  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.
