There isn't a fuel sending unit per se... There's just a thermistor in the tank. When the fuel level goes below the thermistor it heats up, lowers resistance, and allows enough current to flow in the circuit to turn on the lamp.
The brown/black wire that goes to the tank feeds the thermistor.
If you unplug the 3 pin black connector for the fuel pump and the lamp goes off then it's either a short between the brown/black and the fuel tank, or a bad thermistor.
If you unplug the pump and the light stays on then your brown/black is shorted to ground somewhere. It could be in the big grey connector in the front fairing, or somewhere else where the harness could get pinched.