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

RESUMO

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, 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 (and under an assumption expected to be supplied in forthcoming work), we deduce that the Fargues--Scholze correspondence has finite fibers and is surjective onto inertial L-parameters.