Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence

Xue, Ting

doi:10.1090/S1088-4165-09-00364-1

Nilpotent orbits in the dual of classical Lie algebras in characteristic $2$ and the Springer correspondence

Author: Ting Xue
Journal: Represent. Theory 13 (2009), 609-635
MSC (2010): Primary 20G15
DOI: https://doi.org/10.1090/S1088-4165-09-00364-1
Published electronically: November 4, 2009
MathSciNet review: 2558787
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a simply connected algebraic group of type $B$, $C$ or $D$ over an algebraically closed field of characteristic $2$. We construct a Springer correspondence for the dual vector space of the Lie algebra of $G$. In particular, we classify the nilpotent orbits in the duals of symplectic and orthogonal Lie algebras over algebraically closed or finite fields of characteristic $2$.

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 https://doi.org/10.1007/BF01214043
Jens Carsten Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211. MR 2042689
V. Kac and B. Weisfeiler, Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic $p$, Nederl. Akad. Wetensch. Proc. Ser. A 79=Indag. Math. 38 (1976), no. 2, 136–151. MR 0417308
Reinhardt Kiehl and Rainer Weissauer, Weil conjectures, perverse sheaves and $l$’adic Fourier transform, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 42, Springer-Verlag, Berlin, 2001. MR 1855066
David B. Leep and Laura Mann Schueller, Classification of pairs of symmetric and alternating bilinear forms, Exposition. Math. 17 (1999), no. 5, 385–414. MR 1733879
G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205–272. MR 732546, DOI https://doi.org/10.1007/BF01388564
G. Lusztig, Character sheaves on disconnected groups. II, Represent. Theory 8 (2004), 72–124. MR 2048588, DOI https://doi.org/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 https://doi.org/10.1016/0001-8708%2885%2990064-7
G. Lusztig, A class of irreducible representations of a Weyl group, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 3, 323–335. MR 546372
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713

T. Xue, Nilpotent orbits in classical Lie algebras over finite fields of characteristic 2 and the Springer correspondence. Represent. Theory 13 (2009), 371–390.