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There's a well-known series of articles by Lusztig about Character Sheaves. They have important connections to many things in (geometric) representation theory, e.g. 0904.1247

How to understand these for a person with less than excellent representation theory background?

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I would recommend to start with a paper by Mars and Springer "Character sheaves" in Ast\'erisque 173. –  Victor Ostrik Jun 24 '10 at 17:40
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up vote 5 down vote accepted

That's a pretty vague question. The vague answer is that all the operations (like induction from subgroups) that can be done for representations and characters can be done for sheaves, and doing these results in a category of sheaves on a group (like GL_n over a finite field), which are close enough to the characters of representations to tell us something about them, but which also have more structure, since they are sheaves, not just functions.

Here's a somewhat more precise description: if you have a variety X which a group G acts on, then you can take the action of G on the cohomology of X. Better yet, you can get a sheaf on the group G, whose stalk over a group element g is the cohomology of the fixed points of g on X. The function sheaf correspondence sends this sheaf to the character of the representation on the cohomology of X (this follows from Lefschetz). Deligne and Lusztig defined certain varieties (the set of flags over F_q in a given relative position to their conjugates by Frobenius) on which GL(n,F_q) acts (actually, this works for any split simple algebraic group), and the corresponding sheaves (or rather the simple perverse constitutuents) are called character sheaves, and roughly capture the structure of the corresponding representations.

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Is this really true? it's certainly not the standard definition, which doesn't involve Deligne-Lusztig varieties.. I'm sure what you write is true in type A (where characters and almost characters are the same I think?) but I don't know in other types. The definition I understand doesn't involve Frobenius - ie one looks at the Springer correspondence and its W-twisted versions and takes the corresponding sheaves on G/G.. –  David Ben-Zvi Oct 22 '09 at 2:12
    
My understanding is that these definitions are equivalent; I prefer the Deligne-Lusztig theory for making woozy imprecise statements, since it's more obviously related to actual representations (on the cohomology of the DL varieties). –  Ben Webster Oct 22 '09 at 4:15
    
I'm not convinced... far from an expert in this, but I thought the trace functions of character sheaves are the so-called almost characters, and in general they are not characters, while the trace functions of the sheaves you write are just the characters of Deligne-Lusztig representations.. these are two different bases for the K-group I thought (related by Lusztig's nonabelian Fourier transform)?? I'm probably being dense though. –  David Ben-Zvi Oct 22 '09 at 14:23
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Ben's answer and the comments on this old question are useful, but I find myself wishing for a somewhat more detailed account of character sheaves written for a fairly wide audience of people interested in representation theory and Lie theory. An account which is neither too long nor too technical (not easy to write), with references. For the moment the 100+ Math Reviews items you get by searching for "character sheaf" are worth browsing, especially those by Bhama Srinivasan. The reviews are however mostly too technical to give a clear overview of the subject and its applications. –  Jim Humphreys Jun 24 '10 at 13:47
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Well, Jim, I think you may have found your next book project. I know I would read it. –  Ben Webster Jun 24 '10 at 14:53
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