LOCALLY ANALYTIC DISTRIBUTIONS AND p - ADIC REPRESENTATION THEORY

Let L/Qp be a finite extension, and let G be a locally L-analytic group such as the L-points of an algebraic group over L. The categories of smooth and of finite-dimensional algebraic representations of G may be viewed as subcategories of the category of continuous representations of G in locally convex L-vector spaces. This larger category contains many interesting new objects, such as the action of G on global sections of equivariant vector bundles over p-adic symmetric spaces and other representations arising from the theory of p-adic uniformization as studied for example in [ST]. A workable theory of continuous representations of G in Lvector spaces offers the opportunity to unify these disparate examples in a single theoretical framework. There are a number of technical obstacles to developing a reasonable theory of such representations. For example, there are no unitary representations over L, and continuous, or even locally analytic, functions on G are not integrable against Haar measure. As a result, even for compact groups one is forced to consider representations of G in fairly general locally convex vector spaces. In such situations one encounters a range of pathologies apparent even in the theory of representations of real Lie groups in Banach spaces. For this reason one must formulate some type of “finiteness” or “admissibility” condition in order to have a manageable theory. In this paper we introduce a restricted category of continuous representations of locally L-analytic groups G in locally convex K-vector spaces, where K is a spherically complete nonarchimedean extension field of L. We call the objects of this category “strongly admissible” representations and we establish some of their basic properties. Most importantly we show that (at least when G is compact) the category of strongly admissible representations in our sense can be algebraized; it is faithfully full (anti)-embedded into the category of modules over the locally analytic distribution algebra D(G,K) of G over K. We may then replace the topological notion of irreducibility with the algebraic property of simplicity as D(G,K)-modules. Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous K-valued representations of locally L-analytic groups. As an application of our theory, we prove the topological irreducibility of generic members of the p-adic principal series of GL2(Qp). This result was claimed by Morita for general L, not only Qp, in [Mor], Theorem 1(i), but his method to deduce irreducibility from something weaker that he calls local irreducibility is