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

TL;DR

This paper constructs and classifies irreducible representations of certain unipotent subgroups of symplectic and unitary groups over rings, extending understanding of their representation theory in broad algebraic contexts.

Contribution

It introduces a new family of irreducible representations for these unipotent groups, applicable over arbitrary fields and rings, including finite rings and complex numbers.

Findings

Constructed irreducible representations over arbitrary fields.

Classified a broad family of such representations.

Achieved highest degree irreducible representations when rings are finite and field is complex.

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.