$c$-functions and Koornwinder polynomials

TL;DR

This paper advances the theory of Koornwinder polynomials, providing explicit formulas and proofs for their properties, including norm and constant term conjectures, in the context of affine root systems of type $CC_n$.

Contribution

It develops $c$-function formulas and $E$-expansions for Koornwinder polynomials, extending the theory to classical root systems and proving key conjectures.

Findings

Derived $c$-function formulas for symmetrizers

Established norm and constant term formulas for Koornwinder polynomials

Proved the norm and constant term conjectures for the Koornwinder case

Abstract

This paper develops the theory of Macdonald-Koornwinder polynomials in parallel analogy with the work done for the $GL_n$ case in [CR22]. In the context of the type $CC_n$ affine root system the Macdonald polynomials of other root systems of classical type are specializations of the Koornwinder polynomials. We derive $c$-function formulas for symmetrizers and use them to give $E$-expansions, principal specializations and norm formulas for bosonic, mesonic and fermionic Koornwinder polynomials. Finally, we explain the proof of the norm conjectures and constant term conjectures for the Koornwinder case.