Do Iwahori-Hecke algebras come from cohomology classes?

Question

Let $W$ be a Coxeter group. The Iwahori-Hecke algebra $H_q(W)$ is a deformation of $k W$.

Question: is there some way to interpret the deformation $H_q(W)$ as a cohomology class? It doesn't seem like $kW$ twisted by a class in $H^2(W,k^{\times})$...

Answer 1

Not in an interesting way. Cohomology classes describe formal deformations (i.e. deformations over an Artinian ring, or more generally complete local rings), and the Iwahori-Hecke algebra is trivial on a formal neighborhood of $q=1$. In fact, the only points where it is not trivial on a formal neighborhood is when $q$ is a root of unity of small order.