## A topological approach to induction theorems in Springer theory

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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

## Additional Information

**David Treumann**- Affiliation: Department of Mathematics, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, Minnesota 55455
- Received by editor(s): October 14, 2008
- Published electronically: February 6, 2009
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**13**(2009), 8-18 - MSC (2000): Primary 32S30
- DOI: https://doi.org/10.1090/S1088-4165-09-00342-2
- MathSciNet review: 2480385