Quasi-Polynomial Extensions of Nonsymmetric Macdonald-Koornwinder Polynomials

TL;DR

This paper extends quasi-polynomial representations of the double affine Hecke algebra to include Koornwinder polynomials, connecting algebraic structures with metaplectic Iwahori-Whittaker functions in a broader framework.

Contribution

It introduces quasi-polynomial extensions of nonsymmetric Koornwinder polynomials within Sahi's 5-parameter double affine Hecke algebra, expanding previous work on these algebraic objects.

Findings

Extended quasi-polynomial representations to Koornwinder polynomials

Connected quasi-polynomial extensions to metaplectic Iwahori-Whittaker functions

Generalized algebraic framework for nonsymmetric 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 $\mathfrak{p}$-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.