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I'm working on the unramified local Langlands conjecture and there is something that I don't understand if it is true or not. I want to start by saying that I don't care about endoscopic transfer or ...

Consider the projective curve $\mathbb P^1$ over a finite field $k$. Consider $\ell$-adic local systems $E$ on $\mathbb P^1\backslash \{0,\infty\}$ such that a) $E$ is tame at $\infty$ b) The ...