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.

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