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Local geometry of Jordan classes in semisimple algebraic groups

2020; Wiley; Volume: 103; Issue: 2 Linguagem: Inglês

10.1112/jlms.12385

1469-7750

Filippo Ambrosio, Giovanna Carnovale, Francesco Esposito,

Homotopy and Cohomology in Algebraic Topology

We prove that the closure of every Jordan class J in a semisimple simply connected complex algebraic group G at a point x with Jordan decomposition x = r v 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 strata in SL n ( C ) are smooth.