what are the conclusions we can drawn from following data "In photoactive material photoluminescence is at a higher energy than the onset of the absorbance edge, and its FWHM is strikingly narrow, only about 10 nm."
comparison of the energy of emitted photons to that of optical band-edge/near-band-edge absorption can give some information about the carrier recombination process depending on the material doping condition (band-to-band, etc.). Analysis of FWHM can indicate on the strength of electron-phonon coupling or interaction between recombination centers. However, as mentioned colleagues before, to give a more substantive answer the full paper where the statement is given should be analyzed.
As Peter said "photoluminescence measurement is highly suspicious". What is your excitation wavelength? And why the photoluminescence spectrum is recorded/ shown for such a short range (725-775 nm)?
From the absorption spectrum I guess you are working with a film/ dispersion. When working with such medium scattering/ reflection is a huge concern. I hope you have checked that the photoluminescence spectrum (peak position and shape) is excitation wavelength independent.
Is there a chance that a 375 nm excitation source is being used? Because that could be why you see a peak at 750 nm. It may be a second order feature from your diffraction grating system.