The project has two main aspects. Firstly, to investigate the geometry of certain rigid-analytic moduli spaces by analyzing the representations of p-adic groups furnished by the l-adic etale cohomology groups of these spaces. Secondly, we aim at understanding the representation theory of reductive p-adic groups by making use of Rigid-Analytic Geometry.
1) The first aspect concerns the moduli spaces of p-divisible groups and their covering spaces, as defined by M. Rapoport and Th. Zink. A conjecture of R. Kottwitz predicts that the cohomology groups in question realize local Langlands correspondences. We develop and extend methods to show that the l-adic cohomology realizes correspondences of Jacquet-Langlands type, at least on the level of characters. We treat in detail some cases in which the phenomenon of endoscopy plays a significant role.
2) The second aspect is about the theory of locally analytic representations of reductive p-adic groups. Here we aim at developing a localization theory which unifies simultaneously the localization theory of A. Beilinson and J. Bernstein for Lie algebra representations, and the localization theory of P. Schneider and U. Stuhler for smooth representations. The theory of Rigid-Analytic Spaces provides a framework to relate these two theories.
The project's main theme is that of symmetries, families of symmetries (i.e., groups), about relating different kinds of symmetries, and studying symmetries in terms of the geometry of spaces (and vice versa). This is a classic approach, but is done for a class of symmetries and spaces that, so far, have not been well understood. The projects aim at unifying and better understanding theories that have been considered as separate branches so far. This is the case for the so-called smooth representation theory, and the representation theory of Lie algebras. The results will also help to understand the geometry of moduli spaces which are defined by polynomial equations with integral coefficients, e.g., integral models of Shimura varieties, and as such, are expected to deepen our knowledge about these fundamental objects in Geometry and Number Theory.
