A Theorem on Unitary Representations of Semisimple Lie Groups

Abstract

We show that a connected semisimple Lie group G none of whose simple constituents is compact (in particular, any connected complex semisimple group) has no nontrivial measurable unitary representations into a finite factor,-i. e. a factor of type In(n oo ) or II , in the terminology of [3]. This has been known for the case of representations of complex groups into factors of type In X but the existing proofs are not applicable either to real groups or to factors of type II,, and the present proof is therefore necessarily of a different character from the proof for the complex, finite-dimensional case. Our theorem has the relevant consequence that in the reduction of the regular representation of G into factors (see [9] and [5]), those of type In or II, cannot occur. This is in marked contrast with the situations for compact and discrete groups, only I.'s occurring in the compact case (as is well-known) and only In's and II's in the discrete case (loc. cit.). In order to clarify the statement of our theorem, we make the following definitions. A representation U of a locally compact group G by unitary operators on a Hilbert space X3C (of arbitrary dimension) is called measurable if the inner product (U(a)x, y) is a measurable function of a e G, relative to Haar measure, for all x and y in X. (When X is separable, such a representation is necessarily continuous in the strong operator topology, as follows from a modification of the proof by the second-named author of a special case of this result; details, as well as a more precise result, are given below.) U is said to be into a factor iFif Tiis a factor of which U(a) is an element, for all a E G. Now we state our central result. THEOREM 1. Let G be a connected semisimple Lie group none of whose simple constituents is compact. Then the only measurable unitary representation of G into a finite factor is the identity representation. The following proof applies also to the case of any weakly continuous representation of G into an algebra of operators on a Hilbert space, on which a weakly continuous trace is defined. To outline briefly the general plan of the proof, the non-compactness of the simple constituents of G is used to show that G must contain one of a certain class of 2and 3-dimensional solvable Lie groups. A special study of the unitary representations into finite factors of the members of this class of groups shows that such representations must be trivial on certain subgroups. The proof is concluded by showing that G is generated by such subgroups. An analog of the following lemma is valid for arbitrary topological groups in the large, and can be proved in the same way. A trivial modification of the proof shows also that the same conclusion is valid if G is a cross product of groups A