Timeline for How to understand character sheaves
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Jul 18, 2018 at 5:07 | comment | added | Cheng-Chiang Tsai | 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. | |
| Oct 13, 2011 at 12:07 | comment | added | Dror Speiser | So, how's the book coming along? Alternatively, two years after this question was posted, are there any new references? | |
| Jun 24, 2010 at 14:53 | comment | added | Ben Webster♦ | Well, Jim, I think you may have found your next book project. I know I would read it. | |
| Jun 24, 2010 at 13:47 | comment | added | Jim Humphreys | 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. | |
| Oct 22, 2009 at 16:00 | comment | added | Ben Webster♦ | Everything you say is right, but it doesn't conflict with anything I said. On the character issue, what I wrote down is the alternating sum of the D-L characters over the different cohomology groups, so it has no positivity. The almost characters are obtained by convoling against a character for the Weyl group. | |
| Oct 22, 2009 at 14:23 | comment | added | David Ben-Zvi | 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. | |
| Oct 22, 2009 at 4:15 | comment | added | Ben Webster♦ | 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). | |
| Oct 22, 2009 at 2:12 | comment | added | David Ben-Zvi | 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.. | |
| Oct 7, 2009 at 20:15 | vote | accept | Ilya Nikokoshev | ||
| Oct 5, 2009 at 2:48 | history | answered | Ben Webster♦ | CC BY-SA 2.5 |