Unitary representations of SO(n,1)

All unitary representations of SO(n, 1) have been obtained in the group chain SO(n,1)⊃SO(n)⊃SO(n−1)⊃⋯⊃SO(2). The branching laws have been explicitly formulated. These results follow from the observation that the matrix elements of the ``noncompact'' generators of SO(n, 1) differ from the corresponding matrix elements of the same generators of SO(n + 1) by a factor of −1. The branching laws then follow from the unitarity condition. It is also observed that the invariants of SO(n, 1) have the same eigenvalues as the invariants of SO(n + 1). Finally we show that the normalized raising and lowering operators in SO(n + 1), obtained by Pang and Hecht in graphs, and by Wong in algebraic form, can be similarly defined and applied to SO(n, 1).