Orbits and characters associated with rook placements for Sylow p-subgroups of finite orthogonal groups

Let U be a Sylow p-subgroup in a classical group over a finite field of characteristic p. The coadjoint orbits of the group U play the key role in the description of irreducible complex characters of U. Almost all important classes of orbits and characters studied to the moment can be uniformly described as the orbits and characters associated with so-called orthogonal rook placements. In the paper, we study such orbits for the orthogonal group. We construct a polarization for the canonical form on such an orbit and present a semi-direct decomposition for the corresponding irreducible characters in the spirit of the Mackey little group method. As a corollary, we compute the dimension of an orbit associated with an orthogonal rook placement.