ρ'(γ_1), …, ρ'(γ_T+1)] by simultaneous conjugation.
	
	Taking G = SL_2(), we can consider the SL_2()-character variety of π_1(Σ_0, T+1). 
	
	Given a tuple of of matrices [M_1, M_2, …, M_T+1], let 
    t_i = (M_i)      t_ij = (M_iM_j)      t_ijk = (M_iM_jM_k) *for 1≤ i < j< k ≤ T+1

	Since π_1(Σ_0, T+1) = F_T we have a natural inclusion 𝔛(π_1(Σ_0, T+1), SL_2()) ⊆𝔛(F_T, SL_2()). This means for our purposes, M_T+1 will always be (M_1M_2⋯ M_T)^-1 so t_T+1 = (M_1M_2⋯ M_T) as all the matrices M_i will be in SL_2() where matrices have the same trace as their inverse.
	
	
		Let [M_1, M_2, …, M_T+1] ∈ SL_2()^T+1 such that Π M_i = I. Then up to simultaneous conjugation, any such (T+1)-tuple can be identified by the following trace coordinates: 
		
			
  * For T = 3: (t_1, t_2, t_3, t_12, t_13, t_23, t_4) 
			
  * For T = 4: (t_1, t_2, t_3, t_4, t_12, t_13, t_14, t_23, t_24, t_34, t_123, t_124, t_134, t_234, t_5)
		
	
	
		For proof and further discussion of character varieties of the punctured sphere see <cit.> and <cit.>. 
	
	Such trace coordinates are also sometimes referred to as the 𝐬𝐢𝐠𝐧𝐚𝐭𝐮𝐫𝐞 of the tuple. 
	
	

§ METHODS
 
	For the rest of this written work, fix ζ_d = e^2π i/d as a primitive d-th root of unity. 
	
	
	Since the product of [A_1, ⋯, A_T+1] must be the identity, we know any MCG-finite tuple is determined by the first T matrices, up to simultaneous conjugation. As middle convolution does not preserve products (see Remark <ref>), our idea is to take middle convolution of the first T matrices to get 2× 2 matrices and then induce a representation of π_1(Σ_0, T+1) by taking the T+1-th matrix to be the inverse product. The condition for middle convolution being rank 2, (stated in Theorem <ref>) depends on the dimension of the subspace fixed by, or equivalently the multiplicity of the eigenvalue 1 of each matrix in the tuple and the eigenvalues of their product. For much better control on the former, we consider reflections since they only have one nontrivial eigenvalue, of multiplicity 1. 
	
	Thus, for each irreducible finite complex reflection group W of rank n, we want to find all irreducible T-tuples of reflections A = [A_1, A_2, …, A_T] in W such that for a suitable choice of parameter λ, MC_λ(A) has rank 2. Given T, the formula given in Theorem <ref> tells us that the product Π_i = 1^T A_i of reflections in A should have a nontrivial eigenvalue λ^-1 of multiplicity T-2 so that MC_λ(A) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃] is a T-tuple of rank 2. We give a name to tuples satisfying these conditions.
	
		A T-tuple A = [A_1, A_2, …, A_T] of matrices from a finite complex reflection group W is called 𝐧𝐢𝐜𝐞 if A satisfies the following: 
		
			
  * All A_i are reflections in W
			
  * W = A_1, A_2, …, A_T
			
  * (A_1A_2 ⋯ A_T)^-1 has an eigenvalue λ≠ 1 of multiplicity T-2. 
		
	
	
	The condition of generating the whole group is to ensure that we are not “over-counting" as some reflection groups contain other reflection groups as subgroups, and we only want to study irreducible MCG-finite tuples. 
	
	
		Given a nice T-tuple 𝐀 = [A_1, A_2, …, A_T], we know there exists some λ≠ 1 such that MC_λ(𝐀) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃] is a T-tuple of rank 2. taking the T+1-th matrix to be 
    A_T+1 = (Π_i = 1^T Ã_̃ĩ)^-1
 we get a T+1-tuple [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃, A_T+1] whose product is the identity. As 𝐀 comes from a finite group we know it has finite braid group orbit, and since middle convolution is equivariant with the braid group action (see Section <ref>), we know that [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃, A_T+1] must be MCG-finite. Moreover, if {A_1, A_2, …, A_T} generates an irreducible group, then as middle convolution preserves irreducibility, we know our T+1-tuple is irreducible as well. Thus, we have used a nice T-tuple to produce an irreducible MCG-finite T+1-tuple. 
	
	 
		For some nice T-tuples, it is possible that multiple choices of λ have multiplicity T-2 and hence the different choices of parameter λ gives rise to possibly different rank 2 MCG-finite T+1-tuples. 
	
	
	 
		The eigenvalue condition of multiplicity T-2 tells us that in any complex reflection group of rank n, for T > n+2 the middle convolution will not be rank 2 so there are no nice T-tuples.  By Definition <ref>, since we need at least n reflections to generate the whole group, we know there are no nice tuples when T < n. For T = n+2, the product of the T matrices would be a scalar matrix (as there must be an eigenvalue of multiplicity n) which we can take to be the identity, reducing to the T= n+1 case. 	So for each n ∈{3, 4}, we only search the primitive reflection groups for nice tuples of length T = n or T = n+1. 
	
	
		

 §.§ Computing The Braid Group Orbit

	The braid group can be presented as B_T+1 = σ_1, …, σ_T |σ_iσ_i+1σ_i = σ_i+1σ_iσ_i+1, σ_iσ_j = σ_jσ_i where the first relation holds for 1 ≤ i≤ T-3 and the second for i-j ≥ 2. A generator σ_i acts on T+1-tuples in the following way: 
	
    σ_i(M_1, M_2, …, M_T, M_T+1) = (M_1, …, M_i-1, M_iM_i+1M_i^-1, M_i, M_i+2, …, M_T+1)
	
	
	Let F_+ denote the set of all words in positive powers of these generators. Further, let Orb(B_T+1, x) denote the orbit of some tuple x under the action of B_T+1 and Orb(F_+, x) denote the set of images of x under the action of words in F_+. 
	
	
		For a tuple x of matrices, Orb(B_T+1, x) is finite if and only if Orb(F_+, x) is finite. 
	
	
		Since F_+ ⊂ B_T+1, one direction is obvious. For the other direction first note that for any fixed g ∈ B_T+1, if the set {g^i x : i ≥ 0} is finite, then we can rewrite g^-1x = g^n x for some n ≥ 0, so we get that {g^i x: i ∈} must be finite. Further, since the elements of B_T+1 act one at a time, it is enough to study how words in positive powers of the generators act on x. Thus, if Orb(F_+, x) is finite, then the braid group orbit must be finite as well. 
	
	
	To help with computing the braid group orbit, we need a way to capture the equivalence relation of tuples being related by simultaneous conjugation. Using character varieties, we can write down trace coordinates for our tuple, which we know how to do from Theorem <ref> when the matrices are in SL_2().
	
	
		Given a nice tuple 𝐀 = [A_1, A_2, …, A_T], let  MC_λ(𝐀) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃] denote the middle convolution. If (a_1, a_2, …, a_T+1) ∈ (^×)^T+1 is such that a_i Ã_̃ĩ∈ SL_2() for i = 1, … T and a_T+1 ((Ã_̃1̃Ã_̃2̃⋯Ã_̃T̃)^-1) ∈ SL_2() with Π^T+1 a_i = 1, the tuple 
		
    Â = [a_1 Ã_̃1̃, a_2 Ã_̃2̃, …, a_T+1(Ã_̃1̃Ã_̃2̃⋯Ã_̃T̃)^-1]

		is said to be 𝐢𝐧𝐝𝐮𝐜𝐞𝐝 by MC_λ(𝐀) or by  𝐀 if the parameter λ is clear. Â is a tuple in (SL_2())^T+1 which multiplies to the identity.
	
	When we say trace coordinate of a tuple MC_λ(𝐀) we will always mean the trace coordinates of the induced tuple Â unless otherwise specified.
	
	We want to compute the orbit of induced tuples Â up to simultaneous conjugation. This is equivalent to computing the orbit of the point associated to Â in the character variety 𝔛(π_1(Σ_0, T+1), SL_2()). Note that since all of our tuples come from FCRG and middle convolution is equivariant with the action of the braid group we already know all the orbits are finite. 
	
	The code for the algorithm described below can be found online at <https://amalvayalin.github.io/research.html>.
	
	Given a tuple 𝐌 = [M_1, …, M_T+1] of SL_2()^T+1, such that their product is the identity, the braid group B_T+1 orbit is computed as follows. 
	
	We first compute the signature of 𝐌, and then apply each of the T generators to the tuple 𝐌. If σ_i(𝐌) has a new signature, then we store the tuple in To_Test and the signature in Signatures and add 𝐌 to Tested.  We then apply σ_i again to the (atmost T) new tuples in To_Test we got from the action on 𝐌, once again storing only tuples which have new signatures. We repeat this process until the sets To_Test and Tested are the same. The function returns the set Tested, which should contain only one representative per distinct signature, and can optionally return the set Signatures as well. 
	Note that here our definition of signature is the trace coordinates, which  differs from the definition used in Tykhyy's classification in terms of residues <cit.>.
	
	

 §.§ Finding Nice Tuples
 
	
	
	Nice T-tuples correspond to  MCG-finite T+1-tuples in the following way: given a nice tuple [A_1, …, A_T], we get a corresponding representation  [A_1, …, A_T, A_T+1] of the T+1-punctured sphere if we take A_T+1 = (A_1A_2 ⋯ A_T)^-1. 
	Thus, it is enough to search each complex reflection group for nice tuples.
	
	By Theorem <ref>, an eigenvalue of multiplicity T-2 for an element in W must be a d-th root of unity for d dividing at least T-2 degrees of W. 
	Any reflection group can be constructed in Magma by providing the generating reflections; the irreducible finite complex reflection groups can be easily constructed in Magma by providing the Shephard-Todd number or the corresponding parameters for G(m, p, r). In most cases Magma generates the group W via the reflection representation over the trace field of W <cit.>. 
	
	
	One issue is that not all characteristic polynomials will split over the trace field, as Magma computes eigenvalues for matrices over the field of entries. Since our search depends on the eigenvalues of the product of matrices in the tuple(s), we must change the base field of W to the smallest cyclotomic field containing all the possible eigenvalues. The largest this extension will be (ζ_d), where d is the least common multiple of all the degrees of W, but in certain cases smaller fields might suffice. For large d computation time is typically slower, but we have expanded the field to the LCM of the degrees unless otherwise stated. For example, in the case of W = G_32, the degrees are (W) = {12,18,24,30 } and so ((W)) = 360 which slows down computation time considerably. For this group, we prove in Section <ref> that it is enough to search for 4-tuples over (ζ_12) by showing any eigenvalue of multiplicity 2 must be a 12-th root of unity. 
	
	Before outlining the steps of our search, we define some terms. 
	
	 
		Given a tuple of matrices 𝐀 = [A_1, A_2, …, A_T], the 𝐭𝐲𝐩𝐞 of 𝐀 is determined by the eigenvalues of (A_1A_2 ⋯ A_T)^-1 counted with multiplicity. Tuples whose inverse product has the same eigenvalues with multiplicity are said to be of the same type.  An 𝐞𝐱𝐞𝐦𝐩𝐥𝐚𝐫 of a type is a choice of nice tuple of that type. Given a tuple of some type A, the type of the inverse tuple is called the 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐭𝐲𝐩𝐞 of A and the two types together make an 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐩𝐚𝐢𝐫. 
	

		The use of types allows us to partition a set of nice tuples based on the eigenvalues of the inverse product. Since there are many nice tuples of the same type, we choose exemplars for each type to perform the in-depth computations on. Note that the choice of exemplar for any type is arbitrary and it may be possible that different exemplars result in different orbits. To ensure this does not happen, we go back through all the tuples for each type and make sure that it belongs to the braid group orbit we compute for the middle convolution of the exemplar. More details on the braid group computations can be found in Section <ref>.  
	
	In some groups we get types that are inverse pairs. In these cases we only provide the computations for one exemplar from each pair. To ensure the inverse type would not produce a new orbit, we check that the inverse tuple of the middle convolution lies in the orbit of the middle convolution of the exemplar; this is enough by the relationship between inverse tuples and inverse parameters given in Theorem <ref>. 
	The code for how we checked each type is available at <https://amalvayalin.github.io/research.html>.
	 
		Aside from a few  exceptional cases that were computationally too large, we verified that for each choice of the parameter λ, the rank 2 MCG-finite tuples coming from nice tuples of the same type are in the same braid group orbit, as are exemplars from the inverse type. In this sense, types can be viewed as `almost' equivalence classes for nice tuples up to inverse pairs.
	
	
	After coercing the matrix group into a larger field if necessary, the general process for searching for T-tuples is as follows.
	
		
  * Fix a representative Q_i for each conjugacy class of reflections in W and fix an ordering [Q_1, ⋯, Q_n].
		
  * Given T, construct the list M of distinct sets of T reflections in W, using the ordering of Q to eliminate some tuples equivalent up to simultaneous conjugacy. 
		
  * Search M for all sets such the matrices generate W (means the tuple is irreducible) and the product of the matrices, for some order of multiplying, has an eigenvalue of multiplicity T-2. Each such set is then organized into a tuple based on the order of multiplication and stored in a list X of irreducible tuples. 
		
  * For each nice tuple x ∈ X, compute the eigenvalues of the inverse product of x and partition X into different types corresponding to the different sets of eigenvalues of the inverse product. 
		
  * Choose exemplars 𝐀, 𝐁... for each type and compute the middle convolution MC_λ_1(𝐀), MC_λ_2(𝐀), MC_λ(𝐁)... for suitable choice(s) of λ.
		
  * Compute the subgroup of GL_2() generated by MC_λ(𝐀), MC_λ(𝐁), …
		
  * For each middle convolution of the exemplars, find a suitable character [a_1, …, a_T+1] and construct the induced tuple Â
		
  * Compute the braid group orbit of Â and identify the SL_2() subgroup generated by the matrices of Â
		
  * Check the other exemplars and inverse tuple do not produce new orbits
		
  * Use the tuples in the braid group orbit to compute the parameter and residue signatures of the full equivalence class as defined in the previous classifications to try and match our examples to theirs. 
	
	We recorded all the details for our exemplars and the middle convolution computations for each of the primitive reflection groups of rank 3 and 4. The middle convolution tables are listed in the Appendix. The types and exemplars are listed in the section for each group, as is the comparison tables with the other classifications.  
	More details on what we use to compare with the existing classifications of <cit.> and <cit.> are given in Sections <ref> and <ref> .
	
		

§ IMPRIMITIVE REFLECTION GROUPS

	Imprimitive reflection groups G(m, p, n) are parameterized by m, p , n ≥ 1 with p | m. These groups can be viewed as n × n generalized permutation matrices, which each row and column containing only one non-zero entry which is either 1 or an m/p-th root of unity.  For m > 1 and (m, p, n) ≠ (2, 2, 2) these groups are always irreducible, meaning they act irreducibly on ^n (see <cit.>). 
	
	An element A  of G(m, p, n) can be written as A = [a_1, a_2, …, a_n |σ] for σ∈ S_n and p | a_1 + a_2 … + a_n. This element can be identified with the matrix where the (i, σ(i))-th entry of A is ζ_m^a_i and all the other entries are 0 <cit.>. Note that all these a_i should be read modulo m since we are working with powers of an m-th root of unity. 
	Additionally, there is a natural surjective homomorphism : G(m, p, n) ⟶ S_n given by sending A ↦σ. 
	
	One observation that follows from the definition is that for any p,  G(m, m, n) ≤ G(m, p, n) and G(m, p, n) ≤ G(m, 1, n) as subgroups.  
	
	Just as in the primitive case, the degrees of imprimitive groups tell us about the possible eigenvalues.  Writing q = m/p, we have: 
	
    (G(m, p, n))    = {m, 2m, …, (n-1)m, qn}
    
    		|ZG(m, p, n)|    = q (p, n)

	where ZG(m, p, n) is the center of G(m, p, n) and a cyclic subgroup.  
	
	
	Reflections in the group G(m, p, n) will be of one of the following two types:
	
		
  * Type 1: These are elements such that σ = (ij) is a transposition with a_k ≡ 0  m for k≠ i, j and a_i + a_j ≡ 0  m. All reflections of Type 1 are of order 2 and lie in the subgroup G(m, m, n) of G(m, p, n). Using the notation from <cit.> we write a Type 1 reflection as s(i, j; a_i) with the understanding that this is the same as s(j, i; -a_i).
		
  * Type 2: These are elements such that σ = e and for some i, a_i ≢0   m and a_k ≡ 0  m for all k ≠ i. These matrices are diagonal matrices of order m/(m, a_i) and will be denoted s(i; a_i). Type 2 matrices can only exist when p < m. 
	

	
		Let S be a set of reflections with minimal possible cardinality such that S generates G(m,p,n). Then S satisfies the following conditions: 
		
			
  * When p = 1, S contains n-1 reflections of Type 1 and one reflection of Type 2 of order m
			
  * When p = m, S contains n reflections of Type 1
			
  * When p ≠ 1, m, S contains n reflections of Type 1 and one reflection of Type 2 of order m/p
		
	
	
		Stated in <cit.> with proofs in <cit.> and <cit.>.
	
	
	
		For any reflection r ∈ G(m, p, n), the homomorphism : G(m, p, n) ⟶ S_n, one of the following must be true: 
		
			
  * (r) is a transposition and r has order 2
			
  * (r) is the identity.
		 
	
	
		This is true by the definition of reflections and the definition of the homomorphism. For a more thorough examination of related properties, we refer to <cit.>.
	
	
	Given any set of reflections generating an irreducible imprimitive group the permutations associated to the  reflections in the set must generate a transitive subgroup of S_n, or else the group cannot be irreducible. Note that in the case where the associated permutations are transpositions (i.e. the reflections are Type 1), the only transitive subgroup of S_n generated by transpositions is S_n itself <cit.>. 
	
	A subgroup of a reflection group generated by reflections is called a reflection subgroup. We modify some notation from <cit.> to state relevant theorems for identifying reflection subgroups in the imprimitive groups. Let X = {s(a_h, a_h+1; b_k)} be a set of reflections in G(m, m, n). We can associate to X the graph Γ_X with nodes {a_i} and edges (a_h, a_h+1) for s(a_h, a_h+1; b_k) in X. Letting X denote the reflection group generated by the reflections in X, we see that this group is irreducible if and only if the graph Γ_X is connected. 
	Suppose Γ_X is such that the graph contains exactly one cycle, given by r edges corresponding to {s(a_h, a_h+1; b_h)} for 1 ≤ h ≤ r for some 2 ≤ r ≤ n. Then we define the parameter δ(X) to be the absolute value of ∑_1^r b_h. 
	If p ≠ m, we say X is a nice reflection set of G(m, p, n) if X contains exactly one reflection of Type 2.
	
	
		Note that s(a_h, a_h+1; b_h) = s(a_h+1, a_h; -b_h), so if X is such that the graph contains only one cycle of length r, then the cycle corresponds to some permutation (a_1a_2 ⋯ a_r) (in cycle notation) and the signs of b_h must be chosen accordingly when computing δ(X). In particular, if the cycle is of length two, i.e. given as a double edge between two nodes corresponding to reflections s(a_1, a_2; b_1) and s(a_1, a_2; b_2) then δ(X) = |b_1 -b_2|.
	

	
		With the notation as defined above, we have the following relations for the reflection subgroup generated by X.  
		
			
  * Let X be reflection set of G(m, m, n) such that the graph Γ_X is connected and contains exactly one cycle. Then X = G(m, m, n) if and only if the integer δ(X) is coprime to m.
			
  * Let X be a nice reflection set of G(m, p, n) with s(a_1; b) the one reflection of Type 2, such that Γ_X is connected with n_1 nodes and exactly one cycle. Then X = G(m/m_1, (b, m)/m_1, n_1) where m_1 = (b, m, δ(X)).
			
  * Let X be a nice reflection set of G(m, p, n) with s(a_1; b) the unique reflection of Type 2 such that the graph Γ_X is a connected tree with n_1 nodes. Then X = G(m/(b, m), 1, n_1)
		
	
	
		The first is proven in <cit.> and the second is proven in <cit.>. Note that in both these papers they refer to cycles as circles. The third statement is proven in <cit.>, where they define rooted trees to capture the presence of Type 2 reflections. We refer to these paper for more details on the graph theoretic interpretation of generating sets and their connection to reflection subgroups of the imprimitive reflection groups.
	

		
		Among the imprimitive groups, we have to check for n-tuples and n+1-tuples in G(m, p, n). By Theorem <ref>, the only possible groups that can contain nice n-tuples are the well-generated groups G(m, 1, n) and G(m, m, n). We prove there are no nice n-tuples in either of these groups when n ≥ 5 in Section <ref>. By results in <cit.>, we know that n+1-tuples can only exist in G(m, p, n) for n < 5. This work will fully analyze the existence of nice n+1-tuples for 2< n <5.
	
	
		

 §.§ Nice 3-Tuples in G(m, p, 3)

	In this case we are seeking nice tuples [r_1, r_2, r_3] from G(m, p, 3) such that their product has a nontrivial eigenvalue of multiplicity 1. Note that since r_1, r_2, r_3 = G(m, p, 3) we know that p = 1 or p = m are the only possibilities as these are the only well-generated imprimitive groups (see Definition <ref>)
	
		In G = G(m, 1, 3)  for a nice tuple [r_1, r_2, r_3] with r_1, r_2 of Type 1 and r_3 of Type 2 with nontrivial eigenvalue ζ_m' of order m, letting A = r_1r_2r_3, the eigenvalues of A must be the distinct cube roots of ζ_m'. 
	
	
		Say we have three reflections 
    r_1r_2r_3 = A
 for some A ∈ G with Eigv(A) = {λ_1, λ_2, λ_3} with some λ_i ≠ 1. Note that since the r_i generate G, exactly two of them are Type 1 corresponding to distinct transpositions and the third is Type 2 with order m, say with nontrivial eigenvalue ζ_m'. Using the homomorphism : G ⟶ S_3, we know that (A) is even and since it cannot be the identity it must be a 3-cycle.
		Since A is a generalized permutation matrix, we know (A) = 0 and we get the following equalities must be true: 
		
    ζ_m'    =  (r_1r_2r_3) = (A) = λ_1λ_2λ_3 
    
    			0    = (A) = λ_1 + λ_2 + λ_3

		Since λ_i are all roots of unity, we know their sum being zero means the argument differs by 2 π/3.
		Further writing A = [a_1, a_2, a_3 | (123)], without loss of generality, gives us the matrix: 
		
    A = [       0 ζ_m^a_1       0;       0       0 ζ_m^a_2; ζ_m^a_3       0       0 ]

		Thus we can compute the characteristic polynomial as p_A(x) = -(x^3 - ζ_m^a_1 + a_2 + a_3) and since the constant term of the characteristic polynomial is the determinant of A, we get that ζ_m' = ζ_m^a_1 + a_2 + a_3. Thus λ_i must be the distinct cube roots of ζ_m'.
	
	
		There are nice 3-tuples in G(m, 1, 3) for all m >1. 
	
	
		The reflections in the tuple [s(2, 3; 1), s(1, 2; 1), s(3; 1)] will generate G(m, 1, 3) by Theorem <ref>. By the preceding lemma, the product will have eigenvalues corresponding to the three distinct cube roots of ζ_m, all of which cannot be 1.  
	

	
		In G = G(m, m , 3), if we have r_1r_2r_3 = A for reflections r_i generating G, then the eigenvalues of A are {λ^-2, λ, -λ}
	
	
		Since all the r_i are of Type 1, using : G ⟶ S_3 we see that (A) must be a transposition, say (ij) ∈ S_3. Writing A = [a_1, a_2, a_3 | (ij)] we see that (A) = ζ_m^a_k where k ≠ i, j and (A) = -1. Letting {λ_1, λ_2, λ_3} denote the eigenvalues of A, we get that 
		
    ζ_m^a_k   = λ_1 + λ_2 + λ_3 
    
    			-1    = λ_1 λ_2λ_3

		The characteristic polynomial of A is p_A(x) = -(x^2 -ζ_m^a_i + a_j)(x- ζ_m^a_k). Let λ_k = ζ_m^a_k so then we get that λ_i = -λ_j since these are the roots of x^2 - ζ_m^a_i + a_j. Using the determinant equation we get 1 = λ_i^2 λ_k hence λ_i^2 = λ_k^-1 = ζ_m^-a_k = ζ_m^a_i + a_j so both λ_i and λ_j square to the inverse of λ_k.
	
	
	
		Nice 3-tuples exist in G(m, m, 3) for all m >1. 
	
	
		In this case we can consider reflections [s(2, 3; b_1), s(1, 2; b_2), s(1, 2; b_3)] with b_2 and b_3 chosen so that |b_2 -b_3| is coprime to m. Taking b_1 = 1, b_2 = 1 and b_3 = 0, we get δ(X) = |b_2 -b_3| is coprime to m and so by Theorem <ref>, we know these reflections will generate G(m, m, 3). The characteristic polynomial of the product r_1r_2r_3 = A will in this case be p_A(x) = -(x-ζ_m)(x^2 -ζ_m^-1). Since the roots of the quadratic will be distinct, we know we will have at least two distinct eigenvalues, so the only worry is if the eigenvalues might be {ζ_m, ζ_m, 1} as then there is no nontrivial eigenvalue of multiplicity 1. If 1 is a root of the quadratic, then we know ζ_m^-1 = 1 and ζ_m = 1 which is not possible. Hence, such nice tuples always exist. 
	
	
	

 §.§ Nice 4-Tuples in G(m, p, 3)

	We are seeking nice tuples of four reflections from G(m, p, 3) such that their product has an nontrivial eigenvalue λ of multiplicity 2. This can be written as the identity 
	
    r_1r_2r_3r_4 = λ r_5^-1

	where the r_i generate G(m, p, 3) and r_5 is some reflection, not necessarily in G(m, p, 3). 
	
	
		In the notation of equation <ref> above, for any such nice tuple, it must be that r_5^-1 is in G(m, 1, 3) and λ is an m-th root of unity. Moreover, in any G(m, p,3 ), (including p = 1 and p= m) it must be that exactly four of these reflections are Type 1 and only one is of Type 2. 
	
	
		Note that since λ is an eigenvalue of multiplicity 2 for some matrix in G(m, 1, 3) it must be that the order of λ divides at least 2 degrees in G(m, 1, 3) = {m, 2m , 3m}. Thus the order of λ divides m and so λ I is in the center of G(m, 1, 3). This tells us that r_5 is always in G(m, 1, 3). To generate any of G(m, p, 3) we know the set {r_i}_i ≤ 5 must contain at least 2 reflections of Type 1 by Theorem <ref>.  For any p, rearranging the equation and using the homomorphism : G(m, 1, 3) ⟶ S_3 we get that (r_1r_2r_3r_4r_5) = (λ I) = e. Since e is even we know there must be exactly 2 or 4 reflections of Type 1. The product of two transpositions can only be the identity if they are the same transposition and we need at least two distinct transpositions to generate a transitive subgroup of S_3, hence there are exactly 4 reflections of Type 1 and one reflection of Type 2 for any G(m, p, 3).  
	
	Note that this does not prove that λ is primitive m-th root, however in the next three theorems we will conclude this is indeed always the case. 
	Up to rearranging the equation <ref>, we may assume r_5^-1 is the unique reflection of Type 2 and label its nontrivial eigenvalue ζ which we know must be some m-th root of unity, not necessarily primitive. 
	
	 
		To study the tuples in the cases where they might exist, we first study the number of ways to factorize the identity in S_3 into a transitive product of 4 transpositions. In S_3 any two distinct transpositions will generate a transitive subgroup which must be all of S_3 so fixing the first transposition as (12) we only need to find, up to simultaneous conjugacy, tuples of 3 transpositions that multiply to (12). Using computer search to find such conjugacy classes of 3-tuples, then adjoining with (12) on the left yields four options: 
		
			
  * [(12), (12), (23), (23)]
			
  * [(12), (23), (12), (13)]
			
  * [(12), (23), (23), (12)] 
			
  * [(12), (13), (23), (13)]
		
		Note that these tuples are multiplied starting at the leftmost transposition, although that does not matter in this case. 
		Converting these tuples to Type 1 reflections we can compute the product for each option. 
		
			
  * s(1, 2; b_1)s(1, 2; b_2)s(2, 3; b_3)s(2, 3; b_4) = [           ζ_m^b_1 -b_2                      0                      0;                      0 ζ_m^-b_1 +b_2 +b_3-b_4                      0;                      0                      0         ζ_m^-b_3 + b_4 ]
			
  * s(1, 2; b_1)s(2, 3; b_2)s(1, 2; b_3)s(1, 3; b_4) = [  ζ_m^b_1 +b_2 -b_4                  0                  0;                  0      ζ_m^-b_1 +b_3                  0;                  0                  0 ζ_m^-b_2 -b_3+ b_4 ]
			
  * s(1, 2; b_1)s(2, 3; b_2)s(2, 3; b_3)s(1, 2; b_4) = [ ζ_m^b_1 +b_2 -b_3-b_4                     0                     0;                     0         ζ_m^-b_1 +b_4                     0;                     0                     0        ζ_m^-b_2 + b_3 ]
			
  * s(1, 2; b_1)s(1, 3; b_2)s(2, 3; b_3)s(1, 3; b_4) = [  ζ_m^b_1 +b_3 -b_4                  0                  0;                  0 ζ_m^-b_1 +b_2 -b_3                  0;                  0                  0     ζ_m^-b_2 + b_4 ]
		
		Equations for {b_i} can be found by setting each of these matrices equal to λ r_5^-1:
		
    [  λ  0  0;  0  λ  0;  0  0 λζ ]

		for some root ζ with order dividing m. The choice of putting ζ in the third row is arbitrary and ζ could have been placed in any diagonal entry.
	
	 
		For any nice 4-tuple in G(m, p, 3), p≠ 1 it must be that either λ or λζ in the notation of Remark <ref> has the same order as ζ_m^δ(X), where X is any generating reflection set given by the nice 4-tuple.
	
	
		By Lemma <ref>, given any nice 4-tuple in G(m, p, 3) satisfying equation <ref>, we can rearrange all the Type 1 reflections to be one of the factorizations given in Remark <ref> up to simultaneous conjugation. As any reflection set X generating G(m, p, 3) p ≠ 1 will only need at least three Type 1 reflections, we see that for each of the four factorizations above, there are multiple ways to choose X so that the associated graph only contains one cycle. For example, the first factorization gives us the possibilities: X = {s(1, 2; b_1), s(1, 2; b_2), s(2, 3; b_i)} for i = 3, 4 and X = {s(1, 2; b_i), s(2, 3; b_3), s(2, 3; b_4)} for i = 1, 2. So we know δ(X) must be of the form
		
    δ(X) = 
    			|b_1 -b_2| 
    
    			|b_3 -b_4|

		By the equations each factorization must satisfy, we know ζ_m^b_1 -b_2 = λ and ζ_m^-b_3 +b_4 = λζ so one of these must have the same order as ζ_m^δ(X). The argument is similar for the third factorization since the formula for δ(X) only differs in the indices. The possible δ(X) values from the second and fourth factorizations only differ in the indices, so we only justify the result for the second factorization. In this case the options for δ(X) are 
		
    δ(X) = 
    			|b_1 +b_2 -b_4| 
    
    			|b_1 -b_3| 
    
    			|b_2 +b_3 -b_4|

		Up to taking absolute value, the first two options must equal λ and the third must be λζ so one of these has the same order as ζ_m^δ(X).
	

	
		Nice 4-tuples exist in G(m, m, 3) if and only if m ≠ 3. For m ≠ 3, we have: 
		
			
  * If (m, 3) = 1 then order(λ) = order(ζ) = m
			
  * If (m ,3) = 3 then order(λ) = m = 3 order(ζ) with order(ζ) ≠ 1. 
		 
	
	
		To generate G(m, m, 3), we know that our nice 4-tuple must contain a generating reflection set X such that δ(X) is coprime to m, which means either λ or λζ has order m, by Lemma <ref>. If λ does not have order m, then λζ has order m, but taking the determinant of equation <ref>, we know that 1 = λ^3 ζ so λ^2 has order m which means λ has order m. Thus, λ is always a primitive m-th root. Moreover, we know m ≠ 3 since the determinant tells us λ^3 = ζ^-1 and ζ≠ 1 as it is the nontrivial eigenvalue of the Type 2 reflection r_5^-1.
		
		If (m, 3) = 1 then we know λ^3 = ζ^-1 is also a primitive m-th root and hence so is ζ. Using factorization 1, we can take the tuple [s(1, 2; 1), s(1, 2; 0), s(2, 3; 2), s(2, 3; 0)], so λ = ζ_m and ζ = ζ_m^-3. This tuple generates G(m, m, 3)  by Theorem <ref> since taking X = {s(1, 2; 1), s(1, 2; 0), s(2, 3; 2)} gives δ(X) = 1 which is coprime to m.  
		
		If (m, 3) = 3 then λ^3 has order m/3 and thus so too will ζ. Again taking the tuple as above will work, since δ(X) = 1 will still be coprime to m and the product will give λ = ζ_m and ζ = ζ_m^-3 which now is not a primitive m-th root.  
	
	
		
		Nice 4-tuples exist in G(m, 1, 3) if and only if (m, 3) = 1. In this case the order of λ is m.
	
	
		Let [r_2, r_3, r_4, r_5] be a nice 4-tuple in G(m, 1, 3). Let r_5 be the unique reflection of Type 2 with nontrivial eigenvalue ζ^-1 of order m, so we can write ζ^-1 = ζ_m^-b with (-b, m) = 1. Let d = order(λ). We know d| m since λ I is in the center ZG(m, 1, 3). Taking determinants we get the equation
		
    λ^3 = ζ^-1

		so m | d since (ζ^-1)^d = 1, hence d = m. Since ζ^-1 is a primitive m-th root, so too is λ^3 and therefore we know (m, 3) = 1.
		
		Let m be such that (m, 3) = 1, so we know m-3 is coprime to m. Taking b = m-3, we can consider the tuple [s(1, 2; 0), s(2, 3; 1), s(2, 3; -1), s(3; m-3)]. Taking X = {s(1, 2; b_2), s(2, 3; b_3), s(3; -b)} we see by Theorem <ref> that X = G(m, 1, 3) since (m-3, m) = 1.
		Taking the product, we get r_2r_3r_4r_5 = λ r_1^-1 for some Type 1 reflection r_1 = s(1, 2; -1) with λ = ζ_m.
	

	
		Suppose p ≠ 1 and p≠ m. Then there exists a nice 4-tuple in G(m, p, 3) if and only if p = 3. In this case λ must be of order m.
	
	
		Let [r_2, r_3, r_4, r_5] be a nice 4-tuple in G(m, p, 3). Letting r_5 denote the unique reflection of Type 2, we know it must have order q = p^-1m so we can denote its nontrivial eigenvalue as ζ = ζ_q^b = ζ_m^pb with (b, m) = 1. 
		Further, we have the product r_2r_3r_4r_5 = λ r_1^-1 with r_1 ∈ G(m, m, 3) ≤ G(m, p, 3), hence λ I is in the center ZG(m,p , 3). Letting d = order(λ) we see that since |ZG(m, p, 3)| = q (p, 3), we must have that d | q (p, 3). Taking determinants, we get λ^3 = ζ^-1 and hence q | d and d | 3q.  If d ∤ q then since d ≠ 3 (becasue ζ^-1≠ 1), and d | q (p, 3) we must have that (p, 3) = 3.
		This gives us two cases: 
		
			
  * d = q
			
  * d = 3q and (p, 3) = 3
		
		Recall that by Theorem <ref>, for a reflection set X to generate G(m, p, 3), we need (pb, m) = p and (pb, δ(X), m) = 1.  
		
		In the first case d  = q so we can write λ = ζ_q^a = ζ_m^pa and ζ = ζ_q^b = ζ_m^pb with (b, q)= 1 = (a, q). By Lemma <ref>, we see that 
    (δ(X), m) = (pa, m) 
    ((a+b)p, m)

		Since neither of these satisfy (pb, δ(X), m) = 1, it is not possible for a nice 4-tuple in G(m, p, 3) to have d = q. 
		
		Suppose now that p = 3k so m = 3kq. Again we can write ζ = ζ_q^b = ζ_m^pb with (b, q) = 1, and this time since d= 3q we write λ = ζ_m^ka with (a, m) = 1. In this case the possible options for δ(X) gives us
		
    (δ(X), m) = (ka, m) 
    (ka + pb, m)

		In either case we get (pb, δ(X), m) = k so we conclude k = 1. This also gives us that λ must be of order m. 
		Thus, we conclude nice 4-tuples can only exist in G(m, 3, 3). In this case, we can write λ = ζ_m^a with (a, m) = 1 and we still have ζ = ζ_m^3b with (b, m) = 1 for q = p^-1m. We can take our 4-tuple to be [s(1, 2; 0), s(2, 3; 1), s(2, 3; -1), s(3; m-3)], noting that since 3 | m we also have that (m-3, m) =3. Here δ(X) = 2 so (m-3, 2, m) = 1 and we conclude this tuple is indeed nice.  
	
	
	
		Note that in all 3 cases described above, we will have to take middle convolution with parameter λ^-1 since our equations use the tuple product and not the inverse product.
	

	

 §.§ Nice 4-Tuples in G(m, p, 4)
 
	
	We are seeking nice tuples satisfying 
    r_1r_2r_3r_4 = A
 with r_i a reflection in G(m, p,4) and A has a nontrivial eigenvalue of multiplicity 2. 		Again the only possibilities are p = 1 and p = m since the group must be generated by 4 reflections.
	
