A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence

Abstract

We present a conjectural construction of Arthur packets within Fargues-Scholze's framework for the categorical local Langlands correspondence (CLLC). This construction consists of three parts. We first provide an overview the main statement of the CLLC, the underlying moduli stacks -- $\mathrm{Par}_G$ of parameters and $\mathrm{Bun}_G$ of $G$-bundles -- on the two sides of the correspondence, and its relation to representations of reductive p-adic groups. We then review the geometric Satake correspondence in order to define Hecke operators and the spectral action of sheaves on $\mathrm{Par}_G$ on sheaves on $\mathrm{Bun}_G$, and to construct semisimple parameters using excursion data. Finally, we generalize the geometric construction of Arthur packets from pushing-forward skyscraper sheaves on the regular conormal bundle of $\mathrm{Par}_G$ over $\mathbb C$ to a conjectural analogous operation on the stack of singularities on $\mathrm{Par}_G$ over $\overline{\mathbb Q}_\ell$.