Modular functoriality in the Local Langlands Correspondence

We develop a theory of Smith-Treumann localization and relative parity sheaves in the context of Fargues-Scholze’s Geometrization of the Local Langlands Correspondence. We then apply this theory to prove some conjectures of Treumann-Venkatesh concerning mod-$\mathrm{\ell }$ Local Langlands functoriality between a reductive group $G$ and its fixed subgroup under an order $\mathrm{\ell }$ automorphism. As another application, we calculate the Fargues-Scholze parameters of mod-$\mathrm{\ell }$ Howe-unramified toral representations.