Affine hyperplane arrangements and Jordan classes

Abstract

We study the geometry of the stratification induced by an affine hyperplane arrangement $\mathcal{H}$ on the quotient of a complex affine space by the action of a discrete group preserving $\mathcal{H}$. We give conditions ensuring normality or normality in codimension 1 of strata. We apply these results to retrieve the list of the categorical quotients of closures of Jordan classes and of sheets in a complex simple algebraic group $G$ that are normal. In the simply-connected case, we show that normality of such a quotient is equivalent to its smoothness.