Matrix Spherical Functions for $(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times \mathrm{U}(m)))$: Two Specific Classes

We consider the matrix spherical function related to the compact symmetric pair $(G,K)=(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$. The irreducible $K$ representations $(\pi,V)$ in the ${\rm U}(n)$ part are considered and the induced representation $\mathrm{Ind}_K^G\pi$ splits multiplicity free. In this case, the irreducible $K$ representations in the ${\rm U}(n)$ part are studied. The corresponding spherical functions can be approximated in terms of the simpler matrix-valued functions. We can determine the explicit spherical functions using the action of a differential operator. We consider several cases of irreducible $K$ representations and the orthogonality relations are also described.


Generalities 1.Introduction
There is a close relation between representation theory and special functions.In this paper, we consider explicit matrix-valued polynomials, i.e., matrix spherical functions.We use the notion of spherical functions for symmetric pair in [4,28] taking values in a matrix algebra.
The matrix-valued spherical functions of rank one type have been exploited in several cases.In [12,13], the matrix-valued spherical functions for symmetric pair (SU(2) × SU(2), diag) arising from Koornwinder [19] are studied.In [6], the matrix-valued spherical functions for symmetric pair (SU(3), U(2)) are studied, and for the more general case for symmetric pair (SU(m + 1), U(m)) they are given in [23,26].The approach in [23] is to find two different differential operators and the spherical functions are the corresponding eigenfunctions.The approach in [26] is to find the K intertwiner j by Lemma 1.2.The rank two case for symmetric pair (SU(2 + m), S(U(2) × U(m))) has been studied in [11].The approximate spherical functions of this case can be related to the Krawtchouk polynomials, and it is also a matrix analogue of Koornwinder's BC 2 orthogonal polynomials in [15,16,17,18].Moreover, in [25] it shows the relation to mathematical physics and possible applications.
Scalar-valued spherical functions for symmetric pair (SU(n + m), S(U(n) × U(m))) are given by [7].In this paper, we calculate the matrix spherical functions for the same symmetric pair.The approach to calculating the corresponding matrix spherical functions is motivated by [11].Now we introduce the contents of this paper.In Sections 1.2 and 1.3, we briefly recall the definition of a multiplicity free triple and the spherical function.In Section 2, we describe the structure theory and the representation theory in more details for the symmetric pair (G, K) = (SU(n + m), S(U(n) × U(m))).In Section 3, we study the spherical function restricted to a subgroup A of G, since it uniquely determines the spherical function by the Cartan decomposition.In Section 4, we calculate the radial part of the Casimir operator since the spherical function is the eigenfunction of this operator.In Section 5, we give the simplest cases and obtain the approximate functions.It is an intermediate step for calculating the corresponding spherical functions in Sections 6 and 7.In Section 8, we study the orthogonality relations and calculate the matrix weight function.

Multiplicity free triples
Let G be a compact Lie group and K be a compact subgroup of G.We define π G λ as the irreducible G representation with the highest weight λ and V G λ as the corresponding irreducible G module.Also π K µ and V K µ can be defined similarly.We define P + G , P + K as the set of the highest weight of (G, K), and π G λ | K : π K µ as the multiplicity of V K µ in V G λ decomposed as K module.Let µ be a highest weight of K, the triple (G, K, µ) is a multiplicity free triple if and only if π G λ | K : π K µ ≤ 1 for all λ ∈ P + G .We define

