The type I dichotomy for two-step nilpotent locally compact groups
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909816674320384 |
|---|---|
| author | Caprace, Pierre-Emmanuel Carter, Max |
| author_facet | Caprace, Pierre-Emmanuel Carter, Max |
| contents | We address the type I dichotomy for two-step nilpotent locally compact groups. Invoking work of Baggett-Kleppner, we characterize the closed points of the unitary dual of such a group $G$ purely in terms of the group structure. An algebraic criterion characterizing when $G$ is a type I group is derived. We show that this criterion automatically holds if $G$ is a central extension of vector groups over a non-discrete locally compact field $k$ such that the commutator map is $k$-bilinear. As an application, we show that the unipotent radicals of minimal parabolics in simple algebraic groups of $k$-rank one are type I groups. We also discuss the type I dichotomy for $p$-torsion contraction groups, and exhibit, for each prime $p$, uncountably many pairwise non-isomorphic such groups that are not type I. This answers a recently posed question by the second author. Finally, we adapt a recent construction of Chirvasitu to obtain numerous examples of two-step nilpotent torsion locally compact groups that are not type I, but that embed as closed cocompact normal subgroups in two-step nilpotent groups that are type I. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The type I dichotomy for two-step nilpotent locally compact groups Caprace, Pierre-Emmanuel Carter, Max Representation Theory Group Theory Operator Algebras Primary: 22D10, Secondary: 20G05, 22D25, 43A65 We address the type I dichotomy for two-step nilpotent locally compact groups. Invoking work of Baggett-Kleppner, we characterize the closed points of the unitary dual of such a group $G$ purely in terms of the group structure. An algebraic criterion characterizing when $G$ is a type I group is derived. We show that this criterion automatically holds if $G$ is a central extension of vector groups over a non-discrete locally compact field $k$ such that the commutator map is $k$-bilinear. As an application, we show that the unipotent radicals of minimal parabolics in simple algebraic groups of $k$-rank one are type I groups. We also discuss the type I dichotomy for $p$-torsion contraction groups, and exhibit, for each prime $p$, uncountably many pairwise non-isomorphic such groups that are not type I. This answers a recently posed question by the second author. Finally, we adapt a recent construction of Chirvasitu to obtain numerous examples of two-step nilpotent torsion locally compact groups that are not type I, but that embed as closed cocompact normal subgroups in two-step nilpotent groups that are type I. |
| title | The type I dichotomy for two-step nilpotent locally compact groups |
| topic | Representation Theory Group Theory Operator Algebras Primary: 22D10, Secondary: 20G05, 22D25, 43A65 |
| url | https://arxiv.org/abs/2509.26212 |