Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu\_p = {\rm Spec}\ k[\varepsilon]/(\varepsilon^p - 1)$. Denote the quotient $X/\mu\_p$ by $Y$ and the quotient morphism $X\to Y$ by $\pi$. 



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> Let $D$ be a Weil divisor on $X$. Is the sheaf $\pi\_\* (\mathscr{O}\_X(D))$ a direct sum of sheaves of the form $\mathscr{O}\_Y(E)$ for some Weil divisors $E$ on $X$?
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For an action of $\mu\_n$, $n$ not divisible by $p$, we get the above statement because the push-forward decomposes according to the characters of $\mu\_n$, and if we assume that there is an orbit on which $\mu\_n$ acts freely, then all these summands are nonzero - hence all have rank one and are reflexive. So we need a form of ,,diagonalizability'' for $\mu\_p$.
I don't see how the proof for $\mu\_n$ above could be translated to a proof for $\mu\_p$. 



Quotient by p-th roots of unity in characteristic p
If you cannot generalize, you may try to simplify:-)) Representations of $\mu\_p$ are still completely reducible: they are just graded vector spaces by the cyclic group of order $p$. The first part of your argument goes through.


For the second part, you have a simplification as $\mu\_p$ has no subgroup schemes! So your stabilisers are either trivial of full $\mu\_p$. If the action is nontrivial, there is a point with trivial stabiliser, and there you know that each simple character of $\mu\_p$ appears.


Did I miss anything?


