A generalization of Steinberg's cross-section

TL;DR

This paper generalizes Steinberg's cross-section construction from Coxeter elements to elliptic elements in Weyl groups, creating affine subvarieties intersecting unipotent classes finitely.

Contribution

It extends Steinberg's classical construction to a broader class of elements, linking affine subvarieties with unipotent classes for elliptic elements.

Findings

Generalized Steinberg's cross-section to elliptic elements

Constructed affine subvarieties of dimension d

Established finite intersections with unipotent classes

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.