Categorical local Langlands and torsion classes of some Shimura varieties

We study the cohomology of various local Shimura varieties for $GL_n$. This provides an explicit description of the spectral action constructed by Fargues-Scholze in certain cases and allows us to prove some strongly generic part of the categorical local Langlands conjecture for $GL_n$ with torsion coefficients. As applications, we are able to prove an analogue of the Harris-Viehmann conjecture and deduce new vanishing results for the cohomology of Shimura varieties of type A in the torsion coefficient setting.