Abstract
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.
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