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

Let $A$ be a ring with $1\mathrm{\ne }0$, not necessarily finite, endowed with an involution~$\ast$, that is, an anti-automorphism of order $\le 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $\ast$. Let ${\mathcal{U}}_n(A)$ be the subgroup of ${\mathrm{G}\mathrm{L}}_n(A)$ of all upper triangular matrices with 1’s along the main diagonal. Let $P=H_n(A)⋊{\mathcal{U}}_n(A)$, where ${\mathcal{U}}_n(A)$ acts on $H_n(A)$ by $\ast$-congruence transformations. We may view $P$ as a unipotent subgroup of either a symplectic group ${\mathrm{S}\mathrm{p}}_{2n}(A)$, if $\ast =1_A$ (in which case $A$ is commutative), or a unitary group ${\mathrm{U}}_{2n}(A)$ if $\ast \mathrm{\ne }{1}_A$. 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.