Categorical local Langlands on isocrystals

(jt with D. Gaistgory, A. Genestier and V. Lafforgue) Let F be a local field of equal characteristic and G be a reductive group over F. The conjectural categorical form of the local Langlands correspondence proposed by Fargues and Scholze is an equivalence of categories between on one side the category of $\ell$-adic sheaves on $\Bun_G(X_{FF})$ the stack of $G$-torsors on the Fargues-Fontaine curve and and on the other side the category of coherent sheaves on the stack of $L$-parameters. The main result of their work is the construction of a categorical action of the second category on the first one. In this talk, I want to report on an ongoing project whose goal is to construct a similar categorical action on the category of $\ell$-adic sheaves on the category of $G$-isocrystals. In a second part of the talk I will report on a joint work with C. Xue, where we show some property of the cohomology of global chtoucas and how to interpret this property as a strong form of local-global compatibility.