Desingularizations of Schubert Varieties in Double Grassmannians

Let $X=\operatorname{Gr}(k,V)\times\operatorname{Gr}(l,V)$ be the direct product of two Grassmann varieties of $k$- and $l$-planes in a finite-dimensional vector space $V$, and let $B\subset\operatorname{GL}(V)$ be the isotropy group of a complete flag in $V$. We consider $B$-orbits in $X$, which are an analog of Schubert cells in Grassmannians. We describe this set of orbits combinatorially and construct desingularizations for the closures of these orbits, similar to the Bott–Samelson desingularizations for Schubert varieties.