Proof of the l-adic categorical Langlands conjecture for GL_n for Langlands-Shahidi type paramters

Recently Laurent Fargues and Peter Scholze proposed a geometrization of the l-adic Langlands program. This is formulated as an equivalence of two categories linear over the stack of L-parameters. Linus Hamann singled out a suitable subset of L-parameters he calls "Langlands-Shahidi type" for which such an equivalence is also t-exact and the categories admit an easy description as the product of representation categories for inner forms of GL_n. We discuss a proof of this conjecture following an inductive argument of Nguyen and review some applications for the cohomology of local and global Shimura varieties and for sheaves on the stack of G-bundles on the Fargues-Fontaine curve.