Spherical functions
Now we give some preliminaries of spherical functions.We recall some results from [4,8,14,20].
Let C[G] be the algebra of matrix elements of finite-dimensional irreducible representations of the compact group G = SU(n + m).Then we have an action of which is the biregular representation.By restriction, C[G] is a K × K representation.For a fixed K-representation π K µ : K → V K µ of highest weight µ, we also have End V K µ as a K × K representation by Then we consider C[G] ⊗ End V K µ as a space of functions Φ : Definition 1.1.A matrix spherical function is an element of (C[G] ⊗ End(V K µ )) K×K , i.e., (K × K)-invariant elements.So Φ : for any k 1 , k 2 ∈ K and g ∈ G.
Assuming that (G, K, µ) is a multiplicity free triple, we can associate a matrix spherical function to each λ ∈ P + G (µ).
Lemma 1.2.For λ ∈ P + G (µ), we define a matrix-valued function Φ : where j is a K-invariant embedding, i.e., j ∈ Hom K V K µ , V G λ , and j * is the adjoint of j.Then Φ µ λ is a matrix spherical function.
Remark 1.3.Note that Definition 1.1 includes other spherical functions beyond the ones considered in Lemma 1.2.
We recall [14, Section 2].Since all K representations are unitary, we can take j to be unitary.Then λ is independent of the choice of j.We define such spherical functions in Lemma 1.2 as the zonal spherical functions if µ = 0 and λ ∈ P + G (0).We denote the vector space spanned by Φ µ λ 's with λ ∈ P + G (µ) by E µ , i.e., , and the vector space spanned by zonal spherical functions by E 0 .Note that E µ is a module over where ⟨•, •⟩ is a G-invariant Hermitian inner product.
Remark 1.5 (orthogonality).The Schur orthogonality relations for the matrix spherical functions give The focus of this paper is to calculate the spherical functions Φ µ λ as explicitly as possible.To make such a function explicit we need to know the embedding j in Lemma 1.2 explicitly, a notoriously difficult problem.To narrow down the problem we make several assumptions: Only particular irreducible K-representations are considered.The classification in [24] gives roughly two families of irreducible K-representations, one for the first block of K and one for the second.
The module structure of E µ can be understood on a spectral level, where the spectrum has a product structure B(µ) × N n .The set B(µ) is called the bottom and in the transition from the spherical functions to the orthogonal polynomials, the crucial information is captured by the spherical functions with irreducible G-representations from the bottom.
Here is the main idea of this paper.Instead of calculating the spherical functions, the spherical functions Φ µ λ are approximated by functions Q µ λ .They are approximations in the following sense, where a λ is a non-zero constant.The lower order terms can be described by the partial ordering on the weight lattice for G.
If the approximations are known, then the spherical functions can be recovered by means of an extra piece of information, namely that they are eigenfunctions of the quadratic Casimir operator.To fully control this operator, it has to be calculated explicitly which is technically involved.

Structure theory
The goal of this section is to describe the structure theory of the compact symmetric space and to fix notation.We take G = SU(n + m) and K = S(U(n) × U(m)), where m ≥ n and K is the (2 × 2)-block diagonal type with U(n) in the upper left-hand block and U(m) in the down right-hand block.This section is a generalization of [11,Section 2.1]. Let be an n × n matrix.The abelian subgroup A of G is given by where The complexification of G is denoted by G C = SL(m+n, C).The maximal torus T G C ⊂ G C is the subgroup of diagonal matrices, and similarly, the maximal torus T G ⊂ G is the subgroup of diagonal matrices.Also K C , T K C , M C and T M C are the corresponding complex type.Explicitly, The holomorphic characters of T G C form an abelian group by pointwise multiplication and we use the additive notation for this group.For example, We define the orthogonality relation We define g, k, m and a as the corresponding complex Lie algebras of G, K, M and A. In Nα i denote the non-negative integral linear combinations of the simple roots.The partial ordering η ≼ τ is τ − η ∈ Q + .
We define the fundamental weights corresponding to these positive roots.We define ω i 's for i = 1, 2, . . ., m + n − 1 as Then we have Now we consider the highest weight of M -modules.We define η i as the characters of T M C by Then the highest weight of irreducible M -modules can be written as with a 1 , a 2 , . . ., a n ∈ Z and a n+1 , a n+2 , . . ., a m−1 ∈ N. We define the set of the highest weight of M as P + M .
Remark 2.1.We will use the relation given by [1, p. 250], where We consider the relation between the irreducible K-module and G-module.By Kobayashi [10, Theorem 30], see also Deitmar [3,Theorem 3], we have Lemma 2.2.For µ In this paper, we always consider the situation of V K µ | M splitting multiplicity free.

