Motivic integration and the regular Shalika germ.

Abstract

Let G denote an adjoint semisimple algebraic group defined over the integers. There is an analogy between Haar integration on the <italic>p</italic>-adic integer valued points of G and motivic integration on the complex power series valued points of G . A result of D. Shelstad from <italic>p</italic>-adic harmonic analysis states that the regular Shalika germ of such a group is an asymptotically constant function. In the present work it is demonstrated that the complex power series analogue of the regular Shalika germ is similarly an asymptotically constant function. There are three main steps involved in this demonstration. The first step is a description of certain affine schemes, together with morphisms from these schemes to the group. This allows the calculation of the complex power series analogue of the regular Shalika germ via motivic integration over such an affine scheme. The second step is the association of a less complicated shadow scheme to each such affine scheme. This association is achieved by analysis of the coordinate rings of the affine schemes. The third step is the use of the transformation rule from motivic integration. This is used to equate a motivic integral on each affine scheme with one on its associated shadow scheme. It is again used to equate the resulting motivic integrals on two different shadow schemes. This exactly means that the complex power series analogue of the regular Shalika germ is an asymptotically constant function.