The Categorical Fargues-Scholze Correspondence at Generic Parameters

Source abstract

Let $G$ be a quasi-split reductive group over a finite extension $F$ of $\mathbb{Q}_p$. We study the Fargues-Scholze categorical Langlands correspondence at generic parameters $φ$, namely those whose semisimplification does not coincide with that of any parameter with nontrivial monodromy. Extending the theory of quasi-exceptional sets on the nilpotent cone developed by Bezrukavnikov to the setting of disconnected reductive groups which arise as stabilizers of semisimple generic parameters, to any irreducible representation $ρ$ of such a group we construct sheaves labeled $\overline{\nabla}_ρ$, $\overlineΔ_ρ$, $IC_ρ$, and $δ_ρ$ on the moduli stack of L-parameters for $G$. Using forthcoming work of Hamann, Hansen, and Mann on geometric Eisenstein series compatibility, the (conjectural, in general) behavior of the correspondence at generous parameters, and the (conjectural, in general) agreement between Fargues-Scholze and classical semisimple L-parameters we prove that these sheaves correspond under the Fargues-Scholze functor to the sheaves $i_{b!}π$, $i_{b\sharp}π$, $i_{b\sharp!}π$, and $i_{b!*}π$ on $\text{Bun}_G$, where $π$ is a smooth irreducible representation with Fargues-Scholze parameter $φ$. As applications we prove t-exactness properties of the Fargues-Scholze functor specialized at a generic parameter, and that Hecke operators are perverse t-exact on the generic locus. We also construct and study foundational properties of Hecke eigensheaves at generic parameters, compute non-semisimplified Fargues-Scholze parameters on the generic locus, and construct a Fargues-Scholze-theoretic version of the BM-O bijection at generic parameters.