Timeline for Why is the Langlands dual group always taken over $\mathbb{C}$?
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| Jul 15, 2019 at 1:12 | comment | added | paul garrett | Indeed. Probably the tangible aspect of "repn on complex vector space", or "\ell-adic", or... is an un-necessary artifact of needless concreteness, and the essential invariants rise above those specifics. Like "motive" above specific cohomologies. It does strike me that sometimes progress is made by realizing the irrelevance of seemingly natural/irreplaceable concrete details... Not that I know how to do that here... :) | |
| Jul 15, 2019 at 1:01 | comment | added | Will Sawin | @paulgarrett Well in local Langlands, it may be most natural to compare representations of $p$-adic groups into $\ell$-adic vector spaces and $\ell$-adic representations of the Galois groups of $p$-adic fields. If one agrees that Galois representations of this type are interesting, it's natural for there to be one $\ell$ and one $p$ on the others side. Then because the $\ell$-adic topology isn't relevant on the representation side, we can replace it with another algebraically closed fields. Then one has the $p$-adic Langlands program also, of course. | |
| Jul 15, 2019 at 0:43 | comment | added | paul garrett | Certainly what you say is true, but a naive person could also wonder why, for example, $p$-adic principal series are repns of $p$-adic groups on complex vector spaces... which might be construed as reducing the imperative to have "the Langlands dual" be a complex group despite acting on complex vector spaces, etc. (Not that I have any much better rationale, myself...!) | |
| Jul 15, 2019 at 0:07 | history | answered | Will Sawin | CC BY-SA 4.0 |