Compactification of SL(2)

Abstract

We discuss `hd-compactifications' of $\SL(2,\bbK)$ for $\bbK=\bbC$ or $\mathbb{R}.$ These are compact manifolds with boundary on which both the Schwartz and the Harish-Chandra Schwartz spaces are shown to be relatively standard spaces of conormal functions relative to the boundary. Closure under convolution and other module properties are shown to follow from the structure of appropriate generalized product spaces and the functorial properties of conormal functions and smooth maps between manifolds with corners. It is anticipated that a similar approach applies to general real reductive Lie groups, with the additional complications for $\SL(n,\bbK)$ being essentially combinatorial.