On a Frobenius reciprocity theorem for locally compact groups (Q1067151)

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 \(G=S K\) (S\(\cap K\) not necessarily trivial). \(C_ c(G)\) will denote the algebra of all complex-valued continuous functions on G with compact support, endowed with the usual inductive-\(\lim it\) topology. \(C_ c(S)\) will be defined in the same manner. Fix a topologically irreducible representation \(\pi\) of G on a locally convex Hausdorff topological vector space \({\mathcal H}\). We assume that \(\pi (\alpha)=\int_{G}\pi (x) d\alpha (x)\) defines a continuous linear operator on \({\mathcal H}\) for all Radon measures \(\alpha\) on G with compact support. Especially for \(f\in C_ c(G)\) or in \(C_ c(S)\), \(\pi (f)=\int_{G}\pi (x) f(x) dx\) or \(\pi (f)=\int_{S}\pi (x) f(s) d\mu (s)\) are continuous linear operators on \({\mathcal H}\). Here dx is a Haar measure on G, \(d\mu\) (s) a left Haar measure on S. We now impose the condition that (\(\pi\), \({\mathcal H})\) contains an equivalence class \(\delta\) of irreducible representations of K finitely many times. Let \({\mathcal H}(\delta)\) be the subspace of \({\mathcal H}\) consisting of all vectors which transform according to \(\delta\) under \(u\to \pi (u)\) (u\(\in K)\). Put \({\mathcal H}_ 0=\pi (C_ c(G))\cdot {\mathcal H}(\delta)\). Then, fixing \(v\in {\mathcal H}(\delta)\), \(v\neq 0\), \({\mathcal H}_ 0=\{\pi (f) v:\) \(f\in C_ c(G)\}.\) \({\mathcal H}_ 0\) is G-invariant and dense in \({\mathcal H}\). \({\mathcal H}_ 0\) is also a \(C_ c(S)\)-module. 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 \(\rho\). Define \({\mathcal H}_ 0^{\rho}=({\mathcal H}^{\rho})_ 0\) as above (w.r.t. same \(\delta)\). The author proves the following version of the Frobenius reciprocity theorem: \[ Hom_{C_ c(S)}({\mathcal H}_ 0, H)\simeq Hom_{C_ c(G)}({\mathcal H}_ 0, {\mathcal H}_ 0^{\rho \quad}) \] (linearly isomorphic).