Timeline for What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?
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| Feb 10, 2010 at 7:35 | comment | added | Kevin Buzzard | I think there are general conjectures relating sizes of $L$-packets to representation-theoretic data but I am unclear as to what they are. I think that perhaps locally there are precise ones but globally, at least in the SL_2 case, there is some sort of product formula and some subtleties. I am sufficiently ignorant about this to be unable to make precise statements, but I'm pretty sure that at least the local part of the story is well-understood to a certain extent (just not by me). | |
| Feb 9, 2010 at 17:45 | comment | added | Kevin McGerty | Hey, that's a cool idea about L-packets! I wonder if there's a way to see if it squares with what people already know? The Ka-Lu thing was about p-adic groups: via the Borel idea the category of Iwahori reps is equivalent to the category of finite dimensional Hecke algebra reps and then Ka-Lu just go and classify these. Much more recently, this has been "categorified" by Bezrukavnikov and collaborators, but the categorifications only have to do with geometric/function field cases. I suppose you view that as an extension of James's comment about Satake also (going from G(O) to I). | |
| Feb 9, 2010 at 16:27 | comment | added | Kevin Buzzard | Hey Kevin. I was wondering whether somehow the understanding of the packets would somehow be swallowed up by other "finite errors" when one 'categorified'. Explicitly: the L-packets have size 1 when G=GL_n. But it's not hard to find examples of group homs r1,r2:W-->L (think L the L-group, W a Weil group) which have the property that r1(w) and r2(w) are conjugate for all w, but for which r1 isn't conjugate to r2. That's some sort of "finite error" which would be somehow swallowed up if one considered reps L-->GL_n, where that phenomenon doesn't occur. Does Ka-Lu apply even in the p-adic case? | |
| Feb 9, 2010 at 15:21 | history | answered | Kevin McGerty | CC BY-SA 2.5 |