The decomposition of the
reducible unitary principal series of a connected semisimple Lie group having real
rank one and a simply connected complexification is exhibited on a global analytic
level in such a way that it is seen to correspond to a phenomenon in classical Fourier
analysis. This is done by embedding limits of discrete series representations via a
group equivariant passage to boundary values analogous to the classical Hardy space
inclusion used by Bargmann in the case of SL(2,R). The boundary value map is
shown to be a factor of the projection operator given by the Knapp-Stein
intertwining operator. From a representation theoretic view, while these
decompositions are already known, the method of computing the leading term of the
asymptotic expansion of matrix coefficients is new and does not require a
K-finiteness assumption.
