On the frobenius reciprocity theorem for square integrable representations of nonunimodular groups

Abstract

In the unimodular case, the Frobenius reciprocity theorem for irreducible square integrable representations asserts that certain intertwining spaces are canonically isomorphic; the essential analytic point is that square integrability implies the continuity of functions in particular subspaces of $L^2$ spaces on which the group acts and leads to a characterization of these subspaces in terms of reproducing kernels. In the nonunimodular case this is no longer true. There is a canonical isomorphism between proper subspaces of the intertwining spaces, one of which is uniformly dense in the full intertwining space.