Compactification of semi-simple Lie groups

TL;DR

This paper introduces a new compactification of semi-simple Lie groups as manifolds with corners, linking boundary faces to parabolic subgroups and describing Schwartz spaces in this geometric framework.

Contribution

It presents the `hd-compactification' as a real analog of the wonderful compactification, establishing a correspondence with parabolic subgroups and analyzing Schwartz spaces on this structure.

Findings

Boundary faces correspond to conjugacy classes of parabolic subgroups.

The compactification relates to flag varieties and reductive parts.

Harish-Chandra's Schwartz space is characterized as conormal functions with rapid-logarithmic decay.

Abstract

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.