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Here is just one example (I know there are others too):

Just as representations of the Hecke algebra associated to a Weyl group correspond to representations of a finite group of Lie type which are induced from a Borel subgroup, representations of a semisimple p-adic group which occur in its unramified principal series correspond to representations of its Iwahori-Hecke algebra, which is a q-deformation of an affine Weyl group (see Supercuspidal with Iwahori fixed vectorSupercuspidal with Iwahori fixed vector).

Here is just one example (I know there are others too):

Just as representations of the Hecke algebra associated to a Weyl group correspond to representations of a finite group of Lie type which are induced from a Borel subgroup, representations of a semisimple p-adic group which occur in its unramified principal series correspond to representations of its Iwahori-Hecke algebra, which is a q-deformation of an affine Weyl group (see Supercuspidal with Iwahori fixed vector).

Here is just one example (I know there are others too):

Just as representations of the Hecke algebra associated to a Weyl group correspond to representations of a finite group of Lie type which are induced from a Borel subgroup, representations of a semisimple p-adic group which occur in its unramified principal series correspond to representations of its Iwahori-Hecke algebra, which is a q-deformation of an affine Weyl group (see Supercuspidal with Iwahori fixed vector).

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Amritanshu Prasad
Amritanshu Prasad
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Here is just one example (I know there are others too):

Just as representations of the Hecke algebra associated to a Weyl group correspond to representations of a finite group of Lie type which are induced from a Borel subgroup, representations of a semisimple p-adic group which occur in its unramified principal series correspond to representations of its Iwahori-Hecke algebra, which is a q-deformation of an affine Weyl group (see Supercuspidal with Iwahori fixed vector).