### Banach space representations and Iwasawa theory, by Peter Schneider and Jeremy Teitelbaum

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, the abelian group
$G=\dZ$ has an enormous wealth of infinite dimensional, topologically
irreducible Banach space representations, as may be seen in the paper
by Diarra [Dia]. 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.
Peter Schneider

Jeremy Teitelbaum

Peter Schneider and Jeremy Teitelbaum <jeremy@uic.edu>