Affine Macdonald conjectures and special values of Felder–Varchenko functions

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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 \(U_q(\widehat{\mathfrak {sl}}_2)\) 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).

Mathematics Subject Classification

Primary 17B37 Secondary 17B67 33C75 33D80 81R12 

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of MathematicsColumbia UniversityNew YorkUSA
  3. 3.Department of MathematicsUniversity of North Carolina at Chapel HillChapel HillUSA

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