  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.
