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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?

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    $\begingroup$ I think they are called (open) Richardson varieties. Their dimensions are described here: arxiv.org/pdf/1008.3939v2.pdf (page 2). $\endgroup$
    – Piotr Achinger
    Commented Jan 13, 2015 at 22:20
  • $\begingroup$ It is also stated there that their closures are Cohen-Macaulay, so in particular they are equi-dimensional. $\endgroup$
    – Piotr Achinger
    Commented Jan 13, 2015 at 22:23
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    $\begingroup$ @Piotr: Terminology varies, but in the paper you cite (later published in Crelle Journal) they seem to require that the two elements of $W$ involved are related by the Bruhat ordering. Arbitrary intersections must get more complicated to study. Probably the earliest paper in this direction is by Deodhar: ams.org/mathscinet-getitem?mr=782232 $\endgroup$
    – Jim Humphreys
    Commented Jan 13, 2015 at 23:30
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    $\begingroup$ @JimHumphreys The intersection is empty if the elements aren't comparable in Bruhat order. $\endgroup$
    – Ben Webster ♦
    Commented Jan 14, 2015 at 14:24
  • $\begingroup$ @Ben: Yes, that seems right in retrospect. I guess the important point is that the Bruhat ordering comes into play in nontrivial cases, which wasn't expressed in the question itself. $\endgroup$
    – Jim Humphreys
    Commented Jan 14, 2015 at 14:44

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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 .

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  • $\begingroup$ Thanks! Are you referring to this paper of Kleiman 'The Transversality of a General Translate', or another paper? How does this paper immediately imply that Richardson varieties are nice and have dimension $l(w_1) - l(w_2)$? $\endgroup$
    – Qiao
    Commented Jan 15, 2015 at 20:03
  • $\begingroup$ Yes, that's the one. It'll tell you that $S_x \cap g\cdot S_y$ is transverse for general $g$. This intersection changes to something isomorphic if you multiply $g$ on the left by $B_-$, and changes not at all if you multiply it on the right by $B_-$. So we can reduce $g$ from being in the generic Bruhat stratum of $G$ to actually being $w_0$, and still maintain transversality. $\endgroup$
    – Allen Knutson
    Commented Jan 16, 2015 at 6:02

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