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Nonsymmetric difference Whittaker functions


Starting with nonsymmetric global difference spherical functions, we define and calculate spinor (nonsymmetric) global \(q\)-Whittaker functions for arbitrary reduced root systems, which are reproducing kernels of the DAHA-Fourier transforms of Nil-DAHA and solutions of the \(q\)-Toda–Dunkl eigenvalue problem. We introduce the spinor \(q\)-Toda–Dunkl operators as limits of the difference Dunkl operators in DAHA theory under the spinor variant of the Ruijsenaars procedure. Their general algebraic theory (any reduced root systems) is the key part of this paper, based on the new technique of \(W\)-spinors and corresponding developments in combinatorics of affine root systems.

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  1. 1.

    Bezrukavnikov, R., Finkelberg, M.: Equivariant Satake category and Kostant–Whittaker reduction. Preprint arXiv:0707.3799v2 (2007)

  2. 2.

    Borodin, A., Corwin, I.: Macdonald processes. Preprint. arXiv:1111.4408v4 [math.PR] (2013)

  3. 3.

    Bourbaki, N.: Groupes et algèbres de Lie, Ch. 4–6. Hermann, Paris (1969)

  4. 4.

    Braverman, A., Finkelberg, M.: Finite-difference quantum Toda lattice via equivariant \(K\)-theory. Transform. Groups 10, 363–386 (2005)

    Article  MATH  Google Scholar 

  5. 5.

    Casselman, W., Shalika, J.: The unramified principal series of \(p\)-adic groups, II. The Whittaker function. Compos. Math. 41, 207–231 (1980)

    MATH  Google Scholar 

  6. 6.

    Cherednik, I.: Integration of quantum many-body problems by affine Knizhnik–Zamolodchikov equations. Preprint RIMS 776 (1994). Adv. Math. 106, 65–95 (1991)

  7. 7.

    Cherednik, I.: Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and Macdonald’s operators. IMRN 9, 171–180 (1992)

    Article  Google Scholar 

  8. 8.

    Cherednik, I.: Double affine Hecke algebras and Macdonald’s conjectures. Ann. Math. 141, 191–216 (1995)

    Article  MATH  Google Scholar 

  9. 9.

    Cherednik, I.: Nonsymmetric Macdonald polynomials. IMRN 10, 483–515 (1995)

    Article  Google Scholar 

  10. 10.

    Cherednik, I.: Difference Macdonald–Mehta conjecture. IMRN 10, 449–467 (1997)

    Article  Google Scholar 

  11. 11.

    Cherednik, I.: Intertwining operators of double affine Hecke algebras. Sel. Math. New Ser. 3, 459–495 (1997)

    Article  MATH  Google Scholar 

  12. 12.

    Cherednik, I.: Double affine Hecke algebras and difference Fourier transforms. Invent. Math. 152, 213–303 (2003)

    Article  MATH  Google Scholar 

  13. 13.

    Cherednik, I.: Double affine Hecke algebras. London Mathematical Society Lecture Note Series, 319. Cambridge University Press, Cambridge (2006)

  14. 14.

    Cherednik, I.: Non-semisimple Macdonald polynomials. Sel. Math. (N.S.) 14(3–4), 427–569 (2009)

    Article  MATH  Google Scholar 

  15. 15.

    Cherednik, I.: Whittaker limits of difference spherical functions. IMRN 20, 3793–3842 (2009)

    Google Scholar 

  16. 16.

    Cherednik, I.: On Harish-Chandra theory of global nonsymmetric functions. Preprint. arXiv:1407.5260 (2014)

  17. 17.

    Cherednik, I., Feigin, B.: Rogers–Ramanujan type identities and Nil-DAHA. Adv. Math. 248, 1050–1088 (2013)

    Article  MATH  Google Scholar 

  18. 18.

    Cherednik, I., Ma, X.: Spherical and Whittaker functions via DAHA I, II. Sel. Math. (N.S.). (2012). doi:10.1007/s00029-012-0110-6, 10.1007/s00029-012-0116-0

  19. 19.

    Cherednik, I., Orr, D.: One-dimensional nil-DAHA and Whittaker functions I. Transform. Groups 17(4), 953–987 (2012)

    Article  MATH  Google Scholar 

  20. 20.

    Cherednik, I., Orr, D.: One-dimensional nil-DAHA and Whittaker functions II. Transform. Groups 18(1), 23–59 (2013)

    Article  MATH  Google Scholar 

  21. 21.

    Cherednik, I., Feigin, E.: Extremal part of the PBW-filtration and E-polynomials. Preprint. arXiv:1306.3146 [math.QA] (2013)

  22. 22.

    Etingof, P.: Whittaker Functions on Quantum Groups and \(q\)-deformed Toda Operators. AMS Translations Series 2, vol. 194, pp. 9–25. AMS, Providence (1999)

  23. 23.

