Rook placements and coadjoint orbits for maximal unipotent subgroups of finite symplectic groups

TL;DR

This paper explores the structure of irreducible characters of maximal unipotent subgroups in finite symplectic groups using coadjoint orbits and rook placements, providing explicit formulas and a Mackey-style decomposition.

Contribution

It introduces a semi-direct decomposition for irreducible characters associated with coadjoint orbits via rook placements, extending the orbit method in this context.

Findings

Constructed a semi-direct decomposition for irreducible characters.

Provided an explicit formula for characters of maximal orbit dimension.

Unified description of orbits and characters via orthogonal rook placements.

Abstract

Let $U$ be a maximal unipotent subgroup in a symplectic group over a finite field of sufficiently large characteristic $p$. According to the Kirillov's orbit method, 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 construct a semi-direct decomposition for the corresponding irreducible characters in the spirit of the Mackey little group method. As a corollary, we present an explicit formula for the character corresponding to an orbit of maximal possible dimension.