Does the Poincare series of a Coxeter group always describe a "flag variety"? - MathOverflow

Does the Poincare series of a Coxeter group always describe a "flag variety"?

2010-06-16T18:42:03Z

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"?

Answer by Jim Humphreys for Does the Poincare series of a Coxeter group always describe a "flag variety"?

2010-06-16T19:25:32Z

Coxeter groups form a very large class of groups defined by generators and relations, whose Poincare series are unlikely to have a common geometric interpretation. However, the Poincare series of an affine Weyl group has been important in work of Bott on topology of Lie groups, as well as in work of Iwahori-Matsumoto and others on p-adic groups, etc. References are given in section 8.9 of my 1990 Cambridge Press book Reflection Groups and Coxeter Groups, while the seminal 1965 IHES paper by Iwahori-Matsumoto is available online at www.numdam.org (search for article):

Iwahori, Nagayoshi; Matsumoto, Hideya, On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups. Publications Mathématiques de l'IHÉS, 25 (1965), p. 5-48 Full entry | Full text djvu | pdf | Reviews MR 32 #2486 | Zbl 0228.20015

There are Kac-Moody groups in some generality, with associated "Weyl groups" and "Bruhat decompositions"; but Bruhat cells may have finite dimension in some cases, finite codimension in others. It gets complicated.

It's possible that work of Michael Davis and Ruth Charney on hyperbolic Coxeter groups would be relevant to your question, but I don't know enough about that.

Answer by Peter McNamara for Does the Poincare series of a Coxeter group always describe a "flag variety"?

2010-06-17T00:08:43Z

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 m_ij 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.