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.