The case of
Pezzini and van Pruijssen [24] define the extended weight monoid, which consists of the pairs So P + G (µ) consists of those pairs (τ 1 , τ 2 ) for which τ 2 = µ = aω 1 + bω n .Note that we have multiplied the second entry of the pairs of the extended weight monoid by −1 in comparison to [24].
For the special case µ = 0 we obtain the description of the spherical representation P + G (0) = n i=1 N(ω i + ω n+m−i ) for symmetric pair (G, K) as proved by Krämer [20].Note that λ i = ω i + ω n+m−i with i = 1, 2, . . ., n are the generators of the spherical weights and they can be written in terms of simple roots as follows: and we define the corresponding vector as the K-fixed vector.Also we rewrite Φ 0 λ i as ϕ i and it is bi-K-invariant.Proposition 2.5.For the multiplicity free triple (G, K, µ) with the convention ω 0 = ω m+n = 0.
Proof .We prove this proposition by straightforward calculation to find out all the λ ∈ P + G satisfying (λ, It leads to c i being free for 1 ≤ i ≤ n − 1, which corresponds to n−1 i=1 c i λ i , a spherical weight.The non-trivial remaining conditions are Since ξ 2n + ξ 2n+1 = (λ n , 0) leads to the remaining generator of the spherical weights P + G (0), we can additionally assume c 2n c 2n+1 = 0 in order to determine B(µ).
Since b ∈ N we need to take c 2n = 0 and B(µ) is described by 2n−1 i=n c i = a, c 2n+1 = b+c 2n−1 .Relabelling gives the result and the proposition is proved.■ Remark 2.6.Note that for any λ ∈ P + G (µ), we have λ ≽ µ.

2.2
The case of µ = ω s + bω n Let 1 < s < n and b ∈ N. The goal of this subsection is to give some preliminaries for the method to calculate Proof .Let H = {h 1 , h 2 , . . ., h s } ⊂ {1, 2, . . ., n} be an ordered tuple with ≤ 1 for any λ ∈ P + G and the lemma follows. ■ We have the stability result for the multiplicities due to van Pruijssen [27].
For a weight vector Note that in this equation, we have Then we define a set and we have Lemma 2.9.For any λ ∈ P + G (µ), we have Proof .For λ ∈ P + G (µ), since V K µ | M splits multiplicity free, by Remark 2.8 we have It leads to where 2).We assume b i = 0.If d i < 0, then the coefficient of ω i in λ is negative, which contradicts the fact that λ is dominant.Also the situation of b n+m−i = 0 is similar.So d i ≥ 0 for i = 1, 2, . . ., n and we have This lemma is proved.■

Spherical function restricted to A
By Definition 1.1 and the Cartan decomposition G = KAK, we know that the corresponding spherical functions are uniquely determined by the spherical function restricted to A. Let Φ ∈ E µ , for m ∈ M and a ∈ A, we have It leads to We have an M -module decomposition of V K µ such that then by Schur's lemma and V K µ | M splitting multiplicity free, we have where by Cartan decomposition, since the integrand is independent of k 1 and k 2 and µ is a unitary representation.For a = a t ∈ A as in (2.1), the expression of δ(a) = δ(a t ) is given by, see [8, p. 383], Recall the restriction in Section 1, where E µ has a structure B(µ) × N n , we give a general structure of Φ µ λ | A 's where ). E µ is freely and finitely generated by ϕ i 's with i = 1, 2, . . ., n as an E 0 module.Moreover, let , then all F λ 's are linearly independent.
Lemma 3.3.Let λ ∈ P + G (µ), then the corresponding spherical function can be written as Proof .It is true for all the elements in B(µ) since , we assume it is true for all P + G (µ) ∋ λ ′ ⪵ λ.We define where v ν is the K highest weight vector with weight µ in V G ν , and v i 's are the K-fixed vectors in V G ω i +ω n+m−i , i.e., the vectors generating the trivial K module as described in Section 2.1.So u is also a K highest weight vector with weight µ in U G λ , and F λ is the corresponding matrix spherical function.Since by the complete reducibility theorem, we have by the induction hypothesis.If d λ = 0, then F λ can be written as which contradicts the fact that F λ 's are linearly independent.So d λ ̸ = 0 and this lemma is proved.■

