  We discuss the `hd-compactification' of a semi-simple Lie group to a manifold
with corners; it is the real analog of the wonderful compactification of
deConcini and Procesi. There is a 1-1 correspondence between the boundary faces
of the compactification and conjugacy classes of parabolic subgroups with the
boundary face fibering over two copies of the corresponding flag variety with
fiber modeled on the (compactification of the) reductive part. On the
hd-compactification Harish-Chandra's Schwartz space is identified with a space
of conormal functions of rapid-logarithmic decay relative to square-integrable
functions.
