Affine Hecke Algebras via DAHA

Research Contribution


A method is suggested for obtaining the Plancherel measure for Affine Hecke Algebras as a limit of integral-type formulas for inner products in the polynomial and related modules of Double Affine Hecke Algebras. The analytic continuation necessary here is a generalization of “picking up residues” due to Arthur, Heckman, Opdam and others, which can be traced back to Hermann Weyl. Generally, it is a finite sum of integrals over double affine residual subtori; a complete formula is presented for \(A_1\) in the spherical case.


Hecke algebras Fourier transform Spherical functions Plancherel measure Nonsymmetric Macdonald polynomials 



The author thanks RIMS, Kyoto university for the invitation, and the participants of his course at UNC. Many thanks to the referee for important remarks.


  1. Carlitz, L.: A finite analog of the reciprocal of a theta function, Pubblications de la Faculté D’électrotechnique De L’Université À Belgrade. Ser. Math. et Phys. 412–460, 97–99 (1973)Google Scholar
  2. Cherednik, I.: Double Affine Hecke Algebras, London Mathematical Society Lecture Note Series, vol. 319. Cambridge University Press, Cambridge (2006)Google Scholar
  3. Cherednik, I.: Difference Macdonald–Mehta conjecture. IMRN 10, 449–467 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. Cherednik, I.: Nonsemisimple Macdonald polynomials. Selecta Math. 14(3–4), 427–569 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. Cherednik, I.: Whittaker limits of difference spherical functions. IMRN 20, 3793–3842 (2009)MathSciNetMATHGoogle Scholar
  6. Cherednik, I.: Integration of quantum many-body problems by ffine Knizhnik-Zamolodchikov equations. Preprint RIMS 776 (1991) [Advances in Math. 106, 65–95 (1994)]Google Scholar
  7. Cherednik, I.: On Harish-Chandra theory of global nonsymmetric functions. arXiv:1407.5260 (2014)
  8. Cherednik, I., Ma, X.: Spherical and Whittaker functions via DAHA I, II. Selecta Mathematica (N.S.) 19(3), 737–817, 819–864 (2013)Google Scholar
  9. Cherednik, I., Orr, D.: One-dimensional nil-DAHA and Whittaker functions I. Transform. Groups 17(4), 953–987 (2012). arXiv:math/0111130v1 (2011)
  10. Cherednik, I., Orr, D.: Nonsymmetric difference Whittaker functions. Mathematische Zeitschrift 279(3), 879–938 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. Cherednik, I., Ostrik, V.: From double Hecke algebras to Fourier transform. Selecta Math. New Ser. 8, 1–89 (2003). arXiv:math/0111130
  12. Ciubotaru, D., Kato, M., Kato, S.: On characters and formal degrees of discrete series of affine Hecke algebras of classical types. Inventiones mathematicae 187(3), 589–635 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. Enomoto, N.: Composition factors of polynomial representation of DAHA and crystallized decomposition numbers. J. Math. Kyoto Univ. 49(3), 441–473 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. Etingof, P., Stoica, E., with an appendix by Griffeth, S.: Unitary representations of rational Cherednik algebras. Represent. Theory 13, 349–370 (2009)Google Scholar
  15. Heckman, G.J., Opdam, E.M.: Harmonic analysis for affine Hecke algebras. In: Yau, S.-T. (ed.) Current Developments in Mathematics. Intern. Press, Boston (1996)Google Scholar
  16. Ion, B.: Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series. Adv. Math. 201, 36–62 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. Kazhdan, D., Lusztig, G.: Proof of the Deligne–Langlands conjecture for Hecke algebras. Inventiones Math. 87, 153–215 (1987)MathSciNetCrossRefMATHGoogle Scholar
  18. Lusztig, G.: Green functions and character sheaves. Ann. Math. 131, 355–408 (1990)MathSciNetCrossRefMATHGoogle Scholar
  19. Stokman, J.: The c-function expansion of a basic hypergeometric function associated to root systems. Ann. Math. 179(1), 253–299 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. Opdam, E.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175, 75–121 (1995)MathSciNetCrossRefMATHGoogle Scholar
  21. Opdam, E.: Hecke algebras and harmonic analysis. In: Proceedings of the International Congress of Mathematicians -Madrid, vol. II, pp. 1227–1259. EMS Publ. House (2006)Google Scholar
  22. Opdam, E.: A generating formula for the trace of the Iwahori–Hecke algebra. Prog. Math. 210, 301–323 (2003). arXiv:math/0101006
  23. Opdam, E., Solleveld, M.: Discrete series characters for affine Hecke algebras and their formal degrees. Acta Math. 205(1), 105–187 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2018

Authors and Affiliations

  1. 1.Department of MathematicsUNC Chapel HillChapel HillUSA

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