Categorical local Langlands for ${\mathrm{GL}}_n$ for parameters of Langlands-Shahidi type with integral coefficients

Abstract

We prove the categorical form of Fargues' geometrization conjecture for ${\mathrm{GL}}_n$ along $L$-parameters of Langlands-Shahidi type for rational, torsion, and integral coefficients. Additionally, we prove that in this case the categorical equivalence is $t$-exact, which yields new torsion vanishing results in the cohomology of unitary Shimura varieties.