A topological approach to induction theorems in Springer theory

Treumann, David

doi:10.1090/S1088-4165-09-00342-2

A topological approach to induction theorems in Springer theory
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by David Treumann

Represent. Theory 13 (2009), 8-18

DOI: https://doi.org/10.1090/S1088-4165-09-00342-2

Published electronically: February 6, 2009

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Abstract:

We give a self-contained account of a construction due to Rossmann, which lifts Springer’s action of a Weyl group on the cohomology of a Springer fiber to an action on its homotopy type. We use this construction to produce a generalization of an “induction theorem” of Alvis and Lusztig, which relates the Springer representations attached to a reductive group to those attached to a Levi subgroup. Our generalization applies to more general centralizers and to representations of Weyl groups on mod $p$ cohomology.

References

Dean Alvis and George Lusztig, On Springer’s correspondence for simple groups of type $E_{n}$ $(n=6,\,7,\,8)$, Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 1, 65–78. With an appendix by N. Spaltenstein. MR 662961, DOI 10.1017/S0305004100059703
E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 279–284. MR 0437798
Jean-Luc Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque 140-141 (1986), 3–134, 251 (French, with English summary). Géométrie et analyse microlocales. MR 864073
C. De Concini, G. Lusztig, and C. Procesi, Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), no. 1, 15–34. MR 924700, DOI 10.1090/S0894-0347-1988-0924700-2
Pierre Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302 (French). MR 422673, DOI 10.1007/BF01406236
Mikhail Grinberg, A generalization of Springer theory using nearby cycles, Represent. Theory 2 (1998), 410–431. MR 1657203, DOI 10.1090/S1088-4165-98-00053-3
Mark Goresky, Robert Kottwitz, and Robert Macpherson, Homology of affine Springer fibers in the unramified case, Duke Math. J. 121 (2004), no. 3, 509–561. MR 2040285, DOI 10.1215/S0012-7094-04-12135-9
Ryoshi Hotta, On Springer’s representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 863–876 (1982). MR 656061
R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), no. 2, 327–358. MR 732550, DOI 10.1007/BF01388568
D. Juteau. Modular Springer correspondence and decomposition matrices. Ph.D Thesis, Institut de Mathématiques de Jussieu, 2007.
David Kazhdan and George Lusztig, A topological approach to Springer’s representations, Adv. in Math. 38 (1980), no. 2, 222–228. MR 597198, DOI 10.1016/0001-8708(80)90005-5
G. Lusztig. An induction theorem for Springer’s representations. Adv. Studies in Pure Math., 40, in “Representation theory of algebraic groups and quantum groups”, pp. 253–259, 2004.
J. Mather. Notes on topological stability, 1970. Available at www.math.princeton.edu/ facultypapers/mather/.
Bao-Châu Ngô, Fibration de Hitchin et structure endoscopique de la formule des traces, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1213–1225 (French, with English and French summaries). MR 2275642
Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI 10.2307/1970770
W. Rossmann, Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra, Invent. Math. 121 (1995), no. 3, 531–578. MR 1353308, DOI 10.1007/BF01884311
Peter Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MR 584445, DOI 10.1007/BFb0090294