Solid locally analytic representations

2023/05/04 by Jacinto, Joaquín Rodrigues, Camargo, Juan Esteban Rodríguez · 1 citation
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2305.03162

Abstract

We develop the p-adic representation theory of p-adic Lie groups on solid vector spaces over a complete non-archimedean extension of 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 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 GL₁.

Cited by

• Locally analytic completed cohomology of Shimura varieties of Hodge type

Related

• Solid locally analytic representations of p-adic Lie groups
• Solid locally analytic representations in mixed characteristic
• U(g)-finite locally analytic representations
• Locally analytic distributions and p-adic representation theory, with applications to GL₂
• Locally analytic vectors in representations of locally p-adic analytic groups