Compactification of Lie Groups

I will discuss compactifications of semisimple Lie groups, in particular SU(n) over the real and complex fields, as manifolds with corners. The left- and right-invariant vector fields correspond to fibrations of the boundary faces and this allows the Harish-Chandra modules to be identified as rather simple function spaces. This is joint work with Pierre Albin and Panagiotis Dimakis.