Harmonic analysis on spannors

The spannor representation of the universal covering group of the conformal group of Minkowski space-time is studied herein. It is an example of an indecomposable representation. Various parallelizations for spanners are described. Normalized K̃-finite basis fields for the spannors are introduced, and a computation of the actions of the scale generator and other generators of the conformal group on these basis fields is given. The irreducibility of all irreducible composition factors is established, and it is decided which ones are infinitesimally unitary. Two sets of generators for maximal Abelian subalgebras of the enveloping algebra of the conformal group are introduced. The action of one of the sets on K̃-finite basis fields in the spannor representation is studied; this is important for understanding of the problem of degeneracy in the spannor representation, since K̃ types occur with multiplicity 2. Many calculations make use of the lowest (highest) weight module structures of the unitarizable and positive (negative) energy, irreducible composition factors; and, thus, many of our results and techniques can be used to initiate a study of the quantum deformations of the associated Lie algebra representations. Results on the Poincaré content of the unitarizable, irreducible composition factors are stated, and unitary equivalences between certain of these factors are established.