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Affine Macdonald conjectures and special values of Felder–Varchenko functions

Rains, Eric M. and Sun, Yi and Varchenko, Alexander (2018) Affine Macdonald conjectures and special values of Felder–Varchenko functions. Selecta Mathematica - New Series, 24 (2). pp. 1549-1591. ISSN 1022-1824. doi:10.1007/s00029-017-0328-4.

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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).

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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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Simons Foundation336826
Issue or Number:2
Classification Code:MSC: Primary 17B37 Secondary 17B67 33C75 33D80 81R12
Record Number:CaltechAUTHORS:20170607-102914977
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Official Citation:Rains, E.M., Sun, Y. & Varchenko, A. Sel. Math. New Ser. (2018) 24: 1549.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77998
Deposited By: Ruth Sustaita
Deposited On:07 Jun 2017 18:16
Last Modified:15 Nov 2021 17:35

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