Symmetric and nonsymmetric Koornwinder polynomials in the $$q \rightarrow 0$$ q → 0 limit

Abstract

Koornwinder polynomials are a 6-parameter $$BC_{n}$$BCn-symmetric family of Laurent polynomials indexed by partitions, from which Macdonald polynomials can be recovered in suitable limits of the parameters. As in the Macdonald polynomial case, standard constructions via difference operators do not allow one to directly control these polynomials at $$q=0$$q=0. In the first part of this paper, we provide an explicit construction for these polynomials in this limit, using the defining properties of Koornwinder polynomials. Our formula is a first step in developing the analogy between Hall---Littlewood polynomials and Koornwinder polynomials at $$q=0$$q=0. In the second part of the paper, we provide a construction for the nonsymmetric Koornwinder polynomials in the same limiting case; this parallels work by Descouens---Lascoux in type $$A$$A. As an application, we prove an integral identity for Koornwinder polynomials at $$q=0$$q=0.