  We study the interaction of a dark exciton Bose-Einstein condensate with the
nuclei in GaAs/AlGaAs coupled quantum wells and find clear evidence for nuclear
polarization buildup that accompanies the appearance of the condensate. We show
that the nuclei are polarized throughout the mesa area, extending to regions
which are far away from the photoexcitation area, and persisting for seconds
after the excitation is switched off. Photoluminescence measurements in the
presence of RF radiation reveal that the hyperfine interaction between the
nuclear and electron spins is enhanced by two orders of magnitude. We suggest
that this large enhancement manifests the collective nature of the N-excitons
condensate, which amplifies the interaction by a factor of sqrt{N}.
