Induced Representations of Locally Compact Groups I

Abstract

In the theory of representations of a finite group by linear transformations the closely related notions of imprimitivity and of play a prominent role. In [18] the author has generalized these notions to the case in which the group is a separable locally compact topological group and the linear transformations are unitary transformations in Hilbert space. It turns out (and this is the principle theorem of [18]) that the classical theorem of Frobenius according to which every imprimitive representation of a finite group is in a certain canonical fashion by a representation of a subgroup may be reformulated so as to remain true under the more general circumstances indicated above. This connection between representations of groups and representations of their subgroups has many interesting and useful properties in the finite case and it naturally occurs to one to study the extent to which these properties persist in general. The present paper is the first of a projected series in which it is planned to investigate this question in a systematic manner. Our principal results, formulated as Theorems 12.1, 12.2 and 13.1, are closely related; each being essentially a corollary of its predecessor. The first asserts that if L is a representation of the closed subgroup G1 of (M and UL is the corresponding induced representation of (M then the restriction of UL to the closed subgroup G2 is a sum over the G1: G2 double cosets of certain induced representations o G 2 . The second gives a similar decomposition of the Kronecker product UL f? UM where L and M are representations of G1 and G2 respectively. The third provides a usable formula for computing the strong intertwining numbers of the induced representations UL and UM of (M. The in question are ordinary discrete sums only when there are at most countably many non trivial double cosets. In general direct integrals as defined by von Neumann in [26] must be used and we must restrict ourselves to the case in which the relevant double coset decomposition of (M is measurable. As we have shown in detail elsewhere [20] these theorems for finite groups imply certain classical results; in particular the Frobenius reciprocity theorem and the Shoda criteria for the irreducibility and unitary equivalence of monomial representations. Our theorems yield generalizations of these results but these generalizations may be regarded as satisfactory only insofar as they deal with representations whose irreducible constituents are discrete and finite dimensional. More far reaching generalizations must be sought in other directions. We