SPRINGER ISOMORPHISMS IN CHARACTERISTIC p

Let G be a simple algebraic group over an algebraically closed field of characteristic p, and assume that p is a separably good prime for G. Let P be a parabolic subgroup whose unipotent radical U_P has nilpotence class less than p. We show that there exists a particularly nice Springer isomorphism for G which restricts to a certain canonical isomorphism Lie UP→∼UP defined by J.-P. Serre. This answers a question raised both by G. McNinch in [M2], and by J. Carlson et. al in [CLN]. For the groups SL_n; SO_n, and Sp_2n, viewed in the usual way as subgroups of GL_n or GL_2n, such a Springer isomorphism can be given explicitly by the Artin–Hasse exponential series.