The investigator and his colleagues apply a new type of integration, called motivic integration, to the study of representations of p-adic groups and their characters. Motivic integration was introduced in 1995 by M. Kontsevich and developed subsequently by J. Denef, F. Loeser, and others. The arithmetic version of this integral takes values in a ring of virtual Chow motives. Many objects that occur naturally in the representation theory of p-adic groups, including characters of representations, orbital integrals, Shalika germs, and Fourier transforms of orbits have conjectural descriptions in terms of points on varieties over finite fields, or more generally as the trace of Frobenius operators on motives. The research of this proposal will make use motivic integration to affirm that many of these objects have geometric descriptions of the conjectured type. For many years, mathematicians have dreamed that some ofthe fundamental objects of study in algebra should have a uniform description. Until recently, it was not possible to carry out this dream, or even to give precise meaning tothe words. However, by combining three different branches of mathematics -- algebra, geometry, and logic -- it now seems possible to bring this dream to fruition. This field of research relies on methods of mathematical logic to give a geometric interpretation of what was previously considered through pure analysis. Specifically, mathematical logic gives a geometric interpretation (called motivic integration) of integral calculus and measure. The research supported by this grant will use this new tool to give a uniform description of some of the fundamental objects in modern algebra, including symmetry through the mathematics of group representations and their characters.
