Representations of the group of two-diagonal triangular matrices

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.