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.