On the Group of Infinite p-Adic Matrices with Integer Elements

Abstract

Let G be an infinite-dimensional real classical group containing the complete unitary group (or the complete orthogonal group) as a subgroup. Then G generates a category of double cosets (train), and any unitary representation of G can be canonically extended to the train. We prove a technical lemma on the complete group GL of infinite p-adic matrices with integer coefficients; this lemma implies that the phenomenon of an automatic extension of unitary representations to a train is valid for infinite-dimensional p-adic groups.