Air Conditioner Works on shore power not the Gemerator

I haven't read the history (other related threads), but the only relation between the genny and the shore power is the transfer switch.
If everything is 100% OK on shore power, then your genny isn't doing what it should, or your transfer switch is screwed up.

As I din't read any of your other threads, is your rig 30 or 50 amp?
What kind of power management system does it have?
Exactly what genny?