Jan,
If it's been that way since you bought the car, then the charging system is keeping the battery charged.
Connect a volt meter to the battery and read the voltage. It should be ~12.4 volts. Start the car and read the voltage. It should be 14.2 to 14.6 volts.
The wire from the GEN light may have been disconnected during an alternator change or upgrade, and may be grounded somewhere in the car, probably in the engine bay.
John
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