  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.
