Orthogonal subsets of root systems and the orbit method

Abstract: Let $k$ be the algebraic closure of a finite field, $G$ a Chevalley group over $k$, $U$ the maximal unipotent subgroup of $G$. To each orthogonal subset $D$ of the root system of the group $G$ and each set $\xi$ of $|D|$ non-zero scalars from $k$ one can assign the coadjoint orbit of the group $U$. We prove that the dimension of such an orbit does not depend on $\xi$. We also give an upper bound of the dimension in terms of the Weyl group.