Communications in Mathematical Physics

, Volume 347, Issue 2, pp 573–653 | Cite as

Traces of Intertwiners for Quantum Affine \({\mathfrak{sl}}_2\) and Felder–Varchenko Functions

Article
  • 96 Downloads

Abstract

We show that the traces of \({U_q({\widehat{\mathfrak{sl}}}_2)}\)-intertwiners of [ESV02] valued in the three-dimensional evaluation representation converge in a certain region of parameters and give a representation-theoretic construction of Felder–Varchenko’s hypergeometric solutions to the q-KZB heat equation given in [FV02]. This gives the first proof that such a trace function converges and resolves the first case of the Etingof–Varchenko conjecture of [EV00]. As applications, we prove a symmetry property for traces of intertwiners and prove Felder–Varchenko’s conjecture in [FV04] that their elliptic Macdonald polynomials are related to the affine Macdonald polynomials defined as traces over irreducible integrable \({U_q({\widehat{\mathfrak{sl}}}_2)}\)-modules in [EK95]. In the trigonometric and classical limits, we recover results of [EK94,EV00]. Our method relies on an interplay between the method of coherent states applied to the free field realization of the q-Wakimoto module of [Mat94], convergence properties given by the theta hypergeometric integrals of [FV02], and rationality properties originating from the representation-theoretic definition of the trace function.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AAR99.
    Andrews, G., Askey, R., Roy, R.: Special functions. In: Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  2. BF90.
    Bernard D., Felder G.: Fock representations and BRST cohomology in SL(2) current algebra. Commun. Math. Phys. 127(1), 145–168 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. EFK98.
    Etingof, P., Frenkel, I., Kirillov, Jr. A.: Lectures on representation theory and Knizhnik–Zamolodchikov equations. In: Mathematical Surveys and Monographs, vol. 58. American Mathematical Society, Providence (1998)Google Scholar
  4. EK94.
    Etingof P., Kirillov A. Jr.: Representations of affine Lie algebras, parabolic differential equations, and Lamé functions. Duke Math. J. 74(3), 585–614 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. EK95.
    Etingof P., Kirillov Jr. A.: On the affine analogue of Jack and Macdonald polynomials. Duke Math. J. 78(2), 229–256 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. ES95.
    Etingof P., Styrkas K.: Algebraic integrability of Schrödinger operators and representations of Lie algebras. Compositio Math. 98(1), 91–112 (1995)MathSciNetMATHGoogle Scholar
  7. ES98.
    Etingof P., Styrkas K.: Algebraic integrability of Macdonald operators and representations of quantum groups. Compositio Math. 114(2), 125–152 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. ES01.
    Etingof, P., Schiffmann, O.: Lectures on the dynamical Yang–Baxter equations. In: Quantum Groups and Lie Theory (Durham, 1999). London Math. Soc. Lecture Note Ser., vol. 290, pp 89–129. Cambridge University Press, Cambridge (2001)Google Scholar
  9. ESV02.
    Etingof P., Schiffmann O., Varchenko A.: Traces of intertwiners for quantum groups and difference equations, II. Lett. Math. Phys. 62(2), 143–158 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. Eti95.
    Etingof P.: Quantum integrable systems and representations of Lie algebras. J. Math. Phys. 36(6), 2636–2651 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. EV00.
    Etingof P., Varchenko A.: Traces of intertwiners for quantum groups and difference equations, I. Duke Math. J. 104(3), 391–432 (2000)MathSciNetCrossRefMATHGoogle Scholar
  12. EV02.
    Etingof P., Varchenko A.: Dynamical Weyl groups and applications. Adv. Math. 167(1), 74–127 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. EV05.
    Etingof P., Varchenko A.: Orthogonality and the q-KZB-heat equation for traces of \({U_q(\mathfrak{g})}\) -intertwiners. Duke Math. J. 128(1), 83–117 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. FSV03.
    Felder G., Stevens L., Varchenko A.: Elliptic Selberg integrals and conformal blocks. Math. Res. Lett. 10(5–6), 671–684 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. FTV97.
    Felder, G., Tarasov, V., Varchenko, A.: Solutions of the elliptic q-KZB equations and Bethe ansatz. I. In: Topics in Singularity Theory. Amer. Math. Soc. Transl. Ser. 2, vol. 180, pp. 45–75. American Mathematical Society, Providence (1997)Google Scholar
  16. FTV99.
    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)MathSciNetCrossRefMATHGoogle Scholar
  17. FV00.
    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)MathSciNetCrossRefMATHGoogle Scholar
  18. FV01.
    Felder G., Varchenko A.: The q-deformed Knizhnik–Zamolodchikov–Bernard heat equation. Commun. Math. Phys. 221(3), 549–571 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. FV02.
    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)MathSciNetCrossRefMATHGoogle Scholar
  20. FV04.
    Felder G., Varchenko A.: Hypergeometric theta functions and elliptic Macdonald polynomials. Int. Math. Res. Not. 21, 1037–1055 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. HK07.
    Heckenberger I., Kolb S.: On the Bernstein–Gelfand–Gelfand resolution for Kac–Moody algebras and quantized enveloping algebras. Transform. Groups 12(4), 647–655 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. Jos95.
    Joseph, A.: Quantum groups and their primitive ideals. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29. Springer, Berlin (1995)Google Scholar
  23. KK79.
    Kac V., Kazhdan D.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. Math. 34(1), 97–108 (1979)MathSciNetCrossRefMATHGoogle Scholar
  24. Kon94a.
    Konno H.: BRST cohomology in quantum affine algebra \({U_q(\widehat{\mathfrak{sl}}_2)}\). Modern Phys. Lett. A 9(14), 1253–1265 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. Kon94b.
    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. Nucl. Phys. B 432(3), 457–486 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. KT99.
    Kuroki G., Takebe T.: Bosonization and integral representation of solutions of the Knizhnik–Zamolodchikov–Bernard equations. Commun. Math. Phys. 204(3), 587–618 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. Mat94.
    Matsuo A.: A q-deformation of Wakimoto modules, primary fields and screening operators. Commun. Math. Phys. 160(1), 33–48 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. Neg09.
    Negut A.: Laumon spaces and the Calogero–Sutherland integrable system. Invent. Math. 178(2), 299–331 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. Neg11.
    Negut, A.: Affine Laumon spaces and the Calogero–Moser integrable system (2011). (Preprint). arXiv:1112.1756
  30. Rai10.
    Rains E.: Transformations of elliptic hypergeometric integrals. Ann. Math. (2) 171(1), 169–243 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. SV07.
    Styrkas, K., Varchenko, A.: Resonance relations, holomorphic trace functions and hypergeometric solutions to qKZB and Macdonald–Ruijsenaars equations. Int. Math. Res. Pap. IMRP (4):Art. ID rpm008 (2008)Google Scholar
  32. Tsy10.
    Tsymbaliuk A.: Quantum affine Gelfand-Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces. Selecta Math. (N.S.) 16(2), 173–200 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. Wak86.
    Wakimoto M.: Fock representations of the affine Lie algebra \({A^{(1)}_{1}}\). Commun. Math. Phys. 104(4), 605–609 (1986)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.MITCambridgeUSA

Personalised recommendations