Solid locally analytic representations

TL;DR

This work develops a robust foundation for p-adic representation theory on solid vector spaces, unifying continuous, smooth, and locally analytic representations within the condensed/multi-functor framework. It constructs solid categories Rep^{la}, Rep^{sm}, and their geometric incarnations as qc-sheaves on classifying stacks [*/G^{la}] and [*/G^{sm}], establishing equivalences with locally analytic distribution modules $\mathcal{D}^{la}(G,K)$ and smooth distributions $\mathcal{D}^{sm}(G,K)$, and linking representations to comodules over $C^{la}(G,K)$ and $C^{sm}(G,K)$. The paper proves cohomology comparison theorems via formal adjunctions, extending Lazard and Casselman-Wigner results to arbitrary G and solid representations, and applies the framework to a locally analytic p-adic Langlands correspondence for $GL_1$. In addition to the analytic theory, it develops the solid smooth and locally algebraic sides, including dualizing functors and compatibility results. Overall, the results provide a cohesive, six-functor–friendly, categorified perspective on p-adic representation theory with deep geometric and cohomological implications.

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 $\mathrm{GL}_1$.

Theorems & Definitions (203)

• Theorem 1
• Remark 1.2.1
• Theorem 2: Theorem \ref{['TheoremEquivalenceLocAn1']}
• Remark 1.2.2
• Theorem 3: Proposition \ref{['PropCommutativeDiagramDual']}
• Remark 1.2.3
• Definition 1.2.4
• Theorem 4: Theorem \ref{['TheoremComparisonCohomologies']}
• Remark 1.2.5
• Theorem 5: Theorem \ref{['TheoLLGL1Algebraic']}