		In G = G(m, 1, 4) there are no nice tuples of 4 reflections.
	
	
		Suppose we had r_1r_2r_3r_4 = A such that r_i generate G and A has eigenvalue λ≠ 1 of multiplicity 2. Since {r_i} generate G, we know exactly three of them are of Type 1 and exactly one is of Type 2 with nontrivial eigenvalue an m-th root of unity, say ζ_m'. Note that since the three Type 1 reflections should correspond to transpositions that generate all of S_4, they must be three distinct transpositions with each of {1, 2, 3, 4} showing up at least once. This means their product cannot be a transposition. Using : G ⟶ S_4 we get that (A) = σ for some 4-cycle σ since the permutation must be odd and cannot be a transposition. 
		Writing A = [a_1, a_2, a_3, a_4 |σ] we can compute that the character polynomial of A p_A(x) must be 
    p_A(x) = x^4 - ζ_m^a_1+ a_2 + a_3 + a_4

		This gives us that all the eigenvalues of A are the distinct 4th roots of ζ_m^a_1 + a_2 + a_3 + a_4 which means A cannot have an eigenvalue of multiplicity 2. 
	
	
	In the case of G(m, m, 4), we know for any nice 4-tuple, we must have that all the reflections are of Type 1. This means they all map to transpositions in S_4 and hence the permutation associated to A must be even. In S_4 the only options are e, the disjoint product of transpositions and a three cycle. We would need at least 6 transpositions to write e as the transitive product of transpositions, so we only have two choices. Without loss of generality, if A corresponds to disjoint transpositions, we take it to be (12)(34) and if A corresponds to a 3-cycle we take it to be (132).
	
	 
		
		The number of transitive ordered factorizations of (12)(34) into four transpositions in S_4 is 96 <cit.>. Using a computer, we find that there are 12 factorizations up to simultaneous conjugation. We can reduce these 12 even further by eliminating factorizations that are related by the action of one of the generators of B_4, since this means they lie in the same braid group orbit. 
		These preliminary computations give us three equivalence classes of factorizations, (which may still lie in the same braid group orbit) which we write with representatives: 
		
			
  * [(12), (34), (13), (13)]
			
  * [(13), (13), (12), (34)]
			
  * [(13), (24), (23), (14)]
		
		Choosing general reflections in G(m, m, 4) associated to these factorization and computing the products we get:
		
			
  * s(1,2; b_1) · s(3, 4; b_2) · s(1, 3; b_3) · s(1, 3; b_4) = [                0          ζ_m^b_1                0                0; ζ_m^-b_1+b_3-b_4                0                0                0;                0                0                0          ζ_m^b_2;                0                0 ζ_m^-b_2-b_3+b_4                0 ]
			
  * s(1,3; b_1) · s(1,3; b_2) · s(1, 2; b_3) · s(3, 4; b_4) = [                0  ζ_m^b_1-b_2+b_3                0                0;         ζ_m^-b_3                0                0                0;                0                0                0 ζ_m^-b_1+b_2+b_4;                0                0         ζ_m^-b_4                0 ]
			
  * s(1, 3; b_1) · s(2,4; b_2) · s(2, 3; b_3) · s(1, 4; b_4) = [             0   ζ_m^b_1-b_3             0             0;   ζ_m^b_2-b_4             0             0             0;             0             0             0  ζ_m^-b_1+b_4;             0             0 ζ_m^-b_2 +b_3             0 ]
		
	
	
	 
		Similarly, we can consider transitive ordered factorizations of (132) into four transpositions in S_4. Up to simultaneous conjugation and the action of a generator of B_4, we again get three equivalence classes of factorizations (which may not be distinct): 
		
			
  * [(14), (14), (12), (23)]
			
  * [(14), (14), (13), (12)]
			
  * [(14), (14), (23), (13)]
		
		These factorizations give us the following matrices: 
		
			
  * s(1, 4; b_1) · s(1, 4; b_2) · s(1, 2; b_3) · s(2, 3; b_4) = [                   0                   0 ζ_m^b_1-b_2+b_3+b_4                   0;            ζ_m^-b_3                   0                   0                   0;                   0            ζ_m^-b_4                   0                   0;                   0                   0                   0       ζ_m^-b_1 +b_2 ] 
			
  * s(1, 4; b_1) · s(1, 4; b_2) · s(1, 3; b_3) · s(1,2; b_4) = [                0                0 ζ_m^b_1 -b_2+b_3                0;         ζ_m^-b_4                0                0                0;                0    ζ_m^-b_3 +b_4                0                0;                0                0                0     ζ_m^-b_1+b_2 ]
			
  * s(1, 4; b_1) · s(1, 4; b_2) · s(2, 3; b_3) · s(1, 3; b_4) = [               0               0 ζ_m^b_1-b_2+b_4               0;    ζ_m^b_3 -b_4               0               0               0;               0        ζ_m^-b_3               0               0;               0               0               0   ζ_m^-b_1 +b_2 ]
		
	
		 
			In G = G(m, m, 4) if we had r_1r_2r_3r_4 = A with r_i reflections generating G and A has a nontrivial eigenvalue λ of multiplicity 2 then the eigenvalues of A are one of the following: 
			
				
  * {λ, λ, -λ, -λ} for λ a 4-th root of unity
				
  * {λ, λ, λ_1, λ_2} for λ a 4-th root of unity and λ_i ≠λ are the two distinct cube roots of λ^-1 other than λ.  
			
		
		
			We know all the r_i are of Type 1, so using : G ⟶ S_4 we know that (A) must be an even permutation, so it is either a double transposition or a 3-cycle. Let the eigenvalues of A be {λ, λ, λ_1, λ_2}. 
			If (A) is a double transposition then we see that 
			
    0    = (A) = 2λ + λ_1 + λ_2 
    
    				1    = (A) = λ^2 λ_1 λ_2

			Rearranging the equation for trace gives us that 
    1 = |λ| = | λ_1 + λ_2|/2 = |λ_1| + |λ_2|/2 = 1

			By properties of the triangle inequality, we get that λ_1 and λ_2 are colinear, hence λ_1 = ±λ_2. However, if λ_1 = -λ_2 then the trace equation means λ = 0 which is not possible, hence λ_1 = λ_2. Now the trace equation tells us that 2(λ + λ_1) = 0, hence λ_1 = -λ. 
			Next, we can use the determinant equation to see that 1 = λ^4 and so λ must be a 4th root of unity. This gives us that the eigenvalues of A are as described in case 1). 
			
			Further, when (A) is a double transposition, say A = [a_1, a_2, a_3, a_4 | (12)(34)], the characteristic polynomial is p_A(x) = (x^2 -ζ_m^a_1 + a_2)(x^2 - ζ_m^a_3 + a_4). However, since we know the roots of these polynomials agree, we see that ζ_m^a_1 + a_2 = ζ_m^a_3 + a_4, but we already knew that ζ_m^a_1 + a_2 = ζ_m^-a_3 - a_4 since m | a_1 + a_2 + a_3 + a_4. Thus we conclude that ζ_m^a_1 + a_2 = ± 1 and so ζ_m^a_1 = ±ζ_m^-a_2 and similarly, ζ_m^a_3 = ±ζ_m^-a_4 where the signs must agree for both pairs. 
			If (A) is a 3-cycle, we write A = [a_1, a_2, a_3, a_4 | (132)], since taking a different 3-cycle just changes the indices. The determinant condition stays the same as above, however our equation for trace is now: 
			
    ζ_m^a_4 = (A) = 2λ + λ_1 + λ_2

			Computing the characteristic polynomial we get p_A(x) = (x-ζ_m^a_4)(x^3 - ζ_m^a_1+a_2+a_3) and since the cube roots of ζ_m^a_1 + a_2 + a_3 are all distinct, we know that for λ to have multiplicity two, it must be one of the cube roots and λ = ζ_m^a_4. Since ζ_m^a_1 + a_2 + a_3 = ζ_m^-a_4 = λ^-1, we see that λ, λ_1 and λ_2 must be the three distinct cube roots of λ^-1. Now we have that λ^3 = λ^-1 and hence λ^4 = 1, proving that the eigenvalues of A are as described in case 2).
		
	
	 
		Nice 4-tuples exist in G(m, m, 4) iff  m = 2 or m = 4. In either case, λ is of order m. 
	
	
		Say we had a nice tuple in G(m, m, 4) such that r_1r_2r_3r_4 = A. 
		Suppose first that A corresponds to a three-cycle, then our nice 4-tuple is equivalent to one of the factorizations in Remark <ref>.  For all three factorizations, by the previous Lemma, we know that ζ_m^-b_1 +b_2 = λ and by Theorem <ref>, we know that δ(X) = |b_1 - b_2| must be coprime to m. This means λ is a primitive m-th root and by Lemma <ref> we know m = 4 or m = 2 in this case. 
		
		If A corresponds to a double transposition, then our nice 4-tuple is equivalent to one of the factorizations in Remark <ref> and we may write A = [a_1, a_2, a_3, a_4 | (12)(34)]. From the first factorization and the proof of Lemma <ref>, we see that ± 1 = ζ_m^a_1 + a_2 = ζ_m^b_3 -b_4 = λ^2. In this case δ(X) = |b_3 -b_4| which we know must be coprime to m by Theorem <ref>, we get that λ^2 is a primitive m-th root so m = 2. The same argument will also give that m must be 2 if our nice tuple is equivalent to the second factorization in Remark <ref>. For the third possible factorization, we see that the cycle has four edges and so δ(X) = |b_1 -b_3 +b_2 -b_4 since the cycle corresponds to the permutation (1324). But again we have that ± 1 = ζ_m^a_1 + a_2 = ζ_m^b_1 + -b_3 + b_2 -b_4 so λ^2 is a primitive m-th root so m = 2. 
			If m = 4, then we can take r_1 = s(1, 4; -1), r_2 = s(1, 4; 0), r_3 = s(1, 2; -1), r_4 = s(2, 3; 0). Then δ(X) = 1 so these reflections generate G(4, 4, 4). Further, we have 
    A = [  0  0 -1  0;  i  0  0  0;  0  1  0  0;  0  0  0  i ]
 which has eigenvalue i with multiplicity 2.	
		If m = 2, then we can take r_1 = s(1, 4; 1), r_2 = s(1, 4; 0), r_3 = s(1, 2; 0), r_4 = s(2, 3; 0) so again δ(X) = 1 and generates G(2, 2, 4). In this case 
    A = [  0  0 -1  0;  1  0  0  0;  0  1  0  0;  0  0  0 -1 ]
 which has eigenvalue -1 with multiplicity 2.
	
	Note that when taking the middle convolution for such tuples, the parameter should be λ^-1 since these equations are for the tuple product and not the inverse product.
	
		

 §.§ Nice 5-Tuples in G(m, p, 4)

	Searching for 5-tuples G(m, p, 4) so that the middle convolution will be rank 2 means we want the product to have an eigenvalue of multiplicity 3. A 4 × 4 matrix with an eigenvalue of multiplicity 3 can be viewed as a scalar times a reflection, so we want to explore the existence of [r_1, r_2, r_3, r_4, r_5] such that
	
    r_1r_2r_3r_4r_5 = λ r_6^-1
 where λ≠ 1 and r_6 is some reflection, not necessarily in G(m, p, 4). 
	
	
		If we have 5 reflections generating G = G(m , p, 4) such that 
		
    r_1r_2r_3r_4r_5 = λ r_6^-1
 
		then it must be that
		
			
  * r_6 ∈ G and G = G(m, m, 4) with order(λ) | m 
			
  * λ^4 = 1 
		
	
	
	
		Note first that  G(m, m, 4) ≤ G(m, p, 4) ≤ G(m, 1, 4) so r_i ∈ G(m, 1, 4) for i < 6 and so λ r_6^-1∈ G(m, 1, 4). Since this matrix has an eigenvalue of multiplicity 3, we know d = order(λ) is such that d divides 3 of the degrees in G(m, 1, 4) = {m, 2m , 3m, 4m}. This tells us that d | m as if it did not then d | 3m -2m = m. Since λ has order dividing m, λ I is in the center of G(m, 1, 4) and thus we can conclude r_6 ∈ G(m, 1, 4), though not yet necessarily in G(m, p, 4). This changes our problem into studying 6 reflections in G(m, 1, 4) satisfying
		
    r_1r_2r_3r_4r_5r_6 = λ I

		
		Next we use the homomorphism : G(m, 1, 4) ⟶ S_4. By Theorem <ref>, we know (r_1r_2r_3r_4r_5r_6) = (λ I) = e and since e is even, we know an even number of (r_i) must be transpositions.
		
		To conclude all 6 are transpositions, note that {r_i}_i < 6 are supposed to generate G(m, p, 4) which means {(r_i)}_i<6 must generate a transitive subgroup of S_4, hence S_4 itself. This removes the possibility of just 2 transpositions since we need at least 3 distinct transpositions to generate S_4.
		
		While it is possible to generate S_4 with 4 transpositions, our equation also requires that their product is the identity which is not possible with at least 3 distinct transpositions (as can be verified by hand). 
		
		Thus we know all 6 reflections must be Type 1 and so {r_i}_i < 6 can only generate G(m , m, 4), we conclude G = G(m , m ,4).
		
		Using that all reflections of Type 1 have nontrivial eigenvalue -1, taking determinant gives us that 
    1 = (-1)^ 6 = (r_1r_2r_3r_4r_5r_6) =  (λ I) = λ^4
 so we know lambda must be a 4th root of unity. Further, using that order(λ) ∈{2, 4 } must divide at least 3 degrees in G(m, m , 4) = {m, 2m, 3m, 4} we conclude that order(λ) | m. 
	
	
	 
		To study such tuples in G(m, m, 4) we first find the ways to decompose e in S_4 into a transitive product of 6 transpositions up to conjugacy. To make such a list, note that we can assume the first two transpositions are not disjoint, as if they are we can swap them around. This tells us their product is a 3-cycle and up to conjugation we may assume it is (132). We then found all the 4-tuples of transpositions that multiple to (123) up to simultaneous conjugation and generate all of S_4. Adjoining these options to the (12), (13) on the left, we get 27 options for transitive factorizations of the identity , and using computer code to partially compute the braid group orbit we were able to reduce the list further to two options (not necessarily distinct). 
		The tuples in S_4 are: 
		
			
  * [(12), (13), (12), (23), (24), (24)]
			
  * [(12), (13), (24), (24), (12), (23)]
		
		Using the Type 1 notation for the reflections in G(m, m, 4) as r_i =s(j_i, k_i; b_i) we can take the matrix product. The first tuple gives us the equation
		
    [               ζ_m^b_1 -b_3                          0                          0                          0;                          0 ζ_m^-b_1 +b_2-b_4 +b_5-b_6                          0                          0;                          0                          0         ζ_m^-b_2 +b_3 +b_4                          0;                          0                          0                          0              ζ_m^-b_5 +b_6 ]   = [ λ 0 0 0; 0 λ 0 0; 0 0 λ 0; 0 0 0 λ ]

		and the second tuple gives us the equation 
		
    [ ζ_m^b_1 +b_3 -b_4 -b_5                      0                      0                      0;                      0      ζ_m^-b_1 +b_2-b_6                      0                      0;                      0                      0     ζ_m^-b_2 +b_5 +b_6                      0;                      0                      0                      0          ζ_m^-b_3 +b_4 ]   = [ λ 0 0 0; 0 λ 0 0; 0 0 λ 0; 0 0 0 λ ]

	

	
		Nice 5-tuples exist in G(m, m, 4) if and only if m = 2 or m = 4 and in either case λ is of order m. 
	
	
		Suppose we had a nice 5-tuple in G(m,m, 4), so by the previous lemma we know 2 | m. We know it must be equivalent to one of the two factorizations in Remark <ref>. Considering the first factorization and looking at all transitive subsets of 4 reflections we can choose from the 6, we get the following possibilities for δ(X): 
		
    δ(X) = 
    				|b_1 -b_3| 
    
    				|b_6 -b_5| 
    
    				|b_4 +b_3-b_2| 
    
    				|b_4 +b_1 -b_2|

		The first three of these are such that ζ_m^δ(X) equals λ up to the absolute value on δ(X), so if any of those are coprime to m then λ is of order m. So the only way m ≠ 2, 4 is if the only option for δ(X) that is coprime to m is |b_4 +b_1-b_2|. 
		Writing m = 2^sk with (k, 2) =1 and s ≥ 1, we can write λ = ζ_m^k (s-j) for j = 1, 2 depending on the order of λ. The equations for b_i  modulo m then becomes: 
		
    b_1-b_3 ≡ k2^s-j
    
    			-b_1 + b_2-b_4+b_5 -b_6 ≡ k2^s-j
    
    			-b_2 +b_3 +b_4 ≡ k2^s-j
    
    			-b_5 +b_6 ≡ k2^s-j

		The second and fourth equation give us -b_1 +b_2-b_4 ≡ 2 k2^s-j which is not coprime to m. Thus, this option for δ(X) does not work so one of the first three options are coprime to m and hence λ is of order m. Since λ is a 4-th root of unity, we know m = 2 or m = 4. 
		
		A similar argument will give the same conclusion for the second factorization. 
		For m = 4, consider the tuple [s(1, 2; 1), s(1, 3; 0), s(1, 2; 0), s(2,3; 1), s(2, 4; 1)]. Then δ(X) = 1 and so these reflections will generate G(4, 4, 4). The product of this tuple will be i · s(2, 4; 2) as this corresponds to the first factorization.
		For m = 2, we can take the tuple [s(1, 2; 1), s(1, 3; 0), s(1, 2; 0), s(2,3; 1), s(2, 4; 0)]. Then δ(X) = 1 and so these reflections will generate G(2, 2, 4).  The product of this tuple will be -1 · s(2, 4; 1) as this corresponds to the first factorization.
	
		Note that with the examples above, we will have to take middle convolution with parameter λ^-1 since our equations are for the tuple product and not the inverse product.
	
	

 §.§ Nice n-Tuples in G(m, p, n) for n ≥ 5
 
	Recall that the only possible groups when nice n-tuples can exist are when p = 1 and p = m since these are the only well-generated imprimitive groups (See Definition <ref>).
	Fact: A connected graph of n vertices and n-1 edges must be a tree.
	
	
	 The product of n-1 transpositions is an n-cycle if and only if 
		the graph associated to a set of transposition in S_n is a tree, where the edge (a,b) is only present if (ab) is one of the n-1 transpositions. 
	
	
		By induction: if the product of n-1 transpositions is the product of a disjoint k-cycle and an (n-k)-cycle, then each of these correspond to  disjoint trees with k-1 and (n-k-1) edges respectively, so the remaining edge must connect the two trees, resulting in a tree with n-1 edges, hence the product is an n-cycle.
	
	
	
		There are no nice n-tuples in G(m, 1, n) for n ≥ 5. 
	
	
	
		In G(m, 1, n) any nice n-tuple consists of n-1 reflections of Type 1 and one reflection of Type 2, say r_n. Suppose we had r_1r_2⋯ r_n = A with [r_1, …, r_n] a nice tuple. Then by the claim(s) above, the graph associated to the transpositions {(r_i) }, with (r_n) = e must be a tree since the graph is connected by transitivity and therefore (A) is an n-cycle. Thus the characteristic polynomial 
    char_A(x) = (-1)^n (x^n - ζ)
 for some m-th root of unity ζ and therefore has n distinct roots so no A has no eigenvalues of multiplicity n-2 for n > 3.
	
	For G(m, m, n) we need the additional observations provided below.
	
	
		Let σ = (x_1x_2, · x_n) be an n-cycle in S_n and τ = (y_1y_2) be some transposition in S_n with y_1 <y_2. Letting c denote the distance between y_1 and y_2 in σ modulo the length of σ, we have that the product σ·τ is the product of a disjoint c-cycle and a disjoint (n-c)-cycle. 
	
	
		Since σ fixes no points, it is enough to show that multiplying with τ cannot fix two points in {1, …, n}. Letting y_1 = x_i and y_2 = x_j with i <j then we get 
    y_1  ↦ x_j+1↦ x_j+2↦…↦ x_i-1↦ x_i = y_1
 where the subscripts are read modulo n. The length of this cycle is precisely j-i  n. Doing a similar process for y_2 we get a cycle of length n -(j-i)  n. 
	
	Note that since σ is written in disjoint cycle notation the `distance' between, i.e. number of digits between y_1 and y_2 read in the order of multiplication (e.g. left to right if multiplication is left to right) is well defined. 
	
		In the notation of the claim above, if n > 2, then the product στ has at most one fixed point, with this occurring only if y_1 and y_2 are adjacent in the cycle σ. 
	
	
		If y_1 and y_2 are adjacent then c  = 1, so we are left with just an n-1 cycle which has exactly 1 fixed point when n-1 > 1 and 2 fixed points when n = 2. 
	
	
	
		There are no nice n-tuples in G(m, m, n) for n ≥ 5.
	
	
		In G(m, m, n), any nice tuple [r_1, r_2, …, r_n] consists of n reflections of Type 1. Again by transitivity, we know the graph associated to {(r_i)} is connected and since we only need n-1 edges to make a connected graph on n vertices, we know there is some transposition we can remove so that the associated graph is still connected. By cyclic permutation of the r_i via simultaneous conjugation, we can assume the redundant transposition corresponds to r_n, so (A · r_n^-1) is an n-cycle by the claim above. This tells us that (A) is the product of an n cycle and a transposition. 
		By the proofs above, we know (A) is either an (n-1)-cycle or the product of a c-cycle and an (n-c)-cycle. So either the characteristic polynomial is 
    char_A(x) = (-1)^n(x^n-1-ζ_1)(x-ζ_2)
 or 
    char_A(x) = (-1)^n (x^c -ζ_1)(x^n-c-ζ_2)

		We see that in both cases an eigenvalue of A can have atmost multiplicity 2, so when n > 4, there are no eigenvalues of multiplicity n-2, hence no such nice tuples can exist in G(m, m, n). 
	
	
	

§ PRIMITIVE REFLECTION GROUPS

	
	For each rank r ∈{3, 4}, we investigate the existence of nice T-tuples for T = r and T = r+1 in the primitive reflection groups, denoted G_i for i = 23, 24, …, 32. 
	For each group and each choice of T, we provide a table of the different types (up to inverse pairs), including the label for the type, the residues of the eigenvalues, as well as the nontrivial eigenvalues of the exemplar tuple under `Tuple Eigenvalues'.
	We also list the chosen exemplars 𝐀 = [A_1, A_2, …, A_T] for each type. For each exemplar we compute the middle convolution with the suitable choice(s) of parameter λ and this data is available in the appendices <ref>, <ref>, <ref>. These tables also include, for each choice of eigenvalue λ, the residue of λ in the `ξ' column as well as the notation used in Magma to identify the root of unity under `λ'.  We list the 2 × 2 matrices composing MC(𝐀) = [M_1, …, M_T] and record the order of the GL_2() subgroup generated by the these matrices under `S. Size'. Note that a zero in this column means the subgroup is infinite. 
	
	To get the induced tuple Â from MC_λ(𝐀) many choices of character will work, so we provide the one we use under the column `Character' or sometimes abbreviated as `Char.'. 
	In some cases, we may need to go to a larger field to construct a suitable character. We then compute and record the braid group orbit size of the induced tuple (under action of the positive powers of the braid group generators) and record it under `O. Size'. More information about how we compute the braid group orbit is provided in Section <ref>.
	For examples where the braid group orbits are the same size, we check if the signature of one lies in the orbit of the other.
	
		We say two braid group orbits are 𝐞𝐪𝐮𝐢𝐯𝐚𝐥𝐞𝐧𝐭 if the signature of one of the induced tuples lies in the orbit of another induced tuple.
	  
	This is one reason why our braid group code optionally can return the list of signatures in each orbit. 
	Note that this differs from the broader notion of equivalence in other literature such as Tykhyys's classification <cit.>. Note that the work of Lisovy-Tykhyy <cit.> and Tykhyy <cit.> only consider the case where matrices in the tuple do not have a shared eigenvector. In our case, since nice tuples give an irreducible representation and middle convolution preserves irreducibility, we know all our tuples cannot share a common eigenvector. Thus, we expect to identify which tuples in these classifications come from the complex reflection groups via middle convolution. Details on what parameters we use to compare our orbits with these classifications is provided below at the start of Section <ref> and <ref>.   
	
	For each distinct orbit we find, we provide data in the comparison tables on how our orbits relate to other classifications, as well as identify the finite SL_2() subgroups generated by the tuples. We list the subgroups of SL_2() in the Magma notation, given by order, number where order is the order of the finite group and number is the numbering of the group in the Small Groups Database. The numbering used by Magma is the same as that in the Small Group database implemented in GAP (see <https://magma.maths.usyd.edu.au/magma/handbook/text/779>, accessed on July 7th, 2024) and can be readily found online for all the groups we see. For infinite groups, the order is listed as `0', following the notation used in Magma. The notation SL(2, k) means 2× 2 matrices of determinant 1 over the finite field of _k.
	
		

 §.§ Nice 3-Tuples in Rank 3 Groups
 
	Nice tuples in this case will be tuples of 3 reflections generating the whole group such that their inverse product has some nontrivial eigenvalue of multiplicity 1. 
	After middle convolution, our the induced tuple will correspond to SL_2() local systems on the 4-punctured sphere and we can compare our braid group orbits to those in the literature. We compute the parameters (ω_X, ω_Y, ω_Z, ω_4) in the notation of <cit.> and found that the ω parameters for each of our orbits match (up to permutation) with one listed in the work of Lisovy-Tykhyy. Since Tykhyy's classification <cit.> further refines the original classification by Lisovy-Tykhy <cit.>, each tuple of ω parameters gives multiple orbits that Tykhyy identifies uniquely in terms of residues of the trace coordinates  (θ_1, θ_2, θ_3, θ_4) and (σ_12, σ_23, σ_13, σ_24) for some representative. In the nice 3-tuple case, using the ω parameters, we are able to easily identify how our orbit size relates the orbit size listed by Tykhyy for each ℳ^(4) (in Tykhyy's notation) and narrow down the possibilities for which of Tykhyy's orbits our tuple corresponds to. 
	In Tykhyy's classification,  tuples are considered equivalent up to:  multiplying any two matrices by -1, complex conjugation of all elements of all matrices, cyclically permuting the matrices in the tuple, and up to taking inverses and reversing the order of the tuple <cit.>. We perform all these operations and compute the residue signatures for the matrices in each orbit to find one that matches the example provided in Tykhyy's work. In most cases we found exact matches, and in other cases we provide the closest match, marking the differing coordinate with an asterisk (see Section <ref> and <ref>).   
	
	Note that we only list this data for orbits that are considered different by our definition and these may end up being the same orbit in the Lisovy-Tykhyy or Tykhyy classifications (<cit.>, <cit.>). 
	For each orbit we compute, we provide the identifying label used in both the classifications, as well as our orbit size and the orbit size they provided  (which is the same in both works). 
	
	
	

  §.§.§ G_23
 
	The group G_23 of order 120 is also denoted W(H_3) with degrees (G_23) = {2, 6, 10} <cit.>. All reflections in this group are conjugate with nontrivial eigenvalue -1. 
	For nice 3-tuples in G_23, the computation returned three possible sets of eigenvalues for the inverse product. 
	
	With q = ζ_5, we provide one example of a nice tuple such that the inverse product has eigenvalues corresponding to each type. 
	
	For Type A in G_23 one exemplar is:
	
    A_1    = [  1  0  0;  0  1  1;  0  0 -1 ]   A_2 = [            0           -1            0;           -1            0            0; -q^3-q^2 + 1   -q^3-q^2+1            1 ]   A_3 = [       1       0       0; q^3+q^2      -1      -1;       0       0      -1 ]

	Similarly for Type B, an exemplar is: 
	
    B_1 = [  1  0  0;  0  1  1;  0  0 -1 ]   B_2 =[        1 -q^3-q^2 -q^3-q^2;        0        0       -1;        0       -1        0 ]   B_3 = [       1       0       0; q^3+q^2      -1      -1;       0       0       1 ]
 	
	For Type C an exemplar is: 	
    C_1 = [  1  0  0;  0  1  1;  0  0 -1 ]   C_2 = [        1 -q^3-q^2        0;        0       -1        0;        0        1        1 ]   C_3 = [          0         -1          0;         -1          0          0; -q^3-q^3+1 -q^3-q^2+1          1 ]

	The data for all three types for G_23 is compiled in Table <ref>. For all three types, choosing a different exemplar would not have resulted in a new orbit. The two orbits of size 10 arising from Type A are the same, as are the two orbits of size 10 from Type B. However, these two orbits are distinct across the two types in the sense that their signatures did not show up in the others' orbit. For Type C, the two orbits of size 18 are the same. Information about how these 6 distinct orbits relate to the other classifications is provided in Table <ref>. In this group we also find nice 4-tuples, details provided in Section <ref>
	
	
		

  §.§.§ G_24
 
	The group G_24 of order 336 is also denoted W(J_3(4)) with degrees (G_24) = {4, 6, 14} <cit.>. All reflections in this group are conjugate with nontrivial eigenvalue -1. 
	Looking at the degrees, we first expand the base field to be the cyclotomic field of degree 84. There is one inverse pair of nice 3-tuples in this group, so we only provide the details for one in Table <ref>.
	
	
	Letting z = ζ_7, an exemplar of Type A is: 
	
    A_1 = [          1 -z^4-z^2-z -z^4-z^2-z;          0          0         -1;          0         -1          0 ]   A_2 = [  1  1  0;  0 -1  0;  0  1  1 ]   A_3 = [        -1         0         0;         1         1         0; z^4+z^2+z         0         1 ]

	The middle convolution data for G_24 is provided in Table <ref>. These three distinct orbits are the only orbits from Type A and we checked that in all three cases, the inverse of the exemplars do not produce any new orbits. The comparison data for G_24 is provided in <ref>.
	
	
	

  §.§.§ G_25
 
	The group G_25 of order 648 is also denoted W(L_3) with degrees (G_25) = {6, 9, 12} <cit.>. 
	G_25 has two conjugacy classes, consisting of matrices with nontrivial eigenvalue w or w^2 respectively, for w = ζ_3.
	Expanding the base field to be the cyclotomic field of order 36, we get 8 possible types, which actually split into 4 pairs of inverses. We only list one set of inverse eigenvalues for each pair in Table <ref>.
	
	
	With w = ζ_3 the exemplars for each type are provided below. 
	Exemplar of Type A: 
	
    A_1 = [   1   0   0;   0   1 w^2;   0   0   w ]   A_2 = [   w   0   0; w^2   1   0;   0   0   1 ]   A_3 = [   1   w   0;   0 w^2   0;   0   w   1 ]

	Exemplar of Type B: 
	
    B_1 = [   1   0   0;   0   1 w^2;   0   0   w ]   B_2 = [   -w    1  w^2;    1 -w^2    1;  w^2    1   -w ]   B_3 = [   1   w   0;   0 w^2   0;   0   w   1 ]

	Exemplar of Type C: 
	
    C_1 = [    1 -w^2    0;    0    w    0;    0 -w^2    1 ]   C_2 = [   w   0   0; w^2   1   0;   0   0   1 ]   C_3 = [    1    0    0; -w^2    w -w^2;    0    0    1 ]

	Note that for Type D, there is only two choices of eigenvalue to take midlde convolution with and only one of them will result in 2× 2 matrices. Exemplar of Type D: 
	
    D_1 = [    1 -w^2   -w;    0 -w^2    1;    0   -w    0 ]   D_2 = [   w   0   0; w^2   1   0;   0   0   1 ]   D_3 = [    0   -w  w^2;    0    1    0; -w^2   -1 -w^2 ]

	The details for the middle convolution of each type is provided in Table <ref>. For all three types, different exemplars do not produce new orbits nor do their inverses. For Type A, the two orbits of size 12 are distinct, as are all three orbits of size 36 from Type B. For Type C, we had to extend to a field containing ζ_72 to construct the character. The two orbits of size 4 from Type C are the same. Including the one from Type D, we get 8 distinct orbits from G_25. The comparison data is provided in Table <ref>. Comparing with the Lisovy-Tykhyy paper, all the ω parameters seemed to correspond to the family of orbits listed in Lemma 39 in <cit.>, so we have provided the specific values for the variables that work in each case. When comparing with Tykhyy's classification, there was only one orbit (from Type B) that we could not find an exact match for. We listed the closest match, which seemed to be Orbit 5 in <cit.> and distinguish the different coordinate with an asterisk. 
	In this group we also find nice 4-tuples, details provided in Section <ref>
	
	
	
	

  §.§.§ G_26
 
	The group G_26 of order 1296 is also denoted W(M_3) with degrees (G_26) = {6, 12, 18} <cit.>.
	In G_26 there are three conjugacy classes of reflections, corresponding to nontrivial eigenvalues in {ζ_3, ζ_3^2, -1}. In this group the are 6 sets of eigenvalues for the inverse product, which correspond to 3 inverse pairs, so we only list one from each pair.  
	
	The exemplars for this group are all provided over (ζ_36) with z = ζ_36.
	
	Here is an exemplar of Type A:
	
    A_1 = [       1       0       0;      -1  2· z^6     z^6;  -z^6+2 -3· z^6    -z^6 ]   A_2 = [      0    z^6      0;      1 -z^6+1      0;  z^6-2  z^6+1      1 ]   A_3 = [  1  0  0;  0  1  1;  0  0 -1 ]

	An exemplar for Type B: 
	
    B_1 = [  z^6-1 2· z^6    z^6;      0      1      0;      0      0      1 ]   	B_2 = [      0  z^6-1  z^6-1;    z^6      2      1; -z^6-1  z^6-2  z^6-1 ]   B_3 = [  1  0  0;  0  1  1;  0  0 -1 ]

	An exemplar for Type C: 
	
    C_1= [          1          0          0;       -z^6      z^6-1      z^6-1;      z^6+1 -3·(z^6+1) -2·(z^6-1) ]   C_2 = [      1 -z^6+1 -z^6+1;      0      1      0;      0 -z^6-1   -z^6 ]   C_3 = [  1  0  0;  0  1  1;  0  0 -1 ]

	
	In G_26 choice of different exemplars nor the inverses for any type produce new orbits. The middle convolution data for this group can be found in Table 
	<ref>. Further, all three orbits from Type A are distinct, as are all three orbits from Type B. When comparing with Tykhyy's classification, there was only one orbit (from Type B) that we could not find an exact match for. We listed the closest match, which seemed to be Orbit 5 in <cit.> and distinguish the differing coordinate with an asterisk. 
	
	In this group we also find nice 4-tuples, details provided in Section <ref>
	
		 
			λ     ω_X, Y, Z, 4      Lisovy-Tykhyy #     θ_1, 2, 3, 4     σ_12, 23, 13, 24       -90 Tykhyy #       -90 Their O. Size      -90 Our O. Size       -90 SL_2() Subgroup 
  [0pt]0pt0ex
 [0ex] 
			9|c|G_26 Type A Comparisons [0pt]0pt3ex
 
			z^3     (0, √(3), 0, 0)    Type II, X' = 0, X” = √(3)     ( 5/12, 1/4, 5/12, 3/4)     (1/2, 1/2, 5/6, 1/2)      4      2     24     0 [0pt]0pt4ex 
 [1ex] 
			-z^3     (0, -√(3), 0, 0)     Type II, X' = 0, X” = -√(3)     (1/4, 1/12, 1/4, 11/12)     (1/2, 1/2, 1/2 , 5/6)     4       2     24     0 [0pt]0pt4ex
 [1ex] -z^6+1     (0, -z^9+2z^3, 0, 0)     Type II, X' = 0, X” = -z^9+2z^3     (1/6, 1/2, 2/3, 1/2)     (1/2, 1/2, , 5/6, 1/2 )     3      2     24     24, 4 [0pt]0pt4ex
 [1ex] 9|c|G_26 Type B Comparisons [0pt]0pt3ex
 
			z^10     (-z^9+2z^3, -z^9+2z^3, 2, -1)     Type III, ω = -z^9+2z^3     (7/9,7/18, 7/18, 2/3)     ( 1/2, 1/3, 1/2, 5/6^*)     5^*     3     36     0 [0ex]0ex4ex 
 [1ex]
			-z^4     (z^9-2z^3, z^9-2z^3, 2, -1)     Type III, ω = z^9-2z^3     (1/9,1/18, 1/18, 2/3)     (1/2, 1/3, 1/2, 1/6)     5     3     36     0 [0ex]0ex4ex 
 [1ex]
			-z^16     (-z^9+2z^3, -z^9+2z^3, 2, -1)     Type III, ω = -z^9+2z^3     (5/9, 5/18, 5/18, 2/3)     (1/2, 1/3, 1/2, 1/6)     5     3     36     0 [0ex]0ex4ex 
 [1ex]
			9|c|G_26 Type C Comparisons [0pt]0pt3ex
 
			-z^6+1     (0, 0, 2, -1)     Type III, ω = 0     (1, 1/2, 1/2, 2/3)     ( 1/2, 1/3, 1/2, 1/2)     5     3     12     12, 1 [0ex]0ex4ex 
 [1ex]   Here w = ζ_3, z = ζ_36. The correct signature for Tykhyy's orbit 5 would have 3(7/18) ≡1/6 as σ_24 but such a signature was not in the orbit so we provide the closest match. The subgroups 24, 4 and 12, 1 are the dicyclic groups of order 24 and 12 respectively.
	
	

  §.§.§ G_27
	
	This group is of order 2160 and is also denoted W(J_3(5)).
	This group only has one conjugacy class of reflections with nontrivial eigenvalue -1 and (G_27) = {6, 12, 30}. The original base ring of this group in Magma is the cyclotomic field of order 15, and we expanded the base ring to the cyclotomic filed of order 60 to ensure all the eigenvalues are found. We found three distinct inverse pair types in this group.
	
