Abstract: I will give a gentle introduction to two classical theorems about algebraic $\mathcal D$-modules due to Kashiwara and Beilinson-Bernstein. Together, these theorems can be used to give a geometric construction of finitely generated modules over enveloping algebras.
I will then explain the notion of rigid analytic quantisation, which leads to well-behaved categories of $\widehat{\mathcal D}$-modules on smooth rigid analytic varieties. An equivariant version of these categories gives an approach to the localisation of admissible locallly analytic representations of semisimple compact $p$-adic Lie groups. A $p$-adic version of Kashiwara's Theorem then provides a geometric construction of some of these representations.
This is work in progress, joint with Simon Wadsley.