Thin films of lead salt nanocrystals (NCs) offer attractive opportunities as active media for near-infrared optoelectronics but suffer from limiting trade-offs between optical and electrical properties. While NCs separated by nanometer-long ligands are good light emitters, NCs capped with shorter molecules provide a high carrier mobility but degrade the photo- and electroluminescence and broaden the narrow emission spectrum. Here we show that this severe quenching and spectral broadening can be averted with an unconventional use of metallic antennas. The resulting NC-antenna hybridization not only provides a strong boost in luminescence, but also makes it possible to remodel the emission spectrum in radical ways, even at wavelengths where the NC assembly does not emit light. These results cannot be explained with the standard theory of single-emitter luminescence assisted by optical antennas. We propose an alternative model based on a statistical description of light emission by an ensemble of emitters and discuss important consequences of our findings for nano-optics and solution-processed optoelectronics.