The spherical Hecke algebra and motivic integration

In this talk I will explain the main ideas in our proof of the Langlands- Shelstad fundamental lemma for the spherical Hecke algebra for unramified p-adic reductive groups in large positive characteristic. The proof is based on the transfer principle for constructible motivic functions. As an important and innovative step, we encode the entire spherical Hecke algebra into a single constructible function. I will introduce the main objects in both, the theory of reductive p-adic groups and motivic integration. This is joint work with W. Casselman and T. Hales.