The front brake lever should definitely be triggering the brake light.
Check and make sure all of the wires are connected (and connected properly) at the throttle/switch housing.
If everything is connected properly, make sure the trigger for brake light is adjusted correctly (covered in the owners manual).
As far as the wiring goes, I would suggest downloading the service manual (for free - do a search here) and checking the color wiring diagram to determine how the Givi wiring should be connected.
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