On semisimple classes and component groups in type D

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 psq " C H psq ˝¸q A for some complement q 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 pqq and 2 D l,sc pqq 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 [S22]