Here is a quick answer, though I'll have to check more carefully what I remember about the question and its answer.    Roger Richardson amplified Kostant's result in characteristic 0, which in turn led Steve Donkin to work out a closely parallel version in prime characteristic:  *On conjugating representations and adjoint representations of semisimple groups*,  Invent. Math. 91 (1988), no. 1, 137–145.   (This should be freely available by now at the GDZ archive.)    See in particular Donkin's Theorem 2.2.    All serious results of this sort unfortunately require some mild restrictions on the prime involved relative to the root system.    Note too that both the algebraic group and its Lie algebra lead to precise statements.