Lie groupoids, the Satake compactification and the tempered dual, I: The Satake groupoid

TL;DR

This paper introduces the Satake groupoid associated with the Satake compactification of a real reductive group, providing multiple perspectives and laying groundwork for future representation theory applications.

Contribution

It defines and analyzes the Satake groupoid through topological, Lie-theoretic, and geometric approaches, connecting compactification structures with groupoid theory.

Findings

Defined the Satake groupoid as a topological and Lie groupoid.

Connected the Satake groupoid to the geometric $b$-groupoid of the compactification.

Set the stage for a new proof of Harish-Chandra's principle on tempered representations.

Abstract

The (maximal) Satake compactification associated to a real reductive group $G$ is the closure of the symmetric space of all maximal compact subgroups of $G$ within the compact space of all closed subgroups of $G$. We shall present three different views of a groupoid that may be associated to the Satake compactification. To begin, we shall define our Satake groupoid, as we shall call it, as a topological groupoid, and as a special case of a general construction of Omar Mohsen. Then we shall give a Lie-theoretic account of the Satake groupoid, borrowing from work of Toshio Oshima. Finally we shall identify the Satake groupoid with the purely geometric $b$-groupoid of the Satake compactification, using the structure of the compactification as a smooth manifold with corners. In a subsequent paper we shall use the Satake groupoid to present a new proof of Harish-Chandra's principle, that all the tempered irreducible representations of $G$ may be constructed from discrete series representations using parabolic induction.