  In this paper we consider the analytic continuation of the weighted Bergman
spaces on the Lie ball $$\mathscr{D}=SO(2,n)/S(O(2) \times O(n))$$ and the
corresponding holomorphic unitary (projective) representations of SO(2,n) on
these spaces. These representations are known to be irreducible. Our aim is to
decompose them under the subgroup SO(1,n) which acts as the isometry group of a
totally real submanifold $\mathscr{X}$ of $\mathscr{D}$. We give a proof of a
general decomposition theorem for certain unitary representations of semisimple
Lie groups. In the particular case we are concerned with, we find an explicit
formula for the Plancherel measure of the decomposition as the orthogonalising
measure for certain hypergeometric polynomials. Moreover, we construct an
explicit generalised Fourier transform that plays the role of the intertwining
operator for the decomposition. We prove an inversion formula and a Plancherel
formula for this transform. Finally we construct explicit realisations of the
discrete part appearing in the decomposition and also for the minimal
representation in this family.
