A Scholzian Approach to the Local Langlands Correspondence for \(\mathrm{GL}_n\) over function fields

Abstract

Let \(F\) is a local field of characteristic \(p\). Inspired by work of Scholze, we construct a map \(\pi\mapsto\sigma(\pi)\) from irreducible smooth representations of \(\mathrm{GL}_n(F)\) to \(n\)-dimensional Weil representations of \(F\). We prove that this map uniquely satisfies a purely local compatibility condition on traces of a test function \(f_{\tau,h}\), and we also prove that this map is compatible with parabolically inducing tensor products. It is expected that \(\pi\mapsto\sigma(\pi)\) equals the local Langlands correspondence for \(\mathrm{GL}_n\) over \(F\), up to Frobenius semisimplification.