Solid locally analytic representations

TL;DR

This paper develops a new framework for $p$-adic Lie group representations on solid vector spaces, establishing equivalences with modules and sheaves, and extends cohomological results to broader groups.

Contribution

It introduces solid locally analytic representations, generalizes classical equivalences, and extends cohomological comparisons to arbitrary $p$-adic groups.

Findings

Equivalence between solid locally analytic representations and modules over distribution algebras.

Extension of cohomological comparison results to non-compact $p$-adic groups.

Application to the $p$-adic Langlands correspondence for $ ext{GL}_1$.

Abstract

We develop the $p$-adic representation theory of $p$-adic Lie groups on solid vector spaces over a complete non-archimedean extension of $\mathbb{Q}_p$. More precisely, we define and study categories of solid, solid locally analytic and solid smooth representations. We show that the category of solid locally analytic representations of a compact $p$-adic Lie group is equivalent to that of quasi-coherent modules over its algebra of locally analytic distributions, generalizing a classical result of Schneider and Teitelbaum. For arbitrary $G$, we prove an equivalence between solid locally analytic representations and quasi-coherent sheaves over certain locally analytic classifying stack over $G$. We also extend our previous cohomological comparison results from the case of a compact group defined over $\mathbb{Q}_p$ to the case of an arbitrary group, generalizing results of Lazard and Casselman-Wigner. Finally, we study an application to the locally analytic $p$-adic Langlands correspondence for $\mathrm{GL}_1$.