	
	Writing z = ζ_60 and setting u = -z^10-z^8+z^2 and v = z^16 + z^4, we get that z^50v = -z^14+ z^6+ z^4 so we can write the exemplar for Type A as: 
	
    A_1 = [  z^50v      0   z^20;   -v+1      1    v+1;      v      0 -z^50v ]   A_2 = [        0        0    -z^20; u+z^10+2        1   v+z^10;     z^10        0        0 ]   A_3 = [  1  1  0;  0 -1  0;  0  1  1 ]

	Using the same notation, we can write the Type B exemplar as: 
	
    B_1 = [   -1    u    u;  v+1   -u -u-1; -v-1  u+1  u+2 ]   B_2 = [  z^50v      0   z^20;   -v+1      1    v+1;      v      0 -z^50v ]   B_3 = [  1  1  0;  0 -1  0;  0  1  1 ]

	Similarly, the Type C exemplar is:
	
    C_1 = [  z^50v      0   z^20;   -v+1      1    v+1;      v      0 -z^50v ]   C_2= [  1  1  0;  0 -1  0;  0  1  1 ]    C_3 = [    1    0    0; -v-1  u+1  u+2;  v+1   -u -u-1 ]

	In G_27 for all three types, different exemplars do not produce new orbits. Moreover the inverse types do not produce new orbits and we have checked that these 9 orbits are distinct. The middle convolution details for Type A, B and C are listed in Table <ref>, <ref>, and <ref> respectively.  The comparison data is provided in Table <ref>. In this group we also find nice 4-tuples, details provided in Section <ref>
	
	
	

 §.§ Nice 4-Tuples in Rank 3 Groups

	Nice tuples in this group are 4-tuples of reflections generating the whole group such that the inverse product has a nontrivial eigenvalye of multiplicity 2. If (ζ) denotes the trace field for some FCRG W, and we have a matrix A with eigenvalues {λ, λ, λ_1}, then we know (A) = 2λ + λ_1 ∈(ζ). If we further knew that λ∈(ζ) then we can conclude that λ_1 ∈(ζ) and so we do not need to extend the base field to ensure all characteristic polynomials of such matrices A split. By Theorem <ref> we know that any such λ of multiplicity 2 must have order dividing at least two of the degrees of W. Studying degrees, we see that in G_23 and G_24, λ must have order dividing 2, i.e. λ = -1 and so we do not need to expand the base ring. Similarly, for G_25, G_26, G_27 such  λ must have order dividing 6, which means we do not have to expand the base ring in these cases either. 
	Our search found nice 4-tuples in all the primitive rank 3 groups except for G_24. 
	
	In comparing with the existing classifications, we can only compare with Tykhyy's ℳ^(5) orbits <cit.> since our 4-tuples gives us local systems on the 5-punctured sphere. Unfortunately, without the ω parameters from the work of Lisovy-Tykhyy <cit.> like in the 3-tuple cases, it is more difficult to relate the our orbit sizes to Tykhyy's. We still computed all the signatures for the different possible equivalent representatives for each orbit and attempted to find a match, which was successful. 
	
	

  §.§.§ G_23
 
	We only found one type of nice tuple. 
	

	Letting q = ζ_5, here is an exemplar for Type A: 
	
    A_1 = [    q^3+q^2    q^3+q^2    q^3+q^2; -q^3-q^2+1 -q^3-q^2+1   -q^3-q^2;    q^3+q^2         -1          0 ]     A_2 = [ -q^3-q^2+1 -q^3-q^2+1   -q^3-q^2;    q^3+q^2    q^3+q^2    q^3+q^2;         -1    q^3+q^2          0 ]
     A_3 = [   -q^3-q^2          1          0;    q^3+q^2    q^3+q^2          0;   -q^3-q^2 -q^3-q^2+1          1 ]      A_4 = [       -1        0        0; -q^3-q^2        1        0;        0        0        1 ]

	The middle convolution details can be found in Table <ref>. This type is its own inverse. 
	For this tuple we could not compute the orbit size as it was too large, but we were able to find a signature in the orbit (up to Tykhyy's equivalence relations) that agreed with one listed in Tykhyy's paper. Because we cannot compute the full braid group orbit, we also cannot check if different exemplars would result in a different orbit after middle convolution. We provide the comparison details in Table <ref>. 
	
	
	

  §.§.§ G_25

	We found 6 different types which split into 3 inverse pairs, so we only provide the details for one type from each pair in Table <ref>. 
	
	
	For all the exemplars listed below, z = ζ_3. 
	Here is an exemplar for Type A: 
	
    A_1 = [   z   0   0; z^2   1   0;   0   0   1 ]   A_2 = [  -z   1  -z;   0   1   0;   z z^2   0 ]   A_3 = [   1   z   0;   0 z^2   0;   0   z   1 ]   A_4 = [  -z   1 z^2;   1 z+1   1; z^2   1  -z ]

	Here is an exemplar for Type B: 
	
    B_1 = [  -z   1   0; z+1   0   0;   1   z   1 ]   B_2 = [  -z   1  -z;   0   1   0;   z z^2   0 ]   B_3 = [   0 z^2   z;   0   1   0;  -z   1  -z ]   B_4 = [   1   z   0;   0 z^2   0;   0   z   1 ]

	Here is an exemplar for Type C:
	
    C_1 = [   1   z   1;   0   0 z+1;   0   1  -z ]   C_2 = [   z   0   0; z^2   1   0;   0   0   1 ]   C_3 = [ z+1  -1 z+1;   0   1   0; z^2  -z   0 ]   C_4 = [  -z   1   0; z+1   0   0;   1   z   1 ]

	The middle convolution details for G_25 can be found in Table <ref>.
	The orbits for Type A and Type B are the same. All three types do not split into new orbits and their inverses do not produce new orbits. The comparison details for the two distinct orbits can be found in Table <ref>.
	
	

  §.§.§ G_26

	In this group we found two inverse pairs of nice tuples and we provide the details for one type from each pair in Table <ref>. 
	
	With the notation z = ζ_3, here is an examplar for Type A: 
	
    A_1 = [     z   z+2     1;     0     1     0;   z+2 z^2-z    -z ]   A_2 = [   z   0   0; z^2   1   0;   0   0   1 ]   A_3 = [ -z+1 2z^2  z^2;  z^2   -1   -1;    0    0    1 ]   A_4 = [    -z    -1    -1;     z   z+2   z+1; z^2-z z^2-1   z^2 ]

	The exemplar for Type B: 
	
    B_1 = [    1  z^2    0;    0    z    0;    0 -z+1    1 ]   B_2 = [   -z   -1    0;  z^2    0    0; 2z+1  z+2    1 ]   B_3 = [   z   0   0; z+1   1   0;   0   0   1 ]   B_4 = [     1 z^2-z    -z;     0   z^2   z^2;     0  2z+1   z+1 ]

	The middle convolution details for G_26 can be found in Table <ref>. 
	Neither of these types split into multiple orbits nor do their inverse types produce new orbits. The finite group generated by the matrices in SL_2() are the same for both groups, Magma identifies this as SmallGroup<24, 4> which corresponds to the dicyclic group of order 24.
	
	
	

  §.§.§ G_27
 
	In this group we found one inverse pair, and we provide the details for one type of the pair. 
	
	
	Here is an exemplar of Type A, where z = ζ_15, u = z^4 + z, v = z^7-z^3+z^2-1: 
	
    A_1    = [    u  u-1   -1; -u-1    u -v-u;    0    0    1 ]   A_2 = [ -1  0  0;  1  1  0; -u  0  1 ]   A_3 = [          u+2v        v+z^10 -u+z^5(z^5-1);             0             1             0;          -u-1        -v-u-1         -2v-u ]
     A_4    = [ -v-u-z^5   -v-z^5      u+1;  v+u+z^5        v  -u+z^10;   -v-u+1  -v-z^10   u-z^10 ]

	
	The middle convolution details can be found in Table <ref>. In this case we could not compute the orbit size since it was too long. Similar to G_23, we were able to compute some signatures for tuples in the orbit and we found a match with a signature listed in Tykhyy's classification.
	
	
	

 §.§ Nice 4-Tuples in Rank 4 Groups

	In this case we are looking for nice tuples of 4 reflections generating the whole group such that their inverse product has a nontrivial eigenvalue of T-2 = 2. Of the five rank 4 reflection groups, we found nice 4-tuples in G_28, G_30 and G_32. In the case of G_32, the field containing all the eigenvalues would be (ζ_360) which is much too large to perform the computations over. For G_32, in Section <ref> we justify why it is enough to search for such tuples over (ζ_12). 
	
	After taking the middle convolution, multiplying with a suitable character and adjoining the inverse product, our tuples become SL_2() local systems on the 5-punctured sphere. We found that in some cases, one of the 5 matrices in the local system ends up being the identity, which means our tuple comes from some local system on the 4-punctured sphere. In the case where we have 4 nontrivial matrices, we can compute the ω parameters in the notation of Lisovy-Tykhyy <cit.> and identify the corresponding ℳ^(4) orbit in Tykhyy's paper <cit.>. When all 5 of the matrices are nontrivial, we can only compare with Tykhyy's ℳ^(5) orbit signatures from <cit.>.
	
	

  §.§.§ G_28
 
	The group G_28 of order 1152 is also denoted W(F_4) with degrees (G_28) = {2, 6, 8, 12} <cit.>. 
	All reflections in G_28 have nontrivial eigenvalue -1, and form two conjugacy classes of reflections. We only found one type of nice 4-tuples in this group, with two possible choices of λ parameters for middle convolution. 
	
	An exemplar for such a tuple of Type A is: 
	
    A_1 = [  2  3  4  2; -1 -2 -4 -2;  0  0  1  0;  0  0  0  1 ]   A_2 = [  1  2  2  0;  0 -1 -2  0;  0  0  1  0;  0  1  1  1 ]   A_3 = [ -1 -2 -4 -2;  2  3  4  2; -1 -1 -1 -1;  0  0  0  1 ]   A_4 = [ -1  0  0  0;  1  1  0  0;  0  0  1  0;  0  0  0  1 ]
	
	A different choice of exemplar does not produce a new orbit and the two orbits computed above are the same, so we only get one new orbit from this group. The middle convolution details can be found in Table <ref>. After putting the matrices in SL_2(), we see that the inverse product of the middle convolution of this tuple is the identity, so this tuple comes from a local system on the 4-punctured sphere. As in the case of 3-tuples, we provide the details comparing with the results of both Lisovy-Tykhyy and Tykhyy (<cit.>, <cit.>) in Table <ref>. The ω parameters appear to correspond to Orbit 31, but in fact this orbit is of Type III, as described in Lemma 39 in <cit.>.
	

	
	

  §.§.§ G_30
 
	The group G_30 of order 14400 is also denoted W(H_4) with degrees (G_30) = {2, 12, 20, 30} <cit.>.
	All reflections in G_30 have nontrivial eigenvalue -1 and are conjugate to each other. Looking at the degrees, we extend the base field to (ζ_60). We found three distinct types of nice 4-tuples, the details are in Table <ref>.
	
	For all of the exemplars below z = ζ_60 and u = z^14-z^6-z^4. 
	An exemplar of type A is given by: 
	
    A_1 = [  1  0  0  0;  0  1  0  0;  0  0  1  1;  0  0  0 -1 ]   A_2 = [     1     0     0     0;  2u-1  2u-1   u-1    -1; -2u+1 -2u+2  -u+2     1;  -u+1   -2u    -u    -u ]
     A_3 = [     u   u-1   u-1    -1; -2u+1 -2u+2 -2u+1    -u;  2u-1  2u-1    2u     u;    -u    -u    -u    -u ]   A_4 = [ -u+1 -u+1 -u+1   -u;    0    1    0    0;    u  u-1    u    u;   -1    u    u    0 ]
	
	An exemplar of type B is given by:
	
    B_1 = [  1  0  0  0;  0  1  0  0;  0  0  1  1;  0  0  0 -1 ]   B_2 = [    0   -1   -1   -1;   -u -u+1   -u   -u;    0    0    1    0;  u-1  u-1  u-1    u ]
     B_3 = [  1 -u -u -u;  0  0 -1 -1;  0  0  1  0;  0 -1 -1  0 ]   B_4 = [ -2u+2 -3u+2 -2u+1  -u+1;  2u-1  3u-1  2u-1   u-1;    -u  -u+1  -u+1     1;     u   u-1     u     0 ]

	An exemplar of type C is given by: 
	
    C_1 = [  1  0  0  0;  0  1  0  0;  0  0  1  1;  0  0  0 -1 ]   C_2 = [     1     0     0     0;  -u+1 -2u+1  -u+1    -u;   u-1    2u     u     u;  2u-1  2u-2  2u-1     u ]
     C_3 = [   u-1  2u-1    2u     u; -3u+1 -3u+3 -2u+2  -u+1;  3u-1  3u-2  2u-1   u-1;     0     0     0     1 ]   C_4 = [    0   -1   -1   -1;   -u -u+1   -u   -u;    0    0    1    0;  u-1  u-1  u-1    u ]

	The middle convolution details for each type is presented in Table <ref>. For all three types, different exemplars do not produce new orbits and since each type is their own inverse type, these are all the orbits from this group.  Further, the two orbits produced from Type A are distinct, as are the two from Type C. The two orbits from Type B are actually the same. In total G_30 gives us 5 different orbits. 
	
	For all 5 orbits, once we put the matrices in SL_2(), we see that the last matrix is the identity, hence these actually corresponds to orbits of 4 matrices multiplying to the identity. We compared these to the existing classification(s) by Lisovy-Tykhyy and Tykhyy (<cit.>, <cit.>) as we did for the rank 3 complex reflection groups. Within each type, the two orbits are equivalent to the same orbit of ℳ^(4) in Tykhyy's classification, so we only provide the details once in Table <ref>.
	
	

  §.§.§ G_32
 
	The group G_32 of order 155520 is also denoted W(L_4) <cit.>.
	In this group, the nontrivial eigenvalues of reflections are in {ζ_3, ζ_3^2 } corresponding to two conjugacy classes. 
	For G_32, (G_32) = {12, 18, 24, 30} so the eigenvalues of multiplicity 2 must be 12-th roots of unity. Expanding the field to (ζ_12) we found 5 distinct eigenvalue sets for the inverse product. Note that to ensure we found all examples, we would have to run the code over (ζ_360) since there may be matrices with one or two eigenvalues other than the 12-th roots which would not be found, however we claim it is enough to check over (ζ_12). 
	
		If A = A_1A_2A_3A_4, with [A_1, A_2, A_3, A_4] a nice tuple in G_32 so that A has a nontrivial eigenvalue of multiplicity 2, then the other two eigenvalues of A are both m-th roots of unity (not necessarily primitive) for the same m ∈{12, 18, 24, 30}.
	
	
		Let the eigenvalues of A be {λ, λ, λ_1, λ_2}. Taking residues, we write 
    Res(λ) = x/12,    Res(λ_1) = p_1/u_1· 6,    Res(λ_2) = p_2/u_2 · 6
 where u_i ∈{2, 3, 4, 5} which we know must be the case by the degrees of G_32. Suppose λ_i are not 24-th roots of unity. So u_i must be 3 or 5 and (p_i, u_i) = 1. 
		
		Using that (A_i) is a 3-rd root of unity, we know that (A) ∈{1, ζ_3, ζ_3^2}. This tells us that λ_1λ_2 = (λ^2)^-1(A) and so λ_1λ_2 must be a 6-th root of unity since λ^2 is a 6-th root.  
		In terms of the residues, this means 
		
    p_1/u_1 · 6 + p_2/u_2 · 6 = u_2p_1 + u_1p_2/6 · u_1u_2 = y/6
 for some 1≤ y ≤ 6.
		
		This equation tells us that u_1 | u_2 p_1 and since (p_1, u_1) = 1 we know u_1 | u_2. Similar arguemnt shows u_2 | u_1, so u_1 = u_2 if one of them is 3 or 5.
		
		Suppose now that Res(λ_1) = p_1/4 · 6, is a primitive 24-th root of unity, i.e. u_1 = 4, so (p_1, 24) = 1. We know u_2 is not 3 or 5, so it can only be 2 or 4. If u_2 = 4 and (p_2, 24) = 1 then we are done since they are both primitive 24th roots. Else λ_2 is a 12-th root of unity, which means λ_1 = (λ^2)^-1(A) λ_2^-1. Since the right hand side is a product of 12th roots, λ_1 must also be a 12th root, completing the proof.
	
	In the notation of the lemma above, we know that for nice 4-tuples, if the eigenvalues of the inverse product A is {λ, λ, λ_1, λ_2} then λ_1 + λ_2 = -2λ + (A) ∈(ζ_12) and similarly λ_1λ_2 = λ^2 (A) ∈(ζ_12). 
	With the result from the lemma, we do computer search in each of the fields (ζ_18), (ζ_24), (ζ_30) to find a pair of roots of unity (s, t) other than 12-th roots such that s +t, s· t ∈(ζ_12). For (ζ_18) and (ζ_30) no such pair could be found. In (ζ_24) one example that works is the pair (ζ_24, -ζ_24), however no pairs are found if we further require that s ≠ -t. This tells us the only cases we might be missing when searching over (ζ_12) are tuples whose inverse product has eigenvalues {λ, λ, λ_1, -λ_1} with λ a 12th-root (not nec. primitive) and λ_1 a primitive 24th root. In the notation of the lemma above, using that (A) is a 3rd root, we know -λ^2 ·λ_1^2 is also a 3rd root. Note that if λ_1 is a primitive 24th-root then -λ_1^2 is a primitive 12th-root. Again using a computer (or by hand) we can check that no primitive 12th root multiplied with the square of a 12th-root can be a 3rd root. Thus, we conclude it is enough to search for tuples over (ζ_12).
	
	Searching for nice tuples of in G_32 we find 3 inverse pairs. We list only one from each pair, noting that Type B is its own inverse.
	
	
	For all the exemplars listed below, w = ζ_3. 
	An exemplar of type A is given by: 
	
    A_1 = [    1    0    0    0; -w^2    w  w+2    w;   -w  w+2  -2w    1;    0    0    0    1 ]   A_2 = [   1   0   0   0;   0   1 w^2   0;   0   0   w   0;   0   0 w^2   1 ]
     A_3 = [    1    0    0    0; -w^2    w -w^2    0;    0    0    1    0;  w^2 -w+1  w^2    1 ]   A_4 = [      0      w  w^2-1     -w;   -w^2      2 -w+w^2      1;      w   -w^2      w    w^2;      0      0      0      1 ]
	
	An exemplar of type B is given by: 
	
    B_1 = [  -w   1  -w   0;   0   1   0   0;   w w^2   0   0; w^2   1  -w   1 ]   B_2 = [   1   0   0   0;   0   1 w^2   0;   0   0   w   0;   0   0 w^2   1 ]
     B_3 = [    1    w -w^2    w;    0   -w   -1  w^2;    0   -1 -w^2   -1;    0  w^2   -1   -w ]   B_4 = [    0   -w  w^2   -w;    0    1    0    0;    1    w  w+2    w; -w^2   -1    w    0 ]
	
	An exemplar of type C is given by: 
	
    C_1 = [      1      0      0      0;      0      1      0      0;     -1 -w+w^2     -1    w^2;    w^2   -w+1   2w^2   -w+1 ]   C_2 = [      0    w^2    w-1    w^2;     -w      2 -w+w^2      1;    w^2     -w    w^2     -w;      0      0      0      1 ]
     C_3 = [    1    w    1 -w^2;    0    0 -w^2    w;    0    0    1    0;    0   -w   -1   -w ]   C_4 = [   1   w   0   0;   0 w^2   0   0;   0   w   1   0;   0   0   0   1 ]
	
	
	The middle convolution details can be found in Table <ref>. Different exemplars nor the inverse types produce new orbits. The two orbits from Type B are distinct as are the two orbits from Type C.
	Of the five middle convolutions computed in G_32, after putting the matrices into SL_2() three of them (one from Type B and both orbits from Type C) produce tuples where the 5-th matrix is the identity. For these three, we compare with the classification(s) by Lisovy-Tykhyy and Tykhyy (<cit.>, <cit.>) to see which ℳ^(4) orbits they correspond. The orbits from Type C correspond to the same orbit in Tykhyy's classification so we only provide the details for it once. 
	The tuples that correspond to nontrivial local systems on the 5-punctured sphere are those coming from Type A and the other middle convolution from Type B. For the Type B tuple, after putting the matrices into SL_2(), we get that the product of the third and fourth matrix is not of finite order, hence we cannot compute its signature to compare with Tykhyy's table of ℳ^(5).
	The matrices in this tuple MC_-z^2+1(𝐁) are: 
	
    B̂_̂1̂   = [  z^3    z;    0 -z^3 ], B̂_̂2̂ = [  z^3-z      0; -z^3+z     -z ],
    		B̂_̂3̂ = [ -z^3   -z;    0  z^3 ],
    		B̂_̂4̂ = [      z      0;     -z -z^3+z ],
    		B̂_̂5̂ = [ z^2-1     0;     0  -z^2 ]

	
	
		
	
	
	

 §.§ Nice 5-Tuples in Rank 4 Groups

	In this case we are searching for tuples of 5 reflections generating the group such that their inverse product has a nontrivial eigenvalue of multiplicity 3. Searching all the rank 4 groups, we found that such tuples only existed in G_32. 
	

  §.§.§ G_32

	Since (G_32) = {12, 18, 24, 30}, we know that any eigenvalue of multiplicity three must be a 6-th root of unity. Using the trace, we then get that the 4-th eigenvalue must also be a 6-th root, so we do not need to expand the field to compute all the examples. 
	In G_32, there were too many possible tuples to perform an exhaustive search for all the nice tuples. We were able to find at least 4 types, which correspond to two inverse pairs, so we only provide the details for one type from each pair. 
	
	
	An exemplar of Type A is provided below with w = ζ_3: 
	
    A_1    = [    1    0    0    0;  w+1    w  w+1    0;    0    0    1    0; -w-1 -w+1 -w-1    1 ]   A_2 = [    1    0    0    0;    0    1 -w-1   -w;    0    0  w+1   -1;    0    0    w    0 ]   A_3 = [   1   w   1 w+1;   0   0 w+1   w;   0   0   1   0;   0  -w  -1  -w ]
     A_4    = [    w    0    0    0; -w-1    1    0    0;    0    0    1    0;    0    0    0    1 ]   A_5 = [    1    0    0    0;    1 2w+2 -w+1  w+1; -w-1 -w+1 -w-1   -w;    0    0    0    1 ]

	An exemplar of Type A is provided below with w = ζ_3:
	
    B_1    = [    1    w    1    w;    0    0  w+1   -1;    0  w+1 -w+1  w+1;    0   -1  w+1    0 ]   B_2 = [    w    0    0    0; -w-1    1    0    0;    0    0    1    0;    0    0    0    0 ]   B_3 = [   1   w   1   0;   0   0 w+1   0;   0   1  -w   0;   0 w+1  -w   1 ]
    
    		B_4    = [    1    0    0    0;    0    1 -w-1    0;    0    0    w    0;    0    0 -w-1    1 ]   B_5 = [    1  w+1   -w    1;    0  w+1    1  w+1;    0    0    1    0;    0 -w-1    w    0 ]

	The details of the middle convolution for both types is presented in Table <ref>. The SL_2() subgroups generated by both tuple types is the dicyclic group of order 24. Inspecting the 6 SL_2() matrices we get for both of these types, both tuples satisfy the description of `TYPE' A of ℳ^(6) belonging to the dihedral group in Tykhyy's classification <cit.>. In these cases we also could not check if different examplars/ inverses produce different orbits as we could not generate the list of all nice tuples. 
	plain
	 
	
	The notation for all the appendices are the same.
	The `ξ' column gives the residue of the root of unity in the `λ' column used to compute middle convolution. The `Character' column is the character we tensor our tuple with to get matrices in SL_2(). The column `O. Size' is the size of the SL_2() character variety orbit of the tuple (i.e after it is tensored with the character). In these tables `S. Size' denotes the size of the GL_2() subgroup generated by the 2 × 2  matrices  we get after middle convolution but before we tensor with the character. The SL_2() subgroup size after tensoring with the character is provided in each section for each group as part of the comparison tables. For infinite subgroups, the order is listed as `0', following the notation used in Magma.
	

§ MIDDLE CONVOLUTION TABLES FOR NICE 3-TUPLES
 
	

		 
			8|c|G_23 Middle Convolutions	[0pt]0pt3ex

			ξ     λ     M_1     M_2     M_3     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
			8|c|G_23 Type A Middle Convolutions [0pt]0pt3ex
 
			1/2     -1     [            1            0; -q^3-q^2 + 1            1 ]     [  1 -1;  0  1 ]     [ 1 0; 1 1 ]     (1, 1, 1, 1)     40     0 [0pt]0pt4ex 
 
			1/10     -q^3     [  q^3    0; -q^3    1 ]     [ q^3  -1;   0   1 ]     [   1   0; q^3 q^3 ]      (q, q, q, q^2)     10     600 [0pt]0pt4ex
 9/10     -q^2     [  q^2    0; -q^2    1 ]     [ q^2  -1;   0   1 ]     [   1   0; q^2 q^2 ]     (q^4, q^4, q^4, q^3)     10     600 [0pt]0pt4ex
 8|c| G_23 Type B Middle Convolutions [0pt]0pt3ex
 
			1/2     -1     [ 1 0; 1 1 ]     [  1 -1;  0  1 ]     [         1         0; q^3+q^2+2         1 ]     (1, 1, 1, 1)     40     0 [0pt]0pt4ex 
 
			3/10     -q^4     [        q^4          0; -q^4+q^3+1          1 ]     [ q^4  -1;   0   1 ]     [         1         0; q^4-q^3-1       q^4 ]     (q^3, q^3, q^3, q)     10     600 [0pt]0pt4ex
 7/10     -q     [       q       0; q^2-q+1       1 ]     [  q -1;  0  1 ]     [        1        0; -q^2+q-1        q ]     (q^2, q^2, q^2, q^4)     10     600 [0pt]0pt4ex
 8|c| G_23 Type C Middle Convolutions [0pt]0pt3ex
 
			1/2     -1     [  u-1 3u-5;    1 -u+3 ]     [  1 -1;  0  1 ]     [   1   0; u+1   1 ]     (1, 1, 1, 1)     72     0 [0pt]0pt4ex 
 
			1/6     z^5     [ -u-v-z^10      -z^5; u+2v+z^10       u+v ]     [ -z^5   -1;    0    1 ]     [        1        0; -u-v-z^5     -z^5 ]     -(z^5, z^5, z^5, -1)     18     360 [0pt]0pt4ex
 5/6     -z^10     [     z^5+v      z^10; -u-2v-z^5         v ]     [ z^10   -1;    0    1 ]     [      1      0; v+z^10   z^10 ]     (z^10, z^10, z^10, 1)     18     360 [0pt]0pt4ex
  tableHere q = ζ_5, z = ζ_30, u = z^7-z^3-z^2+1 and v = -z^4+z
	

		

			
				8|c|G_24 Middle Convolutions
				[0pt]0pt3ex
 ξ     λ     M_1     M_2     M_3     Character     O.Size     S. Size [0pt]0pt3ex
 [0.5ex]
				8|c|G_24 Type A Middle Convolutions [0pt]0pt3ex
 
				1/14     -z^4     [    z^5+z^4+z+1 -z^5+z^4+z^3+1;           -z^2         -z^5-z ]     [ z^4   1;   0   1 ]     [    1    0; -z^4  z^4 ]     (z^5, z^5, z^5, z^6)     28     0 [0pt]0pt4ex
 9/14     -z     [ -z^6-z^5-z^4    z^6-z^3+1;         -z^4     -z^3-z^2 ]     [ z 1; 0 1 ]     [  1  0; -z  z ]     (z^3, z^3, z^3, z^5)     28     0 [0pt]0pt4ex
 11/14     -z^2     [        -z^5-z^3-z -z^6 +z^5 +z^2 +1;                -z          -z^6-z^4 ]     [ z^2   1;   0   1 ]     [    1    0; -z^2  z^2 ]     -(z^6, z^6, z^6, z^3)     28     0 [0pt]0pt4ex 
 
				 tableHere z = ζ_7 
		

		
		
		
		
		
		
			 
				8|c|G_27 Type A Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     Ã_̃1̃     Ã_̃2̃     Ã_̃3̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				2/60     z^2     [                  z^6u-z^2 2(z^50+z^4)+z^22+z^18-z^6;             (u+1)(z^12+1)                   -z^6u+1 ]     [ -z^2 z^50;    0    1 ]     [            1            0; z^24+2z^12+1         -z^2 ]     (z^14, z^14, z^14, z^18)     60      0 [0pt]0pt4ex 
 
				38/60     -z^8     [                z^24-z^20 (z^28+2z^4)u +2(z^8-z^2);             z^14+u-z^2-1              -z^24-u+z^2 ]     [   z^8 -z^20;     0     1 ]     [          1          0; -2z^18-z^6        z^8 ]     (z^26, z^26, z^26, z^42)      60     0 [0pt]0pt4ex
 50/60     -z^20     [   -u-z^20         u; -u-z^20-1       u+1 ]     [ -z^20  z^20;     0     1 ]     [           1           0; -z^24+z^6+1       -z^20 ]     (z^20, z^20, z^20, 1)     60     360 [0pt]0pt4ex
  Here z = ζ_60 and u = -z^10-z^8+z^2
			

			 	
				8|c|G_27 Type B Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     B̃_̃1̃     B̃_̃2̃     B̃_̃3̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				5/60     z^5     1/3[       z^5f_1-2z^10+1  f_1+2z^15+3z^10-z^5;              f_1-g_1 -z^5f_1+2z^10-3z^5+2 ]     [         -z^5 -z^20f_1-u+2;            0            1 ]     [          1          0; z^25u+z^15       -z^5 ]      (q^25, q^25, q^25, q^45)     96     0 [0pt]0pt4ex 
 
				35/60     -z^5     1/3[     -z^5f_1-2z^10+1 f_1-2z^15+3z^10+z^5;             f_1+g_1 z^5f_1+2z^10+3z^5+2 ]     [          z^5 -z^20f_1-u+2;            0            1 ]     [          1          0; z^25u-z^15        z^5 ]     (q^55, q^55, q^55, q^75)     96     0 [0pt]-5pt4ex
 50/60     -z^20      [ f_1-u-1+2z^10     -f_1+2u+1;   f_1-u+2z^10   -f_1+u-z^20 ]     [      z^20 -f_1+2u+2;         0         1 ]     [       1       0; f_1-u+1    z^20 ]     (z^20, z^20, z^20, 1)      96     360 [0pt]-5pt4ex
  Here z = ζ_60, q = ζ_120, f_1 = -z^14-2z^10-z^8+z^6+z^4+z^2 and g_1 = 4z^15+3z^13-3z^11-3z^9-3z^7-5z^5+3z.
			

			 
				8|c|G_27 Type C Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     C̃_̃1̃     C̃_̃2̃     C̃_̃3̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				10/60     z^10      [ z^50v-z^10-1 z^50v-u+z^10;     2z^50v-3     -z^50v+2 ]     [ -z^10   u+2;     0     1 ]     [                     1                     0; 1/2(3z^20v+u+2z^10+4)                 -z^10 ]     (z^10, z^10, z^10, -1)     60     360 [0pt]0pt4ex 
 
				34/60     -z^4     [           -z^8-z^6    z^18f_1+2z^12+4; z^14-z^50v+2z^10+u          u-z^50v+1 ]     [ z^4 u+2;   0   1 ]     [                          1                          0; 1/2(z^4f_1+3z^24-z^12+z^6)                        z^4 ]     (z^28, z^28, z^28, z^36)     60     0 [0pt]0pt4ex
 46/60     -z^16      [      z^14-z^4+z^2               f_2; -v+z^28v+z^10-z^8    (z^50+z^28)v-u ]     [ z^16  u+2;    0    1 ]     [                          1                          0; 1/2(z^22+2z^16u+z^10-3z^6)                       z^16 ]     (z^22, z^22, z^22, z^54)     60     0 [0pt]0pt4ex
  Here z = ζ_60, v = z^16+z^4, u = -z^10-z^8+z^2, f_1 = -z^14-2z^10-z^8+z^6+z^4+z^2 and f_2 = -4z^14-z^12+z^10+2z^6+3z^4-2z^2
		
		
		

§ MIDDLE CONVOLUTION TABLES OF NICE 4-TUPLES
 
		In these sections the notation is the same as above.
		
		
		

			
			 
				10|c|G_23 4-Tuple Middle Convolutions [0pt]0pt3ex
 
				Type     ξ     λ     Ã_̃1̃     Ã_̃2̃     Ã_̃3̃     Ã_̃4̃      Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				A     1/2     -1     [    u -u+1;  u-1 -u+2 ]     [  0  1; -1  2 ]     [   1 u+2;   0   1 ]     [  1  0; -1  1 ]     (1, 1, 1, 1, 1)     -     0 [0pt]0pt4ex 
 
				 tableHere q = ζ_5 and u = q^3 +q^2
		

		

			 
				10|c|G_25 4-Tuple Middle Convolutions [0pt]0pt3ex
 
				Type     ξ     λ     M̃_̃1̃     M̃_̃2̃     M̃_̃3̃     M̃_̃4̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				A      1/3     z     [  0 -z;  z -z ]     [ 1 1; 0 1 ]     [ 1 1; 0 1 ]     [  1  0; -1  1 ]     (z^2, 1, 1, 1, z)       45     0 [0pt]0pt4ex 
 
				B      1/3     z     [   1 z^2;   0   1 ]     [  1  0; -z  1 ]     [  1  0; -z  1 ]     [  1  0; -z  1 ]     (1, 1, 1, 1, 1)      45       0[0pt]0pt4ex 
 
				C     1/6     z+1     [ -z  0; -1  1 ]     [    -z    -1; z^2-1     z ]     [  -1 z+1;   0   1 ]     [  1  0;  1 -z ]     (u^7, u^3, u^3, u^7, u^4)      120      72 [0pt]0pt4ex 
 
				 tableHere z = ζ_3 and u = ζ_12 
		

		
			 
				10|c|G_26 4-Tuple Middle Convolutions  [0pt]0pt4ex

				Type     ξ     λ     M̃_̃1̃     M̃_̃2̃     M̃_̃3̃     M̃_̃4̃     Character     O. Size     S. Size [0pt]0pt4ex 

				A     1/6     z+1     [ z^2   0; z+2   1 ]     [ -1 -1;  0  1 ]     [   1   0; z^2  -z ]     [   1   0; z^2  -z ]     (u^8, -u^3, u^7, u^7, u^5)      120      72 [0pt]0pt4ex 
 
				B     1/6     z+1     [   -z  z+1; -z+1    z ]     [  1  0;  z -z ]     [ -1 -z;  0  1 ]     [    1    0; 2z+1  z^2 ]     (u^3, u^7, u^3, u^8, u^3)      240       72 [0pt]0pt4ex 
 
				 Here z = ζ_3 and u = ζ_12
			 
				10|c|G_27 4-Tuple Middle Convolutions [0pt]0pt3ex
 
				Type     ξ     λ     M̃_̃1̃     M̃_̃2̃     M̃_̃3̃     M̃_̃4̃     Character     O. S.     S. Size [0pt]0pt4ex
 [0.5ex] 
				A     5/6     -z^5     [   -f_2+1   1/4g_1;      g_2 z^5 +f_2 ]     [  f_1+2 1/2f_2;    g_3    g_4 ]     [    z^5 1/4g_5;      0      1 ]     [   1   0; g_6 z^5 ]     (z^5, z^5, z^5, z^5, z^10)      -     360 [0pt]0pt4ex 
 
				 Here z = ζ_15 f_1 = z^7-z^3+z^2-1, f_2 = z^5-z^4-z +2, g_1 = 2z^7-3z^5+3z^4-2z^3+2z^2+3z-2, g_2 = 2z^7-z^4-2z^3+2z^2-z+1, g_3 = -3z^7+3z^5-2z^4+3z^3-3z^2-2z^2-2z+3, g_4 = -z^7+z^5+z^3-z^2, g_5 =  -z^7-z^5+z^3-z^2-2, g_6 = -z^7+3z^5-z^4+z^3-z^2-z+2
			 
				9|c|G_28 Type A Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     Ã_̃1̃     Ã_̃2̃     Ã_̃3̃     Ã_̃4̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				1/6     z^4     [    1    1;    0 -z^4 ]     [ -z^4+2 -z^4+1;     -2     -1 ]     [ -z^4   -1;    0    1 ]     [     1     0; -2z^4  -z^4 ]     (-z^4, -z^4, -z^4, -z^4, (-z^4)^2)       45     72 [0pt]0pt4ex 
 
				5/6     -z^4+1     [   1   1;   0 z^8 ]     [ z^4+1   z^4;    -2    -1 ]     [ z^8  -1;   0   1 ]     [    1    0; 2z^8  z^8 ]     (z^8, z^8, z^8, z^8,-z^4)       45     72 [0pt]0pt4ex 
 
				 Here z = ζ_24
		
		
			 
				ξ     λ     M_1     M_2     M_3     M_4     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				9|c|G_30 Type A Middle Convolutions [0pt]0pt3ex
 3/10     z^18     [        -z^6v          z^6; -z^24+z^12-1        -z^24 ]     [ -z^18+z^6     -z^18;       v^2        -v ]    [ -z^18    -1;     0     1 ]     [       1       0; -z^18+v   -z^18 ]     (z^6, z^6, z^6, z^6, z^36)      50     600 [0pt]0pt4ex 
 
				7/10     -z^12     [  z^12-v   -z^24; -z^18+v     z^6 ]     [ -z^24+z^12       z^12;  z^24-z^6v     z^24+1 ]     [ z^12   -1;    0    1 ]     [            1            0; -z^24+z^12-1         z^12 ]     (z^24, z^24, z^24, z^24, z^24)    50     600 [0pt]0pt4ex
 9|c|G_30 Type B Middle Convolutions [0pt]0pt3ex
1/6     z^10      [ -z^20-z^16 -z^16-z^10;     z^16-1       z^16 ]    [ -z^10     0;  z^10     1 ]     [ -z^10   u-1;     0     1 ]     [     1     0; -z^10 -z^10 ]     -(z^10, z^10, z^10, z^10, -z^20)     90     360 [0pt]0pt4ex 
 
				5/6     -z^10+1       [   y+z^20 y+z^10-2;       -y     -y+1 ]     [  z^20     0; -z^20     1 ]     [ z^20  u-1;    0    1 ]     [    1    0; z^20 z^20 ]     (z^20, z^20, z^20, z^20, -z^10)     90     360 [0pt]-5pt4ex
 9|c|G_30 Type C Middle Convolutions [0pt]0pt3ex

				1/10     z^6      [    1 -u-1;    0 -z^6 ]     [      -v -z^12+v;       u       0 ]     [ -z^6  u+1;    0    1 ]     [       1       0; -z^12-1    -z^6 ]     (z^12, z^12, z^12, z^12, z^12)     50     600 [0pt]0pt4ex 
 
				9/10     -z^24       [    1 -u-1;    0 z^24 ]     [ z^24+1   z^6v;      u      0 ]     [ z^24  u+1;    0    1 ]     [      1      0; u+z^12   z^24 ]     (z^18, z^18, z^18, z^18, z^48)     50     600 [0pt]-5pt4ex
  Here z denotes a 60th root of unity, u = z^14-z^6-z^4 and v = z^6 -1 and y = z^10 +z^8-z^2.
			 
