1959 Ford Galaxie - Generator Light On?

I recently purchased a 1959 Ford Galaxie. The dealer had the carburetor and the generator replaced prior to purchasing it, then charged the battery all day long.

Aside from it's own problems (minor things, the brake pads need to be redone soon), it drives okay but on the way home the Generator Light came back on. It's been on ever since.

Is there anything it could be? The generator is brand new, and the dealer put the previous one in the trunk and believes that it has nothing wrong with it either, and that there's something else going on. Neither one of us know what.

I bought this knowing it d be a project, I'm just not sure what it is specifically that I need to replace.

The vehicle, so far, starts up okay, it needs to have the gas pumped on it a few times while being started but I've heard that s pretty normal of older vehicles. Everything on it is original, including the engine.

I've only had it for a few days, I'm concerned one day that I ll find it dead in the yard if I don't get this looked at, there's nobody local that restores old cars however so I'm just wondering what y all have to say about it.