A generalization of Steinberg’s cross section

He, Xuhua; Lusztig, George

doi:10.1090/S0894-0347-2012-00728-0

A generalization of Steinberg’s cross section
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by Xuhua He and George Lusztig;

J. Amer. Math. Soc. 25 (2012), 739-757

DOI: https://doi.org/10.1090/S0894-0347-2012-00728-0

Published electronically: January 10, 2012

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Abstract:

Let $G$ be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element $w$ of minimal length $r$ a subvariety $V$ of $G$ isomorphic to an affine space of dimension $r$ which meets the regular unipotent class $Y$ in exactly one point. In this paper this is generalized to the case where $w$ is replaced by any elliptic element in the Weyl group of minimal length $d$ in its conjugacy class, $V$ is replaced by a subvariety $V’$ of $G$ isomorphic to an affine space of dimension $d$, and $Y$ is replaced by a unipotent class $Y’$ of codimension $d$ in such a way that the intersection of $V’$ and $Y’$ is finite.

References

James Ax, Injective endomorphisms of varieties and schemes, Pacific J. Math. 31 (1969), 1–7. MR 251036
Andrzej Białynicki-Birula and Maxwell Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200–203. MR 140516, DOI 10.1090/S0002-9939-1962-0140516-5
P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
Meinolf Geck, Sungsoon Kim, and Götz Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2000), no. 2, 570–600. MR 1769289, DOI 10.1006/jabr.1999.8253
Meinolf Geck and Jean Michel, “Good” elements of finite Coxeter groups and representations of Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 74 (1997), no. 2, 275–305. MR 1425324, DOI 10.1112/S0024611597000105
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
Xuhua He, Minimal length elements in some double cosets of Coxeter groups, Adv. Math. 215 (2007), no. 2, 469–503. MR 2355597, DOI 10.1016/j.aim.2007.04.005
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
G. Lusztig, Study of a $\mathbf Z$-form of the coordinate ring of a reductive group, J. Amer. Math. Soc. 22 (2009), no. 3, 739–769. MR 2505299, DOI 10.1090/S0894-0347-08-00603-6
G. Lusztig, From conjugacy classes in the Weyl group to unipotent classes, Represent. Theory 15 (2011), 494–530. MR 2833465, DOI 10.1090/S1088-4165-2011-00396-4
G. Lusztig, Elliptic elements in a Weyl group: a homogeneity property, arxiv:1007.5040.
G. Lusztig, On certain varieties attached to a Weyl group element, Bull. Inst. Math. Acad. Sinica (N.S.) 6 (2011), 377-414.
Alexey Sevostyanov, Algebraic group analogues of the Slodowy slices and deformations of Poisson $W$-algebras, Int. Math. Res. Not. IMRN 8 (2011), 1880–1925. MR 2806525, DOI 10.1093/imrn/rnq139
Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49–80. MR 180554