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.
Categories: math.RT