# What is the relation between the representation of a Hecke algebra and the representation of the Coxeter group?

Given a Coxeter group $W$, there is a corresponding Hecke algebra (Iwahori-Hecke algebra). There are many results on the representation of the Hecke algebra. But why is this motivated?

How is the representation of the Hecke algebra related to the representation of the Coxeter group? How is the representation of the Hecke algebra related to the represenation of an algebraic group when the Coxeter group $W$ is the Weyl group of this algebraic group? Or, why are we studying Hecke algebras?

Thanks a lot.

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There is no simple short answer to your question. Connection with rep. of Coxeter group: Let $\mathcal{H}_q(W)$ be the Hecke algebra corresponding to $W$, this is a deformation of the group algebra of $W$, i.e. when $q=1$, we get the group algebrw of $W$. Connection to algebraic group, this comes from connection via corresponding Lie algebra's universal envelop, $U_q(\mathfrak{g})$. In the type A case, this is connected through Schur-Weyl duality. I am not expert in this aspect perhaps someone else can give a more elaborate answer. –  Aaron Jul 23 '12 at 17:40
Kazhdan-Lusztig polynomial is the keyword to look for if you want to some connection between universal enveloping algebra/algebraic group. –  Aaron Jul 23 '12 at 17:47
@Aaron: Thank you very much. –  ShinyaSakai Aug 8 '12 at 11:33