Strictly transversal slices to conjugacy classes in algebraic groups

TL;DR

This paper constructs special transversal slices in algebraic groups that intersect conjugacy classes in a way that relates their dimensions, providing a geometric tool for understanding conjugacy class structure.

Contribution

It introduces a method to find transversal slices in algebraic groups that intersect conjugacy classes with dimension equality, enhancing geometric understanding of conjugacy class arrangements.

Findings

Existence of transversal slices intersecting conjugacy classes

Dimension equality between conjugacy classes and slices

Application to algebraic group structure analysis

Abstract

We show that for every conjugacy class O in a connected semisimple algebraic group G over a field of characteristic good for G one can find a special transversal slice S to the set of conjugacy classes in G such that O intersects S and dim O = codim S.