The Weyl group $W$ can be realized as a fundamental group of a topological space. So that its group algebra $\mathbb{C}(W)$ has a geometric meaning. On the other hand, the Hecke algebra is a deformation of $\mathbb{C}(W)$. Does that correspond to maybe (naively!) some kind of deformation of the Weyl group in some sense (or some more sophisticated idea, like a $1$-parameter family of flat connections, etc.)? Does it have a geometric interpretation?