1. Characterising the Haar measure on the $p$-adic rotation groups via inverse limits of measure spaces
- Author
-
Aniello, Paolo, L'Innocente, Sonia, Mancini, Stefano, Parisi, Vincenzo, Svampa, Ilaria, and Winter, Andreas
- Subjects
Mathematical Physics ,Mathematics - Functional Analysis ,Mathematics - Group Theory ,Mathematics - Number Theory - 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., Comment: 42 pages; to appear in Expositiones Mathematicae
- Published
- 2024