Induced Representations

Abstract

Let G be a locally compact group, Ĝ its unitary dual. A first problem in the Harmonic Analysis on G is to ‘identify’ Ĝ, i.e. to find all possible irreducible unitary representations of G. [Of course, when G is abelian or compact, Ĝ is, in principle at least, ‘known’.] Ideally, one would like to have at hand a systematic method for classifying the elements of Ĝ; although this appears to be too much to hope for in general, nevertheless there is a basic construction by means of which it is possible to write down irreducible unitary representations for large classes of groups — this is the notion of (unitarily) induced representation. For certain classes of groups all the irreducible unitary representations can be obtained as unitarily induced representations, while for other classes this is the case for a large number of the irreducible unitary representations. It turns out that whenever G admits a closed normal subgroup H, there is a procedure (using induced representations) whereby one can classify the elements of Ĝ in terms of those of Ĥ and (G/H)^{^}; naturally this technique for reducing the study of irreducible unitary representations of G to those of successively smaller groups will fail as soon as one reaches groups which are ‘simple’, i.e. have no non-trivial closed normal subgroups (nevertheless, even for ‘simple’ groups, the concept of induced representation is of fundamental importance).