Let $W$ be a Coxeter group and let $P\_W(q) = \sum\_{w \in W} q^{\ell(w)}$ be its Poincare series. When $W$ is the Weyl group of a simple algebraic group $G$ (hence $W$ is finite), $P\_W(q)$ is the generating function describing the cells in the Bruhat decomposition of the flag variety $G/B$, $B$ a Borel. 


What happens when $W$ is infinite, e.g. when $W$ is an affine Weyl group or a hyperbolic Coxeter group? Can $P\_W$ still be associated to an infinite-dimensional "flag variety"? 



Does the Poincare series of a Coxeter group always describe a "flag variety"?
I think that Shrawan Kumar's book "Kac-Moody groups, their flag varieties, and representation theory" will contain the flag varieties (which are really ind-varieties in the non-finite case) that you are looking for.


A crystallographic Coxeter group is one of the form $\langle s\_1,\ldots,s\_n| s\_i^2=(s\_i s\_j)^{m\_{ij}}=1\rangle$ where each mij is equal to 2,3,4,6 or infinity. Such Coxeter groups are precisely the Weyl groups associated to arbitrary Kac-Moody algebras. In this case, there is a corresponding Kac-Moody group, together with an associated flag variety and Schubert cells, which seems to me to be what you want.


It is this geometry that is the starting point of the geometric interpretation of Kazhdan-Lusztig polynomials in the crystallographic case. As is to be expected, the finite case is easier than the affine case, which again is easier than the arbitrary KM case.


