Nilpotent Invariants for Generic Discrete Series of Real Groups
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915963308343296 |
|---|---|
| author | Adams, Jeffrey Afgoustidis, Alexandre |
| author_facet | Adams, Jeffrey Afgoustidis, Alexandre |
| contents | Let $G(\mathbb{R})$ be a real reductive group. Suppose $π$ is an irreducible representation of $G(\mathbb{R})$ having a Whittaker model, and consider three invariants of $π$ related to nilpotents elements of the Lie algebra of $G$ (or its dual): the associated variety, the wave-front set, and the set of Whittaker data for which $π$ has a Whittaker model. If $π$ is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related results, including an elementary proof that passage from $π$ to the three invariants defines natural bijections between the generic discrete series in an $L$-packet, the possible Whittaker data for $G(\mathbb{R})$, and the appropriate sets of nilpotent orbits. Given one of the three invariants, we also explain how to reconstruct the other two. Many of the results were known: we give simplified proofs for several of them, for instance a simple proof (for generic discrete series) that the associated variety and the wave-front set are related by the Kostant-Sekiguchi correspondence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nilpotent Invariants for Generic Discrete Series of Real Groups Adams, Jeffrey Afgoustidis, Alexandre Representation Theory 22E50 Let $G(\mathbb{R})$ be a real reductive group. Suppose $π$ is an irreducible representation of $G(\mathbb{R})$ having a Whittaker model, and consider three invariants of $π$ related to nilpotents elements of the Lie algebra of $G$ (or its dual): the associated variety, the wave-front set, and the set of Whittaker data for which $π$ has a Whittaker model. If $π$ is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related results, including an elementary proof that passage from $π$ to the three invariants defines natural bijections between the generic discrete series in an $L$-packet, the possible Whittaker data for $G(\mathbb{R})$, and the appropriate sets of nilpotent orbits. Given one of the three invariants, we also explain how to reconstruct the other two. Many of the results were known: we give simplified proofs for several of them, for instance a simple proof (for generic discrete series) that the associated variety and the wave-front set are related by the Kostant-Sekiguchi correspondence. |
| title | Nilpotent Invariants for Generic Discrete Series of Real Groups |
| topic | Representation Theory 22E50 |
| url | https://arxiv.org/abs/2410.04134 |