				ξ     λ     M_1     M_2     M_3     M_4     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				9|c|G_32 Type A Middle Convolutions [0pt]0pt3ex
 5/6     -z^2 +1     [  0 -1; -1  0 ]     [   1   0;   0 z^2 ]     [ z^2   0;   0   1 ]     [   1   0;   0 z^2 ]      (-z^3, z^5, z^5, z^5, 1)      40     72 [0pt]0pt4ex 

				9|c|G_32 Type B Middle Convolutions [0pt]0pt3ex
 1/6     z^2     [  1 -w;  0 -w ]     [  w^2  w-1;   -1 -w^2 ]     [ -w  w;  0  1 ]     [  1  0; -w -1 ]      (z, z^3, z, z^3, z^2-1)      60     72 [0pt]0pt4ex 
 
				5/6     -z^2 +1     [  1 -w;  0 -1 ]     [    1    0;   -1 -w^2 ]     [ -1  w;  0  1 ]     [  1  0; -1 -w ]     (z^3, z^5, z^3, z, 1)         0 [0pt]0pt4ex
 9|c|G_32 Type C Middle Convolutions [0pt]0pt3ex
 1/12     z     [ -z^3    0; z^11    1 ]     [    1  z^4;    0 -z^3 ]     [ -z^3 -z^4;    0    1 ]     [     1     0; -z^11  -z^3 ]      (q^3, q^3, q^3, q^3, -1)      20     96 [0pt]0pt4ex 
 
				7/12     -z     [   z^3     0; -z^11     1 ]     [   1 z^4;   0 z^3 ]     [  z^3 -z^4;    0    1 ]     [    1    0; z^11  z^3 ]     (q^9, q^9, q^9, q^9, -1)    20     96 [0pt]0pt4ex
  Here z = ζ_12, w = ζ_3 and q = ζ_24
		
		
		

§ MIDDLE CONVOLUTION TABLES FOR NICE 5-TUPLES
 
		The notation here is the same as above.

		
		


	
	
	
	
	
	


 Paper Title: Whitney extension theorems on symmetric spaces, an example.



§ ABSTRACT  H. Whitney introduced in 1934 the problem of extending a function f on a set of points in ℝ^n to an analytic function on the ambient space. In this article we prove Whitney type extension theorems for data on some homogeneous spaces.  We use harmonic analysis on the homogeneous spaces and representation theory of compact as well as noncompact reductive groups. 


[
     R. Zamini[Corresponding author: rahelehzamini1361@yahoo.com] ^1,  S. Ghafouri^2  and F. Goodarzi^3

	^1  Department  of Mathematics, Faculty of Mathematical Sciences and  Computer, Kharazmi University, Thehran, Iran. 

	^2   Department of Mathematics, Faculty of Science, Arak University, Arak, 384817758, Iran.

^3   Department of Statistics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran.

    
===================================================================================================================================================================================================================================================================================================================================================================================================================================






 
  

 









§ INTRODUCTION

In 1934, H. Whitney introduced the problem of extending a function f on a set of points in ^n to an analytic function on the ambient space.  In the 2000s, C. Fefferman took up Whitney's work and proved a sharp form of the Whitney extension theorem <cit.>. In particular, if the set of points in Euclidean space is finite of cardinality n, there is an extension which is continuously differentiable of some degree depending on n.

In this paper, we study a version of Whitney's extension problem where the points have an inherent symmetry.  
More precisely, we consider points x^(i) in a symmetric homogeneous  space X=G/H for reductive Lie groups G, H and a nonconstant function f on these points with values  
f(x^(i)) = y^(i) in the complex numbers ℂ.
We are especially interested not just in proving existence of an analytic function extending f, but to obtain a formula for this function which can be effectively computed by an algorithm. 
Initially, 
we weaken the assumption 

    f(x^(i))= y^(i),    1⩽ i ⩽ n.

Instead, we find for all ε > 0 a complex valued function f_n: X→ℂ so that 

    |f_n (x^(i)) - y^(i)| < ε,     1⩽ i ⩽ n.

Observe that the function f_n  depends on ε.

Furthermore, we show how to modify the proof to obtain an exact Whitney extension theorem. 
The function f_n generates a representation Π_n of G, which we call the Whitney representation of the data. 
By acting with the maximal compact subgroup K in G, 
we modify the Whitney function without changing the K-types occuring in its Whitney representation
to obtain an analytic, square integrable function, still denoted f_n,
such that
f_n(x^(i))=y^(i) for all i⩽ n.


We consider  symmetric spaces  of the form G/H for certain orthogonal  Lie groups G and H. The analysis on symmetric  spaces  was intensively studied starting with the work of M. Flensted-Jensen in the late 1970s and culminating in  the Plancherel theorem around 2005 by H. Schlichtkrull and E. van den Ban. In this article, we use  information about the representation of the symmetry group G on the square integrable functions on G/H and in particular about the irreducible subrepresentations. We also use results about the restriction of these representations to orthogonal subgroups.

The group of symmetries of the quadrics 


X(p,q)_± = { (x_i)∈^p+q | x_1^2 + ... +x_p^2 - x_p+1^2 … - x_p+q^2 = ± 1 }


is the noncompact Lie group
G = SO(p,q)^∘,  where p⩾ 2 and the superscript  denotes the connected component.  

This class of homogeneous spaces contains the spaces known in physics as 

 
de Sitter spaces (dS)


respectively as 


anti de Sitter spaces (AdS)  


<cit.>. The anti de Sitter space is considered in physics  as a model of space time with negative curvature.  



 §.§ Main results
 We prove
 
 Let p⩾ 2.
Let x^(i), 1⩽ i ⩽ n, be a set of points in  X(p,2)_- and let f(x^(i)) = y^(i) be a nonconstant, complex-valued function. 
For all ε > 0, there exists an analytic, square integrable  function f_n: X(p,2)_-→ℂ such that 

    |f_n (x^(i)) - y^(i)| < ε.

We call f_n the Whitney function of the data points. 


We approach the extension problem by realizing  space X(p,2)_- as the homogeneous space

    X(p,2)_- = SO(p,2)^∘/SO(p,1)^∘.

This allows us to obtain an explicit formula for the Whitney function f_n using the representation theory of noncompact Lie groups. 
In particular, we build on the work of M. Flensted-Jensen <cit.>, who published  in 1980 an article in which he constructs a family of square integrable functions on symmetric spaces  which are eigenfunctions of all invariant differential operators.  We refer to these functions as Flensted-Jensen functions.











    
 Let x^(i), 1⩽ i ⩽ n, be a set of points in X(p,2)_- and let f(x^(i)) = y^(i) be a nonconstant complex-valued function. 
In the notation of Theorem <ref>, the Whitney function f_n 
can be chosen as a ℂ-linear combination of Flensted-Jensen functions 

    φ_ℓ (ξ', t)= (ξ'_1 - √(-1)ξ'_2 )^ℓcosh (t)^-ℓ,

for integers ℓ⩾ℓ_0 for some lower bound ℓ_0 depending on p. 
Here, (ξ,ξ') = (ξ_1, …, ξ_p,ξ'_1,ξ'_2) is an angular variable and t a radial variable such that 

    (ξ_1sinh(t), ξ_2 sinh(t), …, ξ_psinh(t), ξ'_1cosh(t), ξ'_2cosh(t)) ∈ X(p,2)_-.



In applications of the theorem to a list of observable data, we may assume
that the results of all measurements  are rational. In the case of the three-dimensional symmetric space X(2,2)_+ = SO(2,2)^∘/SO(1,2)^∘, we  prove therefore also  a rational version of our Whitney extension theorem starting from the assumption that all the data points are rational. 


    Let x^(i)∈ X(2,2)_+(), 1⩽ i ⩽ n, be a list of -valued points in the three-dimensional symmetric space X(2,2)_+ realized as the quadric surface 
    
    X(2,2)_+ = {x∈^4: x_1^2 + x_2^2 -x_3^2 - x_4^2 = 1 }
 
    and suppose f is a nonconstant function with f(x^(i)) = y^(i)∈. Let ε > 0. 
    We can choose coefficients a_ℓ in the number field (√(-1)) such that 
    
    f_n = ∑_2 ⩽ℓ⩽ n+1 a_ℓφ_ℓ

is a Whitney function of the points: f_n is analytic, square integrable and K-finite and |f_n(x^(i)) - y^(i)| < ε for all i. Here, the Flensted-Jensen functions φ_ℓ are given by the formula

    φ_ℓ(x) = ( x_1 + √(-1) x_2 )^-ℓ,    x ∈ X(2,2)_+.





 §.§ Related work

In the 2000s, C. Fefferman proved a sharp form of the Whitney extension theorem <cit.>. In particular, if the set of points in Euclidean space is finite of cardinality n, he showed that there is an extension which is continously differentiable of some degree depending on n.  
In this paper we start with a set of points of finite cardinality n, and using  additional assumptions we prove the existence of a Whitney function which is infinitely often differentiable. 
The reason we are able to get an extension of such high regularity is that we take advantage of the fact that the points live in a subset of Euclidean space obeying a high amount of symmetries.

C. Fefferman et al. <cit.> have also studied Whitney extension problems in situations where the points live on a smooth manifold X. The Whitney extension theorems proved in that context, however, are  existence theorems. In contrast, the results given in this article  are effective.  We give an algorithm computing the extension with an explicit formula. This is possible because we build the extension from functions in the discrete spectrum of  the symmetric space G/H. Such Flensted-Jensen functions are solutions to a system of  differential equations  and hence satisfy a high amount symmetries. 

Whitney extension theorems have found a range of applications in data science <cit.>. There is a growing literature on equivariant machine learning, where the data are invariant with respect to a group of symmetries <cit.>. 
Most results in this direction, however, are only available when the symmetry group of the points is either finite or a compact Lie group <cit.>, <cit.>, <cit.>.  
However, many groups of importance in physics are noncompact, such as the Lorentz group SO(3,1)^∘, the symmetry group  SO(p,1)^∘ of X(p,1)_±, and the symmetry group SO(p,2)^∘ of X(p,2)_±.  One motivation for this work is to generalize the results in harmonic analysis applicable to Lie group equivariant machine learning to include noncompact Lie groups. 



 §.§ Contents

The paper has four parts. In this section, we summarize the contents of each part.



  §.§.§ Data on a sphere

In the first part, we prove Whitney extension theorems for points on a sphere S^2 in ^3, whose group of symmetries is the compact group SO(3).
The proof is instructive for the noncompact case, so we give it in details. 
To do so, we use  spherical harmonics, which  will be used in the later parts of the article as well. Indeed, spherical harmonics also play a role in the representation theory of non compact roups and semisimple symmetric spaces, since Flensted-Jensen functions can be written as products of  a Jacobi polynomial, which is the restriction of a harmonic polynomial, with a hyperbolic function.



  §.§.§ Data in the plane

In this part, we prove a Whitney extension theorem for functions on a set of points in the Euclidean 2-space ^2. In contrast to the results by H. Whitney and C. Fefferman, we do not have any limitations on how differentiable the extension can be: We can compute a Whitney extension which is analytic right away. Notice that in this case, we do not need the points to lie in a symmetric space; all we need is that they are contained in the complement of the unit disk, in order to make certain functions square integrable. We include these results here to illustrate our techniques.

The result we prove is as follows. 


    Let  x^(i), 1 ⩽ i ⩽ n, be a set of points in the plane ^2. Assume x^(i) is not contained in the unit disk, and consider a nonconstant complex-valued function f(x^(i)) = y^(i).
    There exists an  analytic,  square integrable function on ^2 with the closed unit disk removed which extends this function. 


We prove this theorem by observing that it falls out, automatically, of the proof of the Whitney extension theorem for the three-dimensional symmetric space X(2,2)_+, given in the next part. 

   


  §.§.§ Data on the symmetric space X(2,2)_+=SO(2,2)^∘ / SO(1,2)^∘

This part forms the core of the paper. 
We discuss the Whitney extension problem for the three-dimensional manifold X(2,2)_+=SO(2,2)^∘/SO(1,2)^∘ . We are especially interested not just in proving the existence, but also to obtain a formula for this function which can be easily computed. More precisely, we consider the question:


    Does there exist
 a smooth, square integrable function 
f ∈ C^∞(X(2,2)_+) such that
f(x^(i)) = y^(i)
for all i ⩽ n? Can we we obtain a closed formula for f?

Note that the manifold  is a fibration of Minkowski space over the circle, SO(2). In particular, we have no well-defined global time direction.
We make the following assumptions: 


  * We consider only finitely many points x^(i), 1⩽ i ⩽ n, although the number may be large.

  * We assume that x^(i)∈ℚ^4 for all i⩽ n and consider a nonconstant function f on these points with values f(x^(i))=y^(i) in . 

  * We assume that the points are in general position, in the sense that they are not contained in the orbit of a proper subgroup of the symmetry group of the ambient symmetric space (cf. <ref>).




Not only do we find a smooth, square integrable function extending the function on the points, 
but we find a constructive way to obtain a formula for this function using the representation theory of noncompact Lie groups and symmetric spaces.
We show that 

  Under these  assumptions, for all  ε >0, there exists an analytic, square integrable function f ∈ C^∞(X(2,2)_+) such that
    
    |f(x^(i)) - y^(i)| < ε

    for all i ⩽ n.


Under the above assumptions, we consider the representation of G generated by f_n in the representation space L^2(X(2,2)_+).
We show that to each Whitney function, we can associate a reducible, unitary representation, Π_n, which we call the Whitney representation of the data. 

We then consider the problem: 
If we enlarge the set of points from cardinality n to n+1, can we  choose a smooth, square integrable, K-finite function
f_n+1∈ L^2(X(2,2)_+) 
such that f_n+1(x^(i)) = y^(i) for all i 
and such that Π_n is a subrepresentation of Π_n+1?
We give an analytic proof that we can always act on the points by the maximal compact subgroup SO(2) ×SO(2)  to arrange that all irreducible subrepresentations below some fixed degree will occur in the Whitney representation. 



  §.§.§ Data on symmetric spaces X(p,2)_- =SO(p,2)^∘/SO(p,1)^∘

We show that our results extend to the symmetric spaces

    X(p,2)_-=SO(p,2)^∘/SO(p,1)^∘

for all p⩾ 2.
The essential feature is that the homogeneous space X(p,2)_- has a holomorphic structure and holomorphic discrete series represesentations.  



 §.§ Acknowledgements

These results were first presented in a lecture at the AIM virtual seminar “Representation Theory and Noncommutative Geometry”. We are grateful to Bent Ørsted for helpful conversations. We would like to thank  David Vogan and Robert Yuncken for insightful questions and comments related to this work.  

The first author was supported by  grant 89086 from the Simons Foundation.









































    


















 

























































§ LOW DIMENSIONAL EXAMPLES




 §.§ The sphere S^2
 
In this subsection we consider a set of points 

    x^(i)∈ S^2 = { x ∈^3: x_1^2 + x_1^2 + x_3^2= 1},     1⩽ i ⩽ n,

and a function f(x^(i)) = y^(i) (1⩽ i⩽ n) which we assume is not constant. 
We think of this as an observable set of data with an inherent spherical symmetry. From this perspective, it is natural to assume 

    x^(i)∈^3,    y^(i)∈
 
for all i, and we will impose this assumption throughout this section. 
Our goal is to find an explicit function interpolating the points. Since the symmetry group of a sphere is SO(3), 
it is natural to solve this problem using the representation theory of SO(3).




 §.§ Preliminaries

There is a natural action of SO(3) on S^2.  
We endow it with the surface area measure, μ, which is invariant under the action of SO(3), and normalized such that μ(S^2)=1. We consider the Hilbert space 
L^2(S^2, μ) of square integrable functions with respect to μ.
Then SO(3) acts on L^2(S^2) as 

    (g ·ψ)(x) = ψ (g^-1x),    g ∈SO(3), ψ∈ L^2(S^2),    x ∈ S^2.

Let Δ =∑_i=1^3 ∂^2/∂ x_i^2 be the Laplacian on ^3. Recall that a harmonic polynomial on ^3 
is a polynomial p = p(x_1,x_2,x_3) ∈ℂ[x_1, x_2, x_3] in three variables
with complex coefficients satisfying 
    
    Δ p = 0.

Example: Let ℓ∈_⩾ 0. The polynomial  
    
    p(x_1,x_2,x_3 ) = (x_1 + √(-1)x_2)^ℓ

is a  harmonic of degree ℓ. 
Note that p is independent of x_3 and is holomorphic as a function of the complex variable
z=x_1 + √(-1)x_2. By <cit.>, it follows that p is a harmonic function of (x_1,x_2) in the sense that 

    ( ∂^2/∂ x_1^2 + ∂^2/∂ x_2^2) p = 0.

In particular, 
Δ p = 0.
The space of spherical harmonics of degree ℓ (ℓ∈_⩾ 0) is the subspace
V_ℓ of L^2(S^2) consisting of restrictions to S^2 of harmonic polynomials on ^3 that are homogeneous of degree ℓ. 
Recall that the space V_ℓ  of spherical harmonics of degree ℓ is an irreducible representation of SO(3) of degree 2ℓ +1.
If V is any irreducible finite-dimensional representation of SO(3), then there is a unique integer ℓ such that V is isomorphic to V_ℓ. The space L^2(S^2) decomposes as a Hilbert space direct sum 
    
    L^2(S^2) ≃⊕̂_ℓ∈_⩾ 0 V_ℓ.

In the physics literature <cit.>, ℓ is called the spin of the representation V. In Lie theoretic language, the irreducible representations of SO(3) are classified by their highest weight <cit.>, and ℓ is the highest weight of V. 



 §.§ The first interpolation theorem


   




 



Consider a set of points x^(i)∈ S^2 ∩^3, 1⩽ i ⩽ n, 
and a function f on the points with values f(x^(i)) = y^(i)∈.
Suppose x^(i)≠ (0,0,1) and (x_1^(j),x_2^(j)) ≠ (x_1^(i),x_2^(i)) whenever i ≠ j.
There are a unique coefficients a_1,… a_ℓ∈(√(-1))
such that the harmonic polynomial 

    f(x_1,x_2,x_3) = ∑_ℓ = 1^n a_ℓ (x_1 + √(-1) x_2)^ℓ

satisfies
f(x^(i))=y^(i) for all i.


Let p_ℓ (x) = (x_1 + √(-1) x_2)^ℓ be the spherical harmonic of degree ℓ.
The n× n matrix (p_ℓ(x^(i)))_i, ℓ is the product of the diagonal matrix

    [ ( x_1^(1) + √(-1) x_2^(1));                                                     ⋱;                            ( x_1^(n) + √(-1) x_2^(n)) ]

with the Vandermonde matrix 

    [                              1     ( x_1^(1) + √(-1) x_2^(1)) ( x_1^(1) + √(-1) x_2^(1))^2 ⋯ ( x_1^(1) + √(-1) x_2^(1))^n-1;                              1     ( x_1^(2) + √(-1) x_2^(2)) ( x_1^(2) + √(-1) x_2^(2))^2 ⋯ ( x_1^(2) + √(-1) x_2^(2))^n-1;                              ⋮;                              1     ( x_1^(n) + √(-1) x_2^(n)) ( x_1^(n) + √(-1) x_2^(n))^2 ⋯ ( x_1^(n) + √(-1) x_2^(n))^n-1 ]

By the formula for a Vandermonde determinant (which holds over any commutative ring, cf. <cit.>),   

    det (p_ℓ(x^(i)))_i, ℓ 
    = ∏_i=1^n  ( x_1^(i) + √(-1) x_2^(i)) ∏_1 ⩽ i < j ⩽ n( ( x_1^(j) + √(-1) x_2^(j)) -  ( x_1^(i) + √(-1) x_2^(i))
    )

Since x^(i)≠ (0,0,1) and (x_1^(j),x_2^(j)) ≠ (x_1^(i),x_2^(i)) for all i ≠ j, this determinant is nonzero.
In particular, the system 

    [ y^(1);     ⋮; y^(n) ]
    =
    [ p_1(x^(1)) p_2(x^(1))          ⋯ p_n(x^(1));          ⋮; p_1(x^(n)) p_2(x^(n))          ⋯ p_n(x^(n)) ][ a_1;   ⋮; a_n ]

has a unique solution in a_ℓ∈(√(-1)).





 §.§ Harmonic polynomials on ^3


Let 𝒫_ℓ be the complex vector space of homogeneous polynomials on ^3 with complex coefficients.    
Recall that the restriction map  
    
    𝒫_ℓ⟶ L^2(S^2)

is injective <cit.>, so we may identify V_ℓ with a space of functions on ^3.
We endow ^3 with the product measure 

    μ⊗ 4π r^2 dr

of the surface area measure μ on S^2 with the measure 4π r^2 dr on (0,∞). The measure μ⊗ 4π r^2 dr is invariant with respect to the natural action of SO(3) on ^3,
and we define L^2(^3) with respect to this measure.
Then L^2(^3) is a unitary representation of SO(3).

  







 
There is a canonical isomorphism of 
    L^2(^3) with the Hilbert tensor product 
    
    L^2(^3) ≃ L^2(S^2)  ⊗̂ L^2((0,∞), 4π r^2 dr).

    Under this isomorphism, a vector f⊗ g in L^2(S^2)  ⊗̂ L^2(0,∞) corresponds to the function 
    f(θ, ϕ )g(r) in L^2(^3) written in polar coordinates. 



    Let ℓ∈_⩾ 0, and let f be a measurable function on 
(0,∞) such that f(r)r^ℓ∈ L^2(0,∞), i.e. 

    ∫_(0,∞) |f(r)|^2 r^2l 4π r^2 dr < ∞.

If p∈ V_ℓ, then p(x)f(|x|) ∈ L^2(^3).
Define 
V_ℓ, f to be the subspace of L^2(^3) of functions of the form

    φ(x) = p(x) f(||x||),    p∈ V_ℓ.

Then V_ℓ,f is invariant and irreducible under the action of SO(3).  


Note that if φ∈ V_ℓ,f, then under the isomorphism in Lemma <ref>, 

    φ(x) = p(x/||x||)||x||^ℓ f(||x||) ≃ p ⊗ g,

where g(r) = r^ℓ f(r) and
p ∈ V_ℓ.



  Let V be any irreducible representation of SO(3) in L^2(^3). Then V is of the form V_ℓ,f for some ℓ and f 
  satisfying the condition in Lemma <ref>. 


See <cit.>. 



 §.§ An open subset  ℝ^2∖B^1 of the plane

Let B^1 = {x ∈^2: x_1^2 + x_2^2 < 1 } be an open ball of radius 1, and let  B^1 = {x ∈^2: x_1^2 + x_2^2 ⩽ 1 } be its closure. 
In this subsection we consider the domain ^2 ∖B^1 and  a nonconstant function on the set of points
x^(i)∈^2 ∖B^1 for i⩽ n. Assume x^(i)∈^2 and that the function maps any
x^(i) to a rational point y^(i)∈.
We equip ^2 ∖B^1 with the measure 

    μ = dθ⊗ r dr,

which is Lebesgue measure in polar coordinates. 

For any integer ℓ⩾ 2, the functions

    φ_ℓ^±(x) = ( x_1 ±√(-1) x_2 )^-ℓ,    x∈^2∖B^1

are in L^2( ^2∖B^1 , dθ⊗ r dr).


  rCl
∫_^2∖B^1 |φ_ℓ^±(x)|^2  dθ⊗r dr
   =    ∫_-π^π∫_1^∞|r cosθ±√(-1) rsinθ|^-2ℓ dθ⊗r dr 

   =   
2π∫_1^∞r^-2 ℓ r dr

which is finite if and only if 2ℓ-1 > 1 ℓ⩾ 2.


 
For any ℓ⩾ 2,
let

    ψ_ℓ^+ (x)= (x_1 + √(-1)x_2 )^-ℓ,  ψ_ℓ^- (x)= (x_1 - √(-1)x_2 )^-ℓ,
       x = (x_1, x_2) ∈ℝ^2∖{(0,0) },

and let x^(i) (1⩽ i ⩽ n) be a sequence of points with x^(i) = (x^(i)_1, x^(i)_2) ∈^2∖{(0,0) } for all i ⩽ n.
Up to multiplication by a diagonal matrix, 
(ψ_ℓ^+(x^(i)))_i, ℓ
and (ψ_ℓ^-(x^(i)))_i, ℓ
where 1⩽ i ⩽ n and 2 ⩽ℓ⩽ n+1,
are Vandermonde matrices. 
In particular, (ψ_ℓ^+(x^(i)))_i, ℓ is invertible if and only if 
the points x^(i) = (x_1^(i), x_2^(i)) have the properties   

    (x_1^(i), x_2^(i)) ≠ (x_1^(j), x_2^(j)) ,   whenever  i < j,    i,j ⩽ n.



Define
    
    α_i = ( x_1^(i) + √(-1) x_2^(i))^-1∈(√(-1)),    1⩽ i ⩽ n,

    then 
    
    (ψ_ℓ^+(x^(i)))_i, ℓ
        =
        [ α_1^2;           ⋱;       α_n^2 ][       1     α_1 α_1^2 ⋯ α_1^n-1;       1     α_2 α_2^2 ⋯ α_2^n-1;       ⋮;       1     α_n α_n^2 ⋯ α_n^n-1 ].

By the formula for a Vandermonde determinant <cit.>, 

    det (ψ_ℓ^+(x^(i)))_i, ℓ 
    = ∏_i=1^n α_i^2 ∏_1 ⩽ i < j ⩽ n (α_j - α_i)




For any ℓ⩾ 2,
let

    ψ_ℓ^± (x)= (x_1 ±√(-1)x_2 )^-ℓ,    x = (x_1, x_2) ∈ℝ^2 ∖B^1 .

Let x^(i) = (x^(i)_1, x^(i)_2) ∈^2 ∖{ (0,0)} be a sequence of points on which a function f is defined such that 
f(x^(i))=y^(i)∈ for all i ⩽ n.
Suppose that 

    x^(i)≠ x^(j)

whenever i<j.
There exist unique coefficients a_ℓ∈(√(-1)) 
such that if we define the function 

    f_n = ∑_l=2^n+1 a_ℓψ_ℓ^+ ∈ L^2(^2∖B^1 ,
    dθ⊗ r dr
    )

then 
f_n( x^(i) ) = y^(i)
for all i ⩽  n.



By Lemma <ref>, 
the column vector (a_ℓ)_ℓ can be computed in terms of the points (x^(i),y^(i)) as  

    (a_ℓ)_ℓ = ( ψ_ℓ^+(x^(i)) )_i,ℓ^-1 (y^(i))_i.





§ THE SYMMETRIC SPACE X(2,2)_+
 




 §.§ Preliminaries


Let p, q ⩾ 1 be integers.
Let (·, ·)_p,q be the symmetric bilinear form on ^p+q 
of signature (p,q)
defined by

    (x,y)_p,q = ∑_k=1^p x_k y_k - ∑_k=p+1^p+q x_k y_k,    x,y ∈^p+q.

The generalized orthogonal group of signature (p,q) is the group

    O(p,q) = { g ∈GL(p+q,) : (gx,gy)_p,q = (x,y)_p,q ∀ x,y∈^p+q},

and SO(p,q)= { g∈O(p,q):  g = 1 }.
Note that if g∈GL(p+q, ), then g∈O(p,q) if and only if  
rCl
(ge_j, ge_k)_p,q    =    0,   j≠k, 
    
(ge_j, ge_j)_p,q    =    1,   j⩽p, 
    
(ge_j, ge_j)_p,q    =    -1,   p+1 ⩽j ⩽p+q. 
    



 Let 
    
    Q(x)=x_1^2 + x_2^2 - x_3^2 -x_4^2,    x ∈^4.

    Then X ={ x∈^4: (x) =1 } is a manifold endowed with a transitive Lie group action by G=SO(2,2), and there is a
    G-equivariant diffeomorphism
    
    X≃SO(2,2)/ SO(1,2).




There is a smooth, injective group homomorphism 
φ: SO(1,2)⟶SO(2,2) of the form 

    φ: g ↦[ 1 0; 0 g ] = 
    [    1    0    0    0;    0 g_11 g_12 g_13;    0 g_21 g_22 g_23;    0 g_31 g_32 g_33 ],    g∈SO(1,2)

since for x,y∈^4 and g∈SO(1,2),
rCl
( 
[ 1 0; 0 g ] x,
[ 1 0; 0 g ] y
)_2,2
   =    x_1 y_1 + 
( 
g 
[ x_2; x_3; x_4 ],
g 
[ y_2; y_3; y_4 ]
)_1,2 

   =   
x_1 y_1 +
( 
[ x_2; x_3; x_4 ],
[ y_2; y_3; y_4 ]
)_1,2
= (x,y)_2,2.

Let G = SO(2,2). Let e_1 = (1,0,0,0)∈^4.
The stabilizer in G at e_1 is 

    G_e_1  = {g ∈ G: g e_1 = e_1  } ={ g∈ G: 
    g = 
    [  1  0;  0 g̃ ]}  = SO(1,2).

By definition, Q(x) = (x,x)_2,2, and the G-orbit of e_1 is

    Ge_1 = { x∈^4: (x,x)_2,2 = 1} = { x∈^4: Q(x) =1 }.

 In particular, the orbit Ge_1 is a manifold, and 
there is a diffeomorphism  

    SO(2,2)/SO(1,2) ⟶ Ge_1 = { x∈^4: Q(x) =1 }.


We follow the conventions in the literature and work with the connected components, so we consider

    X(2,2)_+ = SO(2,2)^∘/SO(1,2)^∘

respectively

    X(2,2)_- =
    SO(2,2)^∘/SO(2,1)^∘.


    Let 𝔰𝔬(2,2) be the Lie algebra of SO(2,2), and let 𝔰𝔬(2,2;ℂ) be its complexification. 
    There is an isomorphism of Lie algebras
    
    φ: 𝔰𝔬(4;ℂ) ⟶𝔰𝔬(2,2;ℂ)

    In particular, if 𝔤_1 ≃𝔰𝔩(2,ℂ),
    then 
    
    𝔰𝔬(2,2;ℂ) ≃𝔤_1 ⊕𝔤_1

    and 
    
    𝔰𝔬(2,2) ≃𝔰𝔩(2;ℂ)_.

  Hence 𝔰𝔬(2,2) is simple as a real Lie algebra, while
    𝔰𝔬(2,2;ℂ) is not simple (but semisimple.)


    Let 
    
    g = [  1         ;     1      ;       -1   ;    -1 ].

Let M_4() be the space of 4× 4 real matrices, and for any matrix X, let tX be its transpose. 
We have g = g^-1 and 

    𝔰𝔬(2,2) ≃{ X∈M_2() :  g ( tX) g^-1 = -X }.

Note that g has a complex square root:

    g^1/2 = 
    [ 1      ;   1    ;     i  ;       i ]∈GL(4,ℂ).

In particular, 

    tX = -X  g ( t ( g^1/2 X g^-1/2 ))  g^-1 = -g^1/2 X g^-1/2.

It follows that there is a Lie algebra isomorphism 

    Ad_g^1/2: 𝔰𝔬(4;ℂ) ⟶𝔰𝔬(2,2;ℂ)

By <cit.>, 

    𝔰𝔬(4;ℂ) ≃𝔰𝔩(2;ℂ)
    ⊕𝔰𝔩(2;ℂ).

As in the proof of <cit.>, these two copies of 𝔰𝔩(2;ℂ) are complex conjugate. 
If 𝔤_1 denotes the first copy in the decomposition, then 
𝔰𝔬(2,2,ℂ) = 𝔤_1 ⊕𝔤_1.
The map φ: 𝔤_1 →𝔰𝔬(2,2;ℂ)

    φ: X ↦ X + X

is an ℝ-linear isomorphism and respects Lie brackets. 
We conclude that 

    𝔰𝔬(2,2) ≃𝔰𝔩(2;ℂ)_.





 §.§ From rational data to Whitney functions. The first case
 











The functions constructed by M. Flensted-Jensen <cit.>  will in our considerations play the role for the noncompact group SO(2,2)^∘ analogous to the role played by spherical harmonics for the compact group SO(3).


Let X = { x ∈ℝ^4: x_1^2 + x_2^2 -x_3^2 -x_4^2 = 1}.
For any integer ℓ≥ 2, define
ψ_ℓ^+: X →ℂ and ψ_ℓ^-: X →ℂ by

    ψ_ℓ^+(x) = ( x_1 + √(-1) x_2 )^-ℓ,   ψ_ℓ^-(x) = ( x_1 - √(-1) x_2 )^-ℓ.

Then
ψ_ℓ^± are Flensted-Jensen functions for the parameter λ=2ℓ-2, and the (𝔤, K)-modules generated by ψ_ℓ^+ and ψ_ℓ^- form a complete set of discrete series representations for X.




Cf. <cit.> Part 4, section 9, p. 215.    




We say  that the  points x^(i) (1⩽ i ⩽ n) are in general position if not all points in the sequence have the same coordinate equal to zero,  i.e.  not all points x^(i)∈ G/H have representatives in G=SO(2,2)^∘ contained in the same  subgroup G' isomorphic to one of the groups SO(1,2)^∘, SO(2,1)^∘,  SO(1,1)^∘, SO(2,0), or SO(0,2). In this section we assume that the points x^(i) are in general position.





Let x^(i) (1⩽ i ⩽ n) be a sequence of points of X=G/H in general position, and assume x^(i)∈^4 for all i.
Let f be a nonconstant function on the points with f(x^(i))=y^(i)∈. We would like to find coefficients a_ℓ such that 

    y^(i) = ∑_ℓ≥ 2 a_ℓψ_ℓ^± (x^(i))

for all i ⩽  n, and where a_ℓ≠ 0 for finitely many ℓ≥ 2.
Then 

    f = ∑_ℓ≥ 2 a_ℓψ_ℓ^±∈ L^2(X)

is a K-finite vector in the representation space L^2(X) with the property that 
f(x^(i)) = y^(i) for all i⩽ n.
We find (a_ℓ) by solving a linear system of equations 

    [ y^(1);     ⋮; y^(n) ]
    =
    [ ψ_2^±(x^(1)) ψ_3^±(x^(1))            ⋯ ψ_m^±(x^(1));            ⋮; ψ_2^±(x^(n)) ψ_3^±(x^(n))            ⋯ ψ_m^±(x^(n)) ][ a_1;   ⋮; a_m ]

for a choice of m ≥ n.

 
    The matrix  (ψ_ℓ^+(x^(i)))_i, ℓ
    where 1⩽ i ⩽ n, 2 ⩽ℓ⩽ n+1
    is Vandermonde (up to multiplication by a diagonal matrix). Furthermore,   
(ψ_ℓ^+(x^(i)))_i, ℓ is invertible if and only if 
the points x^(i) = (x_1^(i), x_2^(i), x_3^(i), x_4^(i)) in X have the property 

    (x_1^(i), x_2^(i)) ≠ (x_1^(j), x_2^(j)) ,   whenever  i < j,    i,j ⩽ n.

In this case, a_ℓ∈(√(-1)) for all 2 ⩽ℓ⩽ n+1, and
the column vector (a_ℓ)_ℓ can be computed in terms of the points (x^(i),y^(i)) as  

    (a_ℓ)_ℓ = ( ψ_ℓ^+(x^(i)) )_i,ℓ^-1 (y^
    (i))_i.




    Follows from Lemma <ref>.




 §.§ Cayley's rational parametrization of SO(n)

A matrix A is called non-exceptional if (I+A) ≠ 0, where I is the identity matrix, and otherwise A is called exceptional.     

 
    Let A be a non-exceptional matrix. Then the matrix S defined by the equation
    
    I + S = 2(I+A)^-1

    is also non-exceptional,  
    (I+A) = 2(I+S)^-1, and A and S are mutually related by the equations 
    
    S = (I-A)(I+A)^-1  = (I+A)^-1(I-A),    A = (I-S)(I+S)^-1  = (I+S)^-1(I-S).

Since the factors commute, we will follow Weyl <cit.> and use the rational notation 

    A = I-S/I+S.


