## Intersection cohomology of flag varieties/Schubert varieties

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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?

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You're interested in the intersection cohomology of Schubert varieties, so you might want to change your title to reflect that (the flag variety is smooth, and its cohomology is well-known).  It would be quite inaccurate to say this problem isn't studied.  In the KL paper you cite, they show that the coefficients of the KL polynomials are related to intersection cohomology of Schubert varieties.  You are correct that the coefficients of the KL polynomials aren't known in general.  This is considered to be an extremely difficult problem to solve (and important due to KL conjectures). – Mike Skirvin Dec 6 2009 at 1:38
I promise a useful answer once you clarify your question... In particular, please distinguish between a flag (or a Schubert) variety and the flags that its points represent. – Alexander Woo Dec 6 2009 at 2:04
Sorry, you're talking to someone whose only heard Schubert varieties in passing before (sorry about that, I'm still a second year undergrad, don't know basics). So from the wikipedia definition, I understand that one example of the Schubert variety is where $dim(V_{i} \int W_{j}) \geq j$. Is the variety I specified also a Schubert variety?  – Vinoth Dec 6 2009 at 9:47
Alexander, my question is the following: Consider the subspace of the product of the two flag varieties: ${( {V_{j}}, {W_{k}}) | dim V_{j} = a_{j}, dim W_{k} = b_{k}, dim ( V_{j} \cap W_{k} ) = a_{jk} )}$ Compute its IC sheaves and dimensions of stalks etc.  And I think I'd considered the problem roughly "solved" if we can reduce it to Kazhdan-Lusztig polynomials, so my question is not to explicitly find the KL coefficients  – Vinoth Dec 6 2009 at 9:57
vinoth: Given your subvariety, we can project to the first flag variety. The fiber over a point will be a certain Schubert cell (because you're requiring equalities of intersection) which is isomorphic to an affine space. So what you're getting is a locally trivial fibration of a flag variety whose fibers are affine spaces. It's not quite a Schubert variety. – Steven Sam Dec 6 2009 at 17:34
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## 1 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.

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thanks, that is really helpful.  One last question: could you give me a reference that proves intersection cohomology of Schubert varieties are KL polynomials? I haven't had success finding one.  – Vinoth Dec 7 2009 at 6:33
The original reference seems to be Kazhdan and Lusztig, Schubert varieties and Poincare duality, in Geometry of the Laplace operator, Proc. Sympos. Pure Math. 34, AMS, Providence, RI 1980, pp. 185--203.  I must confess I've never actually read that, nor the later expositions in Borel's or Kirwan's books, nor ever understood much of the proof, despite writing a couple combinatorial papers on K-L polynomials from the Schubert point of view. – Alexander Woo Dec 7 2009 at 19:52
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