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How is the central character reflected under the transfer map?
this is just a hallucination, but one could imagine that maybe eigenvarieties are “coarse moduli space” shadows of some more complicated “fine moduli stacks” which actually know about L-packets in some more refined way
oh, where the groups S_phi show up as automorphisms of points?
and which satisfy some local-global compatibility with the moduli stacks of L-parameters constructed by DHKM, Zhu, Fargues-Scholze
but it is just a hallucination
a challenge for the audience :)
I’ve got to go teach, thanks Judith!!!!