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 ${\overline{\mathbb{Q}}}_{\ell }$.

As pointed out by Prof Teruhisa Koshikawa, Conjecture 4.8 is ill-formulated. We will address the issue in the near future and provide a correction