Let $G$ be a complex reductive Lie group, $G/P$ a flag manifold, and $\Phi: T^* G/P \to {\mathfrak g}^*$ the moment map. So $\Phi(T^* G/P)$ is the closure of a nilpotent orbit.

Lots of classes of nilpotent orbits have special names. Is there a name for those nilpotent orbits arising as $\Phi(T^* G/P)$, where $G/P$ is cominuscule (also known as, a compact Hermitian symmetric space)?

I'm trying to generalize some type $A$ examples, where every $G/P$ is cominuscule for $P$ maximal, but still one gets very few nilpotent orbits this way.