Conjugating representations and related results on semi-simple Lie groups

Abstract

Introduction. In applying symmetry considerations to quantum mechanics, one is often forced to consider representations T of a symmetry group G by means of Hilbert space operators which may be either unitary or conjugate-unitary. Indeed, in some situations there is a well-determined subgroup N of G (necessarily normal and of index 2 in G) such that T, must be unitary for x E N and conjugate-unitary for x E G N. Such a representation T will be called a conjugating representation of G (relative to N). More generally one can consider projective conjugating representations T, in which the homomorphism relation TT, = T, is replaced by Tj, = A(x, y)Txy (the A(x, y) being complex scalars). These also are required in the applications to quantum mechanics. In ?1, by a small modification of Mackey's analysis, we shall classify the equivalence classes of irreducible projective conjugating representations of G in terms of the irreducible projective representations of N (provided N is of Type I). As an immediate application of this, we obtain in ?2 a classification of all the irreducible unitary representations of any Type I group acting in real or quaternionic Hilbert space, in terms of those acting in complex Hilbert space. In working out this classification for a given group G, the essential step is to subdivide the self-conjugate irreducible complex representations T of G into two classes-those of real type and those of quaternionic type-according as the conjugate-linear map setting up the equivalence of T with its conjugate has positive or negative square. Exactly the same results are obtained if, instead of unitary (not necessarily finite-dimensional) representations, we consider finite-dimensional (not necessarily unitary) representations. The remaining sections of this paper concern finite-dimensional representations of semisimple groups. Fix a connected semisimple Lie group G. The family G of all finite-dimensional (not necessarily unitary) irreducible complex representations of G is parametrized by the well-known Cartan-Weyl method of dominant weights. Which of the self-conjugate elements of Gi are of real type and which are of quaternionic type? In ??3 and 4 this question is answered in two steps: First, Theorem 5 reduces the general case to the case that G is compact. It turns out that, if G is not compact, each self-conjugate element T of G gives rise to a self-conjugate element