Springer Isomorphisms In Characteristic p

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Let G be a simple algebraic group over an algebraically closed field of characteristic p, and assume that p is a very good prime for G. Let P be a parabolic subgroup whose unipotent radical Uₚ 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 SLₙ, SOₙ, and Sp₂ₙ, viewed in the usual way as subgroups of GLₙ or GL₂ₙ, such a Springer isomorphism can be given explicitly by the Artin-Hasse exponential series.