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fixed some things in my original post.
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Malkoun
Malkoun
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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{C}(W)$$\mathbb{Z}(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$\mathbb{Z}(W)$.)? Does it have a geometric interpretation?

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?

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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Malkoun
Malkoun
  • 5.2k
  • 15
  • 31

Is there a geometric interpretation of the Hecke algebra of a Weyl group?

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?