Symmetric and nonsymmetric Hall-Littlewood polynomials of type BC

Abstract: Koornwinder polynomials are a 6-parameter BC_n-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. 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. Next, we provide an analogous construction for the nonsymmetric Koornwinder polynomials in the same limiting case. The method employed in this paper is a BC-type adaptation of techniques used in an earlier work of the author, which gave a combinatorial method for proving vanishing results of Rains and Vazirani at the Hall-Littlewood level. As a consequence of this work, we obtain direct arguments for the constant term evaluations and norms in both the symmetric and nonsymmetric cases.