Local Systems on Classical Nilpotent Orbits and Maximal Length Elements

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.