Representations of finite pattern groups

Abstract

Let $G=1+A$ be a finite pattern group over the finite field ${\mathbb{F}}_q$. We give a natural bijection between coadjoint orbits of $G$ and its equivalent classes of irreducible representations. More precisely, given any $T\in A^t$, viewed as a representative of associated coadjoint orbit ${\mathfrak{O}}_T$ of $G$, we can explicitly construct a subgroup $H_T $ of $G$, such that ${\mathrm{Ind}}_{H_T}^G \psi_T$ is irreducible and ${\mathrm{Ind}}_{H_T}^G \psi_T \cong {\mathrm{Ind}}_{H_{T'}}^G \psi_{T'}$ if and only if $T$ and $ T'$ are in the same coadjoint orbit. Here $\psi_T(x)=\psi({\mathrm{tr}} Tx)\text{ for }x\in H_T,$ and $\psi$ is a fixed nontrivial additive character of ${\mathbb{F}}_q$.