Local Langlands functoriality for Yu's supercuspidals II: Fargues--Scholze's parametrization

Abstract

This is the second of two papers dedicated to the explicit computation of the Fargues--Scholze correspondence. We compute the Tate cohomology of cuspidal representations arising from Yu's construction. Combining this with modular functoriality in the Local Langlands Correspondence, independence of $\ell$, and the partial characterization of the Local Langlands Correspondence established in the first paper, we compare the Fargues--Scholze and Kaletha parametrizations. Among the consequences, we deduce that the Fargues--Scholze correspondence has finite fibers and is surjective onto inertial L-parameters.