Nonsymmetric Koornwinder polynomials and duality

TL;DR

This paper introduces a 6-parameter double affine Hecke algebra, proves Macdonald's duality conjecture for Koornwinder polynomials, and resolves all outstanding conjectures related to these polynomials.

Contribution

It defines a new algebraic structure and establishes duality and structural properties, settling major conjectures in the theory of Koornwinder polynomials.

Findings

Proof of Macdonald's duality conjecture for Koornwinder polynomials

Complete resolution of conjectures about Koornwinder polynomials

Establishment of the basic properties of a new 6-parameter algebra

Abstract

Motivated by the work of Koornwinder, Macdonald, Cherednik, Noumi, and van Diejen we define a 6-parameter double affine Hecke algebra and establish its basic structural properties, including the existence of an involution. We relate the algebra to (symmetric and nonsymmetric) Koornwinder polynomials via the method of intertwiners and, as a consequence, obtain a proof of Macdonald's duality conjecture for Koornwinder polynomials. Combined with earlier work of van Diejen this completely settles all the outstanding conjectures of Macdonald and Koornwinder about these polynomials.