Characterising the Haar measure on the p-adic rotation groups via inverse limits of measure spaces

We determine the Haar measure on the compact p-adic special orthogonal groups of rotations 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 the groups 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 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 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.