For a matrix R=(r_ij), the  bilinear form ∑_ij r_ijx_i y_j, is A-invariant if and only if 

    tA R A = R,

where tA is the transpose of A.


    Let A and S be a pair of non-exceptional matrices related as in Lemma <ref>, and let R be any matrix. Then tARA = R if and only if 
    
    tS R + RS = 0.

In particular, if R=I and A belongs to the orthogonal group O(n),
then A∈SO(n) and S is skew-symmetric: tS + S = 0. 
Conversely, if S is skew-symmetric matrix, then A ∈SO(n).



    Let A ∈SO(n).
    Then A = A_1A_2 for two non-exceptional matrices
    
    A_1 = I-S_1/I+S_1,    A_2 = I-S_2/I+S_2∈SO(n).

If (I+S_1S_2) ≠ 0, then A is itself non-exceptional and     

    A= I-S/I+S

where 

    S = S_1 +S_2/I+S_1 S_1.



The following theorem implies that there are plenty of rational points in the space X. 


    Let 
    f = a_1 x_1^2 + a_2 x_2^2 + a_3 x_3^2 + a_4 x_4^2 + a_5 x_5^2,    a_i ∈
 
    be a quadratic form over  of rank 5. Then f represents zero in  if and only if it represents zero in ℝ. In particular, there are coordinates x_i ∈ such that
    
    0 =  x_1^2 + x_2^2 - x_3^2 - x_4^2  - x_5^2,

    x_5 ≠ 0, and 
 the point x' = (x_1 / x_5, x_2 /x_5, x_3/ x_5, x_4/x_5) ∈ X(2,2)_+∩^4,
 where 
 
    X(2,2)_+  = {x ∈ℝ^4 :  x_1^2 + x_2^2 - x_3^2 - x_4^2 = 1}.

 For any proper orthogonal matrix A∈SO(2,), the point

    [ A  ;   I ] 
    x' ∈ X(2,2)_+∩^4.
 
In particular, there are infinitely many rational points on X(2,2)_+.


See <cit.> p. 43.



    Let x = (x_1,x_2,x_3,x_4) and x'=(x'_1,x'_2,x'_3,x'_4) be rational points in X(2,2)_+.
    Suppose 
    
    x_1 = x_1',   x_2 = x_2'.

    For any matrix A ∈SO(2, ),
    let 
    
    [ x_1^∘; x_2^∘ ]
    =
    [ a b; c d ][ x'_1; x'_2 ]
    =
    [ ax'_1 + bx'_2; cx'_1 + dx'_2 ]

    and let x^∘_3 =x'_3 and x^∘_4 = x'_4.
    Then the perturbed point x^∘ = (x^∘_1,x^∘_2,x'_3,x'_4) is a rational point on X(2,2)_+.



    By assumption, 
    
    (x'_1)^2 + (x'_2)^2 = 1 + (x'_3)^2 + (x'_4)^2,

    so (x'_1,x'_2) lies on a circle of radius 1 + (x'_3)^2 + (x'_4)^2, and SO(2)
    acts on this circle. 


 
Let ε∈, ε > 0, and define the skew-symmetric matrix
    
    S = [   0   ε; - ε   0;     ].

    Then  
    (I + S) = 1 + ε^2 > 0, so S is non-exceptional, 
and 
    
    A = I-S/I+S = 
    [  1  ε; -ε  1;    ]^-1[  1 -ε;  ε  1;    ]
    =1/1+ε^2[ 1 - ε^2     -2ε;     2 ε 1 - ε^2;         ]

    is a non-exceptional matrix in SO(2,). 
Also, the distance between A and the identity transformation I is

    ||A-I||^2 
    = 8ε^2/1+ε^2,

which tends to 0 as ε tends to zero. 


 
    Let x and x̃ be rational points in X(2,2)_+ whose first and second coordinates coincide: x_1 = x̃_1 and x_2 = x̃_2.
    Then the point 
    
    x̃^(ε) =
    [ 1-ε^2/1+ε^2 x_1 - 2ε/1+ε^2x_2; 2ε/1+ε^2x_1 + 1-ε^2/1+ε^2 x_2;                          x̃_3;                          x̃_4 ]

    is a rational point on X(2,2)_+. Clearly, the points x̃
    and x̃^(ε) are equidistant from x, i.e.
    
    ||x - x̃ ||_ℝ^4 = ||x - x̃^(ε) ||_ℝ^4,
 while  the distance between x̃ and the perturbed point x̃^(ε)
is given by   

    ||x̃ - x̃^(ε) ||^2_ℝ^4 =
    4ε^2/(1+ε^2)^2( (x_2-ε x_1)^2 +(x_1-ε x_2)^2 ),

which tends to zero as ε tends to zero: Indeed, if ε⩽ 1, and 
if C = max_i,j |x_j^(i)|
then 
||x̃ - x̃^(ε) ||_ℝ^4⩽ 2^5/2 C ε.
Finally, 

    x_1 ≠x̃_1^(ε),    
    x_2 ≠x̃_1^(ε).





 §.§ From rational data sets  to Whitney functions: the general case
 

Consider a set of points x^(i) (1⩽ i ⩽ n) and  a nonconstant function 
f(x^(i)) = y^(i)
where x^(i)∈ X(2,2)_+, 
x^(i)_j ∈ for all j ⩽ 4, and y^(i)∈ for all i⩽ n.


  * Define a small error tolerance ε∈, ε > 0. 
    
  * Loop through indices i with 2⩽ i ⩽ n. 
    Let such an i be fixed. By induction, it holds for all j,k < i
    (j≠ k) that 
    
    (x_1^(j),x_2^(j)) ≠ (x_1^(k),x_2^(k)).

    Suppose there is some k < i such that 
      
    (x_1^(k),x_2^(k)) = (x_1^(i),x_2^(i)).

    In particular, k< i is the only index with this property.  
    Apply a non-exceptional proper orthogonal transformation to the point x^(i) 
    to obtain the point x^(i, ε) for ε∈_>0 as in Lemma <ref> with the property  that
     (x_1^(i,ε),x_2^(i,ε)) is distinct from all the points in  the finite list
     
    (x_1^(j),x_2^(j)),   j < i.

    Replace the point x^(i) with the point x^(i, ε).
    
  * After having checked all points and replaced some with small perturbations if necessary, all the assumptions of Proposition <ref> are in place. Compute the column vector 
    
    (a_ℓ)_ℓ=2^n+1
        =
      ( (x_1^(i, ε) + √(-1)x_2^(i,ε))^-ℓ)_1⩽ i ⩽ n, 2 ≤ℓ⩽ n+1^-1 (y^(i))_i=1^n,

    where a_ℓ∈(√(-1)) for all ℓ.
   Then the function 
    f_n = ∑_ℓ=2^n+1 a_ℓψ_ℓ^+
 is a linear combination of n linearly independent Flensted-Jensen functions with the property that 
   
    y_i = f_n(x^(i,ε))
 for all i ⩽ n.


Let (x^(i)) be a data set of n points in G/H in general position and let f(x^(i))=y^(i) (1⩽ i ⩽ n) be a nonconstant function. 
After fixing some ε > 0, 
we associate an analytic, square integrable, K-finite function f_n to the data according to the algorithm in this section <ref>. 
We will refer to f_n as a Whitney function of the data. 




 §.§ From Whitney functions to Whitney representations

  


Let G = SO(2,2)^∘ and H = SO(1,2)^∘
and let f_n be a Whitney function of a data set in G/H.
Let V_f_n be the (𝔤, K)-module in L^2(G/H) generated by f_n. This is the vector space of K-finite functions in L^2(G/H)
obtained by acting on f_n with the universal enveloping algebra U(𝔤_ℂ).
Let Π_n be the representation of G on the completion of V_f_n in L^2(G/H) and let (π_ℓ, V_ℓ) (ℓ⩾ 2) be the Flensted-Jensen representation of G in L^2(G/H) generated by φ_ℓ^+.
The results of M. Flensted-Jensen imply in the special case of X(2,2)_+ that if  ℓ, ℓ' ⩾ 2 are distinct indices, then 
the  representations π_ℓ and π_ℓ'  are inequivalent representations of G. 
Evidently, for a Whitney function f_n = ∑_ℓ⩽ n+1 a_ℓφ^+_ℓ,  

    Π_n = ⊕_ℓ⩽ n+1: a_ℓ≠ 0π_ℓ.

Note that if a_ℓ =0 then the representation π_ℓ is not a direct summand.


Define a hyperplane H_k in ℂ^n as

    H_k ={ z∈ℂ^n: z_k = 0},    1⩽ k⩽ n.

The union ∪_k ⩽ nH_k is a subset of ℂ^n of Lebesgue measure zero. 
Let x^(i) (1⩽ i ⩽ n) be a data set in G/H in general position, and let 
f(x^(i))=y^(i) be a nonconstant function. 
Fix ε >0 and choose modified points x̃^(i) =x^(i,ε)=k_ε. x^(i) such that the matrix 

    T = ( φ_ℓ^+(x̃^(i)))_i,ℓ∈GL(n,ℂ).

Let T^-1 be the inverse of T in GL(n,ℂ).
Suppose 

    (y^(i))_i ⩽ n∈ℂ^n ∖ T(∪_k⩽ n H_k).

Then the Whitney function f_n= ∑ a_ℓφ_ℓ^+ of the data has the property that a_ℓ≠ 0 for all ℓ.
In particular, the  representation Π_nwhich it generates  contains a maximal number of summands:  

    Π_n = ⊕_2 ⩽ℓ⩽ n+1 π_ℓ.



If the vector (y^(i))_i ∈ℂ^n lives in the set 
ℂ^n ∖ T ∪_k ⩽ n H_k
(which has full Lebesgue measure in ℂ^n) 
then
    
    (a_ℓ)_ℓ = T^-1 (y^(i))_i  ∉∪_k ⩽ n H_k

    which means that  a_ℓ≠ 0 for all ℓ.



    Let x^(i) (1⩽ i ⩽ n) be a set of points in G/H in general position, and let 
    f(x^(i)) =y^(i) be a nonconstant function. 
    Fix ε >0 and suppose there is a Whitney function 
    
    f_n = ∑_2⩽ℓ⩽  n+1 a_ℓφ_ℓ^+
 such that |f_n(x^(i))-y^(i)| < ε for all i. 
    There exists δ∈ (-1/m,1/m) for some m∈_⩾ 1 and a Whitney function  
    
    f_n^(δ) = ∑_ℓ a_ℓ^(δ)φ_ℓ^+

    such that 
    
    |f_n^(δ)(x^(i))-y^(i)| < ε  ∀ i ⩽ n,

    and such that 
    a_ℓ^(δ)≠ 0 for all ℓ⩽ n+1. In particular
     the Whitney representation Π_n^(δ) generated by f_n^(δ), is of the form 
    
    Π_n^(δ) = ⊕_2⩽ℓ⩽ n+1π_ℓ.

 

The proof relies on the observation that set of regular points

    (ℂ^n)^reg := { z ∈ℂ^n: z_i ≠ 0  ∀ i ⩽ n}

is dense in ℂ^n.
First, we verify that the function 

    W: z = (z^(i))_i ↦ (a_ℓ(z)_ℓ) ∈ℂ^n,    (z^(i))_i ∈ (G/H)^n

is an open mapping. The map φ: (G/H)^n →ℂ^n × n

    z= (z^(i))_i ↦φ(z) = ( (z_1^(i) + √(-1) z_2^(i))^-ℓ )_i,ℓ

is an analytic, open mapping, and so is the restriction  φ|_φ^-1(GL(n,ℂ)) to the open set φ^-1(GL(n,ℂ)).
Likewise, the evaluation map e_y: GL(n,ℂ) →ℂ^n defined by 
e_y: T ↦ T (y^(i))_i for T∈GL(n,ℂ)
and the inversion map ι: GL(n,ℂ) →GL(n,ℂ) are both open mappings.
By definition, 

    W = e_y ∘ι∘φ|_φ^-1(GL(n,ℂ))

so W is also an open mapping from an open subset of ℂ^n into ℂ^n.

Let 

    S_δ = [  0  δ; -δ  0;    ]

be the non-exceptional skew symmetric matrix whose Cayley transform 
A(S_δ) = (I-S_δ)/(I+S_δ) ∈SO(2).
Define 

    k_δ =
    [ I-S_δ/I+S_δ            ;                       I ]∈ K = SO(2)×SO(2).

Note that x=(x^(i))_i ∈φ^-1(GL(n,ℂ)), so there is some m ∈_⩾ 1
such that k_δ.x ∈φ^-1(GL(n,ℂ)) for all δ∈ (-1/m,1/m). 
The open set 

    U_m,ε = { W( k_δ. x ): δ∈ (-1/m,1/m), | ∑_2 ⩽ℓ⩽ n+1 a_ℓ(k_δ . x)φ_ℓ^+ (x^(i))-y^(i)| < ε}
 
is an open neighborhood of 
(a_ℓ(x))_ℓ in ℂ^n, and therefore it meets the dense set (ℂ^n)^reg.  Choose some δ such that (a_ℓ(k_δ.x))_ℓ∈  U_m,ε∩ (ℂ^n)^reg.
Then the function 

    f_n^(δ) = ∑_2 ⩽ℓ⩽ n+1 a_ℓ(k_δ . x)φ_ℓ^+

has the desired properties. 

We call the representation Π_n^(δ) the  Whitney representation of the data set.   
This answers the question in the introduction and shows


We fix ε > 0.
If we enlarge the set of points from cardinality n to n+1, we can  choose a small real number δ and a
smooth, square integrable, K-finite function
f_n+1^(δ)∈ L^2(X(2,2)_+) such that 
| f_n+1^(δ)(x^(i)) - y^(i) |< ε for all i ⩽ n+1 
and such that the representation  Π_n generated by f_n is a subrepresentation of Π_n+1 ^(δ).





 §.§ An exact Whitney theorem

In  this subsection, we obtain an exact Whitney theorem. 
Let 
    κ_t = [  cos(t)  sin(t); -sin(t)  cos(t) ].

By direct calculation, 
rCl
φ_ℓ^+(κ_t.x)    =    
( x_1cos(t)+x_2sin(t) + √(-1)(x_1(-sin(t)) +x_2cos(t)) )^-ℓ 

   =    ( x_1 (cos(t)-√(-1)sin(t)) + x_2
(sin(t) + √(-1)cos(t)) )^-ℓ 

   =    ( x_1 (cos(t)-√(-1)sin(t)) + √(-1)x_2
( cos(t) - √(-1)sin(t)) )^-ℓ 

   =    e^√(-1)(-t)(-ℓ) ( x_1+√(-1)x_2 )^-ℓ 

   =    e^√(-1)tℓ φ_ℓ^+(x).


As a consequence of the algorithm <ref>, there exists
a real number t_n and a Whitney function
f_n such that 

    f_n(κ_t_n. x^(i))= y^(i)

for all i⩽ n. Then
rCl
y^(i)    =    f_n(κ_t_n.x^(i)) 

   =    ∑_ℓ⩽n+1 a_ℓφ_ℓ^+ ( k_t_n.x^(i)) 

   =    ∑_ℓ⩽n+1 a_ℓe^√(-1)t_nℓ φ_ℓ^+(x^(i)).


In particular, 
if 
    a'_ℓ = e^√(-1)t_nℓa_ℓ,   ℓ⩽ n+1,

then the linear combination 
f_n' = ∑ a'_ℓφ_ℓ^+
of Flensted-Jensen functions is an exact Whitney function for the data. 




§ THE HIGHER DIMENSIONAL SYMMETRIC SPACES  X_±(P,2)



 
  
   

 

 

 

 

 


In this section, we sketch the generalization of  the Whitney extension theorems to the higher dimensional symmetric spaces 

    X(p,2)_-=
    SO(p,2)^∘/SO(p,1)^∘ X(p,2)_+=SO(p,2)^∘/SO(p-1,2)^∘ .
 


 §.§ Generalities

First, we clarify the definitions of certain semisimple symmetric spaces that we will study in the sequel. We refer to <cit.> for further details.
Define 

    X(p,q)_+ = { (x,y) ∈^p+q: |x|^2-|y|^2 = 1  }
 and 

    X(p,q)_- = { (x,y) ∈^p+q: |x|^2-|y|^2 = -1  }.

There is a diffeomorphism 

    X(p,q)_-≃ X(q,p)_+

given by (x,y)↦ (y,x).
The Lie group G=SO(p,q)^∘ acts transitively on X(p,q)_+ and on X(p,q)_-, and there are G-diffeomorphisms

    X(p,q)_+ ≃SO(p,q)^∘/SO(p-1,q)^∘,

and 

    X(p,q)_-≃SO(p,q)^∘/SO(p,q-1)^∘.

In the remaining part of the  paper, we will generalize our results from the space
X(2,2)_- to the spaces X(p,2)_- for all p≥ 2.
We keep q=2 because we want the  maximal compact subgroup K of the symmetry group SO(p,2)^∘ to have an abelian factor which is diffeomorphic to the circle group SO(2), so that our space has a complex structure and we can speak about holomorphic and antiholomorphic  discrete series representations. 

Let p⩾ 2.
Next, we parametrize the symmetric space 

    X(p,2)_-
    = { (x,y) ∈^p+2: |x|^2 - |y|^2 = -1 }

in terms of hyperbolic coordinates. 
The lemma below, which gives the KJH-decomposition of the symmetric space
X(p,q)_-, motivates this reparametrization. 

 
Let p,q⩾ 2 and let  
G=SO(p,q)^∘.
There is a decomposition 

    G = KJH,

where 

    H = [ g  ;   1 ],   g∈SO(p,q-1)^∘,    
    K = [ k_1    ;     k_2 ],    k_1∈SO(p), k_2∈SO(q)

and 

    J = exp[   t;  ; t   ]
    =
    [  cosh(t)           sinh(t);         ; -sinh(t)           cosh(t) ],    t∈.

In particular, G/H ≃ X(p,q)_- is a symmetric space of rank one. 


Let G=SO(p,q)^∘
and let H be the image in G of 
SO(p,q-1)^∘ under the embedding 

    g ↦[ g  ;   1 ],   g∈SO(p,q-1)^∘.

Let

    J/H ⊆ G/H

be a maximal, split, abelian subspace of G/H.
We may decompose G as  

    G = KJH,

where 

    K = [ k_1    ;     k_2 ],    k_1∈SO(p), k_2∈SO(q)

is the maximal compact subgroup of G 









 
 
 


In particular, 

    rank G/H = _ J = 1.



We introduce the coordinates 

    Φ(ξ,ξ',t) = ( ξ_1 sinh(t), ⋯, ξ_psinh(t), ξ_1'cosh(t),
    ξ_2'cosh(t) )

where ∑_j=1^pξ_j^2 = 1 and (ξ'_1)^2 + (ξ'_2)^2 =1.  



  §.§.§ Data points
 
Let G = SO(p,2)^∘ and H = SO(p,1)^∘.
Recall that we say that the points x^(i)∈ G/H  are in general position  if not all points are contained 
in an orbit G'/G'∩ H in G/H of a proper reductive subgroup G' of G isomorphic to one of the orthogonal groups  
SO(p',2)^∘,
SO(p',1)^∘, or 
SO(p',0) for some p'⩽ p.



  §.§.§ Flensted-Jensen functions


If p=2, the Flensted-Jensen functions φ_ℓ^+ on the space
rCl
X    =    X(2,2)_+ = { x∈^4: x_1^2 + x_2^2 -x_3^2 - x_4^2 = 1 } 

   ≃    X(2,2)_- = { (x_3,x_4,x_1,x_2) ∈^4: x_3^2 + x_4^2  - x_1^2 - x_2^2  = -1} 

can be expressed in the new coordinates as follows:
rCl
φ_ℓ^+ (x) = (x_1 + √(-1) x_2)^-ℓ    =    (x_1-√(-1)x_2)^ℓ 1/(x_1^2 + x_2^2)^ℓ 

   =    (ξ'_1-√(-1)ξ'_2)^ℓcosh(t)^ℓ 1/ ( (ξ'_1)^2+(ξ'_2)^2)^ℓcosh(t)^2ℓ 

   =    (ξ'_1-√(-1)ξ'_2)^ℓ cosh(t)^-ℓ,   ℓ⩾2.

We see that φ_ℓ^+ is a product of a spherical harmonic polynomial 

    p_ℓ(ξ'_1,ξ'_2) = (ξ'_1-√(-1)ξ'_2)^ℓ
 of degree ℓ,
and a negative power of a hyperbolic cosine. 
This form of the Flensted-Jensen functions generalizes to symmetric spaces X(p,q)_- = SO(p,q)^∘/SO(p,q-1)^∘ of arbitrary signature (p,q) <cit.>.



Consider the symmetric space X(p,2)_- = SO(p,2)^∘/SO(p,1)^∘.

The functions

    φ_ℓ ( ξ',t) 
    = (ξ'_1-√(-1)ξ'_2)^ℓcosh(t)^-ℓ,   ℓ⩾p+2/2, ℓ∈

are Flensted-Jensen functions for X(p,2)_-.
Any Flensted-Jensen function takes the form 

    p_ℓ(ξ'_1,ξ'_2) cosh(t)^-ℓ,    (ξ'_1)^2 + (ξ'_2)^2 = 1,

for a harmonic polynomial p_ℓ of degree ℓ.
 


 §.§ Whitney functions on X(p,2)_- = SO(p,2)^∘/SO(p,1)^∘




Let p ⩾ 2, and
let K = SO(p) ×SO(2) be the maximal compact subgroup of G = SO(p,2)^∘.
Let H = SO(p,1)^∘ and consider the symmetric space 
X(p,2)_-≃ G/H.
Let x^(i)∈ X(p,2)_-, 1 ⩽ i  ⩽ n, be points in general position
and consider a nonconstant function on these points with values y^(i)∈ℂ, i ⩽ n. 
Let ε > 0.
There exists an analytic, K-finite, square-integrable function f_n on the symmetric space X(p,2)_- such that 

    |f_n(x^(i)) - y^(i)| < ε,   ∀ i ⩽ n.

The function f_n may be chosen as a ℂ-linear combination 

    f_n (ξ', t) = ∑_ℓ a_ℓφ_ℓ (ξ', t)

of Flensted-Jensen functions φ_ℓ(ξ', t) =(ξ'_1-√(-1)ξ'_2)^ℓcosh(t)^-ℓ.



    For each x^(i)∈ G/H, let (ξ^(i),ξ'^(i),t^(i)) be such that 
    
    x^(i)= (ξ_1sinh(t^(i)), …,ξ_psinh(t^(i)) , ξ'^(i)_1cosh(t^(i)), ξ'^(i)_2cosh(t^(i)) ).

    Let ℓ_0 be the smallest integer occurring as the weight 
    of a Flensted-Jensen function φ_ℓ(ξ',t) = (ξ'_1-√(-1)ξ'_2)^ℓcosh(t)^-ℓ (note that ℓ_0 depends on p). 
    Let T be the complex n× n matrix
    
    T =  ( ( (ξ'^(i)_1-√(-1)ξ'^(i)_2 )cosh(t^(i))^-1)^ℓ)_1⩽ i ⩽ n, ℓ_0 ⩽ℓ⩽ℓ_0+n-1

and consider the linear system (a_ℓ)_ℓ_0 ⩽ℓ⩽ℓ_0+n-1 = T(y^(i))_i.    
Since T is the product of the diagonal matrix

    T = 
    [ ( (ξ'^(1)_1-√(-1)ξ'^(1)_2 )cosh(t^(i))^-1)^ℓ_0                                                                                              ;                                                                                             ⋱                                               ;                                                                                               ( (ξ'^(n)_1-√(-1)ξ'^(n)_2 )cosh(t^(i))^-1)^ℓ_0 ]

with the Vandermonde matrix, 

    ( ( (ξ'^(i)_1-√(-1)ξ'^(i)_2 )cosh(t^(i))^-1)^ℓ)_1⩽ i ⩽ n, 0 ⩽ℓ⩽ n-1

the statement follows once the points are sufficiently distinct in X(p,2)_-.
This can be arranged as in the algorithm <ref>: After specifying ε > 0, define the skew symmetric matrix 
S_ε = ( [     ε; -ε    ])
with Cayley transform k_ε = (I-S_ε)/(I+S_ε)
∈SO(2) and move the points x^(i) slightly within their 
K-orbit in G/H by acting with 

    [ I_p × p        ;             k_ε ]∈ K

where I_p × p is the identity matrix in SO(p).

Remark: Holomorphic discrete series representations 
on symmetric spaces of Hermitian type were studied in 
<cit.>, <cit.>.  
The Flensted-Jensen functions on these Hermitian spaces admit closed formulas involving consecutive powers of hyperbolic cosine, indexed by weights <cit.>, consistent with the formulas presented here.  
Consequently, our Whitney type extension theorems can be adapted (at least) to Hermitian spaces whose symmetry group is SU(1,1).




 §.§ Whitney representations
 
As in the classical cases of the sphere in section <ref> and of anti de Sitter space in section <ref> we associate a unitary representation  of the symmetry group SO(p,2)^∘ to a data set in the symmetric space SO(p,2)^∘/SO(p,1)^∘ in general position on which a nonconstant function f is defined. 

Let f_n be the Whitney function associated to this data, and 
let V_f_n be the (𝔤, K)-module  generated by f_n. It defines a unitary admissible representation which we call again the 
Whitney representation of the data set.  It is a direct sum of holomorphic discrete series representations of  SO(p,2)^∘. 

























 §.§ Data in orbits of orthogonal subgroups

Assume first that the orbits have  a holomorphic structure compatible with the holomorphic structure on X(p,,2)_-.
There is a nested sequence of orthogonal symmetric spaces 

    X(2,2)_-⊂ X(3,2)_-⊂⋯  X(p-1,2)_-⊂ X(p,2)_-.

Let p_0 be the smallest integer less than or equal to p such that all the points belong to X(p_0,2)_- and such that 
the data is in general position for the space X(p_0,2)_-.
We may then 
observe that the functions 
φ_ℓ^+
for 
    ℓ=(p+2)/2 + δ, (p+2)/2 + δ + 1,…, (p+2)/2+δ+(n-1),

where δ=1/2 if p is odd and δ =0 if p is even,
form a list of n independent Flensted-Jensen functions for all the spaces X(p',2)_- where p_0 ⩽ p'⩽ p.
In particular, there exists a Whitney function 

    f_n = ∑_(p+1)/2 +δ⩽ℓ⩽ (p+1)/2 +δ+ n-1 a_ℓφ_ℓ^+

for the data which is a Flensted-Jensen function for the ambient space X(p,2)_-: f_n is analytic and square integrable on X(p,2)_- and K-finite for K=SO(p)×SO(2) and f_n(x^(i))=y^(i) for all i.
Furthermore, f_n has the property that when restricted to X(p_0,2), 
f_n can be regarded as a Whitney function for the space X(p_0,2).

Remark: Suppose that the data are in an orbit of an orthogonal subgroup isomorphic to SO(p',1)^∘ for p' < p.
We conjecture that it is  best to proceed in two steps: 
 
     
  * In the first step, we use our algorithm to obtain a Whitney function for the proper orthogonal subspace X(p',1)_- of X(p,2)_-. 
     
  * In the second step, we use the results on holography by Kobayashi and Pevzner <cit.> to obtain an extension to X(p,2)_- and the  group SO(p,2)^∘. The restriction of the Whitney representations to orthogonal subgroups are known by the work of T. Kobayashi <cit.> and we may use this to understand the extension of a spherical Whitney function from a subgroup to the larger group. 
   


 

§ WHITNEY REPRESENTATIONS IN ELEMENTARY PARTICLE PHYSICS

 The Whitney representations of the generalized orthogonal groups introduced in <ref> may be of interest in physics <cit.>. For the sphere in the compact case, the representations are used to describe the hydrogen atom 
<cit.>. In the work of I. E. Segal our Whitney representations are used to describe electrons <cit.>; see also S. Sternberg <cit.>. The proceedings of the biannual conferences in Varna (Bulgaria) organized by V. Dobrev discuss more recent developments in applications of symmetries to physics <cit.>. 

amsplain 


 
*



 Paper Title: Quantum difference equations and Maulik-Okounkov quantum affine algebras of affine type $A$.



 
  
  To my grandpa Zhang Gongfu with highest admiration
  Yau Mathematical Sciences Center
ztq20@tsinghua.edu.cn

  

  
  Quantum difference equations and Maulik-Okounkov quantum affine algebras of affine type A 
    Tianqing Zhu
    
==========================================================================================

§ ABSTRACT  In this paper we prove the isomorphism of the positive half of the quantum toroidal algebra and the positive half of the Maulik-Okounkov quantum affine algebra of affine type A via the monodromy representation for the Dubrovin connection. The main tool is on the proof of the fact that the degeneration limit of the algebraic quantum difference equation is the same as that of the Okounkov-Smirnov geometric quantum difference equation.




§ INTRODUCTION





 §.§ Maulik-Okounkov quantum algebras

Stable envelopes were first introduced by Maulik and Okounkov in <cit.>. It is a Steinberg correspondence class in the equivariant cohomology H_G^*(X× X^A). Here X is a symplectic resolution, X^A is the torus A-fixed point over X, and A⊂ G is a subtorus of a Lie group G. The stable envelope is a unique class determined by the root chamber C of the corresponding Lie group G, the supporting condition and the degree condition.

In the case of the quiver varieties M_Q(v,w) of quiver type Q, stable envelopes can be used to construct the geometric R-matrix R_Q,C_ij(u_i-u_j). The geometric R-matrix satisfies the rational Yang-Baxter equation with spectral parametres. Using the RTT formalism, we can use the geometric R-matrix to construct the Maulik-Okounkov Yangian algebra Y_ħ^MO(g_Q). The MO Yangian algebra Y_ħ^MO(g_Q) contains the Maulik-Okounkov Lie algebra g_Q^MO, which is generated by the first-order expansion coefficients r_Q of the geometric R-matrix:

    R_Q(u)=Id+r_Q/u+O(u^-2)

This gives a natural filtration on Y_ħ^MO(g_Q) such that gr Y_ħ^MO(g_Q)≅ U(g_Q[u]) as mentioned in <cit.>.

The Maulik-Okounkov Lie algebra admits the root decomposition written as:

    g_Q=h⊕⊕_α∈Z^Ig_α=h⊕n_Q^MO⊕n_Q^MO,-


Here n_Q^MO is the Lie subalgebra consisting of all the positive root vectors. It was proved in <cit.> that g_Q^MO is isomorphic to the the Borcherd-Kac-Moody algebra g_Q of the corresponding quiver type Q using the BPS Lie algebra and the perverse filtration over the preprojective cohomological Hall algebra. It was also proved in <cit.> that the positive half of the MO Yangian is isomorphic to the preprojective cohomological Hall algebra of the corresponding quiver type Q.


Moreover, stable envelopes can also be defined in the settings of the equivariant K-theory. It was introduced in <cit.> and <cit.>, and the existence was proved in <cit.> and <cit.>.  It is a class Stab_C,s in K_G(X× X^A) with similar restriction condition as in the case of the equivariant cohomology, and it not only is determined by the root chamber C, but also determined by the slope parametre s∈Pic(X)⊗Q. 

Similarly we can define the K-theoretic geometric R-matrix R_Q,C_ij^s(u_i/u_j) of slope s from the K-theoretic stable envelope of slope s. Also one can use the RTT formalism to construct the Maulik-Okounkov quantum affine algebra U_q^MO(ĝ_Q). It has been proved in <cit.> that there is an algebra embedding of the Drinfeld double A of the preprojective K-theoretic Hall algebra into the MO quantum affine algebra:

    A↪ U_q^MO(ĝ_Q)


In the case of finite ADE, the MO quantum affine algebra U_q^MO(ĝ_Q) is isomorphic to the quantum group U_q(ĝ_Q) of affine ADE type.

In the case when Q is the affine type A, the Drinfeld double A is isomorphic to the quantum toroidal algebra U_q,t(ŝ̂l̂̂̂_n), so we have the algebra embedding:

    U_q,t(ŝ̂l̂̂̂_n)↪ U_q^MO(ĝ_Q)






 §.§ Quantum difference equations



Quantum difference equation was first introduced in <cit.> as the difference equation of the Kahler variable z for the capping operator J(u,z). The capping operator appears in the theory of  quasimap counting from P^1 to the quiver varieties M(v,w). The difference equation is written as:

    J(p^Lz,u)L=M_L(u,z)J(z,u)


The operator M_L(u,z) is called the geometric quantum difference operator. It turns out that <cit.> the geometric quantum difference operator M_L(u,z) is conjugate to the ordered product of the geometric monodromy operators B^MO_w(z) via the stable envelope Stab_C,s of slope s:

    B_L^MO,s(z)=L∏^←_w∈[s,s-L)B_w^MO(z)

with each monodromy operator B_w^MO(z)∈U_q^MO(g_w) an element in the completion of the MO wall subalgebra.

It is also proved in<cit.> that the difference operators A^s_L:=T_L^-1M_L(u,z) commute with each other for arbitrary L∈Pic(M(v,w)), and it is independent of the choice of the path from s to s-L.

There is also the algebraic analog of the quantum difference equation. In <cit.>, we have constructed the algebraic quantum difference equation for the quantum toroidal algebra U_q,t(ŝ̂l̂̂̂_n). It is constructed via constructing the monodromy operators B_m(z)∈B_m in each slope subalgebras B_m. It is proved in <cit.> that the algebraic quantum difference operator:

    B_L^s=L∏^←_m∈[s,s-L)B_m(z)

is independent of the choice of the path from s to s-L. 





 §.§ Main result of the paper

The main result of the paper is the isomorphism of the positive half of the MO quantum affine algebra of affine type A and the positive half of the quantum toroidal algebras:
(See Theorem <ref>)
The positive half of the quantum toroidal algebra U_q,t^+(ŝ̂l̂̂̂_n) is isomorphic to the positive half of the Maulik-Okounkov quantum affine algebra U_q^MO,+(ĝ_Q) for the affine type A.


The strategy of the proof depends on the analysis of the degeneration limit, or cohomological limit, of the geometric quantum difference equation of the affine type A. Then we do the comparison with the degeneration limit of the algebraic quantum difference equation studied in <cit.><cit.>.

Here we give the sketch of the proof of the Theorem <ref> in this paper:


  * The property of both MO quantum affine algebra and quantum toroidal algebra that is used in the proof is the factorisation property <cit.><cit.>, which means that they are all generated by the wall R-matrices R_w^MO and R_w and the Cartan generators. For the positive half, we only need to focus on the wall R-matrices in both quantum toroidal algebra and MO quantum affine algebra. We need to prove that R_w^MO=R_w.

  * In <cit.>, we have proved that the degeneration limit of the algebraic quantum difference equation of affine type A is the Dubrovin connection of the quantum equivariant cohomology of the affine type A quiver varieties. The Dubrovin connection of affine type A is written as:

    ∇_λ:=d_λ-Q(λ)=d_λ-c_1(λ)∪–(∑_i<j(λ·[i,j))/1-q^-[i,j)E_[i,j)E_-[i,j)),     E_±[i,j)∈ŝl̂_n


This connection has the regular singularities around q^-[i,j)=1 and q=0,∞. Using the result from the quantum difference equations, one can show that <cit.> the monodromy representation of the flat section of <ref> is generated by m((1⊗ S_m)(R_m^-)^-1).


  * In this paper we first show that we have the similar degeneration limit of the geometric quantum difference equation to the Dubrovin connections:
(See Theorem <ref>)
The degeneration limit A^s,coh_L of the Okounkov-Smirnov quantum difference equation A^s_L of the affine type A quiver varieties is the Dubrovin connection of the affine type A quiver varieties.


Using the similar analysis of the algebraic quantum difference equations. We can compute the generators of the monodromy representations in terms of the MO quantum affine algebras:
(See Theorem <ref>)
The monodromy representation:

    π_1(P^n\Sing,0^+)→Aut(H_T(M(v,w)))

of the Dubrovin connection is generated by B_w,MO^* with q_1=e^2π iħ_1,q_2=e^2π iħ_2, i.e. the monodromy operators B_w,MO=m((1⊗ S_w)(R_w^-,MO)^-1) in the fixed point basis. The based point 0^+ is a point sufficiently close to 0∈P^n.



  * We can also show that the fundmental solution Ψ_0,∞^(MO)(z) at z=0,∞ of both algebraic and geometric quantum difference equation have the same degeneration limit solution ψ_0,∞(z) of the Dubrovin connection. Using the result, we can obtain that the monodromy representation from algebraic and geometric sides are the same:
(See Theorem <ref>)
As the element in End(K(v,w)), we have that:

    m((1⊗ S_w)R_w^MO)=m((1⊗ S_w)R_w)



Thus using the lemma <ref>, we obtain that R_w^MO=R_w. Since U_q^+,MO(ĝ_Q) is generated by R_w^MO, U_q,t^+(ŝ̂l̂̂̂_n) is generated by R_w because of the slope factorisation. We obtain the main result.







 §.§ Outline of the paper

The paper is organised as follows: In section two we introduce the quantum toroidal algebra U_q,t(ŝ̂l̂̂̂_n), its slope subalgebra and the slope factorisation. We also introduce the wall subalgebra in the slope subalgebra, and the quantum toroidal algebra action on the equivariant K-theory of the Nakajima quiver varieties. 

In section three we introduce the Maulik-Okounkov quantum affine algebra and its corresponding Maulik-Okounkov wall subalgebra. We will explain how to get the degeneration limit, or cohomological limit, of the MO wall subalgebra, to the Lie subalgebra of the MO Lie algebra.

In section four we will provide key technique in the paper, we will show that the wall subalgebra of the quantum toroidal algebra is the Hopf subalgebra of the Maulik-Okounkov wall subalgebra. Using the result we prove that the degeneration limit of the wall subalgebra and the MO wall subalgebra are the same.

