I have done everything wrong....

[Grocerykartman]

I would not ignore her entirely, you should at least respond. Despite everything she's doing right now, it would also be rude on your part not to acknowledge her.

Now, as far whether or not you accept...

I vote FOR going to lunch with her, but maybe picking a different day LATER, and especially if Tuesday is not "all that" convenient for you. Go, but do not discuss the relationship at all. I think she's at least throwing you a social olive branch, and that you should meet her on a "friendly" basis. She may need to discuss important matters with you that you can't avoid forever. But, go only if you can manage to go and not get all emotional on her. If relationship talk comes up from her, you need to simply and calmly say that you just don't want to get into that right now, let's just have a nice lunch. You need an excuse to get out too, kind of like on a first date, if things get too uncomfortable in the event she insists on talking about it.

That's just one vote though...but I should point out it's not contradictory to detaching, or even going dark. You didn't initiate lunch, and your guard will be up at it regarding your relationship, so that's two ways to think about it.