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Affine Hecke algebras and symmetric quasi-polynomial duality

Vidya Venkateswaran

TL;DR

The paper investigates antisymmetric and symmetric quasi-polynomial generalizations of Macdonald polynomials in the $q \to \infty$ limit within a representation-theoretic framework provided by the double affine Hecke algebra. It develops explicit decomposition formulas using Demazure-Lusztig operators and partial symmetrizers, and characterizes (anti-)symmetric quasi-polynomials as partially (anti-)symmetric polynomials, with a parallel description for metaplectic variants. A central application is to metaplectic spherical Whittaker functions, including a precise GL$_r$ duality (parahoric-metaplectic) that aligns with recent lattice-model dualities. The results extend to arbitrary root systems and connect to Macdonald polynomials with prescribed symmetry, via coefficient-function techniques and PBW-type analyses. Overall, the work bridges quasi-polynomial Hecke-algebra representations, symmetric/antisymmetric Macdonald limits, and metaplectic Whittaker theory, suggesting new dualities and explicit formulas across types.

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.

Affine Hecke algebras and symmetric quasi-polynomial duality

TL;DR

The paper investigates antisymmetric and symmetric quasi-polynomial generalizations of Macdonald polynomials in the limit within a representation-theoretic framework provided by the double affine Hecke algebra. It develops explicit decomposition formulas using Demazure-Lusztig operators and partial symmetrizers, and characterizes (anti-)symmetric quasi-polynomials as partially (anti-)symmetric polynomials, with a parallel description for metaplectic variants. A central application is to metaplectic spherical Whittaker functions, including a precise GL duality (parahoric-metaplectic) that aligns with recent lattice-model dualities. The results extend to arbitrary root systems and connect to Macdonald polynomials with prescribed symmetry, via coefficient-function techniques and PBW-type analyses. Overall, the work bridges quasi-polynomial Hecke-algebra representations, symmetric/antisymmetric Macdonald limits, and metaplectic Whittaker theory, suggesting new dualities and explicit formulas across types.

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 , Hecke parameters, and an additional torus parameter. In this paper, we study and quasi-polynomial analogs of Macdonald polynomials in the 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 , this recovers some recent results of Brubaker, Buciumas, Bump, and Gustafsson, and proves a precise statement of their conjecture about a "parahoric-metaplectic" duality.
Paper Structure (7 sections, 20 theorems, 65 equations)

This paper contains 7 sections, 20 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Hecke algebras
  3. Root systems and Weyl groups
  4. Length function and parabolic subgroups
  5. Affine Hecke algebras
  6. Partial (anti-)symmetrizers
  7. Demazure-Whittaker operators

Key Result

Theorem 1.1

Set $v = q^{2}$ and $W = S_{r}$. Let $\theta \in (\mathbb{Z} / n \mathbb{Z})^{r}$ and $\mu - \rho_{GL} \in P^{+}$. Then $\mathbf{y}^{\rho_{\text{GL}} - \lfloor \theta \rfloor_{n}} \tilde{\phi}_{\theta}^{o}(\mathbf{y}; \varpi^{\rho_{\text{GL}} - \mu})$ is a polynomial in $\mathbf{y}^{n}$, and it vani with $\lambda = \lfloor \frac{\mu}{n} \rfloor - \rho_{\text{GL}}$. Here $w,w' \in W$ and $\mathbf{c

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2: SSV2
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 40 more