If it is a check that is pending with the FBI, it is either your name check or your fingerprints. The latter being fully digitized, I suspect your name check is pending. These are the only checks the FBI performs on applicants.
USCIS runs an IBIS check. This 'check' basically tries to figure out if you are on any of the 20+ U.S. gov't watch lists etc. etc.
If they do suspect that something fishy about your green card or any other immigration benefit you were ever granted, they might also refer a case to ICE for further investigation. Yet, I do not think this applies to your case.
Cheers!
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