Some properties of central idempotents of p-adic groups

Let G be a p-adic group. The center 3(G) of the category of smooth representations of G was given a concrete spectral realization by Bernstein in [4]. On another hand, 3(G) embeds canonically as a set of invariant distributions: this is the geometric realization. The natural link between both realizations is Harish-Chandra's Plancherel formula; however, in general, it is not completely explicit. One aim of this paper is to give another formula for the (invariant distribution attached to) idempotents of 3(G). We call it of Plancherel type since ultimately it provides a development of the Dirac measure at 1(G) in terms of spectral data. The first step to this formula is to show that such distributions have support contained in the set of compact elements. The second step is also of independent interest and is concerned with harmonic analysis on the set of compact elements. There is a canonical pairing between the (complexified) K-o of G and the space of invariant distributions with support contained in the compacts. Using the explicit description of K-o (G) x C given in [12] together with Arthur's description of the "elliptic pairings", we can exhibit natural dual bases of both spaces. As an application (this is also the motivation), we study integrality and rationality properties of these idempotents. Through the above duality, the counterpart of the integrality question is essentially the problem of determining what part of the K-o is generated by the compact open subgroups. This problem is a p-adic group analog of a very general question in K-theory. The only case which will be satisfactorily treated in this paper is that of GL(n), via types theory.