Relative Springer isomorphisms

Let G be a simple algebraic group over the algebraically closed field k. A slightly strengthened version of a theorem of T.A. Springer says that (under some mild restrictions on G and k) there exists a G-equivariant isomorphism of varieties phi:U -> N, where U denotes the unipotent variety of G and M denotes the nilpotent variety of g = Lie G. Such phi is called a Springer isomorphism. Let B be a Borel subgroup of G, U the unipotent radical of B and u the Lie algebra of U. In this note we show that a Springer isomorphism phi induces a B-equivariant isomorphism phi: U/M -> u/m, where M is any unipotent normal subgroup of B and m = Lie M. We call such a map phi a relative Springer isomorphisin. We also use relative Springer isomorphisms to describe the geometry of U-orbits in u. (c) 2005 Elsevier Inc. All rights reserved.