2000/05/07 by Peter Schneider, Schneider, Peter, Jeremy Teitelbaum +1 · 1 citation
Mathematics · #11R23 #11S80 #22E35 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #advanced mathematical theories #math.NT #math.RT #msc:11R23 #msc:11S80 #msc:22E35
paper · pdf · doi:10.48550/arxiv.math/0005066
arxiv created 2000/05/07 · openalex publication_date 2000/05/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The lack of a p-adic Haar measure causes many methods of traditional representation theory to break down when applied to continuous representations of a compact p-adic Lie group G in Banach spaces over a given p-adic field K. For example, Diarra showed that the abelian group G=\dZ has an enormous wealth of infinite dimensional, topologically irreducible Banach space representations. We therefore address the problem of finding an additional ''finiteness'' condition on such representations that will lead to a reasonable theory. We introduce such a condition that we call ''admissibility''. We show that the category of all admissible G-representations is reasonable -- in fact, it is abelian and of a purely algebraic nature -- by showing that it is anti-equivalent to the category of all finitely generated modules over a certain kind of completed group ring K[[G]]. As an application of our methods we determine the topological irreducibility as well as the intertwining maps for representations of GL2(\dZ) obtained by induction of a continuous character from the subgroup of lower triangular matrices.