Affine Hecke algebras and symmetric quasi-polynomial duality

TL;DR

This paper explores symmetric and antisymmetric quasi-polynomial analogs of Macdonald polynomials within affine Hecke algebra frameworks, providing explicit formulas, characterizations, and applications to metaplectic Whittaker functions, especially in the $q ightarrow olinebreak \infty$ limit.

Contribution

It introduces explicit decomposition formulas and characterizations for symmetric and antisymmetric quasi-polynomials, linking them to classical objects and applications in metaplectic representation theory.

Findings

Explicit decomposition formulas in terms of Demazure-Lusztig operators

Complete characterization of (anti-)symmetric quasi-polynomials

Formulas for metaplectic spherical Whittaker functions

Abstract

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.