Affine Macdonald conjectures and special values of Felder-Varchenko functions

I will explain how to refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and to prove the first non-trivial cases of these conjectures. Our method applies recent work of the speaker to relate these conjectures for quantum affine sl(2) to evaluations of certaintheta 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 of Spiridonov.These results are joint work with E. Rains and A. Varchenko.