    Feigin, E., Fourier, G., Littelmann, P.: PBW-filtration over \(\mathbb{Z}\) and compatible bases for \(V_{\mathbb{Z}}(\lambda )\) in type \(A_n\) and \(C_n\). Preprint. arxiv:1204.1854v1

  24. 24.

    Feigin, E., Makedonskyi, I.: Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration. Preprint. arXiv:1407.6316 [RT] (2014)

  25. 25.

    Gerasimov, A., Lebedev, D., Oblezin, S.: On \(q\)-deformed \(\mathfrak{gl}_{l+1}\)-Whittaker functions, I. Preprint. arXiv:0803.0145

  26. 26.

    Givental, A., Lee, Y.-P.: Quantum \(K\)-theory on flag manifolds, finite-difference Toda lattices and quantum groups. Invent. Math. 151, 193–219 (2003)

    Article  MATH  Google Scholar 

  27. 27.

    Goodman, R., Wallach, N.R.: Conical vectors and Whittaker vectors. J. Funct. Anal. 39, 199–279 (1980)

    Article  MATH  Google Scholar 

  28. 28.

    Harish-Chandra, : Discrete series for semisimple Lie groups, II. Acta Math. 116, 1–111 (1963)

    Article  Google Scholar 

  29. 29.

    Humphreys, J.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  30. 30.

    Ion, B.: Nonsymmetric Macdonald polynomials and Demazure characters. Duke Math. J. 116(2), 299–318 (2003)

    Article  MATH  Google Scholar 

  31. 31.

    Joseph, A., Letzter, G., Zelikson, S.: On the Brylinski–Kostant filtration. JAMS 13(4), 945–970 (2000)

    MATH  Google Scholar 

  32. 32.

    Knop, F., Sahi, S.: A recursion and a combinatorial formula for Jack polynomials. Invent. Math. 128(1), 9–22 (1997)

    Article  MATH  Google Scholar 

  33. 33.

    Lusztig, G.: Affine Hecke algebras and their graded version. J. AMS 2(3), 599–635 (1989)

    MATH  Google Scholar 

  34. 34.

    Lusztig, G.: A new class of symmetric functions. Publ. I.R.M.A., Strasbourg, Actes 20-e Seminaire Lotharingen, pp. 131–171 (1988)

  35. 35.

    Lusztig, G.: Affine Hecke algebras and orthogonal polynomials. Séminaire Bourbaki 47(797), 01–18 (1995)

    Google Scholar 

  36. 36.

    Lusztig, G.: Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Mathematics, vol. 157. Cambridge University Press, Cambridge (2003)

  37. 37.

    Opdam, E.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175, 75–121 (1995)

    Article  MATH  Google Scholar 

  38. 38.

    Orr, D., Shimozono, M.: Specializations of nonsymmetric Macdonald–Koornwinder polynomials. Preprint. arxiv:1310.0279v2 (2013)

  39. 39.

    Ruijsenaars, S.N.M.: Factorized weight functions vs. factorized scattering. Commun. Math. Phys. 228, 467–494 (2002)

    Article  MATH  Google Scholar 

  40. 40.

    The Sage–Combinat Community: Sage–Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics. (2008)

  41. 41.

    Sanderson, Y.: On the Connection Between Macdonald Polynomials and Demazure Characters. J. Algebraic Comb. 11, 269–275 (2000)

  42. 42.

    Sevostyanov, A.: Quantum deformation of Whittaker modules and the Toda lattice. Duke Math. J. 105(2), 211–238 (2000)

  43. 43.

    Stokman, J.: The c-function expansion of a basic hypergeometric function associated to root systems. Ann. Math. 179, 253–299 (2014)

  44. 44.

    Wallach, N.R.: Real Reductive Groups II. Academic Press, Boston (1992)

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The first author thanks Tetsuji Miwa and the Mathematics Department of Kyoto University (where the paper was started) for the invitation and hospitality, as well as IHES (where the paper was completed). We thank Eric Opdam for useful remarks and Evgeny Feigin for his help with Conjecture 3.7. The second author thanks the organizers of the 5th Southeast Lie Theory Workshop for the invitation, where the results of this paper were reported.

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Correspondence to Ivan Cherednik.

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Ivan Cherednik is partially supported by NSF Grant DMS-1101535.

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Cherednik, I., Orr, D. Nonsymmetric difference Whittaker functions. Math. Z. 279, 879–938 (2015).

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  • Root systems
  • Hecke algebras
  • Whittaker functions
  • Toda operators
  • Macdonald polynomials

Mathematics Subject Classification

  • 20C08
  • 22E35
  • 33D80
  • 22E66
  • 20G44