Let $x$ and $y$ be two elements of $S\_n$. Let $U(x,y)$ be the intersection $(BxB \cap B\_{-} y B)/B$ inside the flag variety. Here $B$ and $B\_{-}$ are the groups of upper and lower-triangular matrices respectively. Kazhdan and Lusztig define the $R$ polynomial $R\_{x,y}(q)$. As explained in Theorem 1.3 of [Deodhar](http://www.ams.org/mathscinet-getitem?mr=782232), one such description is that
$$R\_{x,y}(q) = \# U(x,y)(\mathbb{F}\_q).$$


Now, $U(x,y)$ is an affine variety, so I don't get to know that the weight filtration matches the cohomological filtration. But, looking at examples, it looks like the coefficient of $(-1)^{\ell(x) - \ell(y) - i } q^i$ in $R\_{x,y}(q)$ is the Betti number $\dim H^i(U(x,y))$. For example, with $n=2$, $x=(21)$ and $y$ being the identity, then $U(x,y)$ is the torus $\mathbb{A}^1 \setminus \{ 0 \}$, it has $q-1$ points, and its betti numbers are $\dim H^0 = \dim H^1 = 1$.


This is probably something very standard, but I didn't find it the first few places I tried, and it seemed faster to ask here than to keep looking. Thanks!



Are Kazhdan-Lusztig $R$-polynomials the Poincare polynomials of the corresponding affine varieties
I think that this is one of these things that looks plausible in small examples but is false. For example, this would imply that the coefficients of R polynomials are alternating in $q$. This is implied by another conjecture called the Gabber-Joseph conjecture (roughly: coefficients of R-poynomials give dimensions of Ext groups between Verma modules), which is false. See "A counterexample to the Gabber-Joseph conjecture" by Brian Boe.


