  Let $F$ be a local field of mixed characteristic, let $k$ be a finite
extension 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 an
exact and fully faithful functor from the category of supersingular ${\mathcal
H}$-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 $\sharp$-supersingular
modules and construct an exact and fully faithful functor from the category of
$\sharp$-supersingular ${\mathcal H}^{\sharp}$-modules to the category of ${\rm
Gal}(\overline{F}/F)$-representations over $k$.
