Vanishing integrals for Hall-Littlewood polynomials

It is well known that if one integrates a Schur function indexed by a partition $\lambda$ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of $\lambda$ have even multiplicity (resp. all parts of $\lambda$ are even). In a recent paper of Rains and Vazirani, Macdonald polynomial generalizations of these identities and several others were developed and proved using Hecke algebra techniques. However at $q=0$ (the Hall-Littlewood level), these approaches do not work, although one can obtain the results by taking the appropriate limit. In this paper, we develop a direct approach for dealing with this special case. This technique allows us to prove some identities that were not amenable to the Hecke algebra approach, as well as to explicitly control the nonzero values. Moreover, we are able to generalize some of the identities by introducing extra parameters. This leads us to a finite-dimensional analog of a recent result of Warnaar, which uses the Rogers-Szeg\"o polynomials to unify some existing summation type formulas for Hall-Littlewood functions.Comment: 31 page

Symmetric and nonsymmetric Koornwinder polynomials in the q → 0 limit

Koornwinder polynomials are a 6-parameter BC[subscript 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. 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. As an application, we prove an integral identity for Koornwinder polynomials at q=0

Affine Hecke algebras and symmetric quasi-polynomial duality

In a recent paper with Sahi and Stokman, we introduced quasi-polynomial generalizations of Macdonald polynomials for arbitrary root systems via a new class of representations of the double affine Hecke algebra. These objects depend on a deformation parameter $q$, Hecke parameters, and an additional torus parameter. In this paper, we study $\textit{antisymmetric}$ and $\textit{symmetric}$ quasi-polynomial analogs of Macdonald polynomials in the $q \rightarrow \infty$ limit. We provide explicit decomposition formulas for these objects in terms of classical Demazure-Lusztig operators and partial symmetrizers, and relate them to Macdonald polynomials with prescribed symmetry in the same limit. We also provide a complete characterization of (anti-)symmetric quasi-polynomials in terms of partially (anti-)symmetric polynomials. As an application, we obtain formulas for metaplectic spherical Whittaker functions associated to arbitrary root systems. For $GL_{r}$, this recovers some recent results of Brubaker, Buciumas, Bump, and Gustafsson, and proves a precise statement of their conjecture about a ``parahoric-metaplectic" duality.Comment: 34 page