Local geometry of Jordan classes in semisimple algebraic groups
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.
Keywords
Cite
@article{arxiv.1910.00251,
title = {Local geometry of Jordan classes in semisimple algebraic groups},
author = {Filippo Ambrosio and Giovanna Carnovale and Francesco Esposito},
journal= {arXiv preprint arXiv:1910.00251},
year = {2020}
}
Comments
The post-print version of this paper will appear in the Journal of the London Mathematical Society