Orthogonality relations and Cherednik identities for multivariable Baker-Akhiezer functions

TL;DR

This paper develops orthogonality relations and integral identities for Baker-Akhiezer functions associated with Macdonald difference operators, leading to new insights and models in quantum integrable systems.

Contribution

It establishes orthogonality and Cherednik-Macdonald-Mehta identities for Baker-Akhiezer functions, and introduces twisted functions for advanced integrable models.

Findings

Orthogonality relations for BA eigenfunctions

Cherednik-Macdonald-Mehta integral for BA functions

Construction of new quantum integrable models

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.