  It is shown that exciton condensates exhibit an incredible richness in the
role that fundamental symmetries play whether manifest or broken. We
investigate the appearance of the singlet and the triplet excitonic condensates
under spin-orbit coupling (SOC) and consider realistic couplings to the
radiation field as manipulated by the fundamental symmetries related to spin,
time, orbital and particle degrees of freedom. The role of the strong SOC in
inducing a ground state with coexisting singlet and triplet components is
shown. It is also shown that the ground state of an excitonic condensate, in
the realistic senario of spontaneous coupling to the radiation field, is
dominated by a dark component transparent to photoluminescence.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics;Quantum Gases