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It is known that there is a bijection between dominant weights of a complexreductive Lie group $G$ and the set $\mathcal{N}_{\mathcal{O},r}$ whoseelements are of the form $(\mathcal{O},\rho)$, where $\mathcal{O}$ is anilpotent orbit and $\rho$ is an irreducible, algebraic representation of thestabilizer 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 proveConjecture 3.1 in \cite{AS} and Conjecture 7.4' in \cite{Ac2} in the classicalsetting.

Local Systems on Classical Nilpotent Orbits and Maximal Length Elements