In section five we introduce the Okounkov-Smirnov geometric quantum difference equation and the algebraic quantum difference equation. We will give the analysis of the solution of the Okounkov-Smirnov quantum difference equation.

In section six we will show that the degeneration limit of the geometric quantum difference equation is the Dubrovin connection. Using this fact we will compute the monodromy representation of the Dubrovin connection in terms of the wall R-matrices of the MO wall subalgebra.

In section seven we will combine all the result above to prove the main theorem.



 §.§ Further directions

The main theorem <ref> implies that the algebraic quantum difference equation is the same as the Okounkov-Smirnov quantum difference equation of affine type A. This implies that the regular part of the fundamental solution of the algebraic quantum difference equation is the capping operator of the affine type A quiver varieties.

One further direction is the generalisation of the theorem to arbitrary quiver type. This requires the further understanding about the slope subalgebra for the Drinfeld double of arbitrary preprojective K-theoretic Hall algebra. The slope subalgebra of arbitrary preprojective KHA was defined in <cit.>. The techniques about dealing with the degeneration limit of the slope subalgebra of arbitrary type is different, and we will leave it in the future work.

The second further direction is the computation of the R-matrix of the quantum toroidal algebra valued in the Fock module. The computation via the generators of the quantum toroidal algebra can be complicated via the bosonic operators, while the computation on the geometric side can be reduced via the techniques of "the universal cover of the quiver" mentioned in <cit.>. It turns out that in this case we can write down the geometric R-matrix in terms of the fermionic operators as in <cit.>. The computation on both sides would reveal some boson/fermion correspondence for the R-matrix. This might reveal new structures in the theory of symmetric polynomials. We will put it in our future work.

The third further direction is on the difference equation related to the vertex function associated to the affine type A quiver varieties. There is a strong connection between the qKZ equation and the vertex functions. As it was shown in <cit.><cit.> that the vertex function for the cotangent bundle of flag varieties are related to the trigonometric Ruijsenaars-Schneider model, which can be constructed via the type A trigonometric R-matrix. We expect that the work would give light to the details of the connection between the vertex function and the quantum integrable system related to the quantum toroidal algebra.

The vertex function can also been interpreted as the "inner product" of Whittaker vectors for Verma modules of the quantised Coulomb Branch. It has been conjectured that the equivariant K-theory of the quasimap moduli space from P^1 to quiver varieties is the Verma module for the K-theoretic quantised Coulomb Branch of the corresponding quiver type. The conjecture is partially confirmed for hypertoric varieties in <cit.>. We expect that the isomorphism would provide tools for proving the conjecture.



 §.§ Acknowledgments.
The author would like to thank Andrei Neguţ, Andrei Okounkov, Andrey Smirnov, Hunter Dinkins and Nicolai Reshetikhin for their helpful discussions on shuffle algebras, quantum difference equations and qKZ equations. The author is supported by the international collaboration grant BMSTC and ACZSP grant. Part of this work was done while the author was visiting Department of Mathematics at Columbia University. The author thanks for their hospitality and provisio of excellent working environment.




§ QUANTUM TOROIDAL ALGEBRA AND QUIVER VARIETIES




 §.§ Quantum toroidal algebra U_q,t(ŝ̂l̂̂̂_n)


The quantum toroidal algebra U_q,t(ŝ̂l̂̂̂_n) is a Q(q,t)-algebra defined as:

    U_q,t(ŝ̂l̂̂̂_n)=Q(q,t)⟨{e_i,d^±}_1≤ i≤ n^d∈Z,{φ_i,d^±}_1≤ i≤ n^d∈N_0⟩/( )

The relation between the generators can be described in the generating functions:

    e_i^±(z)=∑_d ∈ℤ e_i, d^± z^-d  φ_i^±(z)=∑_d=0^∞φ_i, d^± z^∓ d


with φ_i,d^± commute among themselves and:

    e_i^±(z) φ_j^±^'(w) ·ζ(w^± 1/z^± 1)=φ_j^±^'(w) e_i^±(z) ·ζ(z^± 1/w^± 1) 
    
    e_i^±(z) e_j^±(w) ·ζ(w^± 1/z^± 1)=e_j^±(w) e_i^±(z) ·ζ(z^± 1/w^± 1) 
    [e_i^+(z), e_j^-(w)]=δ_i^j δ(z/w) ·φ_i^+(z)-φ_i^-(w)/q-q^-1


Here i,j∈{1,⋯,n} and z,w are variables of color i and j. Here:

    ζ(x_i/x_j)=[x_j/q t x_i]^δ_j-1^i[t x_j/q x_i]^δ_j+1^i/[x_j/x_i]^δ_j^i[x_j/q^2 x_i]^δ_j^i


with the Serre relation:

    e_i^±(z_1) e_i^±(z_2) e_i ±^' 1^±(w)+(q+q^-1) e_i^±(z_1) e_i ±^' 1^±(w) e_i^±(z_2)+e_i ±^' 1^±(w) e_i^±(z_1) e_i^±(z_2)+
        +e_i^±(z_2) e_i^±(z_1) e_i ±^' 1^±(w)+(q+q^-1) e_i^±(z_2) e_i ±^' 1^±(w) e_i^±(z_1)+e_i ±^' 1^±(w) e_i^±(z_2) e_i^±(z_1)=0


The standard coproduct structure is imposed as:

    Δ: U_q,t(ŝ̂l̂̂̂_n)⟶ U_q,t(ŝ̂l̂̂̂_n)⊗ U_q,t(ŝ̂l̂̂̂_n)



    [ Δ(e_i^+(z))=φ_i^+(z) ⊗ e_i^+(z)+e_i^+(z) ⊗ 1               Δ(φ_i^+(z))=φ_i^+(z) ⊗φ_i^+(z);  Δ(e_i^-(z))=1 ⊗ e_i^-(z)+e_i^-(z) ⊗φ_i^-(z)               Δ(φ_i^-(z))=φ_i^-(z) ⊗φ_i^-(z) ]




 §.§ Quantum affine algebra U_q(ĝl̂_n)

The quantum affine algebra U_q(ĝl̂_n) is a Q(q)-algebra generated by:

    Q(q)⟨ e_±[i;j],ψ_s^±1,c^±1⟩^s∈{1,⋯,n}_(i<j)∈Z^2/(n,n)Z

The generators e_±[i;j) satisfy the well-known RTT relation:

    R(z/w)T_1^+(z)T_2^+(w)=T_2^+(w)T_1^+(z)R(z/w)


    R(z/w)T_1^-(z)T_2^-(w)=T_2^-(w)T_1^-(z)R(z/w)


    R(z/wc)T_2^-(w)T_1^+(z)=T_1^+(z)T_2^-(w)R(zc/w)


Here T^±(z) are the generating function for e_±[i;j):

    T^+(z)=∑_1 ≤ i ≤ n^i ≤ j e_[i ; j)ψ_i · E_j  n, i z^⌈j/n⌉-1
    
    T^-(z)=∑_1 ≤ i ≤ n^i ≤ j e_-[i ; j)ψ_i^-1· E_i, j  n z^-⌈j/n⌉+1


And R(z/w) is the standard R-matrix for gl_n:

    R(z/w)=∑_1 ≤ i, j ≤ n E_i i⊗ E_j j(z q-w q^-1/w-z)^δ_j^i+(q-q^-1) ∑_1 ≤ i ≠ j ≤ n E_i j⊗ E_j iw^δ_i>j z^δ_i<j/w-z


In addition, we have the relations:

    ψ_k · e_±[i ; j)=q^±(δ_k^i-δ_k^j) e_±[i ; j)·ψ_k   ∀arcs[i ; j)  and  k ∈ℤ


The algebra U_q(ĝl̂_n) has the coproduct structure given by:

    Δ(T^+(z))=T^+(z)⊗ T^+(zc_1),    Δ(T^-(z))=T^-(zc_2)⊗ T^-(z)


and the antipode map:

    S(T^±(z))=(T^±(z))^-1


The corresponding antipode map elements can be written as:

    S^+(z)=∑_1 ≤ i ≤ n^i ≤ j(-1)^j-i f_[i ; j)ψ_j · E_j  n, i z^⌈j/n⌉-1
    
    S^-(z)=∑_1 ≤ i ≤ n^i ≤ j(-1)^j-i f_-[i ; j)ψ_j^-1· E_i, j  n z^-⌈j/n]+1


Unwinding the relation we have that:

    e_±[a,c)e_±[b,d)/q^δ^b_a-δ^d_b+δ^d_a-e_±[b,d)e_±[a,c)/q^δ^b_c-δ^d_b+δ^d_c=(q-q^-1)[∑_a≤ x<c^x≡ de_±[b,c+d-x)e_±[a,x)-∑_a<x≤ c^x≡ be_±[x,c)e_±[a+b-x,d)]



    [e_[a,c),e_-[b,d)]=(q-q^-1)[∑^x≡ b_a≤ x<ce_[c+b-x,d)e_[a,x)/q^δ^b_c+δ^a_c-δ^a_bψ_x/ψ_c-∑_a<x≤ c^x≡ de_[x,c)e_-[b,a+d-x)/q^-δ^b_a+δ^d_b-δ^d_aψ_x/ψ_a]




The isomorphism U_q(ĝl̂_n)≅ U_q(ŝl̂_n)⊗ U_q(ĝl̂_1) is given by the following:

    e_[i;i+1)=x_i^+(q-q^-1)
    
    e_-[i;i+1)=x_i^-(q^-2-1)


For the generators {p_k}_k∈Z of the Heisenberg algebra U_q(ĝl̂_1)⊂ U_q(ĝl̂_n), they satisfy the following commutation relations:

    [p_k,p_-k]=c^nk-c^-nk/n_k,     n_k=q^k-q^-k/k



It can be packaged into the following proposition proved by Neguţ in <cit.>:

There exists constants α_1,α_2,⋯∈Q(q) such that:

    f_±[i ; j)=∑_k=0^⌊j-i/n⌋ e_±[i ; j-n k) g_± k

Here ∑_k=0^∞g_± kx^k=exp(∑_k=1^∞α_kp_± kx^k)




 §.§ Shuffle algebra realization of U_q,t(ŝ̂l̂̂̂_n)

Here we review the reconstruction of the quantum toroidal algebra via the shuffle algebra and the induced slope factorization of the quantum toroidal algebra. For details see <cit.>.

Fix a fractional field F=Q(q,t) and consider the space of symmetric rational functions:

    Sym(V):=⊕_k=(k_1,⋯,k_n)∈N^nF(⋯,z_i1,⋯,z_ik_i,⋯)^Sym_1≤ i≤ n

Here "Sym" means to symmetrize the rational function for each z_i1,⋯,z_ik_i. We endow the vector space with the shuffle product:

    F*G=Sym[F(⋯,z_ia,⋯)G(⋯,z_jb,⋯)/k!l!∏_1≤ a≤ k_i^1≤ i≤ n∏_k_j+1≤ b≤ k_j+l_j^1≤ j≤ nζ(z_ia/z_jb)]


We define the subspace S^±_k⊂Sym(V) by:

    S^+:={F(⋯,z_ia,⋯)=r(⋯,z_ia,⋯)/∏_1≤ a≠ b≤ k_i^1≤ i≤ n(qz_ia-q^-1z_ib)}


where r(⋯,z_i1,⋯,z_ik_i,⋯)^1≤ i≤ n_1≤ a≤ k_i is any symmetric Laurent polynomial that satisfies the wheel conditions:

    r(⋯,q^-1,t^±,q,⋯)=0

for any three variables of colors i,⋯,i±1,i.

The shuffle algebra S^+ has two natural bigrading given by the number of the variables in each color k∈N^n and the homogeneous degree of the rational functions d∈Z:

    S^+=⊕_(k,d)∈N^n×ZS^+_k,d


Similarly we can define the negative shuffle algebra S^-:=(S^+)^op which is the same as S^+ with the opposite shuffle product. Now we slightly enlarge the positive and negative shuffle algebras by the generators {φ_i,d^±}_1≤ i≤ r^d≥0:

    S^≥=⟨S^+,{(φ^+_i,d)^d≥0_1≤ i≤ r}⟩,    S^≤=⟨S^-,{(φ^-_i,d)^d≥0_1≤ i≤ r}⟩

And here φ^±_i,d commute with themselves and have the relation with S^± as follows:

    φ^+_i(w)F=Fφ^+_i(w)∏_1≤ a≤ k_j^1≤ j≤ nζ(w/z_ja)/ζ(z_ja/w)


    φ^-_i(w)G=Gφ^-_i(w)∏_1≤ a≤ k_j^1≤ j≤ nζ(z_ja/w)/ζ(w/z_ja)


The Drinfeld pairing ⟨-,-⟩:S^≤⊗S^≥→Q(q,t) between the positive and negative shuffle algebras S^≥,≠ is given by:

    ⟨φ^-_i(z),φ^+_j(w)⟩=ζ(w/z)/ζ(z/w)
       ⟨ G,F⟩=1/k!∫_| z_ia|=1^| q|<1| p|G(⋯,z_ia,⋯)F(⋯,z_ia,⋯)/∏_a≤ k_i,b≤ k_j^1≤ i,j≤ nζ_p(z_ia/z_jb)∏_1≤ a≤ k_i^1≤ i≤ ndz_ia/2π iz_ia|_p↦ q

for F∈S^+, G∈S^-. Here ζ_p is defined as follows:

    ζ_p(x_i/x_j)=ζ(x_i/x_j)[x_j/p^2x_i]^δ^i_j/[x_j/q^2x_i]^δ^i_j


This defines the shuffle algebra S:=S^≤⊗̂S^≥. Also the coproduct Δ:S→S⊗̂S is given by:

    Δ(φ_i^±(w))=φ^±_i(w)⊗φ^±_i(w)
       Δ(F)=∑_0≤l≤k[∏_1≤ j≤ n^b>l_jφ_j^+(z_jb)]F(⋯,z_i1,⋯,z_il_i⊗ z_i,l_i+1,⋯,z_ik_i,⋯)/∏_1≤ i≤ n^a≤ l_i∏_1≤ j≤ n^b>l_jζ(z_jb/z_ia),     F∈S^+
       Δ(G)=∑_0≤l≤kG(⋯,z_i1,⋯,z_il_i⊗ z_i,l_i+1,⋯,z_ik_i,⋯)[∏_1≤ j≤ n^b>l_jφ_j^-(z_jb)]/∏_1≤ i≤ n^a≤ l_i∏_1≤ j≤ n^b>l_jζ(z_ia/z_jb),     G∈S^-


In human language, the above coproduct formula means the following:For the right hand side of the formula, in the limit | z_ia|<<| z_jb| for all a≤ l_i and b>l_i, and then place all monomials in {z_ia}_a≤ l_i to the left of the ⊗ symbol and all monomials in {z_jb}_b>l_j to the right of the ⊗ symbol. Also we have the antipode map S:S→S which is an anti-homomorphism of both algebras and coalgebras:

    S(φ^+_i(z))=(φ^+_i(z))^-1,     S(F)=[∏_1≤ a≤ k_i^1≤ i≤ n(-φ_i^+(z_ia))^-1]*F
       S(φ^-_i(z))=(φ^-_i(z))^-1,     S(G)=G*[∏_1≤ a≤ k_i^1≤ i≤ n(-φ_i^-(z_ia))^-1]


The following theorem has been proved by<cit.>:

There is a bigraded isomorphism of bialgebras

    Y:U_q,t(ŝ̂l̂̂̂_n)→S

given by:

    Y(φ_i,d^±)=φ^±_i,d,     Y(e_i,d^±)=z_i1^d/[q^-2]





 §.§ Slope subalgebra of the shuffle algebra

One of the wonderful property for the shuffle algebra is that it admits the slope factorization.
For the positive half of the shuffle algebra S^+, we define the subspace of slope ≤m as:

    S^+_≤m:={F∈S^+|lim_ξ→∞F(z_i1,⋯,z_ia_i,ξ z_i,a_i+1,⋯,ξ z_i,a_i+b_i)/ξ^m·b-(a,b)/2<∞∀a,b}


Similarly, for the negative half:

    S^-_≤m={G∈S^-|lim_ξ→0G(ξ z_i1,⋯,ξ z_ia_i,z_i,a_i+1,⋯,z_i,a_i+b_i)/ξ^-m·a+(a,b)/2<∞∀a,b}


It can be checked that for the elements F∈S_≤m^+, G∈S_≤m^-:

    Δ(F)=Δ_m(F)+(anything)⊗(slope<m),    Δ_m(F)=∑_a+b=klim_ξ→∞Δ(F)_a⊗b/ξ^m·b



    Δ(G)=Δ_m(G)+(slope<m)⊗(anything),    Δ_m(G)=∑_a+b=klim_ξ→∞Δ(G)_a⊗b/ξ^-m·a



The slope subalgebra B_m^± is defined as:

    B^±_m:=⊕_(k,d)∈N^n×Z^m·k=dS^±_≤m∩S^±_±k,d


It is easy to see that the morphism Δ_m descends to a true coproduct on B_m^≥,≤:=B_m^±⊗F[φ_i^±1]. Using the Drinfeld double defined for the shuffle algebra. We eventually obtain the slope subalgebra B_m⊂S.



It is known that the slope subalgebra B_m for the quantum toroidal algebra U_q,t(sl_n) is isomorphic to:

    B_m≅⊗^g_h=1U_q(gl_l_h)



For the proof of the above isomorphism <ref> see <cit.>. The isomorphism is constructed in the following way. For m∈Q^n, it is proved in <cit.> that as the shuffle algebra, 
The generators of the slope algebra B_m can be written in the following way. For m·[i;j⟩∈Z, we denote the following elements:

    P_±[i ; j)^±𝐦=Sym[∏_a=i^j-1 z_a^⌊ m_i+…+m_a⌋-⌊ m_i+…+m_a-1⌋/t^ind  d_[i, j⟩^m q^i-j∏_a=i+1^j-1(1-q_2 z_a/z_a-1)∏_i ≤ a<b<jζ(z_b/z_a)]∈S^±


    Q_∓[i ; j)^±𝐦=Sym[∏_a=i^j-1 z_a^⌊ m_i+…+m_a-1⌋-⌊ m_i+…+m_a⌋/t^-ind_[i, j)^m∏_a=i+1^j-1(1-q_1 z_a-1/z_a)∏_i ≤ a<b<jζ(z_a/z_b)]∈S^∓


Here ind^m_[i;j) is defined as:

    ind^m_[i;j)=∑_a=i^j-1(m_i+⋯+m_a-⌊ m_i+⋯+m_a-1⌊)


The positive and negative part of the slope subalgebra B_m^± for the shuffle algebra S^± can be defined as the algebra generated by {P^m_±[i;j)}_i≤ j and {Q^m_±[i;j)}_i≤ j. We can define the slope subalgebra B_m as the Drinfeld double of B_m^± with the Cartan elements {φ_±[i;j)} by the bialgebra pairing defined in <ref> and <ref>.

We can check that the antipode map S_m:B_m→B_m has the following relation:

    S_m(P^m_[i;j))=Q^m_[i;j),     S_m(Q^m_-[i;j))=P^m_-[i;j)



    Δ_𝐦(P_[i ; j)^𝐦)=∑_a=i^j P_[a ; j)^𝐦φ_[i ; a)⊗ P_[i ; a)^𝐦  Δ_𝐦(Q_[i ; j)^m)=∑_a=i^j Q_[i ; a)^𝐦φ_[a ; j)⊗ Q_[a ; j)^𝐦



    Δ_𝐦(P_-[i ; j)^𝐦)=∑_a=i^j P_-[a ; j)^𝐦⊗ P_-[i ; a)^𝐦φ_-[a ; j)  Δ_𝐦(Q_-[i ; j)^𝐦)=∑_a=i^j Q_-[i ; a)^𝐦⊗ Q_-[a ; j)^𝐦φ_-[i ; a)



Here Δ_m is defined as (5.27) and (5.28) in <cit.>. The isomorphism <ref> is given by:

    e_[i;j)=P^m_[i;j)_h,     e_-[i;j)=Q^m_-[i;j)_h,    φ_k=φ_[k;v_m(k))


We have the factorisation of the quantum toroidal algebra proved in <cit.> and <cit.>:

There is an isomorphism of algebras:

    U_q,t^+(ŝ̂l̂̂̂_n)≅⊗_μ∈QB_m+μθ^+,    m∈Q^n,θ∈(Z^+)^n


Moreover, 

    U_q,t(ŝ̂l̂̂̂_n)≅⊗_μ∈Q∪{∞}B_m+μθ,    m∈Q^n,θ∈(Z^+)^n


The isomorphism induces the Hopf algebra embedding B_m↪ U_q,t(ŝ̂l̂̂̂_n).


The theorem gives the factorisation of the universal R-matrix:

    R^∞=∏_m∈Q^→(R_s+mθ^+)R_∞=(∏_m∈Q^→∏_h=1^g^(m)R_U_q(ĝl̂_h^(m))^+)· R_Heisenberg^⊗ n


Now we choose arbitrary m∈Q^n, we can twist the coproduct by:

    Δ_(m)(a)=[∏^→_r∈Q_>0∪{∞}R_m+rθ^+]·Δ(a)·[∏^→_r∈Q_>0∪{∞}R_m+rθ^+]^-1



    R^m=∏^←_μ∈Q_-R_m+μθ^-R_∞∏^←_μ∈Q_+⊔{0}R_m+μθ^+


The antipode map S can be twisted as:

    S_m(a)=U_mS(a)U_m^-1,     U_m=m((1⊗ S)[∏^→_r∈Q_>0∪{∞}R_m+rθ^+])


This makes (U_q,t(ŝ̂l̂̂̂_n),Δ_(m),S_m,ϵ,η) a Hopf algebra over Q(q,t).

It turns out that B_m is a Hopf subalgebra of (U_q,t(ŝ̂l̂̂̂_n),Δ_(m),S_m,ϵ,η):

Δ_(m)(a)=Δ_m(a) when a∈B_m. Morevoer, this means that B_m is a Hopf subalgebra of (U_q,t(ŝ̂l̂̂̂_n),Δ_(m),S_m,ϵ,η).

For the proof see <cit.>.








 §.§ Wall subalgebra for the slope subalgebra


The slope factorisation of the quantum toroidal algebra implies that the wall set for affine type A quiver varieties are given as:

    {m∈R^n|m·[i,j)∈Z}

This means that each wall hyperplane is indexed by w=([i,j),n)∈Z^n×Z. In this way we can naively "define" the wall subalgebra U_q(g_w) as:

    ⊕_k∈N^m·[i,j)=nS^+_≤m∩S^+_k[i,j),kn


It is easy to check that this is a subalgebra of B_m but not a Hopf subalgebra of B_m. So we need some modification of the definition.


To make sense of the wall subalgebra, first we write down the slope factorisation of the universal R-matrix of the quantum toroidal algebra:

    R^m=∏^←_μ∈Q_-R_m+μθ^-R_∞∏^←_μ∈Q_+⊔{0}R_m+μθ^+


We take the universal R-matrix in the representation K(w_1)⊗ K(w_2). Moreover, R is also a linear morphism in the subspace:

    R^m:⊕_v_1+v_2=vK(v_1,w_1)⊗ K(v_2,w_2)→⊕_v_1+v_2=vK(v_1,w_1)⊗ K(v_2,w_2)


and so is each slope R-matrix R_m+μθ^±. Since each slope subalgebra B_m is isomorphic to ⊗_h=1^g U_q(ĝl̂_l_h), the universal R-matrix for B_m can be factorised as:

    R_m^±=q^Ω∏^←_αR_α^±

via using the Khoroshkin-Tolstoy factorisation. Here α are the affine roots corresponding to the wall hyperplane intersecting with the slope point m with some order, i.e. ⟨α,m⟩∈Z. If we restrict the morphism to ⊕_v_1+v_2=vK(v_1,w_1)⊗ K(v_2,w_2), we can see that the ordered product ∏^←_αR_α^± will be reduced to finitely many product terms.

Thus restricted to ⊕_v_1+v_2=vK(v_1,w_1)⊗ K(v_2,w_2), we can formally write down the universal R-matrix R^m of slope m as:

    R^m=∏^←_w<mR_w^-R_∞∏^←_w≥mR_w^+


It is easy to check that q^ΩR_w^± satisfies the Yang-Baxter equation, and the generators in R_w^± generates a Hopf subalgebra.


We define the wall subalgebra U_q(g_w)⊂B_m as the subalgebra generated by the universal R-matrix R_w.


In our examples, since all the slope subalgebras B_m is isomorphic to ⊗_h=1^gU_q(ĝl̂_l_h) generated by φ_[i,j), P^m_±[i,j), an easy computation check for the universal R-matrix can show that all the wall subalgebra U_q(g_w) is either isomorphic to U_q(sl_2) or U_q(ĝl̂_1) generated by φ_[i,j) and P^m_[i,i+1)_h with [i,i+1)_h living on the wall w or p_k the Heisenberg operators in ⊗_h=1^gU_q(ĝl̂_l_h) determined by the formulax of the Proposition <ref>.






 §.§ Geometric action on K_T(M(w))

We first fix the notation for the Nakajima quiver varieties.

Given a quiver Q=(I,E), consider the following quiver representation space

    Rep(v,w):=⊕_h∈ EHom(V_i(h),V_o(h))⊕⊕_i∈ IHom(V_i,W_i)


here v=(dim(V_1),⋯,dim(V_I)) is the dimension vector for the vector spaces at the vertex, w=(dim(W_1),⋯,dim(W_I)) is the dimension vector for the framing vector spaces.

Denote G_v:=∏_i∈ IGL(V_i) and G_w:=∏_i∈ IGL(W_i). There is a natural action of G_v and G_w on Rep(v,w), and thus a natural Hamiltonian action on T^*Rep(v,w) with respect to the standard symplectic form ω, now we have the moment map

    μ:T^*Rep(v,w)→g_v^*



    μ(X_e,Y_e,A_i,B_i)=∑_eX_i(e)Y_i(e)-Y_o(e)X_o(e)+A_iB_i


One can define Nakajima variety:

    M_θ,0(v,w):=μ^-1(0)//_θG_v


with all the θ-stable G_v-orbits in μ^-1(0). Here we choose the stability condition θ=(1,⋯,1).

We will abbreviate M(v,w) as the Nakajima quiver variety defined in <ref>.

In this paper we only consider the affine type A quiver, i.e. the edges are given by e=(i,i+1) with i=imod(n). 




The action of C^*_q×C^*_t× G_w on M(v,w) is given by:

    (q,t,U_i)·(X_e,Y_e,A_i,B_i)_e∈ E,i∈ I=(qtX_e,qt^-1Y_e,qA_iU_i,qU_i^-1B_i)


The fixed points set of the torus action C^*_q×C^*_t on M_θ,0(v,w) is indexed by the |w|-partitions λ=(λ_1,⋯,λ_w). For each box □∈(λ_1,⋯.λ_w), we define the following character function:

    χ_□=u_iq^x+y+1t^x-y,    □∈λ_i



In our context, we will denote M(v,w) as M_θ,0(v,w). We always denote K_G_w×C_q^*×C_t^*(M(v,w)) as the T_w×C_q^*×C_t^*-localised equivariant K-theory of M(v,w). It has the basis denoted by the multipartition |λ⟩.
	



 If we choose the cocharacter σ:C^*→ T_w such that w=u_1w_1+⋯+u_kw_k, the fixed point component can be written as:

    M(v,w)^σ=_v_1+⋯+v_k=vM(v_1,w_1)×⋯× M(v_k,w_k)


We denote:

    K_T(M(w)):=⊕_wK_C_q^*×C_t^*× G_w(M(v,w))_loc


Thus we can choose the fixed point basis |λ⟩ of the corresponding partition λ to span the vector space K_T(M(w)).


Here we briefly review the geometric action of U_q,t(ŝ̂l̂̂̂_n) on K_T(M(w)). The construction is based on Nakajima's simple correspondence. 

Consider a pair of vectors (v_+,v_-) such that v_+=v_-+e_i. There is a simple correspondence

    Z(v_+,v_-,w)↪ M(v_+,w)× M(v_-,w)

parametrises pairs of quadruples (X^±,Y^±,A^±,B^±) that respect a fixed collection of quotients (V^+→ V^-) of codimension δ^i_j with only the semistable and zeros part for the moment map μ for each M(v_+,w). And the variety Z(v_+,v_-,w) is smooth with a tautological line bundle:

    L|_V^+→ V^-=Ker(V_i^+→ V_i^-)

and the natural projection maps:

    Z(v_+,v_-,w)[ld,"π_+"][rd,"π_-"]   
    
    M(v_+,w)      M(v_-,w)


Using this we could consturct the operator:

    e_i,d^±:K_T(M(v^∓,w))→ K_T(M(v^±,w)),     e_i,d^±(α)=π_±*(Cone(dπ_±)L^dπ_∓^±(α))


and take all v we have the operator e_i,d^±:K_T(M(w))→ K_T(M(w)). Also we have the action of φ^±_i,d given by the multiplication of the tautological class, which means that:

    φ_i^±(z)=ζ(z/X)/ζ(X/z)∏^u_j≡ i_1≤ j≤w[u_j/qz]/[z/qu_j]


In particular, the element φ_i,0 acts on K_T(M(v,w)) as q^α_i^T(w-Cv).

The above construction gives the following well-known result <cit.><cit.>:

For all w∈N^r, the operator e_i,d^± and φ^±_i,d give rise to an action of U_q,t(ŝ̂l̂̂̂_n) on K_T(M(w)).


In terms of the shuffle algebra, we can give the explicit formula of the action of the quantum toroial algebra U_q,t(ŝ̂l̂̂̂_n) on K_T(M(w)):

Given F∈S_k^+, we have that

    ⟨λ|F|μ⟩=F(χ_λ\μ)∏_▪∈λ\μ[∏_□∈μζ(χ_▪/χ_□)∏_i=1^w[u_i/qχ_▪]]


Similarly, for G∈S_-k^-, we have

    ⟨μ|G|λ⟩=G(χ_λ\μ)∏_▪∈λ\μ[∏_□∈λζ(χ_□/χ_▪)∏_i=1^w[χ_▪/qu_i]]^-1


Here F(χ_λ\μ) and G(χ_λ\μ) are the rational symmetric function evaluation at the box χ_□ with □∈λ\μ.





§ MAULIK-OKOUNKOV QUANTUM AFFINE ALGEBRA




 §.§ Stable envelopes



We first review the definition of the K-theoretic stable envelopes for the quiver varieties.

Given M(v,w) a Nakajima quiver variety and G_v× A_E acting on M(v,w). Given a subtorus T⊂ G_v× A_E in the kernel of q. By definition, the K-theoretic stable envelope is a K-theory class

    Stab_C^s⊂ K_G(X× X^T)


such that it induces the morphism

    Stab_C,s:K_G(X^T)→ K_G(X)


such that if we write X^T=⊔_αF_α into components:

	
  * The diagonal term is given by the structure sheaf of the attractive space:
	
    Stab_C,s|_F_α× F_α=(-1)^rk T_>0^1/2(det(N_-)/detT_≠0^1/2)^1/2⊗O_Attr|_F_α× F_α


	
  * The T-degree of the stable envelope has the bounding condition for F_β≤ F_α:
	
    deg_TStab_C,s|_F_β× F_α+deg_Ts|_F_α⊂deg_TStab_C,s|_F_β× F_β+deg_Ts|_F_β


	We require that for F_β<F_α, the inclusion ⊂ is strict.



The uniqueness and existence of the K-theoretic stable envelope was given in <cit.> and <cit.>. In <cit.>, the consturction is given by the abelinization of the quiver varieties. In <cit.>, the construction is given by the stratification of the complement of the attracting set, which is much more general.

The stable envelope has the factorisation property called the triangle lemma <cit.>. Given a subtorus T'⊂ T with the corresponding chamber C_T',C_T, we have the following diagram commute:

    K_G(X^T)[rr,"Stab_C_T,s"][dr,"Stab_C_T/T',s"]      K_G(X)
       K_G(X^T')[ur,"Stab_C_T',s"]





 §.§ Maulik-Okounkov quantum algebra and wall subalgebras


Let us focus on the case of the quiver varieties M(v,w). Choose the framing torus σ:C^*→ A_w⊂ G_w such that:

    w=a_1w_1+⋯+a_kw_k


In this case the fixed point is given by:

    M(v,w)^σ=_v_1+⋯+v_k=vM(v_1,w_1)×⋯× M(v_k,w_k)


Denote K(w):=⊕_vK_G_w(M(v,w)), it is easy to see that the stable envelope Stab_s gives the map:

    Stab_C,s:K(w_1)⊗⋯⊗ K(w_k)→ K(w_1+⋯+w_k)


Using the K-theoretic stable envelope, we can define the geometric R-matrix as:

    R^s_C:=Stab_-C,s^-1∘Stab_C,s:K(w_1)⊗⋯⊗ K(w_k)→ K(w_1)⊗⋯⊗ K(w_k)


Written in the component of the weight subspaces, the geometric R-matrix can be written as:

    R^s_C:=Stab_-C,s^-1∘Stab_C,s:⊕_v_1+⋯+v_k=vK(v_1,w_1)⊗⋯⊗ K(v_k,w_k)
       →⊕_v_1+⋯+v_k=vK(v_1,w_1)⊗⋯⊗ K(v_k,w_k)



From the triangle diagram <ref> of the stable envelope, we can further factorise the geometric R-matrix into the smaller parts:

    R^s_C=∏_1≤ i<j≤ kR^s_C_ij(a_i/a_j),    R^s_C_ij(a_i/a_j):K(w_i)⊗ K(w_j)→ K(w_i)⊗ K(w_j)


Each R^s_C_ij(u) satisfy the trigonometric Yang-Baxter equation with the spectral parametres.

In the language of the integrable model, we denote V_i(u_i) as the modules of type K(w) defined above with the spectral parametre u_i. The formula <ref> means that:

    R^s_⊗^←_i∈ IV_i(u_i),⊗^←_j∈ JV_j(u_j)=∏_i∈ I^→∏_j∈ J^←R^s_V_i,V_j(u_i/u_j):


We can also consider the dual module V_i^*(u_i) as the module isomorphic to V_i(u_i) as graded vector space, with the R-matrices defined as:

    R^s_V_1^*,V_2=((R^s_V_1,V_2)^-1)^*_1
       R^s_V_1,V_2^*=((R^s_V_1,V_2)^-1)^*_2
       R^s_V_1^*,V_2^*=(R^s_V_1,V_2)^*_12

*_k means transpose with respect to the k-th factor.


The Maulik-Okounkov quantum affine algebra U_q^MO(ĝ_Q) is the subalgebra of ∏_wEnd(K(w)) generated by the matrix coefficients of R^s_C.


In other words, given an auxillary space V_0=⊗_wK(w), V=⊗_w'K(w') and a finite rank operator

    m(a_0)∈End(V_0)(a_0)

Now the element of U_q^MO(ĝ_Q) is generated by the following operators:

    ∮_a_0=0,∞da_0/2π ia_0Tr_V_0((1⊗ m(a_0))R^s_V,V_0(a/a_0))∈End(V(a))


The coproduct structure, antipode map and the counit map can be defined as follows:

For the coproduct Δ_s on U_q^MO(ĝ_Q) is defined via the conjugation by Stab_C,s, i.e. for a∈ U_q^MO(ĝ_Q) as a:K(w)→ K(w), Δ_s(a) is defined as:

    K(w_1)⊗ K(w_2)[r,"Stab_C,s"]   K(w_1+w_2)[r,"a"]   K(w_1+w_2)[r,"Stab_C,s^-1"]   K(w_1)⊗ K(w_2)


The antipode map S_s is defined as follows. We have the isomorphism of the graded vector space V_i≅ V_i^*, and this isomorphism induces the isomorphism:

    ∏_iEnd(V_i)≅∏_iEnd(V_i^*)


The antipode map S_s:U_q^MO(ĝ_Q)→ U_q^MO(ĝ_Q) is given by:

    ∮_a_0=0,∞da_0/2π ia_0Tr_V_0((1⊗ m(a_0))R^s_V,V_0(a/a_0))↦∮_a_0=0,∞da_0/2π ia_0Tr_V_0((1⊗ m(a_0))R^s_V^*,V_0(a/a_0))


The projection of the module V to the trivial module C induce the counit map:

    ϵ:U_q^MO(ĝ_Q)→C





Since M(0,w) is just a point, we denote the vector in K_0,w as v_∅, and we call it the vacuum vector.  We define the evaluation map:

    ev:U_q^MO(ĝ_Q)→∏_wK(w),     F↦ Fv_∅



We define the positive half of the Maulik-Okounkov quantum affine algebra U_q^MO,+(ĝ_Q) as the quotient by the kernel of the evaluation map:

    U_q^MO,+(ĝ_Q):=U_q^MO(ĝ_Q)/Ker(ev)






  §.§.§ Wall subalgebra


It is known that the K-theoretic stable envelope Stab_s is locally constant on s∈Pic(X)⊗Q. It changes as s crosses certain ratioanl hyperplanes:

The K-theoretical stable envelope Stab_C,s is locally constant on s if and only if s crosses the following hyperplanes.

    w={s∈Pic(X)⊗R|(s,α)+n=0,∀α∈Pic(X)}⊂Pic(X)⊗R




We choose the subtorus σ:C^*→ T_w such that:

    w=u_1w_1+⋯+u_kw_k

such that:

    M(v,w)^A=_v_1+⋯+v_k=vM(v_1,w_1)×⋯× M(v_k,w_k)


So now given the tautological line bundle L_i over M(v,w), the restriction of L_i to M(v_1,w_1)×⋯× M(v_k,w_k) has the A-degree is given by:

    L_i|_M(v_1,w_1)×⋯× M(v_k,w_k)=u_1^v_1^(i)⋯ u_k^v_k^(i)(⋯)


Thus the Newton polytope of the degree A is a point:

    deg_A(L_i|_M(v_1,w_1)×⋯× M(v_k,w_k))=(v_1^(i),⋯,v_k^(i))


Thus given the fractional line bundle L=L_1^m_1⋯L_n^m_n, we have that:

    deg_A(L|_M(v_1,w_1)×⋯× M(v_k,w_k))=(m·v_1,⋯,m·v_k)∈R^k


Thus deg_A gives a linear map deg_A:Pic(M(v,w))⊗R→R^k.


