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$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and l’adic Fourier transform. We spend a lot of time proving sheaves are mixed, and I've just realized we haven't seen any non-mixed ones!

I'll repeat the definitions in case there is ambiguity. Let $X_0$ be a variety over $\FF_q$, with $X$ the base change to an algebraic closure $\FF_q$. I also want $X_0$ to be normal, see discussion below. Let $\cG$ an $\ell$-adic sheaf on $X_0$. Let $\tau: \overline{\QQ_{\ell}} \to \CC$ be an embedding. A sheaf is called $\tau$-pure of weight $\beta$ if, for every closed point $x$ of $X_0$, the eigenvalues $(\alpha_1, \ldots, \alpha_r)$ of Frobenius acting on the stalk $\cG_x$ obey $|\tau(\alpha_i)|^2 = q^{d(x) \beta}$, where $d(x)$ is the extension degree $[k(x):\FF_q]$. A sheaf is called $\tau$-mixed if it has a filtration whose subquotients are $\tau$-pure.

I'm fairly sure I can show that, if such an example exists, then it can be found with $\cG$ irreducible and smooth of rank $r$ and $X_0$ a curve (and, if there is an example where $X_0$ is normal, then there is an example where $X_0$ is a smooth curve). So we just need to write down a continuous irreducible representation of $\pi_1(X_0) \cong \pi_1(X) \rtimes \widehat{\mathbb{Z}}$ where either (a) the Frobenius conjugacy class at some point of $X_0$ acts by two eigenvalues with different archimedean norm or (b) there are two different points of $X_0$ whose Frobenius conjugacy classes act with elements of different norms. And $\pi_1(X)$ could be something as nice as the profinite free group on $2$ generators, which is really easy to make representations of.

But I can't figure out how to make my representation extend to the semidirect product.

What if $X_0$ is not normal? Then we can build an example as follows: Take a nodal cubic with non-split node. For example, if $q \equiv 3 \bmod 4$, take the nodal cubic $Z(X^2+Y^2)=X^3$ in $\mathbb{P}^2$. So $\pi_1(X) \cong \widehat{\mathbb{Z}}$ and $Gal(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ acts on it by negation. Quotient $\pi_1(X_0)$ down to $(\pm 1) \ltimes \widehat{Z}$ (the infinite profinite dihedral group). Take $\alpha$ an $\ell$-adic integer not on the unit circle (for example, $1+\sqrt{2}$, and let $(\pm 1) \ltimes \widehat{Z}$ act by $\left( \begin{smallmatrix} \alpha^k & \\ & \alpha^{-k} \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} & \alpha^k \\ \alpha^{-k} & \end{smallmatrix} \right)$ .

This example is slightly unsatisfying, since it becomes a direct sum of two pure sheaves if we extend the base field to $\mathbb{F}_{q^2}$, but it works.

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    $\begingroup$ For $X_0 = {\rm{Spec}}(\mathbf{F}_q)$ we can define $\mathscr{G}$ corresponding to the homomorphism $\widehat{\mathbf{Z}} \rightarrow {\rm{GL}}_n(\overline{\mathbf{Z}}_{\ell})$ given by $1 \mapsto M$ with $M$ arbitrary. So with $n>1$ pick $M$ whose eigenvalues are algebraic integers whose ratios do not lie on the unit circle under any archimedean absolute value. This example can then be pulled back to any finite type $\mathbf{F}_q$-scheme you wish (and then tensored against whatever nonzero mixed sheaf you like). $\endgroup$
    – nfdc23
    Commented May 31, 2016 at 13:17
  • $\begingroup$ @nfdc23 But then it isn't irreducible (at least over $\overline{\mathbb{Q}}_{\ell}$). It splits as a direct sum of line bundles, one for each eigenvalue, each of which is pure, so the whole bundle is mixed. $\endgroup$
    – David E Speyer
    Commented May 31, 2016 at 13:22
  • $\begingroup$ @nfdc23 But you are making me realize that I need to specify $\mathbb{Q}_{\ell}$ versus $\overline{\mathbb{Q}_{\ell}}$. I intended the latter. $\endgroup$
    – David E Speyer
    Commented May 31, 2016 at 13:23
  • $\begingroup$ Thanks! That sounds like an answer -- conj 1.2.10 says there are no examples. I imagine this conjecture is still open? $\endgroup$
    – David E Speyer
    Commented May 31, 2016 at 14:04
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    $\begingroup$ With $\overline{\mathbf{Q}}_{\ell}$-sheaves there should be no counterexamples: in 1.2.9 of Weil II it is conjectured by Deligne. [I deleted an earlier comment to which you refer, deducing 1.2.9 from Deligne's more refined conjecture 1.2.10 because I couldn't track down where Deligne makes a convention about writing "sheaf" to mean "Weil sheaf".] $\endgroup$
    – nfdc23
    Commented May 31, 2016 at 14:06

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