I am currently reading a set of lecture notes by V. Ostrik and G. Williamson, Character sheaves, tensor categories and non-Abelian Fourier transform. In Theorem 1.1, they make the assertion that the characteristic function $\chi_{\mathcal{L}, \phi}$ is a character of the finite torus $T(\mathbb{F}_q)$. Does anyone know of a reference for a proof of this result, preferably one where the details are all worked out?
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asked Aug 21, 2021 at 1:29
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$\begingroup$ I think this is follows from 2.1.4 (see also 2.3.1) in Mars--Springer. This Prop-2.1.4 says that you can view a Kummer local system as the local system $E_{\theta}$ whose functions--sheaves correspondence image is a character $\theta$ of $T^F$ (the latter is a standard construction which should be found, e.g. in Kiehl--Weissauer's book). One advantage of using Kummer local systems, instead directly with the $E_{\theta}$, is that you are really working with the geometric object $T$ (and hence all $T^{F^i}$), instead of a single finite group $T^F$. $\endgroup$– user148212Aug 26, 2021 at 8:46
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