Representations of affine Hecke algebras and based rings of affine Weyl groups

Xi, Nanhua

doi:10.1090/S0894-0347-06-00539-X

Representations of affine Hecke algebras and based rings of affine Weyl groups
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by Nanhua Xi

J. Amer. Math. Soc. 20 (2007), 211-217

DOI: https://doi.org/10.1090/S0894-0347-06-00539-X

Published electronically: June 19, 2006

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Abstract:

In this paper we show that the Deligne-Langlands-Lusztig classification of simple representations of an affine Hecke algebra remains valid if the parameter is not a root of the corresponding Poincaré polynomial. This verifies a conjecture of Lusztig proposed in 1989.

References

Susumu Ariki and Andrew Mathas, The number of simple modules of the Hecke algebras of type $G(r,1,n)$, Math. Z. 233 (2000), no. 3, 601–623. MR 1750939, DOI 10.1007/s002090050489
Roman Bezrukavnikov and Viktor Ostrik, On tensor categories attached to cells in affine Weyl groups. II, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 101–119. MR 2074591, DOI 10.2969/aspm/04010101
Armand Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259. MR 444849, DOI 10.1007/BF01390139
I. Grojnowski, Representations of affine Hecke algebras (and affine quantum $\textrm {GL}_n$) at roots of unity, Internat. Math. Res. Notices 5 (1994), 215 ff., approx. 3 pp.}, issn=1073-7928, review= MR 1270135, doi=10.1155/S1073792894000243, DOI 10.1155/S1073792894000243
Akihiko Gyoja, Modular representation theory over a ring of higher dimension with applications to Hecke algebras, J. Algebra 174 (1995), no. 2, 553–572. MR 1334224, DOI 10.1006/jabr.1995.1139
Akihiko Gyoja, Cells and modular representations of Hecke algebras, Osaka J. Math. 33 (1996), no. 2, 307–341. MR 1416051
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215. MR 862716, DOI 10.1007/BF01389157
George Lusztig, Singularities, character formulas, and a $q$-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208–229. MR 737932
George Lusztig, Cells in affine Weyl groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 255–287. MR 803338, DOI 10.2969/aspm/00610255
George Lusztig, Cells in affine Weyl groups. II, J. Algebra 109 (1987), no. 2, 536–548. MR 902967, DOI 10.1016/0021-8693(87)90154-2
George Lusztig, Cells in affine Weyl groups. II, J. Algebra 109 (1987), no. 2, 536–548. MR 902967, DOI 10.1016/0021-8693(87)90154-2
George Lusztig, Cells in affine Weyl groups. IV, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 2, 297–328. MR 1015001
George Lusztig, Representations of affine Hecke algebras, Astérisque 171-172 (1989), 73–84. Orbites unipotentes et représentations, II. MR 1021500
Marie-France Vignéras, Modular representations of $p$-adic groups and of affine Hecke algebras, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 667–677. MR 1957074
Nan Hua Xi, Representations of affine Hecke algebras, Lecture Notes in Mathematics, vol. 1587, Springer-Verlag, Berlin, 1994. MR 1320509, DOI 10.1007/BFb0074130
Nanhua Xi, The based ring of two-sided cells of affine Weyl groups of type $\widetilde A_{n-1}$, Mem. Amer. Math. Soc. 157 (2002), no. 749, xiv+95. MR 1895287, DOI 10.1090/memo/0749