The parabolic compactification of homogeneous spaces

The compactification of non-compact homogeneous spaces (including
symmetric spaces), and their quotients by discrete groups, is an
important theme in group representation theory. Such compactifications
and their generalisations to curved geometries are also crucial in
problems in geometric analysis and physics for problems that are, in
fact, closely related to those arising in the representation theory
setting. A notion of compactification that applies in all these settings
needs to have both a geometric and analytic basis.

The talk will begin reviewing, in a very elementary way, the basic
problem of compactification as well as the motivating
applications. Then some basic examples will be treated as a lead into
the presentation of a new general theory which applies to large
classes of homogeneous manifolds as well as their generalisations. In
the homogeneous case this leads to a decomposition of the
compactification into group orbits and a description of these as the
zero sets of smooth solutions to certain invariant differential
operators. An example such as SL(n,R)/SO(r,q) will be used to illustrate. 

This based on joint work with Andreas Cap.