Representations of the group of two-diagonal triangular matrices

TL;DR

This paper explores the representation theory of a family of finite groups derived from upper unitriangular matrices, demonstrating the applicability of the orbit method to these groups.

Contribution

It introduces a broad class of finite groups for which the orbit method provides a complete understanding of their irreducible representations.

Findings

The orbit method applies effectively to the defined family of finite groups.

The paper characterizes the irreducible representations of these groups.

It establishes a correspondence between group representations and coadjoint orbits.

Abstract

Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between points of $\widehat G$ and $G$-orbits in $g*$. For many Lie groups it gives the answers to all major problems of representation theory in terms of coadjoint orbits. Formally, the notions and statements of the orbit method make sense when $G$ is infinite-dimensional Lie group, or an algebraic group over a topological field or ring $K$, whose additive group is self dual (e.g., $p$-adic or finite). In this paper, we introduce the big family of finite groups $G_n$, for which the orbit method works perfectly well. Namely, let $N_n(K)$ be the algebraic group of upper unitriangular $(n+1)\times(n+1)$ matrices with entries from $K$, and $F_q$ be the finite field with $q$ elements. We define $G_n$ as the quotient of of the group $N_{n+1}(F_q)$ over its second commutator subgroup.