MEMOIRS of the American Mathematical Society Volume 248 • Number 1175 • Forthcoming Locally Analytic Vectors in p Representations of Locally -adic Analytic Groups Matthew Emerton ISSN 0065-9266 (print) ISSN 1947-6221 (online) American Mathematical Society MEMOIRS of the American Mathematical Society Volume 248 • Number 1175 • Forthcoming Locally Analytic Vectors in p Representations of Locally -adic Analytic Groups Matthew Emerton ISSN 0065-9266 (print) ISSN 1947-6221 (online) American Mathematical Society Providence, Rhode Island Library of Congress Cataloging-in-Publication Data Cataloging-in-PublicationDatahasbeenappliedforbytheAMS.See http://www.loc.gov/publish/cip/. DOI:http://dx.doi.org/10.1090/memo/1175 Memoirs of the American Mathematical Society Thisjournalisdevotedentirelytoresearchinpureandappliedmathematics. Subscription information. Beginning with the January 2010 issue, Memoirs is accessible from www.ams.org/journals. 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ThispublicationisarchivedinPortico andCLOCKSS. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 222120191817 Contents Introduction 1 0.1. Locally analytic vectors and locally analytic representations 2 0.2. The organization of the memoir 4 0.3. Terminology, notation, and conventions 6 Chapter 1. Non-archimedean functional analysis 11 1.1. Functional analytic preliminaries 11 1.2. Fr´echet-Stein algebras 22 Chapter 2. Non-archimedean function theory 31 2.1. Continuous rigid analytic, and locally analytic functions 31 2.2. Distributions 43 2.3. Change of field 46 Chapter 3. Continuous, analytic, and locally analytic vectors 49 3.1. Regular representations 49 3.2. The orbit map and continuous vectors 51 3.3. Analytic vectors 54 3.4. Analytic vectors continued 63 3.5. Locally analytic vectors 66 3.6. Analytic and locally analytic representations 73 Chapter 4. Smooth, locally finite, and locally algebraic vectors 81 4.1. Smooth and locally finite vectors and representations 81 4.2. Locally algebraic vectors and representations 86 Chapter 5. Rings of distributions 93 5.1. Frobenius reciprocity and group rings of distributions 93 5.2. Completions of universal enveloping algebras 99 5.3. Rings of locally analytic distributions are Fr´echet-Stein algebras 107 Chapter 6. Admissible locally analytic representations 119 6.1. Admissible locally analytic representations 119 6.2. Strongly admissible locally analytic representations and admissible continuous representations 125 6.3. Admissible smooth and admissible locally algebraic representations 129 6.4. Essentially admissible locally analytic representations 133 6.5. Invariant lattices 139 Chapter 7. Representations of certain product groups 145 7.1. Strictly smooth representations 145 iii iv CONTENTS 7.2. Extensions of notions of admissibility for representations of certain product groups 150 Bibliography 157 Introduction Recent years have seen the emergence of a new branch of representation the- ory: the theory of representations of locally p-adic analytic groups on locally con- vex p-adic topological vector spaces (or “locally analytic representation theory”, for short). Examples of such representations are provided by finite dimensional algebraic representations of p-adic reductive groups, and also by smooth represen- tationsofsuchgroups(onp-adicvectorspaces). Onemightcallthesethe“classical” examples of such representations. One of the main interests of the theory (from the point of view of number theory) is that it provides a setting in which one can study p-adic completions of the classical representations [6], or construct “p-adic interpolations” of them (for example, by defining locally analytic analogues of the principal series, as in [23], or by constructing representations via the cohomology of arithmetic quotients of symmetric spaces, as in [9]). Locallyanalyticrepresentationtheoryalsoplaysanimportantroleintheanaly- sisofp-adicsymmetricspaces;indeed,thisanalysisprovidedtheoriginalmotivation foritsdevelopment. Thefirst“non-classical”examplesinthetheorywerefoundby Morita, in his analysis of the p-adic upper half-plane (the p-adic symmetric space attachedtoGL (Q ))[17],andfurtherexampleswerefoundbySchneiderandTeit- 2 p elbaum in their analytic investigations of the p-adic symmetric spaces of GL (Q ) n p (for arbitrary n) [22]. Motivated in part by the desire to understand these exam- ples,SchneiderandTeitelbaumhaverecentlyinitiatedasystematicstudyoflocally