On the exotic Grassmannian and its nilpotent variety

Given a decomposition of a vector space $V=V_1\oplus V_2$, the direct product $\mathfrak{X}$ of the projective space $\mathbb{P}(V_1)$ with a Grassmann variety $\mathrm{Gr}_k(V)$ can be viewed as a double flag variety for the symmetric pair $(G,K)=(\mathrm{GL}(V),\mathrm{GL}(V_1)\times\mathrm{GL}(V_2))$. Relying on the conormal variety for the action of $K$ on $\mathfrak{X}$, we show a geometric correspondence between the $K$-orbits of $\mathfrak{X}$ and the $K$-orbits of some appropriate exotic nilpotent cone. We also give a combinatorial interpretation of this correspondence in some special cases. Our construction is inspired by a classical result of Steinberg and by the recent work of Henderson and Trapa for the symmetric pair $(\mathrm{GL}(V),\mathrm{Sp}(V))$.