Solid locally analytic representations of $p$-adic Lie groups

TL;DR

This paper advances the theory of locally analytic representations of compact p-adic Lie groups using condensed mathematics, generalizing Lazard's isomorphisms and comparing cohomologies of solid representations.

Contribution

It introduces a new framework for locally analytic representations via condensed mathematics and extends Lazard's isomorphisms to solid representations.

Findings

Generalized Lazard's isomorphisms to solid representations

Established a comparison between group cohomology and analytic vectors

Developed a new perspective on p-adic Lie group representations

Abstract

We develop the theory of locally analytic representations of compact $p$-adic Lie groups from the perspective of the theory of condensed mathematics of Clausen and Scholze. As an application, we generalise Lazard's isomorphisms between continuous, locally analytic and Lie algebra cohomology to solid representations. We also prove a comparison result between the group cohomology of a solid representation and of its analytic vectors.