Invariant measures on p-adic Lie groups
We provide a general expression of the Haar measure-that is, the essentially unique translation-invariant measure-on a p-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group...
| Autores: | , , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:293993 |
| Acceso en línea: | https://ddd.uab.cat/record/293993 https://dx.doi.org/urn:doi:10.1007/s11005-024-01826-8 |
| Access Level: | acceso abierto |
| Palabra clave: | Locally compact group Haar measure P-adic Lie group Quaternion algebra |
| Sumario: | We provide a general expression of the Haar measure-that is, the essentially unique translation-invariant measure-on a p-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the p-adic special orthogonal groups in dimension two, three and four (for every prime number p). In particular, the Haar measure on SO(2,Q) is obtained by a direct application of our general formula. As for SO(3,Q) and SO(4,Q), instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain p-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field Q and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the p-adic special orthogonal groups, with potential applications in p-adic quantum mechanics and in the recently proposed p-adic quantum information theory. |
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