On the number of irreducible representations for finite unipotent Heisenberg-type groups

Abstract

Let $U$ be an algebraic subgroup of the group of $n\times n$ upper-triangular matrices with units on the diagonal over a finite field of large enough characteristic, and $\mathfrak{n}$ be the Lie algebra of $U$. The main tool in representation theory of $U$ is the orbit method, which classifies irreducible representations of the group $U$ in terms of coadjoint orbits on the dual space $\mathfrak{n}^*$. We consider two types of generalizations of the Heisenberg group, namely, generalized Heisenberg groups defined with an arbitrary bilinear form, and certain subgroups in maximal unipotent subgroups of classical orthogonal algebraic groups. We provide a way to calculate the number of irreducible representations of such groups. It turned out that this number is a polynomial in $q-1$ with nonnegative integer coefficients, which agrees with Isaacs' conjecture.