Affine Hecke algebras and symmetric quasi-polynomial duality

2023/08/21 by Venkateswaran, Vidya
#05E05 (Secondary) #11F68 #20C08 (Primary) 33D52 #22E50 #FOS: Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2308.10844

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 antisymmetric and symmetric quasi-polynomial analogs of Macdonald polynomials in the q → ∞ 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.

Related

• Quasi-polynomial representations of double affine Hecke algebras
• Quasi-Polynomial Extensions of Nonsymmetric Macdonald-Koornwinder Polynomials
• Row-strict Quasisymmetric Schur Functions, Characterizations of Demazure Atoms, and Permuted Basement Nonsymmetric Macdonald Polynomials
• Non-semisimple Macdonald polynomials
• Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials