## Abstract

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_{p}) is obtained by a direct application of our general formula. As for SO(3,Q_{p}) and SO(4,Q_{p}), 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_{p} 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.

Original language | English |
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Article number | 78 |

Journal | Letters in Mathematical Physics |

Volume | 114 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2024 |

## Keywords

- 11E08
- 11E95
- 11R52
- 22E35
- 28C05
- 28C10
- 81Q65
- Haar measure
- Locally compact group
- Quaternion algebra
- p-adic Lie group