Complete irreducibility and \(\chi\)-spherical representations (Q2367748)

Let \(G\) be a locally compact group, \(K\) a compact subgroup and \(\chi:K \to \{| z |=1\}\) be a continuous homomorphism. Let \(\pi\) be a continuous representation of \(G\) on a complete locally convex space \(V\) such that the subspace \(V(\chi)=\{v \in V \mid \pi(k) v=\chi(k)v,\;k \in K\}\) is one-dimensional. The author shows that \(\pi\) is then necessarily completely irreducible. Furthermore for \(v_ 0\neq v\) in \(V(\chi)\), for some special \(\lambda_ 0 \in V'\) the matrix coefficient map \((\Theta v)(g)=\langle \lambda_ 0,\pi(g^{-1})v \rangle\) is the unique operator intertwining \(\pi\) and the left regular representation \(L\) of \(G\) on \({\mathcal C}(G)\) and the function \(\Phi=\Theta v_ 0\) is \(\overline\chi\)- spherical. Furthermore the closure \(E_ \Phi\) of \(\{L(g) \Phi \mid g \in G\}\) in \({\mathcal C} (G)\) for any \(\overline\chi\)-spherical function \(\Phi\) is a reflexive Fréchet space and \(L_{| E_ \Phi}\) together with \((L_{| E_ \Phi})^ t\) are completely irreducible. As a corollary, consider a connected Lie group \(G\) and the algebra \({\mathcal D}(G/K)\) of \(G\)-invariant differential operators on \(G/K\). Let \(\mu:{\mathcal D} (G/K) \to \mathbb{C}\) be a homomorphism and let \({\mathcal E}_ \mu\) denote the corresponding joint eigenspace. If the left regular representation of \(G\) is irreducible, then it is of class one and completely irreducible.