Lie groupoids, the Satake compactification and the tempered dual, I: The Satake groupoid
Jacob Bradd, Nigel Higson, Robert Yuncken
TL;DR
The paper constructs and interrelates three frameworks for the Satake compactification of a real reductive group: a topological coset groupoid in the spirit of Mohsen, a Lie-theoretic model via Oshima’s space and its associated groupoid, and a geometric $b$-groupoid arising from a normal crossing structure. It shows that the Satake compactification embeds naturally into the Oshima space and that the corresponding Satake groupoid is the reduction of the Oshima groupoid; it then identifies the Oshima groupoid with the $b$-groupoid, providing a unified geometric and algebraic setting. This multi-view construction lays the groundwork for a C*-algebraic proof (in a sequel) of Harish-Chandra’s principle, connecting tempered irreducible representations with discrete series and parabolic induction. By detailing the finite orbit stratification indexed by subsets of simple roots and the normal subgroupoid structure, the work furnishes a robust platform for noncommutative geometric analysis of the tempered dual. Overall, the paper establishes a coherent groupoid-based bridge between Satake compactification, Lie theory, and $b$-calculus, crucial for future analysis of the tempered dual and its C*-algebraic realization.
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.