Radial part of the Casimir operator
Now we give the explicit expression of the radial part R of the Casimir operator in this case for the matrix spherical functions related to the K-representation.We follow the approach of Casselman and Miličić [2], see also Warner [28,Proposition 9.1.2.11].The meaning of the radial part is that such an operator is acting on the function restricted to A ⊂ G.By [14, Section 2.2], we know that the spherical function restricted to A is the eigenfunction of the radial part R of the Casimir operator and with c λ = ⟨λ, λ⟩ + 2⟨λ, ρ⟩.We give the explicit calculation in Appendix A.
We need to prove the following lemma before we calculate spherical function Proof .We have , then the finite-dimensional space of matrix spherical functions spanned by Φ µ λ ′ | A λ ′ ≼λ is also spanned by {F λ ′ } λ ′ ≼λ .Moreover, by Lemma 3.3, the transition between these bases is triangular.Since the basis Φ µ λ ′ | A λ ′ ≼λ is a basis of eigenfunctions for the action of the radial part R of the Casimir operator R, the space is invariant for R. By Lemma 4.1, the eigenspace for the eigenvalue c λ is one-dimensional.Using Lemma 3.3, we find that R acts lower triangularly on F λ , From this result, we can obtain precise information on the matrix spherical functions Φ µ λ | A .

Special cases
The goal of this section is to give some simple cases of the matrix spherical functions for the multiplicity free triple (G, K, ω s + bω n ) with s = 0, 1, . . ., n and b ∈ N.This will be used in Sections 6 and 7 to calculate the approximate functions.
Theorem 5.4.For s = 0, 1, 2, . . ., n and u = 0, 1, . . ., n − s, let where We define a linear map Proof .We have by the Laplace expansion of Remark 5.3, we see that (5.3) equals e N\P ∧ e M\R using (5.1).By a similar calculation and using the Laplace expansion for the action of k 1 , we have the conclusion after renaming det(π(k 1 )) det(π(k 2 )) On the other hand, we have Then h k is a K-intertwiner and this theorem is proved.
Proof .By Corollary 5.2, we have the conclusion In Theorem 5.4, we have found u + 1 irreducible K-modules with highest weight ω s in V G ω s+u ⊗ V G ω m+n−u , so this lemma is proved.■ Remark 5.7.Now we calculate the corresponding coefficients of Lemma 5.6.For u = 0, we have We have a G-intertwiner ρ 0 : where we use the notation of Theorem 5.4 and n+m C n+m being the trivial representation.So we have Since, by Theorem 5.1, we know that there is an irreducible K-module with highest weight ω s in V G ω s+1 +ω m+n−1 by Lemma 5.6.Also the corresponding vector can be written as and v H 1 is orthogonal with all the vectors in V G ωs .So Then we can find the K-module with highest weight ω s in V G ω s+1 +ω m+n−1 .This can be done by induction for more general u, but there seem no nice explicit expressions available.Now we calculate the matrix-valued spherical functions.We consider the corresponding approximate function at first.Definition 5.8.For µ = ω s + bω n and then we define a map (−1) b(P ) e H ∧ e P ⊗ e N\P ∧ e M ⊗ (e N ) ⊗b .
(5.4) Also we define the approximate function (5.5) Lemma 5.9.The map j W is a unitary K-intertwiner.
Proof .This follows from Theorem 5.4 and Corollary 5.5 with u = k = i, and which proves the lemma.■ Remark 5.12.For i = 0, i.e., ν 0 = µ, we have the tensor product decomposition and it leads to Since the weight vector of V K µ is also the weight vector of In this case, we have ν 0 ∈ P + G (µ) and Φ µ ν 0 (a t ) = Q µ ν 0 (a t ).
Remark 5.13.For the matrix spherical function Φ µ λ (a t ) ∈ End M V K µ , we have where P (cos t) is a polynomial in cos t i 's.
For an s-tuple H = {h 1 , h 2 , . . ., h s } with h i < h i+1 , the (n − s)-tuple H is the ordered tuple so that H = {h s+1 , h s+2 , . . ., h n } with h i < h i+1 and H ∪ H = N.Let e H ⊗ (e N ) ⊗b ∈ V K µ and n w ∈ N K ′ (A ′ ) in Lemma A.2 be a representative of and In this equation, w(P )(cos t) is the polynomial in cos t i 's where we let the Weyl group element w acts on P (cos t).
Since all the M -types in V K µ , each M type being 1-dimensional and spanned by e H ⊗ (e N ) ⊗b , are in a single Weyl group orbit for the reduced Weyl group, we only need to calculate the first entry of the corresponding spherical function and we can get the other entries by the action of the reduced Weyl group.Lemma 5.14.We have with c ν i = ⟨ν i , ν i ⟩ + 2⟨ν i , ρ⟩ and we define Q µ ν −1 ≡ 0.
Proof .The expression of R is given in Appendix A.2.This lemma can be proved by using computer algebra, in particular Maxima, and some intermediate calculations are shown in Appendix A.3. ■ Remark 5.15.We have c ν i > c ν i−1 by Lemma 4.1 since ν i − ν i−1 ⪶ 0.

