$c$-functions and Koornwinder polynomials

Abstract

This paper develops the theory of Macdonald-Koornwinder polynomials in parallel analogy with the work done for the $G{L}_n$ case in [CR22]. In the context of the type $C{C}_n$ 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.