Banach space representations and Iwasawa theory

Abstract

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 $GL_2(\dZ)$ obtained by induction of a continuous character from the subgroup of lower triangular matrices.