Vanishing integrals for Hall-Littlewood polynomials

TL;DR

This paper develops a direct method to prove vanishing integrals of Hall-Littlewood polynomials over classical groups, extending known identities and introducing new parameters, especially at the special case q=0.

Contribution

It introduces a novel direct approach for Hall-Littlewood polynomials at q=0, enabling proofs of identities not accessible via Hecke algebra methods and allowing parameter generalizations.

Findings

Proved vanishing integrals for Hall-Littlewood polynomials over classical groups.

Extended identities by introducing additional parameters.

Connected results to finite-dimensional analogs of Warnaar's work.

Abstract

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.