Square-integrability modulo a Subgroup

In the present paper we give a self-contained proof of Imprimitivity theorem for systems of covariance, or generalised imprimitivity systems, based on transitive spaces. The theorem holds for locally compact groups and non-normalised positive operator valued (POV) measures. For projective valued measures, the theorem was proven by Mackey, [21], for separable groups, and by Blattner, [4], in full generality, and it is known as Mackey Imprimitivity theorem. For normalised POV measures, there are many independent proofs. Up to our knowledge, Poulsen, [27], first proves it for Lie groups using elliptic regularity theory, Davies, [9], and Scutaru, [31], for topological groups, but with some unnecessary assumption, Neumann, [24], and Cattaneo, [7], for locally compact groups. These last proofs are based on Neumark dilation theorem in order to reduce the problem to the projective case, and on Mackey imprimitivity theorem. Finally, Castrigiano and Henrichs, [8], show the above result using the theory of positive functions on a C∗-algebra. Our proof is independent both on Neumark dilation and on Mackey Imprimitivity theorems, which are corollaries of the main result. It is based on the proof of Mackey theorem given by Orsted, [25], as suggested by a remark in [8] (compare also with [11, Ch. XXII, Sec. 3, Ex. 10]). In particular, we use a realisation of the induced representation inspired by an exercise of [11, Ch. XXII, Sec. 3, Ex. 10]. Our construction is a variation of the one given by Blattner, [4], and, in our opinion, is very elementary and intrinsic, it does not use the notion of quasi-invariant measure and the Hilbert space where the representation acts is a space of square-integrable functions, compare with Folland, [13, Ch. 6]. As a consequence of this approach, one has a weak characterisation of the space of the intertwining operators of the induced representation. If the group is compact, this result reduces to the Frobenius reciprocity theorem,