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 ${\mathrm{Ind}}_{H_T}^G{\psi }_T\cong {\mathrm{Ind}}_{H_{T^{\prime }}}^G{\psi }_{T^{\prime }}$ if and only if $T$ and $T^{\prime }$ 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 ${\mathbb{F}}_q$.

We found mistakes in the proof of the main theorem