Discrete Quantum Gravity: I. Zonal spherical functions of the representationsof the SO(4,R) group with respect to the SU(2) subgroup and their application to the Barrett-Crane model

Starting from the defining transformations of complex matrices for the $SO(4,R)$ group, we construct the fundamental representation and the tensor and spinor representations of the group $SO(4,R)$. Given the commutation relations for the corresponding algebra, the unitary representations of the group in terms of the generalized Euler angles are constructed. The crucial step for the Barrett-Crane model in Quantum Gravity is the description of the amplitude for the quantum 4-simplex that is used in the state sum partition function. We obtain the zonal spherical functions for the construction of the SO(4,R) invariant weight and associate them to the triangular faces of the 4-simplices.