Let $G = SL\_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel. 


Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S\_{w\_1}$ and $S^-\_{w\_2}$ denote the $B$ and $B^-$ orbit corresponding to $w\_1, w\_2 \in W$ respectively. 


So how much is known about the intersections of $B$ and $B^-$ orbits $S\_{w\_1} \cap S^-\_{w\_2}$ in the flag variety $G/B$? Are these intersections affine? Are they equi-dimensional? What are their dimensions?



Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Everything good happens: they are smooth, irreducible, affine, of the expected dimension $\ell(w\_1)-\ell(w\_2)$; the standard reference is Kleiman 1973. Even their closures, "Richardson varieties", are nice (normal, C-M, rational singularities), which one can blame on similar results for Schubert varieties: nearby any T-fixed point a Richardson variety is locally isomorphic to a product of two Schubert varieties (up to factoring out a vector space), due to Knutson-Woo-Yong in <http://arxiv.org/abs/1209.4146> .


