The 120v ac outlet in the overhead of my 2006 Seneca 34SS operates either on the inverter or on shore power, or on the generator. Any of the three. It is the only outlet which will work on the inverter, because it's the only one wired to the inverter. But it also will work fine on shore power or generator. Something seems wrong if yours won't work on shore power.
I would check in this order:
Circuit breakers associated with the front half of the motorhome. Switch them off, then back on again.
Whether the outlet works on the inverter
If the outlet does not work on the inverter:
Power at the input terminals of the inverter (input side would be 12vdc)
Output of the inverter by switching it on
If the outlet works on the inverter, but not on shore power after checking the circuit breakers, then check the wiring of your outlet itself, and change it out if necessary.
2006 Jayco Seneca 34SS
2006 Jeep Wrangler Unlimited Rubicon