Generator light comes on under two conditions.
1. Engine not running and ignition turned on. This allows positive (I'm assuming negative ground) from the ignition switch to the ignition (generator) light which is grounded in the voltage regulator (probably inside the alternator). The light turns off when the generator starts charging and putting positive into what earlier was a ground in the regulator.
2. Engine running and when the ignition is switched off the generator continues to charge a little letting the positive from the regulator to reach the ignition light, which is grounded through the ignition accessories.
If the light stays on when the engine is running, it is either that the regulator is grounding the light through not charging, or that the alternator is trying to charge and the light is grounding on the switch side. This could come from not being connected properly to the ignition switch.
As above, what is the voltage reading. If it is above 13.5 it is certainly charging. If it is below 12.5 it isn't charging.
If the alternator is not charging, it could be through several reasons. If it is new, chances are likely that it is in good condition and operable - but don't always bet on it. It must have a good ground. This is usually through the mounting bolts. Check them! It must have a good power supply both to the main output cable and to the regulator. The latter might be internal within the alternator. A Delco is external. Japanese are usually internal.
If it is charging, somehow the ignition light is grounding on the ignition switch side.