Nilpotent centralizers and Springer isomorphisms

TL;DR

This paper investigates properties of Springer isomorphisms in semisimple algebraic groups, demonstrating smoothness of centralizer centers and describing normalizers, with implications for understanding group automorphisms.

Contribution

It establishes the smoothness of centers of centralizers and characterizes automorphisms of Lie algebras induced by Springer isomorphisms in semisimple groups.

Findings

Centers of centralizers are smooth group schemes.

Normalizers of regular nilpotent centralizers are described.

Automorphisms of Lie algebras are scalar multiples of the identity.

Abstract

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 C_y = C_G(y) and C_Y= C_G(Y) for the centralizers. We show that the center of C_y and the center of C_Y 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 N_G(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.