TITLE: Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: 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 .