Algebras of p - adic distributions and admissible representations

Introduction In a series of earlier papers, ([ST1-4]) we began a systematic study of locally analytic representations of a locally L-analytic group G, where L ⊆ C p is a finite extension of Q p. Such a representation is given by a continuous action of G on a locally convex topological vector space V over a spherically complete extension field K ⊆ C p of L, such that the orbit maps g → gv are locally analytic functions on G. When G is the group of L-points of an algebraic group, the class of such representations includes many interesting examples, such as the principal series representations studied in [ST2], the finite dimensional algebraic representations, and the smooth representations of Langlands theory. A reasonable theory of such representations requires the identification of a finiteness condition that is broad enough to include the important examples and yet restrictive enough to rule out pathologies. In this paper we present such a finiteness condition that we call " admissibility " for locally analytic representations (provided the field K is discretely valued). The admissible locally analytic representations, among which are the examples mentioned above, form an abelian category. Our approach to the characterization of admissible representations is based on the algebraic approach to such representations begun in [ST2]. As in that paper, we require that the vector space V carrying the locally analytic representation be of compact type, a topological condition whose most important consequence is that V is reflexive. We focus our attention on the algebra D(G, K) of locally analytic distributions on G. This algebra is the continuous dual of the locally analytic, K-valued functions on G, with multiplication given by convo-lution. When G is compact, D(G, K) is a Fréchet algebra, but in general is neither noetherian nor commutative. If V is a locally analytic G-representation, then its continuous dual V ′ b , with its strong topology, becomes a module over D(G, K). We identify a subcategory of the module category of D(G, K) that we call the coadmissible modules. We show that any coadmissible module carries a canonical Fréchet topology. We say that V is admissible if V ′ b is topologically isomorphic to a coadmissible module. If G is not compact, we say that V is admissible if it is admissible as a representation for one (or equivalently any) compact open subgroup of G. To define the category of …