Cominuscule property of nilpotent orbits

Question

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.

Answer 1

Would "cominuscule Richardson orbits" be a satisfactory name ? It is basically a stringing together of adjectives that imply the two properties that were outlined in the question. A similar name (w/o the "co-") has also appeared in the literature : Mark Reeder, "Small representations and minuscule Richardson orbits" . Side remark : Actually, Reeder is using minuscule property on G and the Richardson property on Langlands dual of G. So, I think "co-minuscule Richardson" would actually suit his setup better!