14
$\begingroup$

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?

Improve this question
$\endgroup$
1
  • 3
    $\begingroup$ I would recommend to start with a paper by Mars and Springer "Character sheaves" in Ast\'erisque 173. $\endgroup$
    – Victor Ostrik
    Commented Jun 24, 2010 at 17:40

1 Answer 1

Reset to default
11
$\begingroup$

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.

Improve this answer
$\endgroup$
8
  • $\begingroup$ 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.. $\endgroup$
    – David Ben-Zvi
    Commented Oct 22, 2009 at 2:12
  • 9
    $\begingroup$ 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. $\endgroup$
    – Jim Humphreys
    Commented Jun 24, 2010 at 13:47
  • 5
    $\begingroup$ Well, Jim, I think you may have found your next book project. I know I would read it. $\endgroup$
    – Ben Webster ♦
    Commented Jun 24, 2010 at 14:53
  • 1
    $\begingroup$ So, how's the book coming along? Alternatively, two years after this question was posted, are there any new references? $\endgroup$
    – Dror Speiser
    Commented Oct 13, 2011 at 12:07
  • 2
    $\begingroup$ About what Ben and David was discussing: for GL(2), when $\mathcal{L}$ is trivial and $w$ is non-trivial, I think the sheaf $K_w^{\mathcal{L}}$ Lusztig constructed gives (via function sheaf correspondence) q*[Triv] - [St], while the non-trivial Deligne-Lusztig variety gives [Triv] - [St]. So I'd guess that even without Lusztig's non-abelian Fourier transform, the constructions still have some difference. $\endgroup$
    – Cheng-Chiang Tsai
    Commented Jul 18, 2018 at 5:07

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .