Given a curve C, and a reductive group G, there is a moduli stack Loc_G(C), the stack of Glocal systems. I keep reading that there's a substack of "opers" but am having trouble locating a definition. So what's an oper, and how should I think about them?
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Look at http://arxiv.org/abs/math/0501398. 


Edward Frenkel also has a number of papers which deal with opers (just look at his arxiv papers for example). In particular, I'm fond of the paper he wrote with David BenZvi http://arxiv.org/abs/math/9902068 I think this paper might be of particular interest to Charles given that I've seen him previously give links to the BNR paper. This paper by Frenkel and BenZvi relates spectral curves to opers and also gives some nice historical background on how each is related to solving certain kinds of differential equations (which is the kind of thing I'm glad to know is there, even if I don't study it that way). 

