A semisimple mod $p$ Langlands correspondence in families for $GL_2(mathbb{Q}_p)$

This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $mathbf{GL_2}(F)$. In arXiv:2007.01364v1 we have constructed a parametrization of the $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$-modules by certain $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Satake parameters, together with an antispherical family of $mathcal{H}^{(1)}_{overline{mathbb{F}}_p}$-modules. Here we let $F=mathbb{Q}_p$ (and $pgeq 5$) and construct a morphism from $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Satake parameters to $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-Langlands parameters. As a result, we get a version in families of Breuils semisimple mod $p$ Langlands correspondence for $mathbf{GL_2}(mathbb{Q}_p)$ and of Pav{s}k={u}nas parametrization of blocks of the category of mod $p$ locally admissible smooth representations of $mathbf{GL_2}(mathbb{Q}_p)$ having a central character. The formulation of these results is possible thanks to the Emerton-Gee moduli space of semisimple $widehat{mathbf{GL_2}}(overline{mathbb{F}}_p)$-representations of the Galois group ${rm Gal}(overline{mathbb{Q}}_p/ mathbb{Q}_p)$.