In-rich CuInSe2 films have been submitted to post-growth Cu diffusion. The photoluminescence investigation carried out shows a transition from the commonly observed broad band around 0.94 eV to a much sharper peak around 0.98(5) eV under low excitation density. This recombination appears at a slightly but definitely larger energy than the recombination usually reported around 0.96 eV. The excitation power dependence leads us to ascribing the peak observed here to a donor-acceptor pair band, as evidenced by the 2.5 meV/decade shift of the peak-energy and the temperature-dependence of the photoluminescence signal. From the Arrhenius plot of the luminescence intensity an activation energy of 40 meV is extracted, When these data are analyzed in the frame of the most recent theoretical predictions of Cu- and In-related defects a possible candidate for the center appears to be [Cu-ln-Cu-i]. However Cu-Se could equally be involved.