This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of characteristic $p$. Here there is a natural action of a Frobenius morphism $F$ relative to $q$. Given a distinct prime $\ell$, there is an induced operation of $F$ on etale cohomology groups (with compact support) $H^i_c(X, \overline{\mathbb{Q}_p})$. When $X$ is *projective*, this action is conjectured to be *semisimple* on each of the finite dimensional vector spaces involved. But it seems that semisimplicity can fail when $X$ isn't projective. My basic question is:

Is there an elementary example where the Frobenius action fails to be semisimple? (References?)

Of course, etale cohomology developed in response to the Weil conjectures and related matters in number theory. Here there is a lot of deep literature which I'm unfamiliar with, but I'd like to get some insight into the narrow question of what does or doesn't force semisimplicity for non-projective varieties.

My interest lies mainly in *Deligne-Lusztig* varieties and their role in studying characters of finite groups of Lie type. Such varieties $X_w$ are indexed by Weyl group elements and are locally closed smooth subvarieties of the flag variety for a reductive group $G$, with all irreducible components of equal dimension. Here the finite subgroup $G^F$ acts on the etale cohomology, commuting with $F$, and the resulting virtual characters (alternating sums of characters on cohomology spaces) are the D-L characters.

Characters of finite tori also come into play here, but I'm thinking first about the trivial characters of tori which lead to "unipotent" characters. These include essential but mysterious "cuspidal" unipotent characters which can't be extracted from the usual induced characters obtained by parabolic induction.

For example, the Chevalley group $G_2(\mathbb{F}_q)$ typically has 10 unipotent characters (at the extremes the trivial and the Steinberg characters), with four being cuspidal. Those four appear in etale cohomology groups of a variety $X_w$ with $w$ a Coxeter element: the variety has dimension 2, with four characters (three cuspidal, the other Steinberg) in degree 2, one (cuspidal) in degree 3, and one (the trivial character) in degree 4. Miraculously, it always happens in the Coxeter case that $F$ acts semisimply (here with 6 distinct eigenvalues: the Coxeter number) and its eigenspaces afford distinct irreducible characters. In the year after he and Deligne finished their fundamental paper (*Annals*, 1976), Lusztig worked out the Coxeter case in a deep technical paper here. This was followed by a more complete determination of cuspidal unipotent characters, and then much more. The Coxeter case seems to be unusually well-behaved in this program.

P.S. As I suspected, there's more going on under the surface of my basic question about semisimplicity than meets the eye. As an outsider to much of the algebraic geometry framework I can appreciate the outline of Dustin's answer though not yet the details. My question came from wondering whether there are ways to shortcut some of the older steps taken by Lusztig, but the wider questions here are obviously important. I'll have to see how far my motivation (in a manner of speaking) takes me.

And Wilberd: thanks for the proofreading, which is not one of my favorite things to do. (Though I somehow got "just bonce" into a book that was supposedly proofread.)