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The Weyl group $W$ can be realized as a fundamental group of a topological space. So that its group algebra $\mathbb{Z}(W)$ has a geometric meaning. On the other hand, the Hecke algebra is a deformation of $\mathbb{Z}(W)$. Does it have a geometric interpretation?

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    $\begingroup$ The Hecke algebra is a commutative ring. Also, a key step in Wiles's proof of the modularity theorem was to study the relation between universal deformation rings and Hecke algebras. You might want to look into that. Hope this helps some $\endgroup$
    – geocalc33
    Commented Mar 3, 2020 at 2:41

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