Description of unitary representations of the group of infinite 𝑝-adic integer matrices

Abstract

We classify irreducible unitary representations of the group of all infinite matrices over app-adic field (p≠2p\ne 2) with integer elements equipped with a natural topology. Any irreducible representation passes through a groupGLGLof infinite matrices over a residue ring modulopkp^k. Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.