U(g)-finite locally analytic representations

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)

Algebras of p-adic distributions and admissible representations

Let G be a compact, locally L-analytic group, where L is a finite extension of Qp. Let K be a discretely valued extension field of L. We study the algebra D(G,K) of K-valued locally analytic distributions on G, and apply our results to the locally analytic representation theory of G in vector spaces over K. Our objective is to lay a useful and powerful foundation for the further study of such representations. We show that the noncommutative, nonnoetherian ring D(G,K) "behaves" like the ring of functions on a rigid Stein space, and that (at least when G is Qp-analytic) it is a faithfully flat extension of its subring K\otimes Zp[[G]], where Zp[[G]] is the completed group ring of G. We use this point of view to describe an abelian subcategory of D(G,K) modules that we call coadmissible. We say that a locally analytic representation V of G is admissible if its strong dual is coadmissible as D(G,K)-module. For noncompact G, we say V is admissible if its strong dual is coadmissible as D(H,K) module for some compact open subgroup H. In this way we obtain an abelian category of admissible locally analytic representations. These methods allow us to answer a number of questions raised in our earlier papers on p-adic representations; for example we show the existence of analytic vectors in the admissible Banach space representations of G that we studied in "Banach space representations ...", Israel J. Math. 127, 359-380 (2002). Finally we construct a dimension theory for D(G,K), which behaves for coadmissible modules like a regular ring, and show that smooth admissible representations are zero dimensional