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Quasi-Polynomial Extensions of Nonsymmetric Macdonald-Koornwinder Polynomials

Jasper Stokman

Abstract

In a recent joint paper with S. Sahi and V. Venkateswaran (2025), families of actions of the double affine Hecke algebra on spaces of quasi-polynomials were introduced. These so-called quasi-polynomial representations led to the introduction of quasi-polynomial extensions of the nonsymmetric Macdonald polynomials, which reduce to metaplectic Iwahori-Whittaker functions in the $\mathfrak{p}$-adic limit. In this paper, these quasi-polynomial representations are extended to Sahi's 5-parameter double affine Hecke algebra, and the quasi-polynomial extensions of the nonsymmetric Koornwinder polynomials are introduced.

Quasi-Polynomial Extensions of Nonsymmetric Macdonald-Koornwinder Polynomials

Abstract

In a recent joint paper with S. Sahi and V. Venkateswaran (2025), families of actions of the double affine Hecke algebra on spaces of quasi-polynomials were introduced. These so-called quasi-polynomial representations led to the introduction of quasi-polynomial extensions of the nonsymmetric Macdonald polynomials, which reduce to metaplectic Iwahori-Whittaker functions in the -adic limit. In this paper, these quasi-polynomial representations are extended to Sahi's 5-parameter double affine Hecke algebra, and the quasi-polynomial extensions of the nonsymmetric Koornwinder polynomials are introduced.
Paper Structure (13 sections, 22 theorems, 168 equations)

This paper contains 13 sections, 22 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduced affine root systems
  4. Nonreduced extensions and multiplicity functions
  5. Quasi-polynomials
  6. The affine Hecke algebra
  7. The double affine Hecke algebra
  8. Quasi-polynomial representations
  9. The quasi-polynomial representation of $\boldsymbol{H^X}$
  10. Parabolic data and parabolically induced modules
  11. The quasi-polynomial representation of $\boldsymbol{\mathbb{H}}$
  12. Quasi-polynomial analogs of the nonsymmetric Macdonald--Koornwinder polynomials
  13. The quasi-polynomial representation as $\boldsymbol{Y}$-parabolically induced module

Key Result

Theorem B

For $k,k_r,u_r\in\mathbf{F}^\times$, the linear operators $\mathcal{T}_1,\dots,\mathcal{T}_r$ on $\mathbf{F}[\mathbb{R}^r]$ defined by satisfy the type ${\rm C}_r$ braid relations braidC and the quadratic Hecke relations

Theorems & Definitions (43)

  • Definition A
  • Theorem B
  • Definition C
  • Theorem D
  • Theorem E
  • Remark B
  • Definition C
  • Remark D
  • Definition E: ChKZ2Sa
  • Definition A
  • ...and 33 more