One-dimensional double Hecke algebra and Gaussian sums

Duke Mathematical Journal - Tập 108 Số 3 - 2001
Ivan Cherednik

Tóm tắt

Từ khóa


Tài liệu tham khảo

\lccR. Askey and M. E. H. Ismail, “A generalization of ultraspherical polynomials” in <i>Studies in Pure Mathematics</i>, ed. P. Erdös, Birkhäuser, Basel, 1983, 55–78. MR 87a:33015

\lccR. Askey and J. Wilson, <i>Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials</i>, Mem. Amer. Math. Soc. <b>54</b> (1985), no. 319. MR 87a:05023

\lccK. Chandrasekharan, <i>Elliptic Functions</i>, Grundlehren Math. Wiss. <b>281</b>, Springer, Berlin, 1985. MR 87e:11058

\lccI. Cherednik, <i>Nonsymmetric Macdonald polynomials</i>, Internat. Math. Res. Notices <b>1995</b>, 483–515. MR 97f:33032

––––, <i>Difference Macdonald-Mehta conjecture</i>, Internat. Math. Res. Notices <b>1997</b>, 449–467. MR 99g:33046

––––, “From double Hecke algebra to analysis” in <i>Proceedings of the International Congress of Mathematicians, II (Berlin, 1998)</i>, Doc. Math. <b>1998</b>, 527–531, http://www.mathematick.uni-bielefeld.de/documenta MR 2000b:33013

––––, <i>On $q$-analogues of Riemann's zeta</i>, preprint 1998.

––––, <i>Double Hecke algebras and Gauss-Selberg sums</i>, in preparation.

\lccC. F. Dunkl, “Hankel transforms associated to finite reflection groups” in <i>Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, 1991)</i>, Contemp. Math. <b>138</b>, Amer. Math. Soc., Providence, 1992, 123–138. MR 94g:33011

\lccR. J. Evans, <i>The evaluation of Selberg character sums</i>, Enseign. Math. (2) <b>37</b> (1991), 235–248. MR 93c:11062

\lccG. J. Heckman, <i>An elementary approach to the hypergeometric shift operators of Opdam</i>, Invent. Math. <b>103</b> (1991), 341–350. MR 92i:33012

\lccS. Helgason, <i>Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions</i>, Pure Appl. Math. <b>113</b>, Academic Press, Orlando, Fla., 1984. MR 86c:22017

\lccM. F. E. de Jeu, <i>The Dunkl transform</i>, Invent. Math. <b>113</b> (1993), 147–162. MR 94m:22011

\lccV. G. Kac, <i>Infinite-dimensional Lie Algebras</i>, 3d ed., Cambridge Univ. Press, Cambridge, 1990. MR 92k:17038

\lccD. Kazhdan and G. Lusztig, <i>Tensor structures arising from affine Lie algebras, III</i>, J. Amer. Math. Soc. <b>7</b> (1994), 335–381. MR 94g:17048

\lccA. Kirillov, Jr., <i>Inner product on conformal blocks and Macdonald's polynomials at roots of unity</i>, preprint (1995).

\lccE. Koelink and J. Stokman, <i>The big $q$-Jacobi function transform</i>, preprint, 1999, http://www-ma.u-strasbourg.fr/irma/publications

I. G. Macdonald, <i>Affine Hecke algebras and orthogonal polynomials</i>, Astérisque <b>237</b> (1996), 189–207, Séminaire Bourbaki 1994/95, exp. no. 797. MR 90f:33024

––––, <i>A new class of symmetric functions</i>, Sem. Lothar. Combin. <b>20</b> (1988), http://www.emis.de/journals/SLC/

\lccE. M. Opdam, <i>Some applications of hypergeometric shift operators</i>, Invent. Math. <b>98</b> (1989), 1–18. MR 91h:33024

––––, <i>Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group</i>, Composito Math. <b>85</b> (1993), 333–373. MR 95j:33044

––––, <i>Harmonic analysis for certain representations of graded Hecke algebras</i>, Acta Math. <b>175</b> (1995), 75–121. MR 98f:33025