### U(g)-finite locally analytic representations, by Peter Schneider and Jeremy Teitelbaum

In this paper we continue the study of locally analytic
representations of a $p$-adic Lie group $G$ in vector spaces over a
spherically complete non-archimedean field $K$, building on the algebraic
approach to such representations introduced in our paper "Locally analytic
distributions and p-adic representation theory, with applications to GL_2."
In that paper we associated to a representation $V$ a module $M$ over the
ring $D(G,K)$ of locally analytic distributions on $G$ and described
an admissibility condition on $V$ in terms of algebraic properties of $M$.
In this paper
we determine the relationship between our admissibility condition on locally analytic
modules and the traditional admissibility of Langlands theory.
We then analyze the class of locally analytic
representations with the property that their associated modules are annihilated by
an ideal of finite codimension in the universal enveloping algebra of G, showing under
some hypotheses on G that they are sums of representations of the form X\otimes Y,
with X finite dimensional and Y smooth. The irreducible representations of this type
are obtained when X and Y are irreducible.
We conclude by analyzing the reducible members of the locally analytic principal
series of $SL_2(\Qp)$.
Peter Schneider

Jeremy Teitelbaum

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