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Published April 2018 | Submitted
Journal Article Open

Affine Macdonald conjectures and special values of Felder–Varchenko functions

Abstract

We refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof–Kirillov Jr. (Duke Math J 78(2):229–256, 1995) 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 (Math Res Lett 10(5–6):671–684, 2003). 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(sl_2) to evaluations of certain theta hypergeometric integrals defined by Felder–Varchenko (Int Math Res Not 21:1037–1055, 2004). We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral introduced by Spiridonov (Uspekhi Mat Nauk 56(1(337)):181–182, 2001).

Additional Information

© 2017 Springer International Publishing. Published online: 27 April 2017. Y.S. and A.V. thank the Max-Planck-Institut für Mathematik in Bonn for providing excellent working conditions. Y.S. thanks P. Etingof for many helpful discussions. E.M.R. was partially supported by NSF Grant DMS-1500806. This work was partially supported by a Junior Fellow award from the Simons Foundation to Yi Sun. A.V. was partially supported by NSF Grant DMS-1362924 and Simons Foundation Grant #336826.

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