Example: µ = 0
Now we calculate the zonal spherical function ϕ i corresponding to ω i + ω m+n−i , i = 1, 2, . . ., n, by where v i is the K-fixed vector in V G ω i +ω n+m−i .Recalling Section 4, ϕ i , i = 1, 2, . . ., n, is an eigenfunction of the radial part R of the Casimir operator with eigenvalue Instead of calculating the zonal spherical function ϕ i , we calculate related bi-K-invariant function ψ i as matrix elements of K-fixed vector in a reducible G representation.By calculating R(ψ i ), we can relate them to the zonal spherical function as defined in (5.6).
Corollary 5.16.Let It is a special case of Theorem 5.4.■ By Theorem 5.1, we have V G ω j +ω m+n−j , j = 0, 1, . . ., i, then v ′ i is a linear combination of K-fixed vector in V G ω j +ω m+n−j with j = 0, 1, . . ., i.We define and ψ i is i-th elementary symmetric polynomial in cos 2 t k , k = 1, 2, . . ., n.We use the convention ϕ 0 = ψ 0 = 1, and d 0 = 0.In this case, ψ i 's are linearly independent since the total degree of the ψ i 's as polynomials in (cos t 1 , cos t 2 , . . ., cos t n ) are all different.
Proof .We can prove it by Lemma 5.14, where we let s = b = 0. ■ Proposition 5.18.In this case, ϕ i can be written as linear combination of ψ j 's with j = 0, 1, . . ., i, and the coefficient of ψ i is non-zero.Explicitly, we have where Proof .It is true for i = 0 and ϕ 0 (a t ) = ψ 0 (cos t) = 1.By Corollary 5.17 and then let k 0 = 1 and the expression of k j 's are clear.Also let t = (0, 0, . . ., 0), and we have by Chu-Vandermonde summation.Then this proposition is proved by calculation.■ 6 Matrix-valued spherical functions with µ = aω 1 + bω n The goal of this section is to give the approximation of the matrix spherical function corresponding to µ = aω 1 + bω n with a, b ∈ N. In this case, B(µ) is explicit in Section 2.1.We have the tensor product decomposition, recall n i=1 a i = a, a i ∈ N, We recall the notation in (5.4) and define Proof .It can be proved by using the fact that E i,i+1 v µ = 0 for i = 1, 2, . . ., n − 1, n + 1, . . ., n + m. ■ Remark 6.2.We can calculate other weight vectors in V K µ by the Chevalley basis acting on v µ .Then we have a K-intertwiner j W from V K µ to the G-module W G ν .We have • j W which is the corresponding matrix spherical function restricted to A. Since V K µ | M splits multiplicity free and is a diagonal matrix if we choose the M -weight vectors as the basis of V K µ .We calculate the entry corresponding to v µ and i is i-th symmetric polynomial in cos 2 t 2 , cos 2 t 3 , . . ., cos 2 t n as defined in Appendix A.3.Then Q µ ν (a t )'s are linearly independent for ν ∈ B(µ).Other entries can be calculated analogously.Remark 6.3.Similar to Remark 5.12, we have Φ µ aω 1 +bωn (a t ) = Q µ aω 1 +bωn (a t ).Remark 6.4.
We want to show λ sph = 0. Then plugging the expression of ν, λ ′ , we get where ω 0 = ω m+n = 0 by convention.By summation by parts, we get Lemma 6.5.We have Proof .It is true for ν = aω 1 + bω n since Φ µ aω 1 +bωn (a t ) = Q µ aω 1 +bωn (a t ) by Remark 6.3.We assume it is true for all the elements occurring in the subset and Φ µ ν (a t ) can also be written as linear combination of from the tensor product decomposition If d ν = 0, by Remark 6.4 we have and it contradicts the fact that Q µ λ 's are linearly independent.So this lemma follows.■ Theorem 6.6.
then we have Proof .It can be proved by using Lemma 3.3, Proposition 5.18, and Lemma 6.5.■ Example 6.7.For µ = ω 1 + bω n with b ∈ N, using Proposition 2.5, we have and we have the corresponding approximate function Q µ ν i (a t ) by (5.5).Using Lemma 5.14, we have where ) is the eigenfunction of the radial part R of the Casimir operator with eigenvalue c ν i .The calculation is similar to the proof of Proposition 5.18.
7 Matrix-valued spherical functions with µ = ω s + bω n The goal of this section is to calculate P + G (ω s + bω n ) with s = 0, 1, . . ., n and b ∈ N, and approximate the corresponding spherical functions.We recall some notations and results in Section 2.2 and Section 5.
Lemma 7.2.Define u as a linear map from where Proof .Let E ij be (n + m) × (n + m)-matrix with one non-zero entry 1 at (i, j)-entry.Then can be considered as the Chevalley basis of the complex Lie algebra of K.So we need to prove u acting on V K µ commutes with the Chevalley basis action.
⊗b with permutation σ, then we have two possibilities.
with permutation σ ′ where we flip the order of i and i + 1 in X ′ j 's and l(σ ⊗b is a monomial where we change i + 1 to i in X l and X ′ k .Then Similarly, we have where we change i + 1 to i in H = {h 1 , h 2 , . . ., h s }.Also we have The situation of i ∈ H, i + 1 / ∈ H is similar.■ We define a K-intertwiner w from u V K ωs+bωn to W G λ using Theorem 5.4 such that where Let j W = wu, then we can calculate the approximate function Q µ λ .We let and we choose e H ⊗ (e ,...,σ(x 1 −y 1 )} where the notations are given by Appendix A.3.Other entries can be calculated using Remark 5.13.Remark 7.3.We define the lattice of ω 1 , ω 2 , . . ., ω n , ω m , ω m+1 , . . ., ω n+m−1 as We define Then for n i=1 a i ω i + n+m−1 i=m a i ω i = η ∈ P , we have

