Continuous Group Cohomology and Ext-Groups

Abstract

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 ${\mathrm{Ext}}_G^i(1,\pi )$ 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.

Cited by in corpus (1)

• On homological properties of the Schlichting completion