Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One

Abstract

We present a method to obtain infinitely many examples of pairs (W, D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G, K) of rank one and a suitable irreducible K-representation. The heart of the construction is the existence of a suitable base change Ψ 0 . We analyze the base change and derive several properties. The most important one is that Ψ 0 satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group G as soon as we have an explicit expression for Ψ 0 . The weight W is also determined by Ψ 0 . We provide an algorithm to calculate Ψ 0 explicitly. For the pair (USp(2n), USp(2n − 2) × USp(2)) we have implemented the algorithm in GAP so that individual pairs (W, D) can be calculated explicitly. Finally we classify the Gelfand pairs (G, K) and the K-representations that yield pairs (W, D) of size 2×2 and we provide explicit expressions for most of these cases.

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

“…In this paper, it is reduced to the Clebsch-Gordan decomposition, and there is a nice result by Oblomkov and Stokman [38,Proposition 1.15] on a special case of the branching rule for quantum symmetric pair of type AIII, but in general the lack of the branching rule for the quantum symmetric pairs is an obstacle for the study of quantum analogues of matrix-valued spherical functions of e.g. [17,18,38,44].…”

“…Moreover, we know that the Chebyshev polynomials occur as characters on the quantum SU(2) group, see [48, §A.1]. The approach in this paper is to establish the quantum analogue of the group theoretic approach as presented in [23,24], see also [18,44], for the example of the Gelfand pair G = SU(2) × SU (2) with K ∼ = SU (2). For this approach, we need Letzter's approach [34][35][36] to quantum symmetric spaces using coideal subalgebras.…”

“…However, the focus on the relation with matrix-valued or vector-valued special functions only came later, see e.g. references given in [18,44]. Grünbaum et al [17] give a group theoretic approach to matrixvalued orthogonal polynomials emphasising the role of the matrix-valued differential operators, which are manipulated in great detail.…”

“…The paper [17] deals with the case (G, K ) = (SU (3), U(2)). Motivated by [17] and the approach of Koornwinder [29], the group theoretic interpretation of matrix-valued orthogonal polynomials on (G, K ) = (SU(2) × SU(2), SU(2)) is studied from a different point of view, in particular with less manipulation of the matrix-valued differential operators, in [23,24], see also [18,44]. The point of view is to construct the matrix-valued orthogonal polynomials using matrix-valued spherical functions, and next using this group theoretic interpretation to obtain properties of the matrix-valued orthogonal polynomials.…”

The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

“…In the last years, one can see a growing number of papers devoted to different aspects of this question. For some of these recent papers, see [GPT01, GPT02b, Tir03, GPT04, DG05a, DG05c, DG05b, DG05d, CG05, GPT05, Mir05, PT07] as well as [DdlI08a,DdlI08b,PR08,GdlIMF11,dlI11,KvPR12,KvPR13,CdlI14,vPR14,AKdlR15].…”