On a Frobenius reciprocity theorem for locally compact groups. II (Q1074051)

Part I of the paper under consideration [ibid. 24, 539-555 (1984)] was reviewed in this journal (see Zbl 0579.43013). We refer to it for the notations used in this review. Let G be a locally compact unimodular group, and S a closed subgroup of G. Assume that there exists a compact subgroup K of G with \(G=SK\) (S\(\cap K\) non-necessarily trivial). Fix a topologically irreducible representation \(\pi\) of G on a locally convex Hausdorff topological vector space \({\mathcal H}\). Assume that (\(\pi\),\({\mathcal H})\) contains an equivalence class \(\delta\in \hat K\) finitely many times. Let \(\rho\) be a continuous representation of S on a locally convex Hausdorff space H, with similar properties as (\(\pi\),\({\mathcal H})\). Let Ind \(\rho\) be the representation of G induced by \(\rho\). Call \({\mathcal H}^{\rho}\) the representation space of Ind \(\rho\). In part I the author has introduced the subspaces \({\mathcal H}_ 0\), \({\mathcal H}_ 0^{\rho}\) of \({\mathcal H}\), \({\mathcal H}^{\rho}\), respectively, which play a crucial role in the proof of the following reciprocity theorem: \[ Hom_{C_ c(S)}({\mathcal H}_ 0, H)\simeq Hom_{C_ c(G)}({\mathcal H}_ 0, {\mathcal H}_ 0^{\rho \quad}). \] Part II is concerned with the question whether the relation \[ Hom_{C_ c(S)}(H, {\mathcal H}_ 0)\simeq Hom_{C_ c(G)}({\mathcal H}_ 0^{\rho}, {\mathcal H}\quad_ 0) \] is true or not. H is supposed to be finite-dimensional now. In general there is no isomorphism. If S is unimodular then there is a natural embedding of the l.h.s. into the r.h.s. The author presents an example where this embedding is not an isomorphism.