Luke exists only in the narrator's past. She struggles over whether to try to remember him or let him go. Should she keep up the hope of finding him alive, or give him up for dead? We never find out what happened to Luke, and the "Historical Notes" section says that it's unlikely that he survived. By the end of the book, it seems like the narrator's strategy to hold onto him through her memory isn't working:
I ought to have done that with Luke, paid more attention, to the details, the moles and scars, the singular creases; I didn't and he's fading. Day by day, night by night he recedes, and I become more faithless. (40.15)
When she was with Luke, things were ordinary; they didn't realize how lucky they were, or that their time together would be so short. She's forgotten the details of Luke's body, so she makes a conscious effort to remember Nick's, determined not to make the same mistake twice.
Facts about Luke seep through to us gradually. Both Luke and the narrator wanted a family. Luke worried about their daughter, and he got along with his mother-in-law, despite her notorious dislike of men:
He didn't mind [when the narrator's mother criticized], he teased her by pretending to be macho, he'd tell her women were incapable of abstract thought and she'd have another drink and grin at him. (20.21)
Here, as critic Madonne Miner points out, Luke jokingly makes the kind of statements that are made seriously in Gilead by the Commander (source). The narrator says Luke is "teas[ing]" and "pretending"; it's up to readers to decide if his statements are, in retrospect, harmless or harmful.
Luke was already married when he met the narrator, and it takes him two years to get divorced. We know nothing about his previous relationship, except that his first wife calls angrily from time to time and that he and the narrator became romantic while he was still married. When their life is crumbling and her rights have been stripped away, Luke's consoling words to the narrator fall flat. The narrator suspects that the new dynamic between them doesn't really bother him: "He doesn't mind this, I thought. He doesn't mind it at all. Maybe he even likes it. We are not each other's, anymore. Instead, I am his" (28.116). In her next thought, though, she contradicts herself: "Unworthy, unjust, untrue. But that is what happened" (28.117). "What happened" could mean either what the narrator accused Luke of, or the fact that she had these "unworthy" thoughts.
Whether Luke likes the new setup with himself in the power position or not, he seems committed to leaving Gilead behind and escaping with the narrator and their daughter across the border. By doing so, all three lives are at risk. These seem like the actions of a man committed to keeping his family together, although the ultimate cost is the family's splitting apart. On the other hand, if they'd done nothing, the result probably would have been the same.