I am trying to learn character sheaf theory, and encounter the following question:


(\*) Let $f\colon X\rightarrow Y$ be a morphism of quasi-projective smooth varieties over $\overline{\mathbb{F}}\_q$, and suppose $\mathcal{L}$ is a rank $1$ lisse $\overline{\mathbb{Q}}\_{\ell}$-sheaf on $X$. My question is, usually under what condition on $f$, one can conclude the existence of a rank $1$ lisse $\overline{\mathbb{Q}}\_{\ell}$-sheaf $\mathcal{L}'$ on $Y$ such that $\mathcal{L}\cong f^\*\mathcal{L}'$? 




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The two particular situations in my mind are:


(1) $f$ is a principal fibration by a torus. This is the case happened e.g. in Lusztig's paper Character Sheaves I - 2.4.


(2) $f$ is a principal fibration by a connected unipotent algebraic group. This is the case happened e.g. in Lusztig's new paper <http://arxiv.org/pdf/1508.05015v3.pdf> - 4.2.


So I guess this might be true for any principal fibration by a connected affine algebraic group, or more generally true for a surjective morphism with isomorphic connected fibers?



When is a $\overline{\mathbb{Q}}_{\ell}$-local system the inverse image of a $\overline{\mathbb{Q}}_{\ell}$-local system?
I am answering my own question as I find a general result covering the case in my mind:


This is true when $f$ is a locally trivial principal fibration by a connected algebraic group and the local system on $X$ is equivariant with respect to this group; it can be deduced as a special case of 1.9.3 in Lusztig's Character Sheaves I.


