Continuous group cohomology with coefficients in locally analytic vectors of admissible $ \mathbb{Q}_p $-Banach space representations

TL;DR

This paper proves that for $p$-adic reductive groups, the continuous cohomology with coefficients in locally analytic vectors is topologically equivalent to that with coefficients in the original Banach space representation, ensuring well-behaved topologies.

Contribution

It establishes a topological isomorphism between cohomology groups with locally analytic vectors and the original Banach space coefficients in $p$-adic reductive groups.

Findings

Cohomology groups are homeomorphic with different coefficients.

Topologies on these cohomology groups are Hausdorff.

The topologies are the finest locally convex topologies.

Abstract

We show that the continuous cohomology groups of a $ p $-adic reductive group with coefficients in the locally analytic vectors of an admissible $ \mathbb{Q}_p $-Banach space representation are homeomorphic to those with coefficients in the Banach space representation itself. Moreover, we deduce that the canonical topologies on those continuous cohomology groups are Hausdorff and are the uniquely determined finest locally convex topologies.