Matrix-valued orthogonal polynomials
The goal of this section is to give the matrix weight for the case of µ = aω 1 +bω n and µ = ω s +bω n .Note that for these two cases, we have ♯B(µ) = dim V K µ .We put the diagonal entries of Φ µ τ i (a t ) with τ i ∈ B(µ) and i = 1, 2, . . ., ♯B(µ) in a row and all Φ µ τ i (a t )'s generate a matrix Φ(a t ).Similarly, we put the diagonal entries of Q µ τ i (a t ) in a row and all Q µ τ i (a t )'s generate a matrix Q(a t ).The row of Φ(a t ), respectively Q(a t ), is corresponding to τ i with i = 1, 2, . . ., ♯B(µ) and the column of Φ(a t ), respectively Q(a t ), is corresponding to the weight vector in and Theorem 8.1.Each spherical function can be written as where p ν,λ (ψ 1 , ψ 2 , . . ., ψ n ) is a polynomial in ψ j (cos t)'s.Moreover, we have Φ(a t ) = U Q(a t ) where the entries of U are the polynomials in ψ j (cos t)'s.
Remark 8.2.In this case, the diagonal of Φ µ λ (a t ) can be viewed as a row vector-valued function, which can be written as P (cos t)Φ(a t ), and respectively P (cos t)Q(a t ).In this expression, P (cos t), and respectively P (cos t), is a row vector-valued function and all the entries of P (cos t) and P (cos t) are the polynomials in ψ j (cos t)'s.
Moreover, there is a bijection from the elements in B(µ) to the irreducible M -modules in V K µ | M .(3) For any generator λ r of P + G (0) and any ν ∈ B(µ), we have that if the weight η ∈ P (λ r ) satisfies ν + η ∈ P + G (µ) and hence For the case of µ = aω 1 + bω n , the three conditions are satisfied and moreover, the entries of U are all constants by Lemma 6.5.
For the case of µ = ω s + bω n , the third condition is not satisfied.For instance, we define (G, K, µ) = (SU(9), S(U(4) × U( 5 So this situation contradicts the third condition.Remark 8.4 (differential operator).For Φ µ λ = P Q, the radial part R of the Casimir operator can be rewritten as where the entries of L are polynomials in ψ i 's.
The orthogonality relation in Remark 3.1 can be written as and the (i, j)-th entry of S(cos t) is where i, j = 1, 2, . . ., ♯B(µ) = dim V K µ .Note that S i,j (cos t) is a polynomial in cos 2 t k (k = 1, 2, . . ., n).So we have where we let l i = cos 2 t i and l = (l 1 , l 2 , . . ., l n ).In this case, we let , and c −1 1 is the Selberg integration S n (α, β, γ) with α = 1, β = m − n + 1 and γ = 1.Moreover, we have In this case, we have ψ (cos t) as defined in Appendix A.3, and Lemma 8.5.The matrix weight S is indecomposable, i.e., Proof .Recall that the total degree of S ij (cos t) is 2i + 2j + 2b + 2, the way to prove this lemma is similar to the proof of [11,Proposition 5.1], for which we compare the total degree of the entries between the left-hand and right-hand sides in (8.1).■ Now we describe the restricted root system R. Let f i : a ′ → C, 1 ≤ i ≤ n, be defined by Then the identification of R is given in Table A.1.The roots of Φ not occurring in Table A.1 are zero when restricted to a ′ , i.e., the roots of the form ϵ n+i − ϵ n+j for 1 ≤ i ̸ = j ≤ m − n.These roots are contained in m.Also we have We define We define {f i − f j | 1 ≤ i, j ≤ n, i ̸ = j} ∪ {f i + f j | 1 ≤ i, j ≤ n, i ̸ = j} as the middle roots, {±2f i } n i=1 as the long roots, and {±f i } n i=1 as the short roots.Remark A.1.For m > n, the restricted root system is of BC n type.For m = n, the restricted root system is of C n type.Also for m = n, all the matrices in this section can be written as 2 × 2 block matrices.Then the restricted root system only includes middle roots and long roots since the dimension of the short root space is 2(m − n) = 0, for m = n.
In order to calculate the Weyl group of the restricted root system, we need to calculate

