Orthogonality relations and Cherednik identities for multivariable Baker-Akhiezer functions

Abstract

We establish orthogonality relations for the Baker-Akhiezer (BA) eigenfunctions of the Macdonald difference operators. We also obtain a version of Cherednik-Macdonald-Mehta integral for these functions. As a corollary, we give a simple derivation of the norm identity and Cherednik-Macdonald-Mehta integral for Macdonald polynomials. In the appendix written by the first author, we prove a summation formula for BA functions. We also consider more general identities of Cherednik type, which we use to introduce and construct more general, twisted BA functions. This leads to a construction of new quantum integrable models of Macdonald-Ruijsenaars type.

49 pages, with an appendix. Changes compared to v.1: section 3 expanded with more details; deformed A_n(m) case (formerly section 7) delegated to a separate publication. Final version to appear in Advances in Math