Give some reasons why you (the player) would have to hate your character.
In general I save hate for things more serious than gaming, but I'll try to answer the spirit of the question.
As far as playing Lume, I get frustrated with the fact that I don't get to talk to any NPC's normally and even when I do, I have to absolutely "blunder" my way through conversation to be appropriate to her level of social skills. I guess I can see it as an acting challenge, and so far it hasn't bothered me too much, since the sessions have been a good balance of talking and fighting.
I also get very frustrated, that Lume's five skill points in pistols can equal all her physad training when it comes to damage. I get upset when I feel like pure power is being rewarded, but I didn't know enough to have made my character effective (this is generally true for other games as well). My frustration is offset in this game by the fact that even if I'm ineffective, I know my team still needs me as a heavy and a damage sink.
Additionally, what reasons would someone that hates your character have for their dislike?
I'm guessing my GM hates Lume's nerve strike. ;P
In general, I could see someone disliking Lume's relative two dimensionality or her combat bias. I'm trying to slowly develop Lume into a more three dimensional interesting character, but to start out with I didn't have strong ideas about her personality, and I'm afraid it does show in her sheet. While she does have some amusing knowledge skills, she doesn't have much else to her which is non-utilitarian. I feel a bit bad about that, especially her lack of contacts.
I didn't phrase my question clearly, though, so let me try to follow up. What about if you knew her personally, like if she were a real person?
Lume can be very abrasive at times. She tends to be quieter when running, because she respects Ajax's position as face and realizes she doesn't have the tact to deal with people the way he does. In a more normal life situation or around the team outside of runs she can be an absolute bitch. She sees no problem with swearing or being pushy, especially if she thinks people are lying or just being subtly nasty to her or her friends. She also has rather dark sense of humor and doesn't always understand when her ribbing leaves poor taste and enters downright mean.