Modulo $\mathrm{\ell }$-representations of $p$-adic groups $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(F)$

Let $k$ be an algebraically closed field of characteristic $l\mathrm{\ne }p$. We construct maximal simple cuspidal $k$-types of Levi subgroups ${\mathrm{M}}^{\mathrm{\prime }}$ of ${\mathrm{S}\mathrm{L}}_n(F)$, when $F$ is a non-archimedean locally compact field of residual characteristic $p$. We show that the supercuspidal support of irreducible smooth $k$-representations of Levi subgroups ${\mathrm{M}}^{\mathrm{\prime }}$ of ${\mathrm{S}\mathrm{L}}_n(F)$ is unique up to ${\mathrm{M}}^{\mathrm{\prime }}$-conjugation, when $F$ is either a finite field of characteristic $p$ or a non-archimedean locally compact field of residual characteristic $p$.