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7
votes
0answers
227 views

Do p-compact groups have a sufficiently good notion of “flag variety” and “intersection cohomology”?

This is mostly an idle question, since I don't think I'd be able to do anything with a positive answer. But a positive answer would still be interesting, I think. Background An outstanding problem ...
13
votes
2answers
1k views

Is Soergel's proof of Kazhdan-Lusztig positivity for Weyl groups independent of other proofs?

Let $(W, S)$ be a Coxeter system. Soergel defined a category of bimodules $B$ over a polynomial ring whose split Grothendieck group is isomorphic to the Hecke algebra $H$ of $W$. Conjecturally, the ...
12
votes
2answers
620 views

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

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 ...
1
vote
0answers
480 views

A question on a Davis complex of a Coxeter group

Let us have a look at p. 64 of M. Davis book "The Geometry and Topology of Coxeter Groups". The discussion preceeding Definition 5.1.3. shows that $\mathcal{U}(G, X)/G$ is homeomorphic to $X$. Theorem ...
6
votes
1answer
197 views

Reference for representation of Weyl group using r_alpha + c partial_alpha

Take $W = S_n$ for simplicity, though other Weyl groups work too. Let $r_i$ denote the $i$th simple reflection acting on ${\mathbb A}^n$, and $\partial_i = 1/(x_i - x_{i+1}) (Id - r_i)$ denote the ...
12
votes
1answer
446 views

Coxeter Arrangements and an Identity

Let $\{A_i\}$ be a collection of $m$ hyperplanes in $\mathbb{C}^n$ which all pass through the origin (a central hyperplane arrangement). Such an arrangement is called Coxeter if reflecting across any ...
26
votes
5answers
4k views

Definitions of Hecke algebras

There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are ...