Nilpotent centralizers and Springer isomorphisms

الأصل · EN

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