Orthogonal subsets of root systems and the orbit method

TL;DR

This paper investigates the properties of coadjoint orbits associated with orthogonal subsets of root systems in Chevalley groups over algebraically closed finite fields, revealing invariance and bounds related to the Weyl group.

Contribution

It establishes that the dimension of certain coadjoint orbits is independent of scalar choices and provides an upper bound based on Weyl group data.

Findings

Orbit dimension is independent of scalar set $\xi$.

Provides an explicit upper bound for orbit dimensions.

Connects orbit dimensions to Weyl group properties.

Abstract

Let $k$ be the algebraic closure of a finite field, $G$ a Chevalley group over $k$, $U$ the maximal unipotent subgroup of $G$. To each orthogonal subset $D$ of the root system of the group $G$ and each set $\xi$ of $|D|$ non-zero scalars from $k$ one can assign the coadjoint orbit of the group $U$. We prove that the dimension of such an orbit does not depend on $\xi$. We also give an upper bound of the dimension in terms of the Weyl group.