  In this paper, we obtain explicit branching laws for all irreducible unitary
representations of $\Spin(N,1)$ restricted to a parabolic subgroup $P$. The
restriction turns out to be a finite direct sum of irreducible unitary
representations of $P$. We also verify Duflo's conjecture for the branching law
of tempered representations of $\Spin(N,1)$ with respect to a parabolic
subgroup $P$. That is to show: in the framework of the orbit method, the
branching law of a tempered representation is determined by the behavior of the
moment map from the corresponding coadjoint orbit. A few key tools used in this
work include: Fourier transform, Knapp-Stein intertwining operator,
Casselman-Wallach globalization, Zuckerman translation principle, du Cloux's
results for smooth representations of semi-algebraic groups.
