Local geometry of Jordan classes in semisimple algebraic groups

TL;DR

This paper investigates the local geometric structure of Jordan classes in semisimple algebraic groups, revealing smooth equivalences and classifying smooth sheets, normality, and Cohen-Macaulay properties in these groups.

Contribution

It establishes smooth equivalences of Jordan class closures at points, characterizes smooth sheets, and proves smoothness of all sheets and Lusztig's strata in SL(n,C).

Findings

Closure of Jordan classes is smoothly equivalent to centralizer classes.

Complete classification of smooth sheets in simple groups.

All sheets and Lusztig's strata in SL(n,C) are smooth.

Abstract

We prove that the closure of every Jordan class J in a semisimple simply connected complex group G at a point x with Jordan decomposition x = rv is smoothly equivalent to the union of closures of those Jordan classes in the centraliser of r that are contained in J and contain x in their closure. For x unipotent we also show that the closure of J around x is smoothly equivalent to the closure of a Jordan class in Lie(G) around exp^{-1}x. For G simple we apply these results in order to determine a (non-exhaustive) list of smooth sheets in G, the complete list of regular Jordan classes whose closure is normal and Cohen-Macaulay, and to prove that all sheets and Lusztig's strata in SL(n,C) are smooth.