What was found
IMMEDIATELY AFTER TAKEOFF, WE RECEIVED A “BRAKE OVHT” CAS. WE LEFT THE GEAR DOWN, BRAKES COOLED TO <200, THEN RETRACTED THE GEAR PER THE QRH. IN FLIGHT, “BRAKE OVHT” CAS CAME ON AGAIN. FOLLOWED QRH CHECKLIST, LOWERED GEAR, BRAKE TEMP INDICATIONS ON L/H OUTBOARD BRAKE THEN WENT TO >999*. AN EMERGENCY WAS DECLARED FOR LANDING BY THE CREW. FROM MAINTENANCE: FOUND A DIRTY CONNECTOR, REMOVED AND CLEANED CONNECTOR, OPS CHECK GOOD, NO OTHER DEFECTS NOTED REFERENCE G400 AMM 32-00-00.