Characterising the Haar measure on the [formula omitted]-adic rotation groups via inverse limits of measure spaces

Abstract

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 the groups 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.