Solid locally analytic representations in mixed characteristic

Abstract

The theory of locally analytic representations of $p$-adic Lie groups with $\mathbf{Q}_p$-coefficients is a powerful tool in $p$-adic Hodge theory and in the $p$-adic Langlands program. This perspective reveals important differential structures, such as the Sen and Casimir operators. Rodr\'iguez Camargo and Rodrigues Jacinto developed in \cite{RJRC22} a solid version of this theory using the language of condensed mathematics. This provides more robust homological tools (comparison theorems, spectral sequences...) for studying these representations. In this article, we extend the solid theory of locally analytic representations to a much broader class of mixed characteristic coefficients, such as $\mathbf{F}_p((X))$ or $\mathbf{Z}_p[[X]]\langle p/X\rangle[1/X]$, as well as to semilinear representations. In the introduction, we explain how these ideas could relate to mixed characteristic phenomena in $p$-adic Hodge theory, extend eigenvarieties, and the Langlands program.