Topological Frobenius Reciprocity for Representations of Nilpotent Groups and Motion Groups

Let G be a locally compact group and H a closed subgroup of G, and let pi and tau be irreducible representations of G and H, respectively. If G is compact then, by the classical Frobenius reciprocity theorem, pi is contained in the induced representation ind(H)(G) tau if and only if pi/H contains tau. Topological Frobenius properties, which a general locally compact group may or may not satisfy, are obtained by replacing containment by weak containment of representations. We investigate the 'if' and the 'only if' assertions for nilpotent locally compact groups and for motion groups.