Characterising the Haar measure on the -adic rotation groups via inverse limits of measure spaces
P Aniello, S L'Innocente, S Mancini, V Parisi… - arXiv preprint arXiv …, 2024 - arxiv.org
arXiv preprint arXiv:2401.14298, 2024•arxiv.org
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 …
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 …
We determine the Haar measure on the compact -adic special orthogonal groups of rotations in dimension , by exploiting the machinery of inverse limits of measure spaces, for every prime . We characterise 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 . 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 . Our results pave the way towards the study of the irreducible projective unitary representations of the -adic rotation groups, with potential applications to the recently proposed -adic quantum information theory.
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