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Lusztig's conjecture for finite special linear groups

Author: Toshiaki Shoji
Journal: Represent. Theory 10 (2006), 164-222
MSC (2000): Primary 20G40, 20G05
Published electronically: March 22, 2006
MathSciNet review: 2219112
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Abstract: In this paper, we prove Lusztig's conjecture for $ G^F = SL_n(\mathbf F_q)$, i.e., we show that characteristic functions of character sheaves of $ G^F$ coincide with almost characters of $ G^F$ up to scalar constants, assuming that the characteristic of $ \mathbf F_q$ is not too small. We determine these scalars explicitly. Our result gives a method of computing irreducible characters of $ G^F$.

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  • [C] R. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, A Wiley-Interscience Publication, 1985. MR 0794307 (87d:20060)
  • [DLM1] F. Digne, G.I. Lehrer and J. Michel, On Gel'fand-Graev characters of reductive groups with disconnected centre, J. Reine angew. Math. 491 (1997), 131-147. MR 1476090 (98i:20015)
  • [DLM2] F.Digne, G.I. Lehrer and J. Michel, The space of unipotently supported class functions on a finie reductive group, J. Algebra 260 (2003), no. 1, 111-137. MR 1973579 (2004g:20020)
  • [G] M. Geck, A note on Harish-Chandra induction, Manuscripta Math. 80 (1993), 393-401. MR 1243154 (94m:20037)
  • [HL] R. Howlett and G. I. Lehrer, Induced cuspidal representations and generalized Hecke rings, Invent. Math. 58 (1980), 37-64. MR 0570873 (81j:20017)
  • [K1] N. Kawanaka, Generalized Gelfand-Graev representations and Ennola duality, in ``Algebraic groups and related topics,'' Advanced Studies in Pure Math., 6, Kinokuniya, Tokyo and North-Holland, Amsterdam, 1985, 179-206. MR 0803335 (87e:20075)
  • [K2] N. Kawanaka, Generalized Gelfand-Graev representations of exceptional simple algebraic groups over a finite field, I, Invent. Math. 84 (1986), 575-616. MR 0837529 (88a:20058)
  • [K3] N. Kawanaka, Shintani lifting and Gel'fand-Graev representations, in ``The Arcata conference on representations of finite groups,'' Proceedings of Symposia in Pure Math., Vol.47, Part 1, Amer. Math. Soc., Providence, RI, 1987, 147-163. MR 0933357 (89h:22037)
  • [Le] G. I. Lehrer, The characters of the finite special linear groups, J. of Algebra. 26 (1973), 564-583. MR 0354889 (50:7366)
  • [L1] G. Lusztig, ``Characters of reductive groups over a finite field'', Ann. of Math. Studies, Vol.107, Princeton Univ. Press, Princeton, 1984. MR 0742472 (86j:20038)
  • [L2] G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), 205-272. MR 0732546 (86d:20050)
  • [L3] G. Lusztig, Character sheaves, I Adv. in Math. 56 (1985), 193-237, II Adv. in Math. 57 (1985), 226-265, III, Adv. in Math. 57 (1985), 266-315, IV, Adv. in Math. 59 (1986), 1-63, V, Adv. in Math. 61 (1986), 103-155. MR 0806210 (87m:20118a); MR 0825086 (87m:20118b); MR 0849848 (87m:20118c)
  • [L4] G. Lusztig, On the character values of finite Chevalley groups at unipotent elements, J. of Algebra, 104 (1986), 146-194. MR 0865898 (88c:20051)
  • [L5] G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 157-166. MR 1021495 (90j:20083)
  • [L6] G. Lusztig, Green functions and character sheaves, Annals of Math., 131 (1990), 355-408. MR 1043271 (91c:20054)
  • [L7] G. Lusztig, A unipotent support for irreducible representations, Adv. in Math., 94 (1992), 139-179. MR 1174392 (94a:20073)
  • [LS] G. Lusztig and N. Spaltenstein, On the generalized Springer correspondence for classical groups, Advanced Studies in Pure Math. 6 (1985), pp. 289-316. MR 0803339 (87g:20072a)
  • [M] I.G. Macdonald, ``Symmetric functions and Hall polynomials'', second edition, Clarendon Press, Oxford, 1995. MR 1354144 (96h:05207)
  • [S1] T. Shoji, Character sheaves and almost characters of reductive groups, II, Adv. in Math. 111 (1995), 314-354. MR 1318530 (95k:20069)
  • [S2] T. Shoji, Shintani descent for special linear groups, J. Algebra 199 (1998), 175-228. MR 1489361 (99b:20078)
  • [S3] T. Shoji, Generalized Green functions and unipotent classes for finite reductive groups, I. To appear in Nagoya Math. J.
  • [ShS] T. Shoji and K. Sorlin, Subfield symmetric spaces for finite special linear groups, Representation theory, 8, (2004) 487-521. MR 2110358 (2006a:20090)
  • [SpS] T.A. Springer and R. Steinberg, Conjugacy classes, in ``Seminar on Algebraic groups and related topics'', Lecture Notes in Math., Vol. 131, Part E, Springer-Verlag, 1970. MR 0268192 (42:3091)

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Additional Information

Toshiaki Shoji
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

Received by editor(s): February 16, 2005
Received by editor(s) in revised form: January 24, 2006
Published electronically: March 22, 2006
Dedicated: To Noriaki Kawanaka on his sixtieth birthday
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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