On semisimple classes and component groups in type D

2022/09/28 by Cabanes, Marc, Späth, Britta
#FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2209.13922

Abstract

In adjoint reductive groups H of type D we show that for every semisimple element s, its centralizer splits over its connected component, i.e., C_H(s) = C_H(s)^∘ \rtimes \check A for some complement \check A with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups D_l,sc(q) and ²D_l,sc(q) and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conjecture for the prime 3, see [S21], [S23].

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