Quantum zonal spherical functions and Macdonald polynomials

Abstract

A unified theory of quantum symmetric pairs is applied to q-special functions. Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions. Here a distinguished family of such functions, invariant under the Weyl group associated to the restricted roots, is shown to be a family of Macdonald polynomials, as conjectured by Koornwinder and Macdonald. Our results place earlier work for Lie algebras of classical type in a general context and extend to the exceptional 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 )

“…The existence of two Casimir elements in U q (g) leads to the matrix-valued orthogonal polynomials being eigenfunctions of two commuting matrix-valued q-difference operators, see [23] for the group case. This extends Letzter [35] to the matrix-valued set-up for this particular case. The q-difference operators are the key to determining the entries of the matrix-valued orthogonal polynomials explicitly in terms of scalar-valued orthogonal polynomials from the q-Askey scheme [26], namely the continuous q-ultraspherical polynomials and the q-Racah polynomials.…”

“…spaces have been introduced and studied in detail by Letzter [34][35][36], see also Kolb [27]. In particular, Letzter has shown that Macdonald polynomials occur as spherical functions on quantum symmetric pairs motivated by the works of Koornwinder, Dijkhuizen, Noumi and others.…”

“…First we study explicitly the case = 0, motivated by the works of Koornwinder [30], Letzter [34][35][36] and others. Secondly, we show that taking traces of a matrix-valued spherical function of type associated to ( 1 , 2 ) times the adjoint of a spherical function of type associated to ( 1 , 2 ) gives, up to an action by an invertible group-like element of U q (g), a polynomial in the generator for the case = 0.…”

“…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. We stick to the conventions as in Kolb [27] and we refer to [28, §1] for a broader perspective on quantum symmetric pairs.…”

“…[28,[34][35][36][37][38] and references given there. 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.…”