Nilpotent centralizers and Springer isomorphisms
Let GG be a semisimple algebraic group over a field KK whose characteristic is very good for GG, and let σσ be any GG-equivariant isomorphism from the nilpotent variety to the unipotent variety; the map σσ is known as a Springer isomorphism. Let y∈G(K)y∈G(K), let Y∈Lie(G)(K)Y∈Lie(G)(K), and write Cy=CG(y)Cy=CG(y) and CY=CG(Y)CY=CG(Y) for the centralizers. We show that the center of CyCy and the center of CYCY are smooth group schemes over KK. The existence of a Springer isomorphism is used to treat the crucial cases where yy is unipotent and where YY is nilpotent. Now suppose GG to be quasisplit, and write CC for the centralizer of a rational regular nilpotent element. We obtain a description of the normalizer NG(C)NG(C) of CC, and we show that the automorphism of Lie(C)Lie(C) determined by the differential of σσ at zero is a scalar multiple of the identity; these results verify observations of J.-P. Serre.