Local Systems on Classical Nilpotent Orbits and Maximal Length Elements

TL;DR

This paper investigates the relationship between dominant weights and local systems on classical nilpotent orbits, proving two conjectures related to this bijection in the classical case.

Contribution

It proves two conjectures concerning the bijection between dominant weights and local systems on classical nilpotent orbits.

Findings

Confirmed Conjecture 3.1 in  and Conjecture 7.4' in 2 for classical groups.

Established a detailed correspondence between weights and local systems.

Enhanced understanding of the structure of nilpotent orbits in classical Lie groups.

Abstract

It is known that there is a bijection between dominant weights of a complex reductive Lie group $G$ and the set $\mathcal{N}_{\mathcal{O},r}$ whose elements are of the form $(\mathcal{O},\rho)$, where $\mathcal{O}$ is a nilpotent orbit and $\rho$ is an irreducible, algebraic representation of the stabilizer group $G^e$ of an element $e$ in the nilpotent orbit $\mathcal{O}$. We would like to study the above bijection when $G$ is classical and $\rho$ corresponds to a local system of $\mathcal{O}$. In particular, we will prove Conjecture 3.1 in \cite{AS} and Conjecture 7.4' in \cite{Ac2} in the classical setting.