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Birman and Series (J. Lond Math Soc 1984) characterize non-self-intersection of a closed loop via a group-theoretic property of the corresponding element in the fundamental group. Their characterization depends on the Nielesen generators available for $\pi_1$ of cpt orientable surfaces with non-empty bdry. Wild out of the blue idea: try something similar, with Frobenius at $p$, and the usual generators of the Grothendieck-Teichmuller group.
The slightly longer answer is that proving such a result using traditional methods would be to check nice analytic properties for L-functions of twists, e.g. L(s, f x g x h x j) where j is an automorphic form on GL(7) (or a bit lower if you're lucky). That's out of range for now, as far as I can tell.
A maximal torus T in GL(E) over S (with E a vector bundle) gives a decomposition of the vector bundle into line bundles. You can see this, even working with S a smooth variety over the complex numbers, since the eigenspaces for T give a local decomposition of E into 1-dim spaces.
That one should work too, with U the (say) upper-triangular unipotent radical. It's equivalent, I think after substitution, to the one I wrote down with the lower-triangular unipotent radical. The important thing is to use whatever unipotent radical is opposite of the one used for parabolic induction in $I(\chi_1, \chi_2)$.