Now given F_β=M(v_1,w_1)×⋯× M(v_k,w_k), F_α=M(v_1+a_1,w_1)×⋯× M(v_k+a_k,w_k) such that ∑_ia_i=0. By the degree bounding condition:

    deg_AStab_C,L|_F_β× F_α+deg_AL|_F_α⊂deg_AStab_C,L|_F_β× F_β+deg_AL|_F_β


Here we describe the Newton polytope:

    deg_AStab_C,L|_F_β× F_β+deg_AL|_F_β-deg_AL|_F_α


Let us describe:

    Stab_C,L|_F_β× F_β⊗L|_F_β⊗L^-1|_F_α=∑_μ∈Z^ka_μu_1^μ_1+m·a_1⋯ u_k^μ_k+m·a_k,    ∑_ia_i=0

This equations stands for the Newton Polytope:

    P_A(Stab_C,L|_F_β× F_β⊗L|_F_β⊗L^-1|_F_α)
    
    =   Z^k∩[Convex hull of(μ∈Z^k|a_μ≠0)+translation as (m·a_1,⋯,m·a_k)]


We say that the Newton polytope P_A is special if the boundary of the Newton polytope intersects with the integer points For now we choose some special m such that P_A is special. It is clear that in this case P_A is at the "critical" level such that it contains an integer point or not. In this way if we fix the Newton polytope P_A such that it contains the fixed integer points at the boundary, we can move m in certain directions such that the boundary integer points are still on the boundary. 

Also we suppose that ∂ P_A consists of the union of several linearly independent different hyperplanes H_1,⋯, H_l in R^k up to translation, and each hyperplane is determined by a vector n_i∈R^k. In order to satisfy the above conditions, this means that the fractional line bundle m should be chosen on the intersection of the hyperplanes H_1∩⋯∩ H_s containing the integer points. which means that:

    ((m'-m)·a_1,⋯,(m'-m)·a_k)∈ H_1∩⋯∩ H_s


In conclusion, we can see that the critical fractional line bundles m lives in the set of hyperplanes such that:

    {m∈R^n|(m·a_1,⋯,m·a_k)∈Z^k,∑_ia_i=0}

for some a_1,⋯,a_k. The choice of a_1 depends on the boundary hyperplane of the Newton polytope ∂ P_A, thus the choice of (a_1,⋯,a_k) is finite.

Now if we choose A⊂ T_w such that w=u_1w_1+u_2w_2, we can have that the critical fractional line bundle should be in:

    {m∈R^n|(m·a,-m·a)∈Z^2}≅{m∈R^n|m·a∈Z}


This finishes the proof of the proposition.


Now for each quiver variety M(v,w), we can associate the wall set w(v,w), it is clear that via the identification Pic(M(v,w))⊗R=R^n for different v, we can define the wall set of M(w):=⊔_vM(v,w) as w(w):=⊔ w(v,w).

Now fix the slope m and the cocharacter σ:C^*→ A_w. We choose an ample line bundle L∈Pic(X) with X=M(v,w_1+w_2) and a suitable small number ϵ such that m and m+ϵL are separated by just one wall w, we define the wall R-matrices as:

    R_w^±:=Stab_σ,m+ϵL^-1∘Stab_σ,m:⊕_v_1+v_2=vK(v_1,w_1)⊗ K(v_2,w_2)→⊕_v_1+v_2=vK(v_1,w_1)⊗ K(v_2,w_2)


It is an integral K-theory class in K_T(X^A× X^A). Note that the choice of ϵ depends on M(v,w_1+w_2) just to make sure that there is only one wall between m and m+ϵL corresponding to w. By definition it is easy to see that R_w^+ is upper-triangular, and R_w^- is lower triangular.

Given a finite-rank operator m∈End(V_0). We define the positive half of the wall subalgebra U_q^MO,+(g_w) as the algebra generated by the operators:

    Tr_V_0((1⊗ m)(R^-_w)_V,V_0^-1|_a_0=1)∈End(V)


Similarly, the negative half of the wall subalgebra U_q^MO,-(g_w) is defined as the algebra generated by the operators:

    Tr_V_0((1⊗ m)(R^+_w)_V,V_0|_a_0=1)∈End(V)


Since R_w^± is a strictly upper(lower) triangular matrix, we can see that the algebra U_q^MO,±(g_w) is generated by the operators m such that:

    m:K(v,w)→ K(v±n,w),    n∈N^I


Moreover, it can be shown that the operators q^ΩR_w^± satisfy the Yang-Baxter equation. In this way, similarly we can define the non-negative(non-positive) half of the wall subalgebra U_q^MO,≥(≤)(g_w) and also the whole wall subalgebra U_q^MO(g_w). Obviously we have U_q^MO,±(g_w)⊂ U_q^MO,≥(≤)(g_w).

One could define the graded pieces of U_q^MO(g_w) as:

    U_q^MO,±(g_w)=⊕_n∈N^IU_q^MO,±(g_w)_±n

with a∈ U_q^MO,±(g_w)_±n the elements such that a:K(v,w)→ K(v±n,w).



Each graded pieces U_q^MO,±(g_w)_±n is finite-dimensional.


The fact that R_w^± is an integral K-theory class implies that each component of R_w^± is written as Au^k with the multiplication A given as the Laurent polynomial of q and t, thus each graded pieces U_q^MO,±(g_w)_±n is finite-dimensional.


The following proposition was proved in <cit.>:


    R_w^∓=(R_w^±)_21|_u=u^-1


This implies that U_q^MO,≥(g_w)
is isomorphic to U_q^MO,≤(g_w)
as graded vector spaces.

Usually we take R_w^± as the universal R-matrix for U_q^MO(g_w) without spectral parametres with respect to the coproduct Δ_w and Δ_w^op. Now if we write:

    R_w^+=∑_ia_i⊗ b_i∈ U_q^MO(g_w)⊗̂ U_q^MO(g_w)


since R_w^-(u)=(R_w^+)_21(u^-1), this implies that:

    R_w^-=∑_ib_i⊗ a_i∈ U_q^MO(g_w)⊗̂ U_q^MO(g_w)


This implies that one could realize U_q^MO(g_w) as the bialgebra pairing between U_q^MO,≥(g_w) and U_q^MO,≤(g_w) such that:

    ⟨ a_i,b_j⟩=δ_ij




In this way we could define the wall subalgebra U_q^MO(g_w) as:

    U_q^MO(g_w):=U_q^MO,≥(g_w)⊗ U_q^MO,≤(g_w)/(q^Ω⊗ q^-Ω=1)


Since R_w^MO is an integral K-theory class, this is an algebra over Z[q^±1]. It has the universal R-matrix R^univ_w which can be written as ∑_iF_i⊗ G_i with {F_i} and {G_i} the dual bases, and the evaluation in End(K(w_1)⊗ K(w_2)) is equal to the MO wall R-matrix R_w^MO.

Throughout the paper we will assume that U_q^MO(g_w) is localised over Q with q being valued outside of the roots of unity.




 §.§ Factorisation of geometric R-matrices


Fix the stable envelope Stab_σ,m and Stab_σ,∞, we can have the following factorisation of Stab_±,m near u=0,∞:

    Stab_σ,m=   Stab_σ,-∞⋯Stab_σ,m_-2Stab_σ,m_-2^-1Stab_σ,m_-1Stab_σ,m_-1^-1Stab_σ,m
    
    =   Stab_σ,-∞⋯ R_m_-2,m_-1^+R_m_-1,m^+



    Stab_-σ,m=   Stab_-σ,∞⋯Stab_-σ,m_2Stab_-σ,m_2^-1Stab_-σ,m_1Stab_-σ,m_1^-1Stab_-σ,m
    
    =   Stab_-σ,∞⋯ R_m_2,m_1^-R_m_1,m^-


Here R_m_1,m_2^±=Stab_±σ,m_1^-1Stab_±σ,m_2 is the wall R-matrix. For simplicity we always choose generic slope points m_i such that there is only one wall between m_1 and m_2. In this case we use R_w^± as R_m_1,m_2^±. Note that this notation does not mean that R_w only depends on the wall w, but we still use the notation for simplicity.


This gives the factorisation of the geometric R-matrix:

    R^s(u)=∏_i<0^←R_w_i^-R_∞∏_i≥0^←R_w_i^+


This has been proved in <cit.> that this factorisation is well-defined in the topology of the Laurent formal power series in the spectral parametre u.

The factorisation <ref> gives the generators of the positive half of the MO quantum affine algebra:

U_q^MO,+(ĝ_Q) is generated by the positive half of the wall subalgebras U_q^MO,+(g_w) for arbitrary walls w.


Note that the elements of U_q^MO(ĝ_Q) are of the form:

    ∮_a_0=0,∞da_0/2π ia_0Tr_V_0((1⊗ m(a_0))R^s_V,V_0(a/a_0))∈End(V(a))

Now via the mapping a↦ av_∅ we have that:

    ∮_a_0=0,∞da_0/2π ia_0Tr_V_0((1⊗ m(a_0))R^s_V,V_0(a/a_0))v_∅
    
    =   ∮_a_0=0,∞da_0/2π ia_0Tr_V_0((1⊗ m(a_0))∏_i<0^←R_w_i^-R_∞∏_i≥0^←R_w_i^+)v_∅
    
    =   ∮_a_0=0,∞da_0/2π ia_0Tr_V_0((1⊗ m(a_0))∏_i<0^←R_w_i^-v_∅


This result implies that the image of ev in <ref> contains only the coefficients of R_w_i^-, which implies that the lemma.





 §.§ Cohomological stable envelope and Maulik-Okounkov Yangian


In this subsection we review the original definition of the stable envelope in the case of the equivariant cohomology and the definition of the Maulik-Okounkov Yangian algebra in <cit.>.

We still fix the notation as above, given the Nakajima quiver variety M(v,w) with the group action G_v× A_E and T⊂Ker(q)⊂ G_v× A_E. The cohomological stable envelope is a cohomlogy class

    Stab_C⊂ H_G(X× X^T)

which induces the morphism 

    Stab_C:H_G(X^T)→ H_G(X)


such that:

	
  * The support of Stab_C is on Attr^f.
	
  * 
    Stab_C|_F_α× F_α=± e(N^-_F_α/X)

	
  * 
    deg_TStab_C|_F_α× F_β<1/2codim_C(F_β),    ∀ F_β<F_α



The cohomological stable envelope also has the triangle lemma as <ref>

Similarly, we can choose σ:C^*→ A_w in some chamber C such that:

    w=u_1w_1+⋯+u_kw_k


We use the notation H(w):=⊕_vH_G_w×C_ħ^*^*(M(v,w)). This gives the stable envelope map of the form:

    Stab_±C:H(w_1)⊗⋯⊗ H(w_k)→ H(w_1+⋯+w_k)


Similarly we can define the cohomological geometric R-matrix:

    R_C:=Stab_-C^-1∘Stab_C

Similar as the case of the K-theoretical geometric R-matrix, the cohomological geometric R-matrix can also be factorised as:

    R_C=∏_1≤ i<j≤ kR_C_ij(u_i-u_j),    R_C_ij(u_i-u_j):H(w_i)⊗ H(w_j)→ H(w_i)⊗ H(w_j)

via the triangle lemma. Each R_C_ij(u_i-u_j) satisfy the rational Yang-Baxter equation with the spectral parametres.






  §.§.§ Maulik-Okounkov Yangian algebra

The Maulik-Okounkov Yangian algebra Y_ħ^MO(g_Q) is defined in the following way: We consider arbitrary finite-rank operator m(u)∈End(F_0)[u] with F_0=⊗_wH(w), the Maulik-Okounkov Yangian algebra Y_ħ(g_Q) is generated by the operators in ∏_wEnd(H(w)) of the form:

    E(m(u))=-1/ħRes_u=∞tr_F_0((m(u)⊗1)R_F_0(u),W)∈End(W),     W=∏_wEnd(H(w))


i.e. Y_ħ(g_Q) is generated by the matrix coefficients of the geometric R-matrix R_F_0,W under the expansion of u→∞.


It has also been proved in <cit.> that the geometric R-matrix admits the expansion around u=∞:

    R_C_ij(u_i-u_j)=1+ħ/u_i-u_jr_C_ij+O((u_i-u_j)^-2)


where the operator r_C_ij:H(w_i)⊗ H(w_j)→ H(w_i)⊗ H(w_j) satisfies the classical Yang-Baxter equation. In this way similarly, we could define the Maulik-Okounkov Lie algebra g_Q^MO as the Lie algebra generated by the operator E(m_0) for m_0∈End(F_0) constant in u. This is equivalent to say that g_Q^MO is generated by the matrix coefficients of the classical r-matrix r_C=∑_i<jr_C_ij by the following formula of brackets:

    [E(m_1),E(m_2)]=E((tr⊗1)[r_VV,m_1⊗ m_2]),     m_1,m_2∈End(V)


The Maulik-Okounkov Lie algebra g_Q^MO admits the root decomposition:

    g_Q^MO=h⊕⊕_α∈Z^Ig_α^MO

with g_α^MO spanned by the vector e_α such that:

    e_α:H(v,w)→ H(v+α,w)


We denote the set Φ={α∈Z^I|g_α≠0} as the set of roots. It has been known that if α∈Φ, -α∈Φ, and thus Φ=Φ^+⊔-Φ^+. We define the nilpotent Maulik-Okounkov Lie algebra n_Q^MO as:

    n_Q^MO:=⊕_α∈Φ^+n_α^MO

It is a Lie subalgebra of g_Q^MO spanned by the root vectors of positive roots.

The theorem given by Botta and Davison <cit.> states that the nilpotent Maulik-Okounkov Lie algebra is isomorphic to the corresponding nilpotent Lie algebra of quiver type Q:
(Botta-Davison,<cit.>)
There is an isomorphism between the nilpotent Maulik-Okounkov Lie algebra and the nilpotent Lie algebra of quiver type Q:

    n_Q^MO≅n_Q



This theorem will be the key to construct the isomorphism between the positive half of the quantum toroidal algebras and the positive half of the Maulik-Okounkov quantum affine algebras.









 §.§ Wall subalgebra and cohomologial limit


The geometric R-matrix at slope s can be written as:

    R^s(u)=∏^←_w<sR_w^-(u)·R^∞(u)·∏_w≥ s^←R_w^+(u)


Here R_∞:=Stab_-C,-∞^-1∘Stab_C,∞ such that it is diagonal with the basis given by:

    R_∞|_F_α× F_α=i_*∧^*[N_F_α^-]/∧^*[N_F_α^+]=(-1)^codimF_α/2∏_v_α<0(v_α^1/2-v_α^-1/2)/∏_v_α>0(v_α^1/2-v_α^-1/2)

Here v_α are the Chern roots of the normal bundle N_F_α.


We mainly concern the connection between the geometric R-matrix in both K-theory and cohomology theory. We denote the Chern character map ch:K_T(M(v,w))→ H_T(M(v,w)) as sending u to e^κ z, and q to e^κħ.


If we take u=e^κ z, q=e^κħ and take κ→0, we obtain that:

    lim_κ→0R^s(e^κ z)=R(z)

which is the geometric R-matrix for the corresponding Maulik-Okounkov Yangian Y_ħ(g_Q).



First note that the cohomological stable envelope Stab_σ is the lowest degree part of the limit of the K-theoretic stable envelope ch(Stab_σ,m) with κ→0. Thus by the definition of the both cohomological and K-theoretic stable envelope, the degeneration limit of ch(Stab_-σ,m^-1Stab_σ,m) coincides with Stab_-σ^-1Stab_σ.


Now we want to compare the cohomological limit of U_q(g_w) with the Maulik-Okounkov Lie algebra g_Q. To do this, first note that the wall R-matrix satisfy the following translation formula:

    R_w+L^±(u)=LR_w^±L^-1

here w+L⊂R^I is the wall such that translates from w to the direction L∈Z^I.

Now we suppose that the wall in the product can be written as:

    ∏_n≥0^←Ad_L^n(∏_w∈[s,s+L)^←R_w^+(u))


Here L∈Pic(X) is an ample line bundle. We further decompose R_w^+(u)=Id+U_w^+(u) into the identity matrix and the strictly upper-triangular matrix U_w(u), and U_w(u) is nilpotent by definition. It is known that in this case:

    ∏_n≥0^←Ad_L^n(∏_w∈[s,s+L)^←R_w^+(u))=∏_n≥0^←(∏_w∈[s,s+L)^←(Id+u^nLU_w^+(u)))


It has its matrix coefficients as the rational function over u with the rational function of the form:

    Id+∑_w∈[s,s+L)1/1-u^LU_w^+(u)+∑_w_1,w_2∈[s,s+L)1/(1-u^k_1L)(1-u^k_2L)U_w_1^+(u)U_w_2^+(u)+⋯


It is known that U_w(u)^+ has its matrix coefficients as the monomials in u=e^κ z. So take the cohomological limit κ→0 , we can obtain that:

    Cohomological limit of ∏_w≥ s^←R_w^+(u)=Id+1/z∑_w,k≥0a_w^(k)r_w,k^++O(1/z^2),     a_w,k∈N

Here a_w^(k) are some non-negative integers.
Combining the computation,

Here

    lim_κ→0R_w^±-Id/q-1=r_w^±=∑_k≥0r_w,k^±,     q=e^κħ

r_w,k^± gives a root decomposition of the degeneration limit r_w^± with the wall w corresponding to the root α. For the r_∞:

    lim_κ→0R^∞(e^κ z)-1/e^κħ-1=r_∞=Ω,    Ω(γ)=codim(F)/4γ

is the classical limit of the wall R-matrix R_w^± and the classical limit of the diagonal part R^∞(u). Ω is the codimension scaling operator with F⊂ X a component of the fixed point subset X^A⊂ X.

 we can see that:

    Cohomological limit of R^s(u)=Id+1/z(∑_w≥ s,k≥0a_w^(k)r_w,k^++r_∞-∑_w<s,k≥0a_w^(k)r_w,k^-)+O(1/z^2)∈End(H(w))




Now we define r:=∑_w≥ sa_wr_w^++r_∞-∑_w<sa_wr_w^-.


If we do the z^-1-expansion of the Yangian geometric R-matrix, note that:

    R(z)=Id+ħ/zr_Q+O(1/z^2)∈End(H(w))



By definition, r_Q is the classical r-matrix corresponding to the Maulik-Okounkov Lie algebra g_Q.

Also by definition, we can see that r̃_w^+:=r_w^++r_∞ satisfies the classical Yang-Baxter equation:

    [r̃_w,12^+,r̃_w,13^+]+[r̃_w,12^+,r̃_w,23^+]+[r̃_w,13^+,r̃_w,23^+]=0

This follows from the fact that q^ΩR_w^± satisfies the quantum Yang-Baxter equation.
This means that r_w^+ generates a Lie subalgebra n_w^MO⊂n_Q^MO of the positive part of n_Q^MO over Z (over Q after being localised). In conclusion, r_w^± generates the Lie subalgebra g_w^MO⊂g_Q^MO. Generally we call the Lie subalgebra g_w^MO as the root subalgebra of g_Q^MO.

This concludes the following proposition:

The degeneration limit r̃_w^±,MO=r_∞+r_w^±,MO of the wall R-matrices q^ΩR_w^±,MO satisfies the classical Yang-Baxter equation, and it generates a Lie subalgebra g_w^MO of the Maulik-Okounkov Lie algebra g_Q.




§ WALL SUBALGEBRAS AND THE SLOPE SUBALGEBRAS



 §.§ Stable basis and the Pieri rule

A good set of basis for K(w) under the action of the slope subalgebra B_m of slope m is the stable basis s^m_λ of slope m.



The stable basis is constructed via the following way: We choose the stable envelope as:

    Stab_σ,m:K_T(M(v,w)^T)→ K_T(M(v,w))

Here T=A×C^*_q×C^*_t is the product of the maximal torus and the two-dimensional torus C^*_q×C^*_t. It is known that the fixed points M(v,w)^T of T-torus action are isolated, which is labeled by the multi-partitions λ. Fixing the character σ:C^*→ T that corresponds to the isolated fixed point subset. We denote the image of σ by u_i↦ t^N_i, we would always denote s^m_λ:=Stab_σ,m(|λ⟩).

The diagonal part of s^m_λ is given by:

    s^m_λ|_λ=k_λ=∏_□∈λ[∏_▪∈λζ(χ_▪/χ_□)∏_k=1^w[u_k/qχ_□][χ_□/qu_k]]^(-)

Here (-) stands for the product of [x] such that the t-degree of [x] is smaller than 0. The notation for (+) and (0) has the similar meaning.

It is known that the slope subalgebra B_m is generated by P^m_[i,j) and Q^m_-[i,j). 

Now given an operator F:K(v',w)→ K(v,w) whose matrix coefficients in terms of fixed point basis is given by:

    F|μ⟩=∑_|λ|=vF^λ_μ·|λ⟩


If we choose the stable basis s^m_λ of the slope m, we write down the coefficients with respect to the stable basis as:

    F· s^m_μ=∑_|λ|=vγ^λ_μ· s^m_λ



The good property about the stable basis s^m_μ is that it subtracts the special information of the action of the slope subalgebra B_m.



<cit.>
For F∈B_m^±, we have that:

    γ^λ_μ=(l.d.F^λ_μ)·l.d.(k_μ)/l.d.(k_λ)


Here l.d. stands for the lowest t-degree part. 






 §.§ The Hopf algebra embedding


The advantage of the Pieri rule is that it can be used to construct the Hopf algebra embedding of B_m into the Maulik-Okounkov quantum affine algebra U_q^MO(ĝ_Q).


There is a Hopf algebra embedding 

    B_m↪ U_q^MO(ĝ_Q)

such that the restriction to U_q(g_w) gives the Hopf algebra embedding U_q(g_w)↪ U_q^MO(g_w).


We have shown the proof of the fact in <cit.>. Here we give some detailed version of the proof.

The compatibility of the antipode map and the counit is obvious. We only need to prove the compatibility of the coproduct.
First we prove that the coproduct Δ_m on B_m intertwine the Stab_m, i.e.

    B_m[r,hook][d,"Δ_m"]   ∏_wEnd(K_T_w(M(w)))[d,"Stab_m^†∘(-)∘Stab_m"]
    B_m⊗̂B_m[r,hook]   ∏_w_1,w_2End(K_T_w_1(M(w_1)))⊗̂(K_T_w_2(M(w_2)))


For simplicity, we only show the proof for P^m_[i;j), and this means that we need to prove the following identity:

    P^m_[i;j)∘Stab_m(s^m_λ_1⊗ s^m_λ_2)=Stab_m(Δ_m(P^m_[i;j))(s^m_λ_1⊗ s^m_λ_2))


For simplicity, using the formula for the coproduct, we can reduce the proof to the case P^m_[i;i+1)_h.

Recall the formula of P^m_[i;i+1)_h on the stable basis s^m_λ in the Lemma <ref>:

    P^m_[i;i+1)_hs^m_μ=∑_λl.d.(P^m_[i;i+1)_h(λ\μ)Θ_λ\μ)l.d.(K_μ)/l.d.(K_λ)· s^m_λ


Here:

    K_λ=∏_□∈λ[∏_▪∈λζ(χ_▪/χ_□)∏_k=1^w[u_k/qχ_□][χ_□/qu_k]]^(-)


    Θ_λ\μ=∏_▪∈λ\μ[∏_□∈μζ(χ_▪/χ_□)∏_k=1^w[u_k/qχ_▪]]


Observe that the lowest degree part of P^m_[i;i+1)_h(λ\μ) does not affect the term, so we only consider the following part:

    l.d.(Θ_λ\μ)l.d.(K_μ)/l.d.(K_λ)
    
    =   l.d.(∏_□∈μ^▪∈λ\μζ(χ_▪/χ_□)∏_i=1^w[u_i/qχ_▪])^(+)/l.d.(∏_□∈μ^▪∈λ\μζ(χ_□/χ_▪)∏_i=1^w[χ_▪/qu_i])^(-)·∏_▪∈λ\μ(∏_□∈μζ(χ_▪/χ_□)∏_i=1^w[u_i/qχ_▪])^(0)/∏_▪,▪'∈λ\μζ(χ_▪/χ_▪')^(-)


The superscripts (+), (0) and (-) refer to that we only retain those factors [x] such that deg(x)>0, deg(x)=0 and deg(x)<0. We can see that the degree 0 part does not contribute to the lowest degree. 


We want to analyze the behavior of the formula in terms of the power of q. For the first fraction, by calculation we can see that:

    l.d.∏_□∈μ^▪∈λ\μ(∏_i=1^w[u_i/qχ_▪])^(+)/l.d.(∏_□∈μ^▪∈λ\μ∏_i=1^w[χ_▪/qu_i])^(-)=∏_□∈μ^▪∈λ\μq^-∑_i=w_▪+1^w


by the following formula:


    l.d.∏_□∈μ^▪∈λ\μ(∏_i=1^w[u_i/qχ_▪])^(+)=l.d.∏_□∈μ^▪∈λ\μ(∏_i=w_▪+1^w[u_i/qχ_▪])=∏_□∈μ^▪∈λ\μq^-1/2∑_i=w_▪+1^w



    l.d.(∏_□∈μ^▪∈λ\μ∏_i=1^w[χ_▪/qu_i])^(-)=l.d.(∏_□∈μ^▪∈λ\μ∏_i=w_▪+1^w[χ_▪/qu_i])=∏_□∈μ^▪∈λ\μ∏_i=w_▪+1^wq


Also for

    l.d.(∏_□∈μ^▪∈λ\μζ(χ_▪/χ_□))^(+)/l.d.(∏_□∈μ^▪∈λ\μζ(χ_□/χ_▪))^(-)=∏_▪∈λ\μq^(∑_□∈μ^σ_□=σ_▪+1,,c_□>c_▪+∑_□∈μ^σ_□=σ_▪-1,c_□>c_▪-2∑_□∈μ^σ_□=σ_▪,,c_□>c_▪)


Here c_□=x-y+N_i stands for the content of the box □∈λ. σ_□ stands for the color of the box □∈λ. The formula is due to the following formula:

    l.d.(∏_□∈μ^▪∈λ\μζ(χ_▪/χ_□))^(+)=   l.d.(∏_▪∈λ\μ∏_□∈μ^σ_□=σ_▪+1[tχ_□/qχ_▪]∏_□∈μ^σ_□=σ_▪-1[χ_□/qtχ_▪]/∏_□∈μ^σ_□=σ_▪[χ_□/χ_▪][χ_□/q^2χ_▪])^(+)
    
    =   ∏_▪∈λ\μq^1/2(∑_□∈μ^σ_□=σ_▪+1,,c_□>c_▪+∑_□∈μ^σ_□=σ_▪-1,c_□>c_▪-2∑_□∈μ^σ_□=σ_▪,,c_□>c_▪)


For the 1/∏_▪,▪'∈λ\μζ(χ_▪/χ_▪')^(-) part, this does not affect the comparison since:

    1/∏_▪,▪'∈λ\μζ(χ_▪/χ_▪')^(-)=1/∏_▪,▪'∈λ_1\μ_1ζ(χ_▪/χ_▪')^(-)∏_▪,▪'∈λ_2\μ_2ζ(χ_▪/χ_▪')^(-)


So we only need to do the comparison for the coefficients of the form <ref>.

For the general P^m_[i,j), using the coproduct formula:

    Δ_m(P^m_[i;j)_h)=∑_a=i^jP^m_[a,j)_hφ_[i,a)_h⊗ P^m_[i,a)_h


Now use the formula for the stable envelope:

    P^m_[i,j)_hStab_m(s^m_μ_1⊗ s^m_μ_2)=P^m_[i,j)_hs^m_(μ_1,μ_2)
    
    =   ∑_λ(⋯)∏_▪∈λ\μq^(∑_□∈μ^σ_□=σ_▪+1,,c_□>c_▪+∑_□∈μ^σ_□=σ_▪-1,c_□>c_▪-2∑_□∈μ^σ_□=σ_▪,,c_□>c_▪)s^m_λ,    λ=(λ_1,λ_2)



    Stab_m(Δ_m(P^m_[i,j)_h)s^m_μ_1⊗ s^m_μ_2)=∑_a=i^jStab_m(P^m_[a,j)_hφ_[i,a)_hs^m_μ_1⊗ P^m_[i,a)_hs^m_μ_2)
    
    =   ∑_a=i^j∑_λ_1,λ_2(⋯)∏_▪∈λ_1\μ_1q^(∑_□∈μ_1^σ_□=σ_▪+1,,c_□>c_▪+∑_□∈μ_1^σ_□=σ_▪-1,c_□>c_▪-2∑_□∈μ_1^σ_□=σ_▪,,c_□>c_▪)q^v_1·[i,a)_h
       ×∏_▪∈λ_2\μ_2q^(∑_□∈μ_2^σ_□=σ_▪+1,,c_□>c_▪+∑_□∈μ_2^σ_□=σ_▪-1,c_□>c_▪-2∑_□∈μ_2^σ_□=σ_▪,,c_□>c_▪)s^m_(λ_1,λ_2)


Here the terms (⋯) in both formula are the same term that does not affect the formula in both, which means that they are the same. 


Now given our chosen ordering the order (μ_1,μ_2), in the formula P^m_[a,j)_hs^m_μ_1 in <ref> must have some remaining power of q left, since we have higher order of μ_1 than μ_2, which coincides  with the one given by φ_[i,a)_h. thus completing the proof of the above commutative diagram.



For the algebra map B_m→ U_q(g_m), we claim the following thing: 

If a∈ U_q(g_w) such that Δ_m(a)⊂B_n⊗̂B_n, n=m . Then similarly for b∈ U_q^MO(g_w) such that Δ_m(b)∈ U_q^MO(g_w)⊗̂ U_q^MO(g_w), w contains the slope point m.


Now suppose that a∈B_n such that Δ_m(a)⊂B_n⊗̂B_n, now choose a path from n to m, and now use the formula:

    Δ_m(a)=T_m,nΔ_n(a)T_m,n^-1


And since Δ_n(a)⊂B_n⊗̂B_n, now we write it as:

    Δ_n(a)=∑ a^(1)⊗ a^(2)

And without loss of generality, we could reduce the case to P^n_[i;j) and thus by the coproduct formula one could further reduce the proof to P^n_[i;i+1)_h, and in this case:

    Δ_n(P^n_[i;i+1)_h)=P^n_[i;i+1)_h⊗φ_[i;i+1)_h+1⊗ P^n_[i;i+1)_h


Then the first part of the lemma is true following from the fact that [a,P^n_[i;i+1)_h]∉B_n if a∈B_m with m≠n without φ_i kind of elements.

For the second part of the lemma, recall that U_q^MO(g_m) is defined as the matrix coefficients of the MO wall R-matrix R_m,m+ϵθ:=lim_ϵ→0Stab_m+ϵθ^-1∘Stab_m with θ∈Z^r. By simple calculation, one can show that the matrix elements of R_m,m+ϵθ acting on the corresponding stable basis s^m_λ_1⊗ s^m_λ_2 depends on the value over the fixed point geometric R-matrix, i.e.

    R_m,m+ϵθ(s^m_λ_1⊗ s^m_λ_2)=lim_ϵ→0∑_μ_1,μ_2(R^σ_m,m+ϵθ)_(λ_1,λ_2),(μ_1,μ_2)s^m+ϵθ_μ_1⊗ s^m+ϵθ_μ_2


For the precise formula, one could use the result of <cit.> to write down the explicit formula of the fixed point R-matrix.

The R-matrix has the root decomposition:

    R_m,m+ϵθ=1+∑_⟨α,θ⟩>0(R_m,m+ϵθ)_α,     (R_m,m+ϵθ)^-1=1+∑_⟨α,θ⟩>0(R_m,m+ϵθ^-1)_α

We have the formula of the coproduct acting on the R-matrices:

    Δ_m⟨λ|((R_m,m+ϵθ)_α)|μ⟩∈⊕_β+γ=α(R_m,m+ϵθ)_β⊗ (R_m,m+ϵθ)_γ


Thus:

    Δ_n⟨λ|(R_m,m+ϵθ)_α|μ⟩=⊕_β_1+β_2+β_3+β_4=α(T_n,m)_β_1(R_m,m+ϵθ)_β_2(R_m,m+ϵθ)_β_3(T_n,m^-1)_β_4


Here T_n,m:=Stab_n^-1∘Stab_m. Then one can show that given an order on the set of roots {α}, U_q(g_m) is generated by elements of R_α with α of the lowest order, i.e. those elements corresponding to the simple roots. So we have reduced the proof to the case for the lowest order elements in (R_m)_α. And in this case, the above formula implies that:

    Δ_n⟨λ|(R_m,m+ϵθ)_α|μ⟩=⊕_β_1+β_2=α(T_n,m)_β_1(T_n,m^-1)_β_2+(R_m,m+ϵθ)_α⊗1+1⊗(R_m,m+ϵθ)_α


The vanishing of ⊕_β_1+β_2=α(T_n,m)_β_1(T_n,m^-1)_β_2 implies that there is a path from m to n such that there is no wall, which means that n is either in the same wall as the one contains m, or n does not belong to any wall, which is contradiction. This means that n=m.



Now via the embedding U_q(g_w)↪ U_q^MO(ĝ_Q), since Δ_m(a)∈B_m⊗B_m for a∈ U_q(g_w)⊂B_m. By the result in <cit.>, the restriction of Δ_m of U_q^MO(ĝ_Q) to U_q^MO(g_w) with m on the wall w is the coproduct Δ_w on U_q^MO(g_w).


Now if a∉ U_q^MO(g_w), this means that Δ_m(a)∉ U_q^MO(g_w)⊗ U_q^MO(g_w), which is a contradiction.





 §.§ Root subalgebra and dimension counting

It can be proved that n_w^MO is the classical limit of the wall subalgebra U_q(n_w^MO):

There is an isomorphism:

    U_q(n_w^MO)/(q-1)U_q(n_w^MO)=U(n_w^MO)



By definition U_q(n_w^MO) is generated by the matrix coefficients of R_w^+=Id+U_w^+. The left-ideal (q-1)U_q(n_w^MO) makes sense since R_w^MO is an integral K-theory class and it is generated by the matrix coefficients of (q-1)R_w^+, and now by definition:

    lim_q→1R_w^+-Id/q-1=r_w^+


which generates the Lie algebra n_w^MO.


It is easy to see that U_q(n_w^MO) and U(n_w^MO) has the grading:

    U_q(n_w^MO)=⊕_k∈N^IU_q(n_w^MO)_k,     U(n_w^MO)=⊕_k∈N^IU(n_w^MO)_k


It is easy to see that each graded pieces U_q(n_w)_k and U(n_w)_k are finite-dimensional. From the Lemma <ref> we can have that:

    dim(U(n_w)_k)≤dim(U_q(n_w)_k)


Moreover, for the Lie algebra g_Q, we have the decomposition:

    g_Q^MO=⋃_w∈Walls_0g_w^MO

with the walls defined as:

    w={s∈R^I|⟨α,s⟩=0}


The wall set for the K-theoretic stable envelope is actually the "affinisation" of the roots for the Lie algebra n_Q^MO:

The wall set for the K-theoretic stable envelope is of the following type:

    w⊃{s∈R^I|⟨α,s⟩=n}

with α the roots of the Maulik-Okounkov Lie algebra n_Q^MO.


Since R_w^± is an integral K-theory class in K_T(X^A× X^A), it has the coefficients as the Laurent polynomial of q. This means that by the Chern character map ch:K_T(X^A× X^A)→ H^*_T(X^A× X^A), passing to the substitution q=e^ħ and taking the smallest degree part. This means that we have transformed U_q^MO(g_w) as a Q[[ħ]]-algebra with U_q^MO(g_w)/ħ U_q^MO(g_w)≅ U(g_w^MO). Since g_w^MO⊂g_Q^MO. This means that w must be in the set of roots of g_Q^MO corresponding to either the empty subset or root subalgebra of g_Q^MO.


Now we study the generators for the wall Lie algebra g_w^MO.

Recall that given a fractional line bundle L_w on the wall w. It has been proved in <cit.> that R_w^+ is triangular with monomial in spectral parametres u matrix elements:

    R_w^+|_F_2× F_1=
    
    1   F_1=F_2
    
    Au^⟨μ(F_2)-μ(F_1),L_w⟩   F_1≥ F_2
    
    0   otherwise


Since the wall w is defined as ⟨α,s⟩=0, we need to require that v_1-v_2=kα. After the degeneration to the Lie algebra n_w, it means that for e∈n_w, we have:

    e:K(M(v,w))→ K(M(v+kα,w))



This makes the Lie algebra n_w^MO as a subalgebra of ⊕_k≥0n_kα^MO. Here n_kα^MO is the space of root vectors of n^MO of the form e_kα.



If the quiver Q is of the affine type A, then there is an isomorphism

    n_w^MO≅⊕_k≥0n_kα^MO


for the wall w corresponding to the root α such that w={s|⟨ s,α⟩∈Z}.


