The categorical local Langlands conjecture

Abstract

We formulate a program to prove the categorical local Langlands conjecture (CLLC) of Fargues-Scholze, for all quasisplit $p$-adic groups where the Fargues-Scholze $L$-parameters agree with the semisimplification of a known "automorphic" local Langlands parametrization. A key working hypothesis - which we expect to prove elsewhere jointly with Hamann - is the compatibility of the enhanced Whittaker coefficient functor $c_\psi$ with Eisenstein series. For $\mathrm{GL}_n$, we show that this hypothesis alone implies the full CLLC. For more general groups $G$, we prove an induction principle which reduces CLLC for $G$ to CLLC for all proper Levi subgroups together with a very small amount of information about $G$. This principle applies unconditionally to many classical groups with current technology. Along the way, we establish many foundational results. In particular: - We prove a very strong finiteness theorem for spectral constant term functors. - We prove a spectral analogue of Bernstein's finite global dimension theorem for $p$-adic Hecke algebras. - We introduce and develop the theory of admissible ind-coherent sheaves and admissible duality on derived stacks. - We prove a duality theorem for the spectral action. Using all of these results, we unconditionally define a new and explicit functor $t_{\psi}$ from the spectral side to the automorphic side, which is defined on enough ind-coherent sheaves to control the entire conjecture.