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.)
Best Answer
Actually, the semisimplicity should hold with no hypotheses on X, so no example should exist. In fact it is generally expected that, with char. 0 coefficients and over a finite field (both hypotheses being necessary), every mixed motive is a direct sum of pure motives -- so the question for arbitrary varieties reduces to that for smooth projective ones.
The reason is as follows: the different weight-pieces have no frobenius eigenvalues in common (by the Weil conjectures), so the weight filtration can be split by a simple matter of linear algebra. (And the splitting will even be motivic since frobenius is a map of varieties.)
Edit: In response to Jim's comment, let me try to provide a clearer argument (2nd edit: no longer using the Tate conjecture). I claim that if we assume the existence of a motivic t-structure over F_q w.r.t. the l-adic realization in the sense of Beilinson's article http://arxiv.org/pdf/1006.1116v2.pdf, then provided that H^i_c(X-bar) is Frobenius-semisimple for smooth projective X, it is in fact so for aribtrary X.
Indeed, given a motivic t-structure, its heart is an artinian abelian category where every irreducible object is a summand of a Tate-twist of an H^i(X) for X smooth an projective, and furthermore there are no extensions between such irreducibles of the same weight (this is all in Beilinson's article).
That's all true over a general field. But now let's argue that, in the case of a finite field, there also can't be extensions between such irreducibles of different weights; then in the motivic category all of our H^i_c(X-bar) of interest will be direct sums of summands of H^i(X)'s, and we'll have successfully made the reduction to the smooth projective case.
So suppose M and N are irreducible motives of distinct weights over F_q, and say E is an extension of M by N. Consider the characteristic polynomials p_M and p_N of Frobenius acting on the l-adic cohomologies of M and N. By Deligne, they have rational coefficients and distinct eigenvalues, so we can solve q * p_N == 1 (mod p_M) for a rational-coefficient polynomial q. But then (q*p_N)(frobenius) acting on E splits the extension (recall from Beilinson's article that the l-adic realization is faithful under our hypothesis), and we're done.
Later commentary: apparently, when I wrote this I was a little too excited about the perspectives offered by motives. I should emphasize the point essentially made by Minhyong Kim, that the reduction from the general case to the proper smooth case likely doesn't require any motivic technology, and should even be independent of any conjectures. One just needs to know that there's a weight filtration on l-adic cohomology of the standard type where the pure pieces are direct sums of direct summands of appropriate cohomology of smooth projective varieties. As Minhyong says, this probably follows from Deligne's original pure --> mixed argument, via use of compactifications and de Jong alterations. Or at least that's what it seems to me without having gone into the details. I'm sure someone else knows better.