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 , where denotes the unipotent variety of G and denotes the nilpotent variety of . Such is called a Springer isomorphism. Let B be a Borel subgroup of G, U the unipotent radical of B and the Lie algebra of U. In this note we show that a Springer isomorphism induces a B-equivariant isomorphism , where M is any unipotent normal subgroup of B and . We call such a map a relative Springer isomorphism. We also use relative Springer isomorphisms to describe the geometry of U-orbits in .