  We study the homogeneous interacting hole gas in $p$-doped bulk III-V
semiconductors. The structure of the valence band is modelled by Luttinger's
Hamiltonian in the spherical approximation, giving rise to heavy and light hole
dispersion branches, and the Coulomb repulsion is taken into account via a
self-consistent Hartree-Fock treatment. As a nontrivial feature of the model,
the self-consistent solutions of the Hartree-Fock equations can be found in an
almost purely analytical fashion, which is not the case for other types of
effective spin-orbit coupling terms. In particular, the Coulomb interaction
renormalizes the Fermi wave numbers for heavy and light holes. As a
consequence, the ground state energy found in the self-consistent Hartree-Fock
approach and the result from lowest-order perturbation theory do not agree. We
discuss the consequences of our observations for ferromagnetic semiconductors,
and for the possible observation of the spin-Hall effect in bulk $p$-doped
semiconductors. Finally, we also investigate elementary properties of the
dielectric function in such systems.
