Joint Seminar Dijon-Freiburg-Strasbourg
Summary : (joint work with D. Patel, T. Schmidt, M. Strauch). Let G be a reductive group, Lie(G) its Lie algebra; X the flag variety of G. In the complex case, Beilinson-Bernstein and Brylinski-Kashiwara proved in the 80's that there is an equivalence of categories between the central representations of Lie(G) and the D-modules over the flag variety X. In this talk I will explain a p-adic analogous of this theorem. In this case G is a split reductive group, and representations we are considering are the central locally analytic representations of the Qp-points of the group G. On the geometric side I will explain how to contruct differential operators over the rigid flag variety of the group G.
I will present a work in collaboration with Wiesia Niziol which aims to incorporate p-adic Hodge theory into the framework of modules over ring spectra, in the sense of Morel-Voevodsky's motivic homotopy theory. Our main result is the identification of "modules over syntomic cohomology" as a full subcategory of the derived category of potentially semi-stable representations, making use of ideas of Beilinson and Drew. I will then present an ongoing project to extend Fontaine semi-stable comparison to a suitable notion of syntomic modules. The later should be compared to Saito mixed Hodge modules, and our objective is to get some kind of p-adic Riemann-Hilbert correspondence.
Abstract: A variant of the formalism of motivic sheaves, where the Tate objects do not extend among one another, can explain the phenomenon of graded versions of categories of representations underlying the character formulas of Kazhdan-Lusztig. This is joint work with Matthias Wendt.