Let me start with some background to set the notation before I ask my question.
Let G be a semisimple algebraic group over some algebraically closed field K, and suppose we have fixed a Borel subgroup B and an opposite Borel subgroup B'. Let P be a parabolic subgroup containing B and consider the partial flag variety X = G/P. The closures of the B-orbits on X are the Schubert varieties, and the closures of the B'-orbits on X are the opposite Schubert varieties. Their intersections are Richardson varieties.
As far as I am aware, all of this stuff can be defined over the integers (anyway I'm only interested in the case when G is a symplectic or orthogonal group), so we can ask about how many F_q-rational points these various varieties have. For a Schubert varieties Z I know how to do this: consider the poset of Schubert varieties contained in Z. It's graded, and specializing q into the rank generating function of this poset gives me the number of points. For opposite Schubert varieties it's similar.
My question is what can one say about Richardson varieties? What I want to say is to just do the same thing: the Schubert varieties correspond to lower intervals in the Bruhat order, and the Richardson varieties correspond to arbitrary intervals. So it's tempting to say just take the rank generating function of this interval and specialize at q. Does this work? What if I assume something like G/P is minuscule? If that's not enough, I really only care about the case when the opposite Schubert variety is replaced by the open B' orbit on X.
A related question: do the Richardson varieties have nice cell decompositions like the Schubert varieties?
My feeling is that something like this is relatively easy and known (I just can't find it), or hopelessly complicated. Please inform me that I'm in the first case.
Best Answer
The intersections of opposite Schubert cells have a very nice decomposition into products of tori and affine spaces due to Deodhar which, of course, induces such a decomposition of the Richardson. This decomposition is defined over $\mathbb{Z}$ (actually it works in any building), so it lets you count points, and the strata are combinatorially described by special subwords of a reduced decomposition of one of the words. I recommend reading the paper of Marsh and Rietsch.