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.