Complete Irreducibility and X-Spherical Representations

Let G be a locally compact group, K a compact subgroup, and χ: K → {|z| = 1} a continuous homomorphism. Let π be a continuous irreducible representation of G on a complete locally convex space V such that the subspace {v ∈ V | π(k)v = χ(k)v, ∀k ∈ K} has dimension 1. Then π is completely irreducible. Furthermore π is Naimark related to a representation on a reflexive Fréchet space which is a closed subspace of C(G) of the form span {L(g)φ | g ∈ G} where φ is a χ-spherical function. A corollary is that irreducible eigenspace representations are completely irreducible.