Beilinson–Bernstein localization via wonderful asymptotics

We explain how a doubled version of the Beilinson–Bernstein localization functor can be understood using the geometry of the wonderful compactification of a group, as well as the associated Vinberg semigroup. Specifically, bimodules for the Lie algebra give rise to monodromic D-modules on the horocycle space, and to filtered D-modules on the group that respect a certain matrix coefficients filtration. These two categories of D-modules are related via an associated graded construction in a way compatible with localization, Verdier specialization, and additional structures. This is joint work with David Ben-Zvi and David Nadler.