Appears in collection : Tropical Geometry, Berkovich Spaces, Arithmetic D-Modules and p-adic Local Systems

Let G be a split reductive p-adic group and let P be a parabolic subgroup of G. Let X be the rigid analytic flag variety of G and Y a P -stable closed subset. After a brief review of the theory of equivariant D-modules on rigid analytic spaces, I will discuss the geometric induction functor, due to K. Ardakov, which relates P -equivariant DX -modules M with support Y to G-equivariant DX -modules ind(M ) with support GY . I will then explain how to establish the irreducibility of the induced module ind(M ) for M the (push forward of the) structure sheaf on certain smooth Schubert varieties Y . As an application, we reprove geometrically some irreducibility results for locally analytic G-representations in the image of the Orlik-Strauch functor. This is work in progress with Konstantin Ardakov.