Operating equipment on a generator is lot different than running it on grid power. This is due to both the maximum current capacity and the source impedance. Generators have much higher impedance (think of it as series resistance) than the power coming from the power company.
Technical info:
Most electronic devices these days use electronic power supplies, instead of the old style transformer type power supplies. This is cost driven. The electronic power supplies take the 120V, run it through a bridge rectifier to make DC, then step it down to the required voltages, 5V, 12V, 24V, etc. The bridge rectifier only draws current near the peaks of the AC voltage waveform. For easy math, say it draws current for 25% of the waveform time. The other 75% of the time it is drawing no current, however the average power must be maintained, thus during that 25% time, the current is 4X what it would be if it was drawing current 100% of the time. This causes high peak currents, even though the average current is low. High peak currents cause voltage distortion on the power source, and the higher the source impedance, the more the voltage is distorted.
More modern electronic equipment has PFC (Power Factor Corrected) power supplies. These use more sophisticated (and more costly) electronics to draw power throughout the entire sine wave. This is becoming common in consumer electronics due to UL and European requirements. From my experience, it is rare to find variable speed motor drives (at least in the US) that have a PFC front end. Having a PFC front end nearly doubles the cost of the speed controller.
The OP has an AC motor, so if it's variable speed, then the speed controller takes the 120V, converts it to DC, then inverts it back to the required voltage and frequency to achieve the proper speed.
Attached are some pictures of scope measurements of a 3500W briggs generator purchased from a big box store. One shows the voltage unloaded. Another shows it loaded with a resistive load to 4.87A (584W). The next shows fairly bad voltage distortion when powering a 12Vdc power supply at 7.36A (854W). This demonstrates the problems that can occur when operating NON power factor corrected electronic loads on a generator. The last file shows the same 12V power supply plugged into grid power with the load now 9.1A (1101W). For a more detailed discussion, we should probably move this to a new thread.