Nilpotent orbits in classical Lie algebras over finite fields of characteristic 2 and the Springer correspondence

Xue, Ting

doi:10.1090/S1088-4165-09-00357-4

Nilpotent orbits in classical Lie algebras over finite fields of characteristic 2 and the Springer correspondence
HTML articles powered by AMS MathViewer

by Ting Xue

Represent. Theory 13 (2009), 371-390

DOI: https://doi.org/10.1090/S1088-4165-09-00357-4

Published electronically: September 3, 2009

PDF | Request permission

Abstract:

Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.

References

A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
Wim H. Hesselink, Nilpotency in classical groups over a field of characteristic $2$, Math. Z. 166 (1979), no. 2, 165–181. MR 525621, DOI 10.1007/BF01214043
S. Kato, On the geometry of exotic nilpotent cones. (2006) arxiv: 0607478v1.
G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205–272. MR 732546, DOI 10.1007/BF01388564
G. Lusztig, Character sheaves on disconnected groups. II, Represent. Theory 8 (2004), 72–124. MR 2048588, DOI 10.1090/S1088-4165-04-00238-9
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
G. Lusztig, A class of irreducible representations of a Weyl group, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 3, 323–335. MR 546372, DOI 10.1016/1385-7258(79)90036-2
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
Nicolas Spaltenstein, Nilpotent classes and sheets of Lie algebras in bad characteristic, Math. Z. 181 (1982), no. 1, 31–48. MR 671712, DOI 10.1007/BF01214979
Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610, DOI 10.1007/BFb0096302
N. Spaltenstein, Nilpotent classes in Lie algebras of type $F_{4}$ over fields of characteristic $2$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 517–524. MR 731515
T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207. MR 442103, DOI 10.1007/BF01390009
T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
Robert Steinberg, Conjugacy classes in algebraic groups, Lecture Notes in Mathematics, Vol. 366, Springer-Verlag, Berlin-New York, 1974. Notes by Vinay V. Deodhar. MR 0352279, DOI 10.1007/BFb0067854
Robert Steinberg, On the desingularization of the unipotent variety, Invent. Math. 36 (1976), 209–224. MR 430094, DOI 10.1007/BF01390010
Ting Xue, Nilpotent orbits in classical Lie algebras over $\textbf {F}_{2^n}$ and the Springer correspondence, Proc. Natl. Acad. Sci. USA 105 (2008), no. 4, 1126–1128. MR 2375447, DOI 10.1073/pnas.0709626104