Hecke Algebras on p-adic Loop Groups

We will explain how to construct certain commutative and non-commutative convolution algebras on p-adic loop groups. In the non-commutative case, this leads to a group theoretical construction of (a slight variant of) Cherednik’s Double Affine Hecke algebra. In the commutative case, the algebra obtained can be identified by an analogue of the Satake isomorphism for loop groups. We will also explain how to obtain an explicit formula for this Satake map, generalizing a classical formula of Macdonald.

Joint work with A. Braverman, H. Garland, and D. Kazhdan.