Affine Macdonald conjectures and special values of Felder-Varchenko functions

TL;DR

This paper refines and proves initial cases of affine Macdonald polynomial conjectures, linking them to elliptic integrals and conformal blocks, advancing understanding of their structure and evaluations.

Contribution

It provides the first non-trivial proofs of affine Macdonald conjectures and connects them to elliptic Selberg integrals and theta hypergeometric evaluations.

Findings

Proved initial cases of affine Macdonald conjectures.

Connected conjectures to elliptic Selberg and theta hypergeometric integrals.

Evaluated integrals using elliptic beta integral techniques.

Abstract

We refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and prove the first non-trivial cases of these conjectures. Our results provide a q-deformation of the computation of genus 1 conformal blocks via elliptic Selberg integrals by Felder-Stevens-Varchenko. They allow us to give precise formulations for the affine Macdonald conjectures in the general case which are consistent with computer computations. Our method applies recent work of the second named author to relate these conjectures in the case of $U_q(\widehat{\mathfrak{sl}}_2)$ to evaluations of certain theta hypergeometric integrals defined by Felder-Varchenko. We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral introduced by Spiridonov.