Year: 2021
Type: preprint
Abstract: We prove that the continuous group cohomology groups of a locally profinite group $ G $ with coefficients in a smooth $ k $-representation $ π$ of $ G $ are isomorphic to the $ \mathrm{Ext}$-groups $ \mathrm{Ext}^i_G(\mathbb{1},π) $ computed in the category of smooth $ k $-representations of $ G $. We apply this to show that if $ π$ 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 $ π$ 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 $ Π$, 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.
Source: arXiv (Cornell University)
Author: Fust, Paulina
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W4285546146