Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings

Abstract

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$. Let ${\mathcal U}_n(A)$ be the subgroup of $\mathrm{GL}_n(A)$ of all upper triangular matrices with 1's along the main diagonal. Let $P=H_n(A)\rtimes {\mathcal U}_n(A)$, where ${\mathcal U}_n(A)$ acts on $H_n(A)$ by $*$-congruence transformations. We may view $P$ as a unipotent subgroup of either a symplectic group $\mathrm{Sp}_{2n}(A)$, if $*=1_A$ (in which case $A$ is commutative), or a unitary group $\mathrm{U}_{2n}(A)$ if $*\neq 1_A$. In this paper we construct and classify a family of irreducible representations of $P$ over a field $F$ that is essentially arbitrary. In particular, when $A$ is finite and $F=\mathbb C$ we obtain irreducible representations of $P$ of the highest possible degree.