Abstract: In this talk we will discuss some geometric objects associated to not necessarily finite-dimensional representations of finite-dimensional complex semi-simple Lie algebras and their relationship with the geometry of $D$-modules on the flag variety. In particular we will explain how to use these geometric ideas to reprove a theorem of S.P. Smith on the possible growth rates of such modules with respect to certain filtrations. If there is time we will discuss generalisations of these results to the study of $p$-adic Banach space representations of $p$-adic semi-simple Lie algebras and their associated pro-$p$ groups.
All the above is joint work with Konstantin Ardakov. (see http://annals.math.princeton.edu/2013/178-2/p03 or http://arxiv.org/abs/1102.2606).