Luke is a guy who had sexual relationships with Offred before the appearance of Gilead. In the reality which is described by the narrator, he is already dead. He embraces the mind of the poor narrator who doesn't know whether she has to remember him or begin to live a new life. There is no information about the reason for his death. There is even an assumption that he could be alive, but the narrator doesn't know it for sure. She thinks about Luke only positively and feels the nostalgia for their past routine life. The relationships which Luke and Offred symbolize the forbidden family ties in the Gilead society.
Luke Castellan in the Essays