The usual Casselman-Shalika formula relates the Weyl character to unramified Whittaker functions on the p-adic points of a reductive group. We will explain how to define Whittaker functions on a p-adic loop group (i.e, the group of p-adic points of an affine Kac-Moody group), present the analogue of the Casselman-Shalika formula in this context, and then sketch its proof.
Based on joint work with A. Braverman, H. Garland, and D. Kazhdan.