Continuous Group Cohomology and Ext-Groups

We prove that the continuous group cohomology groups of a locally profinite group $G$ with coefficients in a smooth $k$-representation $\pi$ of $G$ are isomorphic to the $\mathrm{Ext}$-groups $\mathrm{Ext}^i_G(\mathbb{1},\pi)$ computed in the category of smooth $k$-representations of $G$. We apply this to show that if $\pi$ is a supersingular $\overline{\mathbb{F}}_p$-representation of $\mathrm{GL}_2(\mathbb{Q}_p)$, then the continuous group cohomology of $\mathrm{SL}_2(\mathbb{Q}_p)$ with values in $\pi$ vanishes. Furthermore, we prove that the continuous group cohomology groups of a $p$-adic reductive group $G$, with coefficients in an admissible unitary $\mathbb{Q}_p$-Banach space representation $\Pi$, are finite dimensional. We show that the continuous group cohomology of $\mathrm{SL}_2(\mathbb{Q}_p)$ with values in non-ordinary irreducible $\mathbb{Q}_p$-Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ vanishes.