Exponential motives on the affine Grassmannian

We will start by recalling the general role of motives in geometric representation theory. Then we will introduce categories of exponential motives on partial affine flag varieties, following Gaitsgory's Kirillov model, as a generalization of the category of Iwahori-Whittaker motives. Specializing to the affine Grassmannian, we will use this to construct a motivic refinement of the geometric asselman--Shalika formula, which exists for very general base schemes and coefficients. This is joint work with Robert Cass and Jakob