Compactification of SL(2)

TL;DR

This paper introduces `hd-compactifications' of SL(2) over real or complex numbers, showing how Schwartz spaces behave on these compactifications and exploring their algebraic properties, with implications for broader Lie groups.

Contribution

It develops a new framework for compactifying SL(2) that clarifies the structure of Schwartz spaces and their algebraic properties, potentially extendable to other reductive Lie groups.

Findings

Schwartz and Harish-Chandra Schwartz spaces are conormal functions on the compactification.

Closure under convolution is established using generalized product spaces.

The approach is potentially applicable to all real reductive Lie groups.

Abstract

We discuss `hd-compactifications' of $\SL(2,\bbK)$ for $\bbK=\bbC$ or $\bbR.$ 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.