Equivalence classes of $\mathrm{GL}(p, \mathbb{C})\times \mathrm{GL}(q, \mathbb{C})$ orbits in the flag variety of $\mathfrak{gl}(p+q, \mathbb{C})$

Abstract

We consider the pair of complex Lie groups

$$\left(G,K\right)=\left(GL\left(p+q,ℂ\right),GL\left(p,ℂ\right)\times GL\left(q,ℂ\right)\right)$$

and the finite set $\left\{\mathcal{Q}:K\text{-orbits on the flag variety }\mathfrak{B}\right\}$. The moment map $\mu$ of the $G$-action on the cotangent bundle $T^{\ast }\mathfrak{B}$ maps each conormal bundle closure $\overline{T_{\mathcal{Q}}^{\ast }\mathfrak{B}}$ onto the closure of a single nilpotent $K$-orbit, ${\mathcal{O}}_K$. We use combinatorial techniques to describe ${\mu }^{-1}\left({\mathcal{O}}_K\right)=\left\{\mathcal{Q}\in \mathfrak{B}:\mu \left(T_{\mathcal{Q}}^{\ast }\mathfrak{B}\right)={\mathcal{O}}_K\right\}$.