Shimura Varieties and the Bernstein Center

The local Langlands conjecture (LLC) seeks to parametrize irreducible smooth representations of a p-adic group G in terms of Weil-Deligne parameters. Bernstein's theory describes the category of smooth representations of G in terms of points on a (disconnected) complex algebraic variety; the ring of regular functions is called the Bernstein center of G . I will review these theories and explain how the LLC enables one to define many interesting functions in the "stable" Bernstein center of G . I will explain how these functions play a role as test functions in the Langlands-Kottwitz approach to the study of Shimura varieties modulo p.