Macdonald polynomials and algebraic integrability

Abstract

We construct explicitly non-polynomial eigenfunctions of the difference operators by Macdonald in case $t=q^k$, $k\in \mathbb{Z}$. This leads to a new, more elementary proof of several Macdonald conjectures, first proved by Cherednik. We also establish the algebraic integrability of Macdonald operators at $t=q^k$ ($k\in \mathbb{Z}$), generalizing the result of Etingof and Styrkas. Our approach works uniformly for all root systems including $B{C}_n$ case and related Koornwinder polynomials. Moreover, we apply it for a certain deformation of $A_n$ root system where the previously known methods do not work.

54 pages

Cited by in corpus (2)

• Refined Chern-Simons theory and (q,t)-deformed Yang-Mills theory: Semi-classical expansion and planar limit
• Generalized Macdonald-Ruijsenaars systems