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

Let U U be a Sylow p p -subgroup in a classical group over a finite field of characteristic p p . The coadjoint orbits of the group U U play the key role in the description of irreducible complex characters of U 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.