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

TL;DR

This paper explicitly describes low-dimensional coadjoint orbits of maximal nilpotent subalgebras in classical Lie algebras, facilitating the classification of certain irreducible representations and confirming a polynomial counting conjecture.

Contribution

It provides an explicit description of low-dimensional coadjoint orbits for nilradicals of Borel subalgebras in classical types, linking orbit structure to representation counts.

Findings

Explicit formulas for low-dimensional coadjoint orbits

Polynomial count of irreducible representations in finite groups

Confirmation of Isaac's conjecture on representation enumeration

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 ${\rm U}(\mathfrak n)$ and unitary representations of the unipotent group $N=\exp(\mathfrak n)$ in terms of coadjoint orbits on the dual space $\mathfrak n^*$. 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.