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Let $F$ be a local field of mixed characteristic, let $k$ be a finiteextension of its residue field, let ${\mathcal H}$ be the pro-$p$-Iwahori Hecke$k$-algebra attached to ${\rm GL}_{d+1}(F)$ for some $d\ge1$. We construct anexact and fully faithful functor from the category of supersingular ${\mathcalH}$-modules to the category of ${\rm Gal}(\overline{F}/F)$-representations over$k$. More generally, for a certain $k$-algebra ${\mathcal H}^{\sharp}$surjecting onto ${\mathcal H}$ we define the notion of $\sharp$-supersingularmodules and construct an exact and fully faithful functor from the category of$\sharp$-supersingular ${\mathcal H}^{\sharp}$-modules to the category of ${\rmGal}(\overline{F}/F)$-representations over $k$.

Supersingular Hecke modules as Galois representations