Equivariant Derived Category

Question

If $G$ is an connected unipotent group over $k$,and $X$ a scheme of finite type over $k$, (an algebraic closed field of positive characteristic) then we can define the bounded derived categorie of constructible complexes of $\overline{\mathbb{Q}}_l$-sheaves on $X$.

And if $G$ is acting on $X$ we can define the equivariant derived categories, in:

http://www.math.lsa.umich.edu/~mityab/12.pdf (page 5)

Remark 1.5. By Definition 1.3, we have a faithful forgetful functor DG(X) −→ D(X). If G is a connected unipotent group over k, one can show that the forgetful functor is fully faithful. In other words, being G-equivariant becomes a property of an adic complex on X in this case.

So this means, in the condition of the remark 1.5 ( G is a connected unipotent group over k) that if for a complex $M$ we have $\alpha^* M \simeq \pi ^*M$ where $\alpha$ is the action and $\pi$ is the projection, then we have that $M$ is an equivariant sheaf (complex)?

Answer 1

That's not what the remark says, but I believe it's true. For any sheaf, the "averaging" $a_*\pi^*M$ is automatically equivariant (for the same reason that the corresponding integral transform gives invariant functions). However, this usually doesn't give a canonical equivariant structure on a given sheaf because $a_*\pi^*M\ncong M$ in most cases. However, if the group is unipotent, then $a_*a^*$ is isomorphic to the identity functor. Thus, under your assumptions, $a_*\pi^*M\cong a_*a^*M\cong M$, so $M$ is equivariant, not quite canonically, but canonically given the isomorphism you've stipulated.