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 ϕ:U→N, where U denotes the unipotent variety of G and N denotes the nilpotent variety of g=LieG. Such ϕ 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 ϕ induces a B-equivariant isomorphism ϕ˜:U/M→u/m, where M is any unipotent normal subgroup of B and m=LieM. We call such a map ϕ˜ a relative Springer isomorphism. We also use relative Springer isomorphisms to describe the geometry of U-orbits in u.