Let *U_1* be a unipotent group inside some Chevalley group G. For now, think of G as being *SL_n(K)* where K is a field; then we can take *U_1* to be a bunch of strictly upper triangular matrics. Assume if you like that K is algebraically closed.
Now suppose that *U_1* is normalized by a non-trivial torus *T_1*. Are there any general statements that can be made about the structure of *U_1*?
For instance: let us assume that *T_1* is 1-dimensional, as this is the limiting case. I suspect that the following is true: if r(t) is not equal to s(t) for all positive roots r,s, and all elements t in *T_1*, then *U_1* is a product of root subgroups.
I haven't written down a proof of this statement, but doodling suggests that it is true! Indeed I suspect it is true of *T_1* contains ANY element t satisfying the given condition. I would like a more general statement though: covering the case where r(t)=s(t) for particular positive roots r and s.
I should note that I tend to automatically ignore small characteristic cases! Funny things can happen in this situation...