Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups

TL;DR

This paper studies the structure of coadjoint orbits of unipotent groups associated with classical root systems, constructing polarizations and determining orbit dimensions, which helps classify irreducible representations over finite fields.

Contribution

It introduces a method to construct polarizations of Lie algebras at canonical forms of coadjoint orbits and computes their dimensions using Weyl groups, linking orbit geometry to representation theory.

Findings

Constructed polarizations at canonical forms of coadjoint orbits.

Derived formulas for orbit dimensions in terms of Weyl group data.

Classified possible dimensions of irreducible representations over finite fields.

Abstract

Let $\Phi$ be a classical root system and $k$ be a field of sufficiently large characteristic. Let $G$ be the classical group over $k$ with the root system $\Phi$, $U$ be its maximal unipotent subgroup and $\mathfrak{u}$ be the Lie algebra of $U$. Let $D$ be an orthogonal subset of $\Phi$ and $\Omega$ be a coadjoint orbit of $U$ associated with $D$. We construct a polarization of $\mathfrak{u}$ at the canonical form on $\Omega$. We also find the dimension of $\Omega$ in terms of the Weyl group of $\Phi$. As a corollary, we determine all possible dimensions of irreducible complex represenations of the group $U$ for the case of finite field $k$.