Beijing notes on the categorical local Langlands conjecture

TL;DR

This paper refines the categorical local Langlands conjecture by proposing compatibilities with Eisenstein series and duality, and introduces a new t-structure on the automorphic side to facilitate spectral matching.

Contribution

It introduces the hadal t-structure on the automorphic side and discusses its expected correspondence with a perverse coherent t-structure, advancing the categorical local Langlands program.

Findings

Proposes compatibility of the conjecture with Eisenstein series and duality.

Introduces the hadal t-structure with good finiteness properties.

Begins matching t-structures on both sides of the conjecture.

Abstract

We formulate some refinements and complements to the categorical local Langlands conjecture of Fargues-Scholze. In particular, we state the expected compatibilities with Eisenstein series and duality, and explain some of their consequences. We also begin the process of matching t-structures on both sides. Notably, we introduce the so-called hadal t-structure on the automorphic side, which has good finiteness properties, and which conjecturally matches with a suitable perverse coherent t-structure on the spectral side.