U(g)-FINITE LOCALLY ANALYTIC REPRESENTATIONS

In this paper we continue our algebraic approach to the study of locally analytic representations of a p-adic Lie group G in vector spaces over a non-Archimedean complete field K. We characterize the smooth representations of Langlands theory which are contained in the new category. More generally, we completely determine the structure of the representations on which the universal enveloping algebra U(g) of the Lie algebra g of G acts through a finite dimensional quotient. They are direct sums of tensor products of smooth and rational G-representations. Finally we analyze the reducible members of the principal series of the group G = SL2(Qp) in terms of such tensor products. 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. In [ST], we began with an algebraic approach to this type of representation theory based on a duality functor that replaces locally analytic representations by certain topological modules over the algebra D(G,K) of locally analytic distributions. As an application, we established the topological irreducibility of generic locally analytic principal series representations of GL2(Qp) by proving the algebraic simplicity of the corresponding D(GL2(Qp),K)-modules. In this paper we further exploit this algebraic point of view. We introduce a particular category of “analytic” D(G,K)-modules that lie in the image of the duality functor and therefore correspond to certain locally analytic representations. For compact groups G, these are finitely generated D(G,K)-modules that allow a (necessarily uniquely determined) Fréchet topology for which the D(G,K)-action is continuous. For more general groups, one tests analyticity by considering the action of D(H,K) for a compact open subgroup H in G. The category of analytic modules has the nice property that any algebraic map between such modules is automatically continuous. The concept of analytic module is dual to the concept of strongly admissible G-representation introduced in [ST]. The actual definition can and will be given in a way that avoids any mention of a topology on the module. Next, we study the modules dual to the traditional smooth representations of Langlands theory. We show that a smooth representation gives rise, under duality, to an analytic module precisely when it is “strongly admissible”; this is a condition on the multiplicities with which the irreducible representations of a compact open subgroup of G appear in the representation. In particular, if L is a finite extension of Qp and G is the group of L-points of a connected reductive algebraic group over L, then any smooth representation of finite length is strongly admissible. This is basically a theorem of Harish-Chandra ([HC]), although we must use, in addition, Received by the editors August 2, 2000 and, in revised form, September 25, 2000. 2000 Mathematics Subject Classification. Primary 17B15, 22D12, 22D15, 22D30, 22E50. c ©2001 American Mathematical Society 111 112 P. SCHNEIDER, J. TEITELBAUM, AND DIPENDRA PRASAD results of Vigneras ([Vig]) to deal with some complications arising from the fact that we do not assume that our coefficient field K is algebraically closed. Given these foundational results, suppose that G is the group of L-points of a split, semisimple, and simply connected group over L. We completely determine the structure of analytic modules M that are U(g)-finite, i.e., that are annihilated by a 2-sided ideal of finite codimension in the universal enveloping algebra U(g) of the Lie algebra g of G. Such a module can be decomposed into a finite sum of modules of the form E⊗Hom(V,K) where E is irreducible, finite dimensional, and algebraic, and V is smooth and strongly admissible. The dual representations E∗ ⊗ V are irreducible—in fact, simple as K[G] modules—if and only if V is irreducible. Some of the technical hypotheses on the group G in this section are consequences of the fact that our coefficient field is not algebraically closed. We conclude the paper by studying the reducible members of the locally analytic principal series of SL2(Qp). The corresponding modules contain a simple submodule such that the quotient is U(g)-finite, and we use our methods to determine the structure of this quotient. In particular, we obtain the result that the topological length of the locally analytic principal series is at most three—a fact that is due to Morita ([Mor]) by a different method. In the appendix by Dipendra Prasad, a global variant of the U(g)-finite representations, called locally algebraic representations, is introduced and studied. This point of view allows us to simplify and generalize the argument for the irreducibility of tensor products.