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