Some applications of motivic integration to the representation theory of p -adic groups.

Abstract

Let <italic>X</italic> be a variety over a field <italic>k</italic>. Motivic integration, introduced by M. Kontsevich in 1995, is a formal procedure that associates virtual Chow motives (or virtual equivalence classes of varieties) with subsets of <italic>X</italic> (<italic>k</italic>[[<italic>t</italic>]]), where <italic>k</italic>[[<italic>t</italic>]] is the ring of formal Taylor series with coefficients in <italic>k</italic>. It is analogous to a measure theory in every way except that the values of the motivic measure are not numbers but formal symbols associated with geometric objects such as Chow motives. In Part I of the thesis, a motivic Haar measure on groups of the form <italic> G</italic>(<italic>k</italic>((<italic>t</italic>))) is defined, where <italic> G</italic> is a reductive algebraic group defined over an algebraically closed field <italic>k</italic> of characteristic zero. The difficulty is contained in extending the original definition of motivic measure from the objects over formal Taylor series to the objects over formal Laurent series. This result is motivated by the hope that it would eventually help to develop a theory for groups of the form <italic>G</italic>(<italic>k</italic>((<italic>t</italic>))) that would be analogous to the representation theory of <italic>p</italic>-adic groups. Part II, which is independent of Part I, uses a different kind of motivic integration that specializes to <italic>p</italic>-adic integration for almost all primes <italic>p</italic>. Let <italic>G</italic> be the group Sp₂<italic>_n</italic> or SO₂<italic>_n</italic>₊₁. Let <italic>F</italic> be a number field. Let o be a logical formula that defines a subset of <italic>G</italic>(<italic>F_v</italic>) for any finite place <italic>v</italic>. In Part II, the arithmetic motivic integration theory, which is due to J. Denef and F. Loeser, is used to deduce the existence of virtual Chow motives such that the trace of <italic> v</italic>-Frobenius action on them gives the values of the distribution characters of a certain class of representations of <italic>G</italic>( <blkbd>Fv</blkbd> ) on the set of regular topologically unipotent elements, for almost all finite places <italic>v</italic> of <italic>F</italic>. A variant of this statement for function fields is proved also. This gives a uniform (independent of <italic>p</italic>) way of thinking of the distribution characters, and illustrates their geometric nature.