Transformation Groups

, Volume 23, Issue 1, pp 119–147 | Cite as

CHEREDNIK ALGEBRAS AND ZHELOBENKO OPERATORS

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Abstract

We study canonical intertwining operators between induced modules of the trigonometric Cherednik algebra. We demonstrate that these operators correspond to the Zhelobenko operators for the affine Lie algebra Open image in new window . To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain Open image in new window -modules to modules of the Cherednik algebra.

References

  1. [1]
    T. Arakawa, T. Suzuki, A. Tsuchiya, Degenerate double affine Hecke algebras and conformal field theory, in: Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progress in Mathematics, Vol. 160, Birkhäuser Boston, Boston, MA, 1998, pp. 1–34.Google Scholar
  2. [2]
    M. Balagović, Irreducible modules for the degenerate double affine Hecke algebra of type A as submodules of Verma modules, J. Comb. Theory A133 (2015), 97–138.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    I. Cherednik, A unification of KnizhnikZamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), 411–431.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    I. Cherednik, Lectures on KnizhnikZamolodchikov equations and Hecke algebras, in: Quantum Many-Body Problems and Representation Theory, MSJ Mem., Vol. 1, Math. Soc. Japan, Tokyo, 1998, pp. 1–96.Google Scholar
  5. [5]
    В. Г. Дринфельд, Вырожденные аффинные алгебры Гекке и янгианы, Функц. анализ и его прил. 20 (1986), вып. 1, 69–70. Engl. transl.: V. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), no. 1, 58–60.Google Scholar
  6. [6]
    P. Etingof, V. Ginzburg, Symplectic reection algebras, CalogeroMoser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    S. Khoroshkin, M. Nazarov, Yangians and Mickelsson algebras I, Transformation Groups 11 (2006), no. 4, 625–658.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    S. Khoroshkin, M. Nazarov, On the functor of Arakawa, Suzuki and Tsuchiya, in: Representation Theory, Special Functions and Painleυé Equations (Kyoto, 2015), Advanced Studies in Pure Mathematics, Vol. 76, Math. Soc. Japan, Tokyo (to appear).Google Scholar
  9. [9]
    S. Khoroshkin, M. Nazarov, E. Vinberg, A generalized Harish-Chandra isomorphism, Adv. Math. 226 (2011), 1168–1180.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    S. Khoroshkin, O. Ogievetsky, Mickelsson algebras and Zhelobenko operators, J. Algebra 319 (2008), 2113–2165.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599–635.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    J. Rogawski, On modules over the Hecke algebras of a p-adic group, Invent. Math. 79 (1985), 443–465.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    T. Suzuki, Classification of simple modules over degenerate double affine Hecke algebras of type A, Int. Math. Res. Notices (2003), 2313–2339.Google Scholar
  14. [14]
    T. Suzuki, Double affine Hecke algebras, conformal coinvariants and Kostka polynomials, C. R. Acad. Sci. Paris I 343 (2006), 383–386.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    V. Tarasov, A. Varchenko, Duality for KnizhnikZamolodchikov and dynamical equations, Acta Appl. Math. 73 (2002), 141–154.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Д. П Желобенко, Экстремальные коциклы на группах Вейля, Функц. анализ и его прил. 21 (1987), вып. 3, 11–21. Engl. transl.: D. Zhelobenko, Extremal cocycles of Weyl groups, Funct. Anal. Appl. 21 (1987), no. 3, 183–192.Google Scholar

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Authors and Affiliations

  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Higher School of EconomicsNational Research UniversityMoscowRussia
  3. 3.Department of MathematicsUniversity of YorkYorkUK

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