Orthogonal Subsets of Classical Root Systems and Coadjoint Orbits of Unipotent Groups

We consider a specific class of coadjoint orbits of maximal unipotent subgroups in classical groups over a finite field, i.e., orbits associated with orthogonal subsets in root systems. We derive a formula for the dimension of these orbits in terms of the Weyl group and construct polarizations for canonical forms on the orbits. As a consequence, we describe all possible dimensions of irreducible representations of such unipotent groups.