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arXiv · 1910.00251

Local geometry of Jordan classes in semisimple algebraic groups

Filippo Ambrosio·Giovanna Carnovale·Francesco Esposito
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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.

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Filippo Ambrosio, Giovanna Carnovale, Francesco Esposito. 2020-10-08. Local geometry of Jordan classes in semisimple algebraic groups. https://doi.org/10.1112/jlms.12385

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