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. 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.