Lusztig’s conjecture for finite special linear groups

Shoji, Toshiaki

doi:10.1090/S1088-4165-06-00275-5

Lusztig’s conjecture for finite special linear groups
HTML articles powered by AMS MathViewer

by Toshiaki Shoji

Represent. Theory 10 (2006), 164-222

DOI: https://doi.org/10.1090/S1088-4165-06-00275-5

Published electronically: March 22, 2006

PDF | Request permission

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 $G^F$.

References

Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
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
F. Digne, G. Lehrer, and J. Michel, The space of unipotently supported class functions on a finite reductive group, J. Algebra 260 (2003), no. 1, 111–137. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1973579, DOI 10.1016/S0021-8693(02)00635-X
Meinolf Geck, A note on Harish-Chandra induction, Manuscripta Math. 80 (1993), no. 4, 393–401. MR 1243154, DOI 10.1007/BF03026560
R. B. Howlett and G. I. Lehrer, Induced cuspidal representations and generalised Hecke rings, Invent. Math. 58 (1980), no. 1, 37–64. MR 570873, DOI 10.1007/BF01402273
N. Kawanaka, Generalized Gel′fand-Graev representations and Ennola duality, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 175–206. MR 803335, DOI 10.2969/aspm/00610175
N. Kawanaka, Generalized Gel′fand-Graev representations of exceptional simple algebraic groups over a finite field. I, Invent. Math. 84 (1986), no. 3, 575–616. MR 837529, DOI 10.1007/BF01388748
Noriaki Kawanaka, Shintani lifting and Gel′fand-Graev representations, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 147–163. MR 933357, DOI 10.1090/pspum/047.1/933357
Gustav Isaac Lehrer, The characters of the finite special linear groups, J. Algebra 26 (1973), 564–583. MR 354889, DOI 10.1016/0021-8693(73)90015-X
George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205–272. MR 732546, DOI 10.1007/BF01388564
George Lusztig, Character sheaves. II, III, Adv. in Math. 57 (1985), no. 3, 226–265, 266–315. MR 806210, DOI 10.1016/0001-8708(85)90064-7
George Lusztig, On the character values of finite Chevalley groups at unipotent elements, J. Algebra 104 (1986), no. 1, 146–194. MR 865898, DOI 10.1016/0021-8693(86)90245-0
G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 10, 157–166. Orbites unipotentes et représentations, I. MR 1021495
George Lusztig, Green functions and character sheaves, Ann. of Math. (2) 131 (1990), no. 2, 355–408. MR 1043271, DOI 10.2307/1971496
George Lusztig, A unipotent support for irreducible representations, Adv. Math. 94 (1992), no. 2, 139–179. MR 1174392, DOI 10.1016/0001-8708(92)90035-J
G. Lusztig and N. Spaltenstein, On the generalized Springer correspondence for classical groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 289–316. MR 803339, DOI 10.2969/aspm/00610289
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Toshiaki Shoji, Character sheaves and almost characters of reductive groups. I, II, Adv. Math. 111 (1995), no. 2, 244–313, 314–354. MR 1318530, DOI 10.1006/aima.1995.1024
Toshiaki Shoji, Shintani descent for special linear groups, J. Algebra 199 (1998), no. 1, 175–228. MR 1489361, DOI 10.1006/jabr.1997.7174
Toshiaki Shoji and Karine Sorlin, Subfield symmetric spaces for finite special linear groups, Represent. Theory 8 (2004), 487–521. MR 2110358, DOI 10.1090/S1088-4165-04-00233-X
T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192