One-dimensional double Hecke algebra and Gaussian sums
Ivan Cherednik
Source: Duke Math. J. Volume 108, Number 3 (2001), 511-538.
Abstract
We introduce and describe one-dimensional cyclotomic Gauss-Selberg sums generalizing the classical Gaussian sums. They correspond to irreducible self-dual unitary spherical representations of the one-dimensional double affine Hecke algebra.
Primary Subjects: 11L05
Secondary Subjects: 20C08, 33D80
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Alternatively, the document is available for a cost of $25. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1091737182
Mathematical Reviews number (MathSciNet):
MR1838660
Digital Object Identifier: doi:10.1215/S0012-7094-01-10834-X
Zentralblatt MATH identifier:
1007.11050
References
[AI] R. Askey and M. E. H. Ismail, ``A generalization of ultraspherical polynomials'' in Studies in Pure Mathematics, ed. P. Erdös, Birkhäuser, Basel, 1983, 55--78. MR 87a:33015
Mathematical Reviews (MathSciNet):
MR820210
Zentralblatt MATH:
0532.33006
[AW] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319. MR 87a:05023
Mathematical Reviews (MathSciNet):
MR783216
[Ch] K. Chandrasekharan, Elliptic Functions, Grundlehren Math. Wiss. 281, Springer, Berlin, 1985. MR 87e:11058
Mathematical Reviews (MathSciNet):
MR808396
Zentralblatt MATH:
0575.33001
[C1] I. Cherednik, Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices 1995, 483--515. MR 97f:33032
Mathematical Reviews (MathSciNet):
MR1358032
Digital Object Identifier: doi:10.1155/S1073792895000341
[C2] —, Difference Macdonald-Mehta conjecture, Internat. Math. Res. Notices 1997, 449--467. MR 99g:33046
[C3] —, ``From double Hecke algebra to analysis'' in Proceedings of the International Congress of Mathematicians, II (Berlin, 1998), Doc. Math. 1998, 527--531., http://www.mathematick.uni-bielefeld.de/documenta MR 2000b:33013
[C4] —, On $q$-analogues of Riemann's zeta, preprint 1998, arXiv:QA/9804099
[C5] —, Double Hecke algebras and Gauss-Selberg sums, in preparation.
[D] C. F. Dunkl, ``Hankel transforms associated to finite reflection groups'' in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, 1991), Contemp. Math. 138, Amer. Math. Soc., Providence, 1992, 123--138. MR 94g:33011
Mathematical Reviews (MathSciNet):
MR1199124
Zentralblatt MATH:
0789.33008
[E] R. J. Evans, The evaluation of Selberg character sums, Enseign. Math. (2) 37 (1991), 235--248. MR 93c:11062
Mathematical Reviews (MathSciNet):
MR1151749
[He] G. J. Heckman, An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), 341--350. MR 92i:33012
Mathematical Reviews (MathSciNet):
MR1085111
[H] S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Pure Appl. Math. 113, Academic Press, Orlando, Fla., 1984. MR 86c:22017
Mathematical Reviews (MathSciNet):
MR754767
Zentralblatt MATH:
0543.58001
[J] M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), 147--162. MR 94m:22011
Mathematical Reviews (MathSciNet):
MR1223227
[K] V. G. Kac, Infinite-dimensional Lie Algebras, 3d ed., Cambridge Univ. Press, Cambridge, 1990. MR 92k:17038
Mathematical Reviews (MathSciNet):
MR1104219
Zentralblatt MATH:
0716.17022
[KL] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, III, J. Amer. Math. Soc. 7 (1994), 335--381. MR 94g:17048
Mathematical Reviews (MathSciNet):
MR1239506
JSTOR: links.jstor.org
[Ki] A. Kirillov, Jr., Inner product on conformal blocks and Macdonald's polynomials at roots of unity, preprint (1995).
Mathematical Reviews (MathSciNet):
MR1635934
[KS] E. Koelink and J. Stokman, The big $q$-Jacobi function transform, preprint, 1999, http://www-ma.u-strasbourg.fr/irma/publications
Mathematical Reviews (MathSciNet):
MR1957384
Digital Object Identifier: doi:10.1007/s00365-002-0498-x
[M1] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), 189--207., Séminaire Bourbaki 1994/95, exp. no. 797. MR 90f:33024
Mathematical Reviews (MathSciNet):
MR1423624
[M2] —, A new class of symmetric functions, Sem. Lothar. Combin. 20 (1988), http://www.emis.de/journals/SLC/
[O1] E. M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), 1--18. MR 91h:33024
Mathematical Reviews (MathSciNet):
MR1010152
[O2] —, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Composito Math. 85 (1993), 333--373. MR 95j:33044
Duke Mathematical Journal
