Skip to main content
SpringerLink
Log in

Spherical and Whittaker functions via DAHA I

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

This paper begins with an exposition of the classical \(p\)-adic theory of the Macdonald, Matsumoto, and Whittaker functions aimed at the affine generalizations. The major directions are as follows: (1) extending the theory of DAHA to arbitrary levels and (2) the affine Satake map and Hall functions via DAHA. The key result is the proportionality of the two different formulas for the affine symmetrizer, the Satake-type formula and that based on the polynomial representation of DAHA. The latter approach results in two important formulas for the affine symmetrizer generalizing the relations between the Kac–Moody characters and Demazure characters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Google Scholar 

  2. Braverman, A., Kazhdan, D.: The spherical Hecke algebra for affine Kac-Moody groups I, preprint arXiv:0809.1461v2 (2008)

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Cherednik, I.: Kadell’s two conjectures for Macdonald polynomials. MRL 3, 418–427 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Cherednik, I.: Nonsemisimple Macdonald polynomials. Selecta Mathematica 14, 3–4, 427–569 (2009); arXiv:0709.1742 [math.QA]

    Google Scholar 

  10. Cherednik, I.: Toward Harmonic Analysis on DAHA (Integral formulas for canonical traces), Notes of the lecture delivered at University of Amsterdam (May 30, 2008), http://math.mit.edu/etingof/hadaha.pdf

  11. Cherednik, I.: Whittaker limits of difference spherical functions. IMRN 20, 3793–3842 (2009) arXiv:0807.2155 (2008)

    Google Scholar 

  12. Cherednik, I.: Difference-elliptic operators and root systems. IMRN 1, 43–59 (1995)

    Article  MathSciNet  Google Scholar 

  13. Cherednik, I., Ma, X.: A new take on spherical, Whittaker and Bessel functions (v5). preprint arXiv:0904.4324v5 [math.QA] (2009)

  14. Etingof, P., Felder, G., Ma, X., Veselov, A.: On elliptic Calogero-Moser systems for complex crystallographic reflection groups. J. Algebra, 31 (2010), arXiv:1003.4689v2 [math.QA]

  15. Fishel, S., Grojnowski, I., Teleman, C.: The strong Macdonald conjecture and Hodge theory on the loop Grassmannian. Ann. Math. 168, 175–220 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Heckman, G., Opdam, E.: Harmonic analysis for affine Hecke algebras. In: Yau, S.-T. (ed.) Current Developments in Mathematics. International Press, Boston (1996)

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Ion, B.: Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series. Adv. Math. 201, 36–62 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kac, V.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  21. Kapranov, M.: The elliptic curve in the \(S\)-duality theory and Eisenstein series for Kac-Moody groups, preprint arXiv:math/0001005v2 (2000)

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

    MathSciNet  MATH  Google Scholar 

  23. Macdonald, I.: Spherical functions on a \(\mathfrak{p}\)-adic Chevalley group. Bull. Am. Math. Soc. 74(3), 520–525 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  24. Macdonald, I.: Spherical functions on a group of p-adic type. Publ. Ramanujan Institute, No. 2 (1971)

  25. Macdonald, I.: A formal identity for affine root systems, in Lie Groups and Symmetric Spaces, in memory of F.I. Karpelevich (ed. S.G. Gindikin). Amer. Math. Soc. Translations Ser. 2, 210: 195–211 (2003)

  26. Macdonald, I.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1999)

    Google Scholar 

  27. Macdonald, I.: Affine Hecke Algebras and Orthogonal Polynomials. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  28. Matsumoto, H.: Analyse harmonique dans les systèmes de Tits bornologiques de type affine. Lecture Notes in Mathematics 590 (1977)

  29. Opdam, E.: On the spectral decomposition of affine Hecke algebras. Journal de l’Institut de Mathematiques de Jussieu 3(4), 531–648 (2004)

    Google Scholar 

  30. Opdam, E.: A generating function for the trace of the Iwahori-Hecke algebra. In: Joseph, A., Melnikov, A., Rentschler, R. (eds.) Studies in Memory of Issai Schur. Progress in Mathematics, pp. 301–324 Birkhäuser, Boston (2001)

  31. Opdam, E., Stokman, J.: MathSciNet review on I. Macdonald’s book Affine Hecke algebras and orthogonal polynomials (2005), MR1976581 (2005b:33021)

  32. Rains, E.: \(BC_n\)-symmetric abelian functions. Duke Math. J. 135(1), 99–180 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sanderson, Y.: On the connection between Macdonald polynomials and Demazure characters. J. Algebr. Combin. 11, 269–275 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shintani, T.: On an explicit formula for class \(1\) Whittaker functions on \(GL_n\) over \(p\)-adic fields. Proc. Jpn. Acad. 52, 180–182 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  35. Stokman, J.: Difference Fourier transforms for nonreduced root systems. Sel. Math. New ser. 9, 409–494 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Viswanath, S.: Kostka-Foulkes polynomials for symmetrizable Kac-Moody algebras, preprint arxiv:math/0610246v1 (2006)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, UNC, Chapel Hill, NC, 27599, USA

    Ivan Cherednik

  2. 632 CAB, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Xiaoguang Ma

Authors
  1. Ivan Cherednik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoguang Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Cherednik.

Additional information

Partially supported by NSF grant DMS–0800642.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cherednik, I., Ma, X. Spherical and Whittaker functions via DAHA I. Sel. Math. New Ser. 19, 737–817 (2013). https://doi.org/10.1007/s00029-012-0110-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-012-0110-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation