Affine hyperplane arrangements and Jordan classes

TL;DR

This paper investigates the geometric structure of stratifications induced by affine hyperplane arrangements on quotient spaces, providing conditions for normality and classifying normal quotients in algebraic groups.

Contribution

It offers new criteria for normality of strata and classifies normal categorical quotients of Jordan classes and sheets in complex simple algebraic groups.

Findings

Normality conditions for strata in hyperplane arrangements

Complete classification of normal quotients of Jordan classes

Normality is equivalent to smoothness in simply connected cases

Abstract

We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in codimension 1 of strata. As an application, we provide the list of those categorical quotients of closures of Jordan classes and of sheets in all complex simple algebraic groups that are normal. In the simply connected case, we show that normality of such a quotient is equivalent to its smoothness.