Macdonald polynomials and algebraic integrability

Details Macdonald polynomials and algebraic integrability Macdonald polynomials and algebraic integrability

2002-12-22

arxiv.org/abs/math/0212313v1

Adv. in Math. 166(2), 193-259 (2002)

Mathematics

Quantum Algebra

54 pages

Scientific paper

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 $BC_n$ 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.

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