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).
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Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997). Computational Algebra and Number Theory, London (1993)
Braverman, A., Finkelberg, M., Kazhdan, D.: Affine Gindikin–Karpelevich formula via Uhlenbeck spaces. In: Contributions in Analytic and Algebraic Number Theory, Springer Proceedings in Mathematics, vol. 9, pp. 17–29. Springer, New York (2012)
Braverman, A., Finkelberg, M., Kazhdan, D.: Affine Gindikin–Karpelevich formula via Uhlenbeck spaces. In: Contributions in Analytic and Algebraic Number Theory, Springer Proceedings in Mathematics, vol. 9, pp. 17–29. Springer, New York (2012)
Braverman, A., Kazhdan, D.: Representations of affine KAC-Moody groups over local and global fields: a survey of some recent results. In: European Congress of Mathematics, pp. 91–117. European Mathematical Society, Zürich (2013)
Braverman, A., Kazhdan, D., Patnaik, M.: Iwahori-Hecke algebras for \(p\)-adic loop groups. Invent. Math. 204(2), 347–442 (2016)
Etingof, P., Kirillov Jr., A.: Macdonald’s polynomials and representations of quantum groups. Math. Res. Lett. 1(3), 279–296 (1994)
Etingof, P., Kirillov Jr., A.: Representations of affine Lie algebras, parabolic differential equations, and Lamé functions. Duke Math. J. 74(3), 585–614 (1994)
Etingof, P., Kirillov Jr., A.: On the affine analogue of Jack and Macdonald polynomials. Duke Math. J. 78(2), 229–256 (1995)
Etingof, P., Kirillov Jr., A.: Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials. Compos. Math. 102(2), 179–202 (1996)
Etingof, P., Schiffmann, O., Varchenko, A.: Traces of intertwiners for quantum groups and difference equations, II. Lett. Math. Phys. 62(2), 143–158 (2002)
Frenkel, I., Reshetikhin, N.: Quantum affine algebras and holonomic difference equations. Comm. Math. Phys. 146(1), 1–60 (1992)
Felder, G., Stevens, L., Varchenko, A.: Elliptic Selberg integrals and conformal blocks. Math. Res. Lett. 10(5–6), 671–684 (2003)
Felder, G., Tarasov, V., Varchenko, A.: Solutions of the elliptic \(q\)-KZB equations and Bethe ansatz. I. In: Topics in Singularity Theory, American Mathematical Society. Translate Series 2, vol. 180, pp. 45–75. American Mathematical Society, Providence, RI (1997)
Felder, G., Tarasov, V., Varchenko, A.: Monodromy of solutions of the elliptic quantum Knizhnik–Zamolodchikov-Bernard difference equations. Internat. J. Math. 10(8), 943–975 (1999)
Felder, G., Varchenko, A.: Algebraic Bethe ansatz for the elliptic quantum group \(E_{\tau,\eta }({\rm sl}_2)\). Nuclear Phys. B 480(1–2), 485–503 (1996)
Felder, G., Varchenko, A.: Algebraic integrability of the two-particle Ruijsenaars operator. Funct. Anal. Appl. 32(2), 8–25 (1998)
Felder, G., Varchenko, A.: The elliptic gamma function and \({\rm SL}(3,\mathbb{Z})\ltimes {\mathbb{Z}^3}\). Adv. Math. 156(1), 44–76 (2000)
Felder, G., Varchenko, A.: The \(q\)-deformed Knizhnik–Zamolodchikov-Bernard heat equation. Comm. Math. Phys. 221(3), 549–571 (2001)
Felder, G., Varchenko, A.: \(q\)-deformed KZB heat equation: completeness, modular properties and \({\rm SL}(3,\mathbb{Z})\). Adv. Math. 171(2), 228–275 (2002)
Felder, G., Varchenko, A.: Hypergeometric theta functions and elliptic Macdonald polynomials. Int. Math. Res. Not. 21, 1037–1055 (2004)
Felder, G., Varchenko, A.: Multiplication formulae for the elliptic gamma function. In: Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemporay Mathematics, vol. 391, pp. 69–73. American Mathematical Society, Providence, RI (2005)
Konno, H.: Free-field representation of the quantum affine algebra \(U_q(\widehat{\mathfrak{sl}}_2)\) and form factors in the higher-spin \(XXZ\) model. Nuclear Phys. B 432(3), 457–486 (1994)
Kato, A., Quano, Y., Shiraishi, J.: Free boson representation of \(q\)-vertex operators and their correlation functions. Commun. Math. Phys. 157(1), 119–137 (1993)
Macdonald, I.: A formal identity for affine root systems. In: Lie groups and symmetric spaces, American Mathematical Society. Translate Series 2, vol. 210, pp. 195–211. American Mathematical Society, Providence, RI (2003)
Matsuo, A.: A \(q\)-deformation of Wakimoto modules, primary fields and screening operators. Commun. Math. Phys. 160(1), 33–48 (1994)
Spiridonov, V.P.: On the elliptic beta function. Uspekhi Mat. Nauk 56(1(337)), 181–182 (2001)
Sun, Y.: Traces of intertwiners for quantum affine \({\mathfrak{sl}}_2\) and Felder-Varchenko functions. Commun. Math. Phys., 347, 573–653, 2016. arXiv:1508.03918
Sun, Y.: Traces of intertwiners for quantum affine algebras and difference equations (after Etingof-Schiffmann-Varchenko). Transform. Groups (2017). arXiv:1609.09038 (to appear)
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Rains, E.M., Sun, Y. & Varchenko, A. Affine Macdonald conjectures and special values of Felder–Varchenko functions. Sel. Math. New Ser. 24, 1549–1591 (2018). https://doi.org/10.1007/s00029-017-0328-4
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DOI: https://doi.org/10.1007/s00029-017-0328-4