Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types

Abstract

Let $\mathfrak{g}$ be a classical simple Lie algebra over an algebraically closed field $\mathbb{F}$ of characteristic zero or large enough, and let $\mathfrak{n}$ be a maximal nilpotent subalgebra of $\mathfrak{g}$. The main tool in representation theory of $\mathfrak{n}$ is the orbit method, which classifies primitive ideals in the universal enveloping algebra $\mathrm{U}(\mathfrak{n})$ and unitary representations of the unipotent group $N=\exp (\mathfrak{n})$ in terms of coadjoint orbits on the dual space ${\mathfrak{n}}^{\ast }$. In the paper, we describe explicitly coadjoint orbits of low dimension for $\mathfrak{n}$ as above. The answer is given in terms of subsets of positive roots. As a corollary, we provide a way to calculate the number of irreducible complex representations of dimensions $q$, $q^2$ and $q^3$ for a maximal unipotent subgroup $N(q)$ in a classical Chevalley group $G(q)$ over a finite field ${\mathbb{F}}_q$ with $q$ elements. It turned out that this number is a polynomial in $q-1$ with nonnegative integer coefficients, which agrees with Isaac's conjecture.