It might be helpful to explain what you mean by "reductive Lie group", or else to use a more common name lkke "semisimple". The term "reductive" comes from the theory of linear algebraic groups; it suggests complete reducibility of finite dimensional representations, which is only true in characteristic 0, etc.

Have you looked at Lusztig's lifting of a Frobenius map to a quantied enveloping algebra in characteristic 0?

@Sushil: Sorry for the long delay in answering, but it's been a busy week. I did try to add to my answer but didn't succeed. Anyway, I think your formulation is correct. But it's unhelpful in terms of computability. More details to follow.

Keep in mind the fact that the known result aready covers all cases in which the group is a "Weyl group", though your question is a natural one to raise. (By the way, I just expanded the tags and fixed a couple of small linguistic errors.)

@Johan: As you say, $\mathfrak{g}$ should be simple No, $\rho$ isn't always proportional to a root, as seen in the tables at the end of Chapter VI in Bourbaki.

@Victor: Thanks for the details.