Beijing notes on the categorical local Langlands 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.