It would be easier if you say something more specific. Is your nanomaterial a semiconductor? if your answer is yes, migth is the exciton´s energy, the peak with more intensity is usually due to the exciton, so when you have a polydisperse sample, with different sizes you will have different exciton energies, also different morphologies in your sample could be influce multi-peak. It´s possible that you have a wide range of multi-peaks, if it is your case, maybe the cause of the multi-peaks are some deffects in the surface of the nanomaterial. Finally, you can check about the intrisect deffects. I hope that information will be useful for you.
As M. Vidal said, it could be size/shape polydispersity or emission from defects in the material. If you have the equipment, it would be a good idea to analyze your sample by TEM. If you have enough fluorescence signal, it could be interesting to measure the fluorescence lifetime of both emission peaks. A long fluorescence lifetime would be more characteristic of emission from defects. If you are working with binary compounds, it is most likely size polydispersity, but if you are working with ternary materials such as I-III-VI2, you probably have emission from trap states.
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