Poor Izîl—not only does he have a name that we have no clue how to pronounce, he also has a giant, lecherous fallen angel clinging to his shoulders at all times of the day. Don't you just hate it when that happens?
By the way, the name "Izîl" means "the sublime" in Berber. But Izîl is the opposite of sublime. He is "a time-lapse of decline" (13.16), a slow descent into decay personified. It's actually really sad to watch this man (d)evolve in the book; he's a scholar who knows too much, and now he's forced into a life of graverobbing.
We think he's around to show us the dark consequences of wish magic-gone-wrong; the monkey on his back (a.k.a. Razgut) was his punishment for wishing for knowledge. At least he's got an awesome mustache, though. Got to look on the bright side, you know?
When Izîl wished for knowledge, he was saddled with Razgut, an invisible seraph who was cast out of his own world. For every ounce of compassion and kindness that Izîl shows—he kills himself to protect Karou for goodness' sake—Razgut responds with creepiness and lechery.
Razgut wants to lick Karou, and he does, describing her as "a piece of empty candy" (22.70)—an observation that really touches a nerve with Karou's abandonment issues. Razgut is crafty, though. He will do anything to get back to his homeland.
When he realizes that Karou wants to go there, he tells her about it... and convinces her to take him with her. We cannot imagine that this will end well. He has to be up to something, right?
He also serves as a foil to Akiva in interesting ways. Just by looking at him, Razgut teaches us that not all seraphim are sexy studs. But they're not all killers like Akiva either. Razgut tells him:
My soul is clean. I've never killed anyone. But you, oh you. Look at your hands. (18.47)
Hm. Razgut's high-and-mighty attitude kind of makes us wonder what he did to get expelled from "heaven" when murder is, apparently, a-okay. Mysterious. Guess we'll have to read the sequel to find out.