In this paper we study the analytic realization of the discrete series representations for the group $G=Sp(1,1)$ as a subspace of the space of square integrable sections in a homogeneous vector bundle over the symmetric space $G/K:=Sp(1,1) /(Sp(1) \times Sp(1))$. We use the Szegő map to give expressions for the restrictions of the $K$-types occurring in the representation spaces to the submanifold $AK/K$.
