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In the context of p-adic local Langlands correspondence:

Is it possible to define Casimir eigenvalues for p-adic automorphic representations? If a local representation arises from a global Galois representation, are there specific expected eigenvalues, analogous to the situation in global Langlands for elliptic curves? References would be appreciated.

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    $\begingroup$ Q1: Sort of, but it depends what you mean by p-adic automorphic representation. "U_p eigenvalues" of overconvergent p-adic automorphic forms are roughly analogous, as are "infinitesimal characters" of locally analytic representations of p-adic groups. Q2 (in any reasonable interpretation): Definitely not. The set of local Galois representations which are globalizable is a countably infinite and somewhat random-looking subset of the space of all local Galois representations (say with fixed residual representation, HT weights and inertial type). $\endgroup$
    – Satan's Minion
    Commented Aug 2 at 18:52
  • $\begingroup$ 1/2. Regarding A1: I was thinking of a more direct analogue. Perhaps using the definition of the Casimir as a set of generators for the center of the Lie algebra, and considering their induced Lie derivatives. On A2: I imagine that if we restrict to the set of isomorphism classes of reps coming from pure global reps with prescribed weight and ramification, this set is even finite. In a reasonable definition of a p-adic local Langlands correspondence, I would expect it to look like a restriction of the Global case. $\endgroup$
    – kindasorta
    Commented Aug 3 at 11:16
  • $\begingroup$ 2/2. For a global rank $2$ automorphic rep. of $\text{GL}_2(\mathbb{A}_{\mathbb{Q}}$, coming from a global Galois rep., we know the rep. should look like a discrete series, generated by the Maass $L$ and $R$ operators, on certain automorphic forms related to modular forms, both in terms of their Casimir eigenvalues, and their Fourier-Whittaker coefficients. This gives strong restrictions on their Fourier-Whittaker coefficients (contained within a finite dimensional vector space of expansions). I wonder if anything replaces Fourier-Whittaker coefficients in the local case, meaningfully. $\endgroup$
    – kindasorta
    Commented Aug 3 at 11:27
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    $\begingroup$ My second answer to A1 was an extremely direct analogue, which makes me think you don't know the relevant definitions; btw, Casimir comes from center of the universal enveloping algebra, which is very different from the center of the Lie algebra itself. $\endgroup$
    – Satan's Minion
    Commented Aug 4 at 7:46
  • $\begingroup$ Thanks for the correction, and the answers! $\endgroup$
    – kindasorta
    Commented Aug 4 at 10:28

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