How do you compute in characteristic $0$, intersection cohomology of partial flag varieties (corresponding to a fixed partition $\lambda$)? I understand the answer involves Kazhdan-Lusztig polynomials; all I can find is a reference for characteristic $p$ (http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.0207v2.pdf), I'm looking for the paper by Kazhdan & Lusztig: Schubert varieties and Poincare duality, which I cannot find.
I'm specifically trying to compute the intersection cohomology of a subspace of the product of two flag varieties $(V_{i}), (W_{j})$ where the intersections $dim(V_{i} \cap W_{j})$ have fixed dimension. This problem isn't known or studied right? Is there anything to be said about intersection cohomology of homogeneous spaces?
Best Answer
First, let me rephrase your question in a slightly pedantic manner.
To establish some notation, for a point $p$ on the flag variety $G/B$, let $V_1(p)\subset\cdots V_{n-1}(p)$ be the flag in $\mathbb{C}^n$ that it corresponds to. (Be careful. There are no flags actually in the flag variety, just points. Rather, the points in the flag variety correspond to flags. If this confuses you you need a live person to straighten you out.)
You are asking for the intersection cohomology of the subvariety $X\subset G/B \times G/B$ consisting of points $(p,q)$ such that $\dim(V_i(p)\cap V_j(q))=a_{ij}$ (for some specified $a_{ij}$).
Now an answer:
Your variety $X$ has a projection onto the second factor, and this map is a fiber bundle whose base space is smooth (since it is the entire flag variety). Therefore, the local intersection cohomology for the whole space is determined entirely by the local intersection cohomology of the fibers.
If the conditions $a_{ij}$ are conditions that determine a Schubert variety, then the fibers are Schubert varieties, and hence local intersection cohomolgy Betti numbers are precisely given by Kazhdan--Lusztig polynomials.
If the conditions $a_{ij}$ are not conditions determining a Schubert variety, then your fibers will be unions of Schubert varieties. I don't know if anyone has bothered to do this, but I would think that if you take any of the definitions of Kazhdan--Lusztig polynomials $P_{u,v}(q)$ and modify it in the obvious way (if there is one) to allow $v$ to be an arbitrary lower ideal in Bruhat order rather than a principal lower ideal you should get the right thing.