The Reduction O(3,1)⊃O(2,1)⊃O(1,1)

We set up a representation of the principal series of the group SL(2, C) in terms of the sequence of noncompact subgroups $\text{SL(2,C)⊃SU(1,1)⊃O(1,1)}$ . The basis functions are just the cross‐basis matrix elements of SU(1, 1) between O(2) and O(1, 1) bases, and we derive much material on these before using them to calculate the representation functions. The latter have two pairs of discrete labels distinguishing between equivalent representations occurring in the reduction, and these are related to two discrete reflection operators that are introduced in order to obtain a maximal Abelian set.