Characterising the Haar measure on the $p$-adic rotation groups via inverse limits of measure spaces
Abstract
We determine the Haar measure on the compact $p$-adic special orthogonal groups of rotations $\mathrm{SO}(d)_p$ in dimension $d=2,3$, by exploiting the machinery of inverse limits of measure spaces, for every prime $p>2$. We characterise $\mathrm{SO}(d)_p$ as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each $\mathrm{SO}(d)_p$. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on $\mathrm{SO}(d)_p$. Our results pave the way towards the study of the irreducible projective unitary representations of the $p$-adic rotation groups, with potential applications to the recently proposed $p$-adic quantum information theory.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2024
- DOI:
- 10.48550/arXiv.2401.14298
- arXiv:
- arXiv:2401.14298
- Bibcode:
- 2024arXiv240114298A
- Keywords:
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- Mathematical Physics;
- Mathematics - Functional Analysis;
- Mathematics - Group Theory;
- Mathematics - Number Theory
- E-Print:
- 42 pages