Abstract
Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of (mathrm{SU}(2) times mathrm{SU}(2), mathrm{diag}) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials.
Highlights
After the introduction of quantum groups, it was realised that many special functions of basic hypergeometric type [15] have a natural relation to quantum groups, see e.g. [9, Chap. 6], [20,31] for references
For special cases with G compact, we obtain orthogonality relations and differential operators for the spherical functions, which can be identified with orthogonal polynomials from the Askey scheme
For the special case G = SU(2) × SU(2) with K ∼= SU(2) embedded as the diagonal subgroup, the zonal spherical functions are the characters of SU(2), which are identified with the Chebyshev polynomials Un of the second kind by the Weyl character formula
Summary
After the introduction of quantum groups, it was realised that many special functions of basic hypergeometric type [15] have a natural relation to quantum groups, see e.g. [9, Chap. 6], [20,31] for references. The approach to matrix-valued orthogonal polynomials from this quantum group setting leads to identities in the quantised function algebra. When considering other quantum symmetric pairs in relation to matrix-valued spherical functions, the branching rule of a representation of the quantised universal enveloping algebra to a coideal subalgebra seems to be difficult. 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. The convention on notation follows Kolb [27] for quantised universal enveloping algebras and right coideal subalgebras and we follow Gasper and Rahman [15] for the convention on basic hypergeometric series and we assume 0 < q < 1
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