On the unitary irreducible representations of the universal covering group of the 3 + 2 deSitter group

ABSTRACT The unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R 2 × R 3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G , called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases. Let ( j , µ) be an irreducible unitary representation of K , where j denotes the (ray) representation of R 3 involved (2 j is a non-negative integer) and µ that of R 2 (µ is a real number). Then in an ‘irreducible case’ representation of G , each ( j , µ) that occurs at all occurs with multiplicity j + ½ if 2 j is odd, or j + ½ ± ½ if 2 j is even. In a ‘reducible case’ representation of G , the multiplicity of the ( j , µ)'s is smaller, and may even be 1 for all ( j , µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R 4 . However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R 4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.) In an irreducible representation of G , only those irreducible representations ( j , µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G . This fractional part of µ helps to characterize the representations of G , as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G . Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.