  Based on the Lie theoretical methods of algebraic Fourier transformation, we
classify in the case of generic values of inducing parameters the scalar
singular vectors corresponding to the diagonal branching rules for scalar
generalized Verma modules in the case of orthogonal Lie algebra and its
conformal parabolic subalgebra with commutative nilradical, thereby realizing
the diagonal branching rules in an explicit way. The complicated combinatorial
structure of singular vectors is conveniently determined in terms of recursion
relations for the generalized hypergeometric function ${}_3F_2$. As a
geometrical application, we classify bilinear conformally equivariant
differential operators acting on homogeneous line bundles on the flag manifold
given by conformal sphere $S^n$.