A.2 Radial part R of the Casimir operator
We recall the Casimir element in the universal enveloping algebra U(g) and it can be written as where X i is the basis of g and X i is the corresponding dual basis.We define the dual basis by the Killing form B(X, Y ) = Tr(XY ).We use the orthogonal decomposition g = m ⊕ a ′ ⊕ n.
In this equation, n is spanned by the root space vector corresponding to the roots in ∆.Note that B| m×m and B| a ′ ×a ′ are non-degenerate.Moreover, denoting the Casimir element of m, respectively a ′ , by Ω m , respectively Ω a ′ , we have and for ϵ i − ϵ j ∈ ∆, we define Y ϵ i −ϵ j = E ij which spans the root space g ϵ i −ϵ j .
For α ∈ ∆, we define We have H ii H ii − We separate the radial part R of the Casimir operator, i.e., Γ −1 a (Ω), into five parts which are M -scalar part, second order differential operator part, short root part, middle root part, and long root part.We have In this operator, the short root part is π K µ (X α )π K µ (X −α )F (a t ) + F (a t )π K µ (X α )π K µ (X −α ) − cos(2t j ) 2π K µ (X α )F (a t )π K µ (X −α ) + sin(2t j ) 2∂ ∂t j F (a t ) .

Figure A. 1 .
Figure A.1.Dynkin diagram with n nodes for the restricted Weyl group.