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Is there an abelian variety $A/\mathbb{F}_q$ and an embedding $\mathrm{SL}_2(\mathbb{F}_q)\to \mathrm{Aut}_{\mathbb{F}_q}(A)$ such that $H^1(A\otimes \overline{\mathbb{F}_{q}}, \mathbb{Q}_l)$ contains the discrete series as a subrepresentation?

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    $\begingroup$ I think the Jacobian of the Deligne-Lusztig curve $x^q y - x y^q = 1$ works for this. $\endgroup$
    – Will Sawin
    Sep 18, 2021 at 14:54
  • $\begingroup$ @WillSawin You need to choose a compactification of the DL curve to define a jacobian, right? (like $xy^q-x^qy-z^{q+1}=0$.) $\endgroup$
    – user148212
    Sep 18, 2021 at 15:10
  • $\begingroup$ @WillSawin What about the irreps of the other finite groups of Lie type? Is there a legitimate need for compactly supported cohomology? I guess in higher dimensions you would need something like intermediate Jacobians which may not be well-defined. $\endgroup$
    – DDL
    Sep 18, 2021 at 15:11
  • $\begingroup$ Yes,the usual Deligne-Lusztig construction uses etale cohomology of degree higher than $1$ for other groups, and higher cohomology doesn't really have a nice relationship to abelian varieties. This doesn't rule out the possible existence of another abelian variety with a suitable group action, though I can't think of any possible idea for constructing one explicitly. $\endgroup$
    – Will Sawin
    Sep 18, 2021 at 15:24

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