Obviously by definition we have the inclusion

    n_w^MO⊂⊕_k≥0n_kα^MO


Also we have the inclusion n_w⊂n_w^MO from the Proposition <ref>. But we know that:

    n_w≅⊕_k≥0n_kα


By the result of Botta and Davison <cit.>, we have the isomorphism:

    ⊕_k≥0n_kα^MO≅⊕_k≥0n_kα


which gives a chain of inclusion:

    ⊕_k≥0n_kα≅n_w⊂n_w^MO⊂⊕_k≥0n_kα


Thus we have the isomorphism n_w≅n_w^MO, and this finishes the proof.






§ QUANTUM DIFFERENCE EQUATION

In this section we review the construction of the quantum difference equations in both algebraic and geometric ways. For details see <cit.><cit.>.

The basic ingredients of the construction is given by the monodromy operators. For the Okounkov-Smirnov geometric monodromy operators, it is given by B_w(z)∈U_q^MO(g_w), which is from the Maulik-Okounkov wall subalgebra U_q^MO(g_w). For the algebraic one, it is given by the algebraic monodromy operators B_m(z)∈B_m with B_m the slope subalgebra.


 §.§ Okounkov-Smirnov quantum difference equation

The Okounkov-Smirnov geometric quantum difference equation was proposed in <cit.> as the difference equation of the capping operator J(u,z)∈ K_G(M(v,w))^⊗ 2⊗Q[[z^d]]_d∈Pic_eff(M(v,w)) over the Kahler variable z:

    J(u,p^Lz)L=M_L(z)J(u,z)


Here we review the construction of the geometric quantum difference operator M_L(z) given by Okounkov and Smirnov in <cit.>:

Here we denote λ as q^λ=z, p=q^τ as in <cit.>, and we abbreviate R_w as R_w^MO the MO wall R-matrix. Using the notation, we denote F(λ):=F(q^λ) as the function F(z) of the Kahler variable.
For each MO wall subalgebra U_q^MO(g_w) there are corresponding ABRR equations:

    J_w^+(λ)q_(1)^-λq^-ΩR_w^+=ħ_(1)^-λq^ΩJ_w^+(λ),     q^-ΩR_w^-q_(1)^-λJ_w^-(λ)=J_w^-(λ)q^Ωq_(1)^-λ




We define:

    J_w^±(λ)=J_w^±(λ-τL_w)=J_w^±(zp^-L_w),    κ=Cv-w/2



and thus the monodromy operators are defined as:

    B_w(λ)=m((1⊗ S_w)(J^-_w(λ)^-1))|_λ→λ+κ



The first ABRR equation can be written as:

    Ad_q_(1)^λq^-Ω(J_w^+(λ))=J_w^+(λ)(R_w^+)^-1∈ U_q(g_w)⊗̂U_q(g_w)


This gives the formal solution for the fusion operators J_w^+(λ):

    J_w^+(λ)=∏_k=0^→
    ∞Ad_(q^λ_(1)q^-Ω)^k(R_w^+)


Now choose a path from s to s-L, the Okounkov-Smirnov quantum difference operator B^s_L(λ) is defined as:

    B^s_L(λ)=L∏^←_wB_w(λ)


Also we denote the p-difference operator as A^s_L=T_L^-1B^s_L(λ) with T_Lf(z)=f(p^L). It has been proved in <cit.> that the difference operators {A^s_L} form a holonomic module over the Picard torus Pic(X)⊗C^*, i.e.

    [A^s_L,A^s_L']=0

and A^s_L is independent of the choice of the path from s to s-L. Moreover, for the slope s being in a subset of the opposite of the ample cone ∇⊂-C_ample, A^s_L is conjugate to T_L^-1M_L(z) via Stab_C,s.

In this paper we will focus on the case of the affine type A. We will always use the expression <ref> as the geometric quantum difference operator. To distinguish from the algebraic quantum difference operators, we will denote them all by B_L^s,MO(λ) and B_w,MO(λ).



 §.§ Algebraic quantum difference equation

Here we review the construction of the algebraic quantum difference equation of affine type A constructed in <cit.>.

The quantum toroidal algebra U_q,t(ŝ̂l̂̂̂_n) admits the slope factorization:

    U_q,t(ŝ̂l̂̂̂_n)=⊗^→_μ∈QB_m+μθ


For each slope subalgebra B_m, one can associate an element J_m^±(λ)∈B_m⊗̂B_m such that they satisfy the ABRR equation:

    J_m^+(λ)q_(1)^-λq^ΩR_m^+=q_(1)^-λq^ΩJ_m^+(λ),     q^ΩR_m^-q_(1)^-λJ_m^-(λ)=J_m^-(λ)q^Ωq_(1)^-λ



the monodromy operator is defined as:

    B_m(λ)=m(1⊗ S_m(J_m^-(λ)^-1))|_λ→λ+κ


Here κ=Cv-w/2.

Let L∈ Pic(X) be a line bundle. Now we fix a slope s∈ H^2(X,R) and choose a path in H^2(X,R) from s to s-L. This path crosses finitely many slope points in some order {m_1,m_2,⋯,m_m}.  And for this choice of a slope, line bundle and a path we associate the following operator:

    B_L^s(λ)=LB_m_m(λ)⋯B_m_1(λ)


We define the q-difference operators:

    A^s_L=T_L^-1B^s_L(λ)


It has been proved in <cit.> that the q-difference operator A^s_L is independent of the choice of the path from s to s-L if we choose the generic path.




 §.§ Review of the analysis of the algebraic quantum difference equations


The algebraic quantum difference equations has been analysed in detail in <cit.>. Here we briefly review the result in there.

First note that at z=0∈Pic(X)⊗C^×, i.e. the corresponding chamber in Pic(X)⊗R corresponds to the chamber such that all the parametres a_1,⋯,a_n goes to -∞. Let M(0)=M_n(0)M_n-1(0)⋯ M_1(0)=L_n⊗⋯⊗L_1 and this matrix is diagonal in K_T(X) with respect to the fixed point basis. And let P be the matrix with columns given by fixed point eigenvectors [λ]. We denote by E_0 the diagonal matrix of eigenvalues, so that:

    M(0)P=PE_0


The fundamental solution around z=0 can be written as:

    Ψ_0(z)=PΨ^reg_0(z)exp(∑_iln(E_0^(i))ln(z_i)/ln(p))


For the solution around z=∞, by the Lemma 3.2 in <cit.>, the operator M_O(1)(∞) consists of the product of the form m((1⊗ S_m)(R_m^-)^-1). It is a diagonalizable matrix over Q((q,t)) with eigenvalues given by the formal power series of q and t, i.e. it has distinct eigenvalues for the generic value of q and t. We shall denote the corresponding matrix of eigenvectors as H, and E_∞ be the digaonal matrix of eigenvalues:

    M_O(1)(∞)H=HE_∞

the solution around z=∞ can be written as:

    Ψ_∞(z)=HΨ_∞^reg(z)exp(∑_iln(E_∞^(i))ln(z_i)/ln(p))

such that

    Ψ^reg_∞(z)E_∞=H^-1M_O(1)(z)HΨ^reg_∞(z)



The transition matrix between two fundamental solutions Ψ_0(z) and Ψ_∞(z) is defined as:

    Mon(z):=Ψ_0(z)^-1Ψ_∞(z)



Now we choose our path such that B_m(z) are all of the generic point m. It remains to analyze B_w(p^sz), by the result above, 

    lim_p→0B_m(p^sz)=
    m((1⊗ S_m)(R_m^+)_21^-1)   s<m,m∈ U_s
    B_m(z)|_p=1   s=m
    B_m(z_I,θ_J)   s_I=m_I,s_I^c≠m_I^c
    
    1   Otherwise


Now we choose a generic representation of the quantum difference operator M_O(1)(z)=O(1)∏^→_m∈WallsB_m(z) such that each monodromy operator B_m(z) are either of the U_q(sl_2)-type or of the U_q(ĝl̂_1)-type. The following theorem gives the expression for the p→0 limit of the connection matrix:

For generic s∈Q^n the connection matrix as the following asymptotic at p→0:

    lim_p→0Mon^reg(p^sz)=
    ∏^←_0≤m<s(B_m^*)^-1·T,     s≥0
    ∏_s<m<0B_m^*·T,     s<0


Here T:=P^-1H, 0≤m<s means the slope points m in one generic path from 0 to the point s without intersecting s. Here s≥0 and s<0 stands for s=(s_1,⋯,s_n) such that s_i≤ 0 or s_i>0 for every component s_i. The B_m^* is defined as P^-1B_mP.


The main advantage of using the theorem is that we can use it to compute the monodromy representation for the Dubrovin connection of the affine type A quiver varieties:


The degeneration limit of the quantum difference operator M_L(z) coincides with the quantum multiplication operator Q(L) up to a constant operator.


This has been proved in <cit.>. A direct consequence is that we can use this result to compute the monodromy representation.

Let Ψ_0,∞(q_1,q_2,p,z) be the solution to the quantum difference equation described above. Then we take z=e^2π s, q_i=e^2π iħ_iτ, q=e^-2π iτ. By the Theoreom <ref>, for the degeneration limit of the solution ψ_0,∞(z), which is defined as follows:

    ψ_0,∞(z)=lim_τ→0Ψ_0,∞(e^2π iħ_1τ,e^2π iħ_2τ,e^-2π iτ,z)∈H_T(M(v,w))_loc


It is the solution to the corresponding Dubrovin connection. The solution lies in the completion of H_T(M(v,w))_loc. Using this, we can define the transport of the solution of the Dubrovin connection:

    Trans(s):=ψ_0(e^2π is)^-1ψ_∞(e^2π is)∈H_T(M(v,w))_loc

Similarly as a result of the Theorem <ref>, the transport of the solution is a limit the monodromy:

    Trans(s)=lim_τ→0Mon(z=e^2π is,q_1=e^2π iħ_1τ,q_2=e^2π iħ_2τ,q=e^-2π iτ)
 

The monodromy corresponds to the element Trans(s')^-1Trans(s), and if we denote the slope point of the difference by m, we can see that:

    Trans(s')^-1Trans(s)=B_m^*



In conclusion, we have the following monodromy representation:

The monodromy representation:

    π_1(P^n\Sing,0^+)→End(H_T(M(v,w)))

of the Dubrovin connection is generated by B_m^* with q_1=e^2π iħ_1,q_2=e^2π iħ_2, i.e. the monodromy operators B_m in the fixed point basis. The based point 0^+ is a point sufficiently close to 0∈P^n.




 §.§ Asymptotic behavior of the geometric monodromy operators



Before we analyze the solution of the difference equation, we first analyze the asymptotic behavior of the monodromy operators B_w(z):

Suppose that B_w(z) corresponds to the wall w such that there is only one connected component, i.e. w corresponds to only one root α∈Z^I. Then we have:

    lim_p→0B_w(zp^s)=
    m((1⊗ S_w)(R_w^-)^-1)   ⟨α,s⟩>-m for the integer m=⟨L_w,α⟩
    
    1   otherwise




We can see this via the analysis of the fusion operator J_w^+(λ). We write down the ABRR equation in the form:

    Ad_q_(1)^λq^-Ω(J_w^+(z))=J_w^+(z)(R_w^+)^-1

Since both J_w^+(z) and R_w^+ are upper-triangular, they admit the root decomposition:

    J_w^+=1+∑_α>0J_α,    
    (R_w^+)^-1=1+∑_α>0R_α

The α-component of J_w^+(z) can be solved as:

    J_α(z)=1/z^αq^m-1∑_γ+δ=α
    γ<αJ_γ(z)R_δ


We substitute z↦ p^sz and we have:

    J_α(p^sz)=1/z^αq^mp^s·α-1∑_γ+δ=α
    γ<αJ_γ(p^sz)R_δ


Now since w contains only one hyperplane corresponding to α, this means that all the roots of R_w are of the form kα, and this means that:

    J_kα(p^sz)=1/z^kαq^mp^ks·α-1∑_k_1+k_2=k,k_1<kJ_k_1α(p^sz)R_k_2α


Specifically for k=1:

    J_α(p^sz)=1/z^αq^mp^s·α-1R_α

and when p→0:

    lim_p→0J_α(p^sz)=
    
    R_α   s·α>0
    1/z^αq^m-1R_α   s·α=0
    
    0   otherwise


In conclusion we have that:

    lim_p→0J_w^+(p^sz)=
    
    R_w^+        s·α>0
    
    J_w^+(z)        s·α=0
    
    1       otherwise


Now for J_w^+(z), it is defined via the shift z↦ zp^L_w, we have that:

    lim_p→0B_w(p^sz)=
    m((1⊗ S_w)(R_w^-)^-1)        (s+L_w)·α>0
    B_w(z)|_p=1        (s+L_w)·α=0
    
    1        (s+L_w)·α<0







 §.§ Analysis of the Okounkov-Smirnov quantum difference equations

The analysis of the Okounkov-Smirnov quantum difference equations is similar to the analysis given in <cit.> for the algebraic quantum difference equations. The origin of the analysis of the quantum difference equation can be dated back to Smirnov's paper<cit.> on the analysis for the case of the Hilbert scheme Hilb_n(C^2). For simplicity, we can analyze the solution for the quantum difference operator B^s,MO_L(z) at L=O(1).



First we analyze the fundamental solution Ψ^MO_0(z) around z=0. When z=0, B^s,MO_L(0)=L, and it has the eigenvectors given by the fixed point basis |λ⟩ with the eigenvalues E_0. In this case the fundamental solution Ψ^MO_0(z) can be written as:


    Ψ_0^MO(z)=PΨ^reg,MO_0(z)exp(∑_iln(E_0^(i))ln(z_i)/ln(p))


Here Ψ^reg,MO_0(z)=1+∑_d∈N^Ia_dz^d is the regular solution satisfying the following difference equations:

    Ψ_0^reg,MO(zp^L_i)E_0^(i)=P^-1B_O(1)^s,MO(z)PΨ_0^reg,MO(z)


For simplicity, we denote M^*(z)=P^-1B_O(1)^s,MO(z)P. Now the regular solution can be formally written as:

    Ψ_0^reg(z)=∏^←
    ∞_k=0M^*_k(z)^-1,    M_k^*(z)=E_0^k-1M^*(p^kz)E_0^-k


For the fundamental solution around z=∞, note that for z=∞:

    B^s_L(∞)=L∏^←_wB_w


In this case one could also find the eigenvector H^MO with the distinct eigenvalues E_∞ the proof is the same as the proof in the Appendix II of <cit.>.

the solution around z=∞ can be written as:

    Ψ_∞^MO(z)=H^MOΨ_∞^reg(z)exp(∑_iln(E_∞^(i))ln(z_i)/ln(p))

such that

    Ψ^reg,MO_∞(z)E_∞=(H^MO)^-1M_O(1)(z)H^MOΨ^reg_∞(z)


We use the similar procedure to compute the regular part of Ψ^reg_∞(z), and it is easy to see that the solution is of the form:

    Ψ^reg_∞(z)=∏^→
    ∞_k=0M^*_-k(z)

with

    M^*_-k(z)=E_∞^k-1M^*
    (zp^-k)E_∞^-k



The transition matrix between two fundamental solutions Ψ_0(z) and Ψ_∞(z) is defined as:

    Mon^MO(z):=Ψ_0^MO(z)^-1Ψ_∞^MO(z)


Furthermore we consider the regular part of the connection matrix:

    Mon^MO,reg(z):=Ψ_0^MO,reg(z)^-1Ψ_∞^MO,reg(z)



Similar analysis as in <cit.> can show that the degeneration limit of Mon^MO,reg(p^sz) as p→0 is similar to that of Mon^MO(p^sz):

For generic s∈Q^n the connection matrix as the following asymptotic at p→0:

    lim_p→0Mon^reg,MO(p^sz)=
    ∏^←_0≤ w<s(B_w,MO^*)^-1·T^MO,     s≥0
    ∏_s<w<0B_w,MO^*·T^MO,     s<0


Here T^MO:=P^-1H^MO, 0≤ w<s means the wall w that intersects with the generic path from 0 to the point s without intersecting s. Here s≥0 and s<0 stands for s=(s_1,⋯,s_n) such that s_i≤ 0 or s_i>0 for every component s_i. The B_w,MO^* is defined as P^-1B_w,MOP.



We want to use this result to compute the monodromy representation of the Dubrovin connection. In the next section we are going to show how to give the degeneration limit of the Okounkov-Smirnov geometric quantum difference equation to the Dubrovin connection. 








§ COHOMOLOGICAL LIMIT TO THE DUBROVIN CONNECTION




 §.§ Degeneration limit of the geometric quantum difference equations

Now we focus on the case of the affine type A.

From <ref> one could have the general formula for the geometric monodromy operators :

    B_w,MO(λ)=∑_l≥0∑_k_1+⋯+k_l=k1/(z^-k_1αq^-mp^-k_1α·L_w-1)⋯(z^-k_lαq^-mp^-k_lα·L_w-1)m((1⊗ S_w)(R_w,k_1α^-,MO⋯ R_w,k_lα^-,MO)^-1)


Now we take the cohomological limit of the quantum difference operators A^s_L:

    A^s,coh_L=-d_L+c_1(L)∪+∑_w∈[s,s+L)B_w,MO^coh(λ)


Now we take p=e^-2π iτ, q_i=e^2π iħ_iτ, and we let τ→0. Since we have the asymptotics of the wall R-matrices as q→1:

    R_w^-,MO=Id+(q-1)r_w^-,MO+O((q-1)^2),    as q→1


We have the degeneration limit of B_w,MO(λ):


    B^coh_w,MO(λ)=   -∑_k≥0ħ/1-z^-kαm((1⊗ S_w)r_w,kα^-,MO)
    =ħ/1-Ad_q_(1)^λm(r_w^-)


By construction, r_w^- corresponds to the root subalgebra g_w of g_Q. By definition, the wall w is defined as 

    w={s∈ H^2(X,R)|(s,α)+n=0}


We have known from the Theorem <ref> that each root pieces n_w^MO is isomorphic to ⊕_k≥0n_kα⊂sl_n^+. 

In this we can write down the orthogonal basis of g_Q^± as {e_±α^(i)}. It is known that r_w+L^±=r_w+L, so in this way we have that:

    A^s,coh_L=   -d_L+c_1(L)∪+∑_w∈[s,s+L)B_w,MO^coh(λ)
    
    =   -d_L+c_1(L)∪+∑_w∈[s,s+L)ħ/1-Ad_q_(1)^λm(r_w^-,MO)
    
    =   -d_L+c_1(L)∪+∑_w∈[s,s+L)∑_i∈ wħ/1-Ad_q_(1)^λe_α^(i)e_-α^(i)
    
    =   -d_L+c_1(L)∪+ħ∑_α⟨α,L⟩/1-q^-αe_αe_-α


Here ⟨α,L⟩ counts the number of root vectors e_±α appearing in the path s and s+L. The roots α corresponds to the vector [i,j) in the affine type A case, and e_±α coincides with E_±[i,j) via the comparison of the classical r-matrix:

    r_Q^+=∑_α>0e_α⊗ e_-α=∑_j>iE_[i,j)⊗ E_-[i,j)


Combining these facts we can see that the degeneration limit of the geometric quantum difference equation coincides with the Dubrovin connection.


In this way we obtain the following theorem:

The degeneration limit A^s,coh_L of the Okounkov-Smirnov quantum difference equation A^s_L of the affine type A quiver varieties is the Dubrovin connection of the affine type A quiver varieties.




 §.§ Monodromy representations for the Dubrovin connection

Now we can express the monodromy representation of the Dubrovin connection in terms of the degeneration limit of the connection matrix of the Okounkov-Smirnov quantum difference equation.

Let Ψ_0,∞^MO(q_1,q_2,p,z) be the solution of the Okounkov-Smirnov quantum difference equation as constructed in the subsection <ref>. We take p=e^-2π iτ, q_i=e^2π iħ_iτ and z=e^2π is, and take the degeneration limit of the solution as:

    ψ_0,∞^MO(z)=lim_τ→0Ψ_0,∞^MO(e^2π iħ_1τ,e^2π iħ_2τ,e^-2π iτ)


The transport of the two solutions is defined as:

    Trans^MO(s):=ψ_0^MO(e^2π is)^-1ψ_∞^MO(e^2π is)


As a result of the Theorem <ref>, the transport of the solution can be written as:

    Trans^MO(s)=lim_τ→0Mon^MO(z=e^2π is,q_1=e^2π iħ_1τ,q_2=e^2π iħ_2τ,q=e^-2π iτ)


Similar as the argument in the Proposition 7.4 in <cit.>, we have that:

    Trans^MO(s)=lim_p→0Mon^MO,reg(p^s,e^2π iħ_1,e^π iħ_2,p)

for generic s∈R^n.


Also the similar argument in <cit.> applies for the case of the geometric quantum difference equations. The monodromy representation corresponds to the difference of the transport matrix Trans^MO(s')^-1Trans^MO(s). Using the similar computation as in <cit.>, one can show that if there is a wall w between s and s', we have:

    Trans^MO(s')^-1Trans^MO(s)=B_w,MO^*


Here we denote B_w,MO^* as B_w^* to distinguish the geometric and the algebraic monodromy operators.

In this way we obtain the similar result as in the Theorem <ref>


The monodromy representation:

    π_1(P^n\Sing,0^+)→Aut(H_T(M(v,w)))

of the Dubrovin connection is generated by B_w,MO^* with q_1=e^2π iħ_1,q_2=e^2π iħ_2, i.e. the monodromy operators B_w,MO in the fixed point basis. The based point 0^+ is a point sufficiently close to 0∈P^n.




§ PROOF OF THE OKOUNKOV'S CONJECTURE

In this section we prove the following statement:


The positive half of the quantum toroidal algebra U_q,t^+(ŝ̂l̂̂̂_n) is isomorphic to the positive half of the Maulik-Okounkov quantum affine algebra U_q^MO,+(ĝ_Q) for the affine type A.



The statement can be proved once we prove the following statement:

Restricted to the morphism space End(⊕_v_1+v_2=vK(v_1,w_1)⊗ K(v_2,w_2)):

    R_w^MO=R_w





The Theorem <ref> implies the Theorem <ref>.


By the Neguţ's result <cit.>, we have an algebra embedding:

    U_q,t(ŝ̂l̂̂̂_n)↪ U_q^MO(ĝ_Q)↪∏_wEnd(K(w))

Thus the equality R^MO_w=R_w on End(⊕_v_1+v_2=vK(v_1,w_1)⊗ K(v_2,w_2)) implies the equality in U_q^MO(ĝ_Q). The equality of the universal R-matrix implies the isomorphism of the Hopf algebra U_q(g_w)≅ U_q^MO(g_w). This implies <ref> by the factorisation property of the universal R-matrix in the Theorem <ref> and the formula <ref>, and thus implies the Theorem <ref>.






 §.§ Proof of the main theorem

Since we know that The monodromy representation of the Dubrovin connection for the affine type A quiver varieties is generated by m((1⊗ S_w)R_w^MO) and m((1⊗ S_w)R_w). We can check that actually these two generators are equal:

As the element in End(K(v,w)), we have that:

    m((1⊗ S_w)R_w^MO)=m((1⊗ S_w)R_w)



First we prove that given two quantum difference equations Ψ(p^Lz)=M_L(z)Ψ(z), Ψ^MO(p^Lz)=M^MO_L(z)Ψ^MO(z). The degeneration limit of their solutions Ψ_0,∞(z) and Ψ_0,∞^MO(z) are the same. This implies that the transport of solutions Trans^MO(s) and Trans(s) from both geometric and algebraic quantum difference equations are the same.

Since the solution of both difference equations are of the form:

    Ψ_0,∞(z)=P_0,∞Ψ_0,∞^reg(z)z^(⋯)


Here P_0,∞ are the matrix of eigenvectors of M_L(0,∞). Ψ_0,∞^reg(z) are the regular part of the solution around z=0,∞ such that Ψ_0,∞^reg(z)=1+O(z^±1).

Now since M_L^(MO)(0)=L, P_0^(MO) can be chosen as the fixed point basis. In fact, they have the same degeneration limit:

 The degeneration limit of Ψ_0(z) and Ψ_0^(MO)(z) are the same.


The proof is based on the analysis of the expression for Ψ_∞(z) and Ψ_∞^(MO)(z). It is known that they are expressed in terms of the infinite product of the form:

    ∏_k≥0^∞E_∞^-k-1H^-1B_m_1(p^kz)^-1⋯B_m_l(p^kz)HE_∞^k,    Algebraic QDE
       ∏_k≥0^∞E_∞,MO^-k-1H_MO^-1B_w_1,MO(p^kz)⋯B_w_r,MO(p^kz)HE_∞,MO^k,    Okounkov-Smirnov QDE


It is known that both for generic B_m(z) and B_w,MO(z) has the expression of the form:

    ∑_α_1+⋯+α_r=kα
    r≥0(-1)^r/(1-q^-mz^-α_1p^-k|α_1|)⋯(1-q^-mz^-α_rp^-k|α_r|)m((1⊗ S_w)(R_α_1,MO^-⋯ R_α_r,MO^-)^-1)


Thus when written into the infinite product of the form <ref>, the contribution of R_α after doing the degeneration is determined by the leading term of the form:

    -∑_k≥01/1-q^-mz^-αp^-k|α|E_∞,(MO)^-k-1H^-1m((1⊗ S_w)(R_α,MO^-)^-1)HE_∞,MO^k


Doing the taylor expansion of 1/1-q^-mz^-αp^-k|α|, we have that this leading term is also determined by the following terms:

    E_∞,(MO)^-1∑_k≥0q^-mz^-αp^-k|α|Ad_E_∞,(MO)^kHm((1⊗ S_w)R_α)=q^-mz^-α/1-p^-|α|Ad_E_∞,(MO)Hm((1⊗ S_w)R_α)

Now if we do the degeneration limit p,q→1 with ln(p)/ln(q) fixed. This implies that if R_α has the degeneration limit as the root pieces of the classical wall r-matrix r_α. Then the term <ref> survives.  If not, it will be zero. 

Also since the degeneration limit of both E_∞,E_∞,MO and H,H_MO are the same, it is reduced to analyze the degeneration limit of R_m=∏_αR_α for slope subalgebras and R_w^MO=∏_αR_α^MO for the Maulik-Okounkov wall subalgebras. But as stated before, the degeneration limit of R_w^MO which is not zero coincides with the degeneration limit of R_m for some slope point m on the wall w. Thus this means that the fundamental regular solutions Ψ_0^reg(z) and Ψ_0^reg,MO(z) have the same degeneration limit ψ_0(z).



It remains to check that the degeneration limit of Ψ_∞(z) and Ψ_∞^(MO)(z) are the same. Since:

    M_L(∞)=L∏^←_mB_m,    M_L^MO(∞)=L∏^←_wB_w^MO


the degeneration limit of M_L(∞) and M_L^MO(∞) are of the form:

    M_L^coh(∞)=c_1(L)+∑_mm(r_m),    M_L^(MO),coh(∞)=c_1(L)+∑_wm(r_w)


Since for generic choice of m, r_m=r_w, we have that M_L^coh(∞)=M_L^(MO),coh(∞). It means that we can choose P_∞ and P_∞^MO such that their corresponding degeneration limit are the same P_∞^coh.

In this case we can see that we can construct the solutions Ψ_∞(z) and Ψ_∞^MO(z) around z=∞ such that they have the same degeneration limit ψ_∞(z) when z=∞. Using the proof in the Proposition <ref>, we can also prove the following statement:


 The degeneration limit of Ψ_∞(z) and Ψ_∞^(MO)(z) are the same.





Now from the definition of Trans(s)=ψ_0^-1(e^2π is)ψ_∞(e^2π is) and the fact that the degeneration limit of the solution from the algebraic QDE and Okounkov-Smirnov QDE are of the same form.


 


Using the fact that U_q,t(ŝ̂l̂̂̂_n)↪ U_q(g_Q)↪∏_wEnd(K(w)), a direct consequence is that as the element of U_q^MO(g_w), we also have the equality:

    m((1⊗ S_w)R_w^MO)=m((1⊗ S_w)R_w)

This means that:

    m((1⊗ S_w)(R_w^MO-R_w))=0

which implies that R_w^MO=R_w from the following general lemma. 


Let A^≤, A^≥ be two finite-dimensional F-Hopf algebra with a non-degenerate bilinear pairing ⟨-,-⟩:A^≤⊗ A^≥→F. Let D(A):=A^≤⊗ A^≥ be the Drinfeld double. Let B^≤⊂ A^≤ and B^≥⊂ A^≥ be the Hopf subalgebra such that the restriction of the bilinear pairing ⟨-,-⟩ to B^≤⊗ B^≥ is still non-degenerate, and using the bilinear pairing we can also define the Drinfeld double D(B):=B^≤⊗ B^≥. Denote R_A and R_B the universal R-matrix for D(A) and D(B). Then denote m the product map for D(A) and D(B). If m(R_A)=m(R_B), R_A=R_B.


We choose {F_i}_i∈ I_A and {G_i}_i∈ I_A the specific dual bases in A^≤ and A^≥ with respect to the non-degenerate bilinear pairing ⟨-,-⟩ such that for D(B)⊂ D(A) as Hopf algebra with respect to the same bilinear pairing, the dual bases {F_i}_i∈ I_B and {G_i}_i∈ I_B in B^≤ and B^≥ are the same such that I_B⊂ I_A. The existence of such basis will be proved in the appendix. Now it can be seen that the basis for D(A) can be given as {F_i⊗ G_j}_i,j∈ I_A, and similarly {F_i⊗ G_j}_i,j∈ I_B for D(B).

By definition of the universal R-matrix we have the formula:

    R_A=∑_i∈ I_A(F_i⊗1)⊗(1⊗ G_i)∈ D(A)⊗ D(A)


    R_B=∑_i∈ I_B(F_i⊗1)⊗(1⊗ G_i)∈ D(B)⊗ D(B)


So we have that:

    m(R_A)-m(R_B)=∑_i∈ I_A\ I_BF_i⊗ G_i

Now {F_i⊗ G_i}_i∈ I_A\ I_B are also bases which are linearly independent. So if ∑_i∈ I_A\ I_BF_i⊗ G_i=0, then F_i⊗ G_i=0 for i∈ I_A\ I_B.


Now for our case, since S_w is an isomorphism of vector spaces. <ref> implies that m(R_w^MO)=m(R_w), which means that R_w^MO=R_w.
Hence finish the proof of the Theorem <ref>, and now we have our main result the Theorem <ref>. 






§ APPENDIX: BASICS ABOUT THE DRINFELD DOUBLE

In this appendix we prove some facts about the Drinfeld double of the Hopf algebra which is useful in the computation of the article.

Given two bialgebras A^- and A^+ over the field F, a bilinear pairing ⟨-,-⟩:A^+⊗ A^-→F is called a bialgebra pairing if it satisfies:

    ⟨ aa',b⟩=⟨ a⊗ a',Δ(b)⟩,    ⟨ a,bb'⟩=⟨Δ^op(a),b⊗ b'⟩

for all a,a'∈ A^+ and b,b'∈ A^-. All our pairings will be Hopf, which means that ⟨ Sa,b⟩=⟨ a,S^-1b⟩. Given such a bialgebra pairing, we can construct the Drinfeld double of the two bialgebras:

    D(A):=A^+⊗ A^-

as a vector space. We further require that the pairing is nondegenerate, which implies that A^+≅ A^- as vector spaces. 

There are natural Hopf algebra embeddings A^±↪ D(A) as a↪ a⊗ 1 for a∈ A^+ and b↪ 1⊗ b for b∈ A^-. D(A) satisfies the following extra relation:

    ⟨ a_1,b_1⟩(a_2⊗1)·(1⊗ b_2)=(a_1⊗1)·(1⊗ b_1)⟨ a_2,b_2⟩,    ∀ a∈ A^+,b∈ A^-

This is equivalent to:

    (a⊗1)·(1⊗ b)=⟨ Sa_1,b_1⟩(1⊗ b_2)·(a_2⊗ 1)⟨ a_3,b_3⟩,    ∀ a∈ A^+,b∈ A^-


Furthermore we can write the product structure as:

    (a⊗ b)·(a'⊗ b')=⟨ Sa_1,b'_1⟩⟨ a_3,b_3'⟩ a_2a'⊗ bb'_2




Now we suppose that A^± are finite-dimensional, and we choose the dual bases {F_i}_i∈ I {G_i}_i∈ I with respect to the bialgebra pairing. Define the universal R-matrix as:

    R=∑_i∈ I(1⊗ G_i)⊗ (F_i⊗ 1)∈ A^-⊗ A^+⊂ D(A)⊗ D(A)


The following lemma shows that R is independent of the choice of the basis:

The definition of the universal R-matrix is independent of the choice of the basis.




Now we choose another set of dual bases {F_i'} and {G_i'} of A^±. We write down the transformation of the bases:

    F_i'=∑_j∈ Ia_ijF_j,     G_i'=∑_j∈ Ib_ijG_j


In terms of matrix:

    F'=AF,    G'=BG


The condition that ⟨ F_i',G_l'⟩=δ_il, we have the condition that:

    ⟨ F_i',G_l'⟩=∑_j,k∈ Ia_ijb_lk⟨ F_j,G_k⟩=∑_j∈ Ia_ijb_lj=∑_j∈ I(A)_ij(B^t)_jl=δ_il

This is equivalent to say that AB^t=Id.

Now for the universal R-matrix, note that:

    ∑_i∈ I(1⊗ G_i')⊗(F_i'⊗1)=∑_i,j,k∈ Ib_ija_ik(1⊗ G_j)⊗(F_k⊗1)
    
    =   ∑_j,k∈ I(1⊗ G_j)⊗ (F_k⊗1)(∑_i∈ Ib_ija_ik)=∑_j,k∈ I(1⊗ G_j)⊗ (F_k⊗1)(∑_i∈ I(A^t)_ki(B)_ij)
    
    =   ∑_j,k∈ I(1⊗ G_j)⊗ (F_k⊗1)(A^tB)_kj=∑_j,k∈ I(1⊗ G_j)⊗ (F_k⊗1)δ_kj
    
    =   ∑_j∈ I(1⊗ G_j)⊗ (F_j⊗1)



Now given the Hopf subalgebras B^±⊂ A^± such that the bialgebra pairing on B^± is the restriction of the bialgebra pairing on A^±. Without loss of generality, we assume that dim_F(A^±)=n+m, dim_F(B^±)=n.


There exists dual bases {F_i}_i=1,⋯,n+m, {G_i}_i=1,⋯,n+m of A^± such that {F_i}_i=1,⋯,n, {G_i}_i=1,⋯,n are the dual bases for B^±.


Now suppose that we have the dual bases {F_i'}_i=1,⋯,n+m, {G_i'}_i=1,⋯,n+m of A^±, and {F_i}_i=1,⋯,n, {G_i}_i=1,⋯,n are the dual bases for B^±. By the requirement of the proposition, we want to construct the dual bases {F_i}_i=1,⋯,n+m, {G_i}_i=1,⋯,n+m of A^± such that {F_i}_i=1,⋯,n, {G_i}_i=1,⋯,n are the dual bases for B^±.

Now we write {F_i}_i=1,⋯,n, {G_i}_i=1,⋯,n of B^± in terms of
{F_i'}_i=1,⋯,n+m, {G_i'}_i=1,⋯,n+m of A^±:

    F_i=∑_j=1^n+ma_ijF_j',     G_i=∑_j=1^n+mb_ijG_j'


Written in matrix:

    [      F_[1,n]; F'_[n+1,n+m] ]=[  A  T;  0 Id ][     F'_[1,n]; F'_[n+1,n+m] ]



    [      G_[1,n]; G'_[n+1,n+m] ]=[  B  S;  0 Id ][     G'_[1,n]; G'_[n+1,n+m] ]


Now we can suppose the transfer matrix:

    [     F_[1,n]; F_[n+1,n+m] ]=[  Id   0; U_1 V_1 ][      F_[1,n]; F'_[n+1,n+m] ]



    [     G_[1,n]; G_[n+1,n+m] ]=[  Id   0; U_2 V_2 ][      G_[1,n]; G'_[n+1,n+m] ]


So combining these we obtain that:

    [     F_[1,n]; F_[n+1,n+m] ]=   [  Id   0; U_1 V_1 ][      F_[1,n]; F'_[n+1,n+m] ]=[  Id   0; U_1 V_1 ][  A  T;  0 Id ][     F'_[1,n]; F'_[n+1,n+m] ]
    
    =   [        A        T;     U_1A U_1T+V_1 ][     F'_[1,n]; F'_[n+1,n+m] ]



    [     G_[1,n]; G_[n+1,n+m] ]=   [  Id   0; U_2 V_2 ][      G_[1,n]; G'_[n+1,n+m] ]=[  Id   0; U_2 V_2 ][  B  S;  0 Id ][     G'_[1,n]; G'_[n+1,n+m] ]
    
    =   [        B        S;     U_2B U_2S+V_2 ][     G'_[1,n]; G'_[n+1,n+m] ]


By the orthogonality condition we need that:

    [        A        T;     U_1A U_1T+V_1 ][            B^t       B^tU_2^t;            S^t S^tU_2^t+V_2^t ]=[ Id  0;  0 Id ]
    
    =   [                                                AB^t+TS^t                               AB^tU_2^t+TS^tU_2^t+TV_2^t;                                   U_1AB^t+U_1TS^t+V_1S^t U_1AB^tU_2^t+U_1TS^tU_2^t+U_1TV_2^t+V_1S^tU_2^t+V_1V_2^t ]


The matrix relation gives us the equation:

    AB^t+TS^t=Id


which reduce the matrix to:

    [                   Id         U_2^t+TV_2^t;           U_1+V_1S^t V_1S^tU_2^t+V_1V_2^t ]=[ Id  0;  0 Id ]


This gives us the equation V_1(S^tU_2^t+V_2^t)=V_1(-S^tT+Id)V_2^t=Id. Thus for the generic choice of S and T, there exists solution for V_1 and V_2. Thus finish the proof.





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