Local geometry of Jordan classes in semisimple algebraic groups

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