Lie groupoids, the Satake compactification and the tempered dual, I: The Satake groupoid
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917110692708352 |
|---|---|
| author | Bradd, Jacob Higson, Nigel Yuncken, Robert |
| author_facet | Bradd, Jacob Higson, Nigel Yuncken, Robert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22637 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lie groupoids, the Satake compactification and the tempered dual, I: The Satake groupoid Bradd, Jacob Higson, Nigel Yuncken, Robert Representation Theory Operator Algebras 22A22, 22E45 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. |
| title | Lie groupoids, the Satake compactification and the tempered dual, I: The Satake groupoid |
| topic | Representation Theory Operator Algebras 22A22, 22E45 |
| url | https://arxiv.org/abs/2511.22637 |