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 $\mathit{\text{antisymmetric}}$ and $\mathit{\text{symmetric}}$ quasi-polynomial analogs of Macdonald polynomials in the $q\to \mathrm{\infty }$ 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 $G{L}_r$, this recovers some recent results of Brubaker, Buciumas, Bump, and Gustafsson, and proves a precise statement of their conjecture about a “parahoric-metaplectic” duality.