Induced Representations of Locally Compact Groups II. The Frobenius Reciprocity Theorem

Abstract

In the study of the relationship between the representation theory of a group and those of its various subgroups an important role is played by Frobenius's notion of induced representation. To every representation L of the subgroup G of the finite group 65 there is assigned a well defined representation UL of ( called the representation of M induced by L. In I of this series [10] the author began a systematic study of a generalization of this notion in which (5 is a separable locally compact topological group and the spaces of L and UL are possibly infinite dimensional Hilbert spaces. In particular [10] contains a generalization of the Frobenius reciprocity theorem; that is the theorem asserting that UL contains the irreducible representation M of (D just as many times as the restriction of M to G contains the irreducible representation L of G. The generalization contained in [10] is unsatisfactory in that it deals only with the discrete finite dimensional irreducible components of the representations concerned and becomes vacuous when these representations decompose in a continuous fashion or have no finite dimensional irreducible components. A more satisfactory generalization has been obtained by Mautner [13, 14]. It deals in a consequent fashion with continuously decomposable representations of 5 but is restricted by the requirement that G be compact. This means in particular that only discretely decomposable representations of G need be dealt with. The principal result of the present article is a generalization of the Frobenius reciprocity theorem which deals effectively with continuously decomposable representations of both M and G. Since in compensation for the hypothesis that G be compact, we require that both G and 5 have regular representations which are of type I, our theorem does not quite include that of Mautner. However we show in addition that whenever the regular representation of G is not only of type I but also discretely decomposable then the requirement that the regular representation of (M be of type I can be eliminated. Thus our methods also yield a result which includes that of Mautner. In neither of our results is it necessary to assume that the groups concerned are unimodular. A noteworthy feature of our principal theorem is that it includes a reciprocity not only for the multiplicities but for the measures involved in the continuous decompositions as well. The basic idea in our approach is a new proof of the Frobenius reciprocity theorem in the finite case which has the advantage of generalizing significantly