analyticrepresentationtheory[22, 23, 24, 25, 27]. Inparticular, theyhaveintro- duced the important notions of admissible and strongly admissible locally analytic representations of a locally p-adic analytic group (as well as the related notion of admissible continuous representations of such a group). The goal of this memoir is to provide the foundations for the locally analytic representation theory that is required in the papers [8, 9, 10]. In the course of writing those papers we have found it useful to adopt a particular point of view on locally analytic representation theory: namely, we regard a locally analytic representation as being the inductive limit of its subspaces of analytic vectors (of various “radii of analyticity”), and we use the analysis of these subspaces as one of the basic tools in our study of such representations. Thus in this memoir we present a development of locally analytic representation theory built around this point of view. Some of the material that we present is entirely new (for example, the notion of essentially admissible representation, which plays a key role in [8], and the results of chapter 7, which are used in [9]); other parts of it can be found (perhaps up to minor variations) in the papers of Schneider and Teitelbaum cited above, or in the thesis of Feaux de Lacroix [12]. We have made a deliberate effort to keep our exposition reasonably self-contained, and we hope that this will be of some benefit to the reader. 1 2 INTRODUCTION 0.1. Locally analytic vectors and locally analytic representations Wewillnowgiveamoreprecisedescriptionoftheview-pointonlocallyanalytic representation theory that this memoir adopts, and that we summarized above. LetLbeafiniteextensionofQ ,andletK beanextensionofL,completewith p respect to a discrete valuation extending the discrete valuation of L.1 We let G denote a locally L-analytic group, and consider representations of G by continuous operators on Hausdorff locally convex topological K-vector spaces. Our first goal is to define, for any such representation V, the topological K-vector space V of la locallyanalyticvectorsinV. Asavectorspace,V isdefinedtobethesubspaceof la V consisting of those vectorsv for which the orbit map g (cid:3)→gv is alocally analytic V-valuedfunctiononG. Thenon-trivial point inthedefinitionistoequipV with la an appropriate topology. In the references [22, 27], the authors endow this space (which they denote by V , rather than V ) with the topology that it inherits as a an la closed subspace of the space Cla(G,V) of locally analytic V-valued functions on G (eachvectorv ∈V beingidentifiedwiththecorrespondingorbitmapo :G→V). la v We have found it advantageous to endow V with a finer topology, with respect la to which it is exhibited as a locally convex inductive limit of Banach spaces. (In some important situations — for example, when V is of compact type, or is a Banach space equipped with an admissible continuous G-action — we prove that the topology that we consider coincides with that considered by Schneider and Teitelbaum.) Suppose first that G is an affinoid rigid analytic group defined over L, and that G is the group of L-valued points of G. If W is a Banach space equipped with a representation of G, then we say that this representation is G-analytic if for each w ∈ W the orbit map o : G → W given by w extends to a W-valued w rigid analytic function on G. For any G-representation V, we define VG−an to be the locally convex inductive limit over the inductive system of G-equivariant maps W →V, where W is a Banach space equipped with a G-analytic action of G. We now consider the case of an arbitrary locally L-analytic group G. Recall that a chart (φ,H,H) of G consists of an open subset H of G, an affinoid space H isomorphic to a closed ball, and a locally analytic isomorphism φ:H −∼→H(L). If H is furthermore a subgroup of G, then the fact that H(L) is Zariski dense in H implies that there is at most one rigid analytic group structure on H inducing the given group structure on H. If such a group structure exists, we refer to the chart (φ,H,H)asananalyticopensubgroupofG. Wewilltypicallysuppressreferenceto the isomorphism φ, and so will speak of an analytic open subgroup H of G, letting Hdenote thecorresponding rigidanalyticgroup, andidentifying H withthe group of points H(L). For any G-representation on a Hausdorff convex K-vector space V, and any analytic open subgroup H of G, we can define as above the space VH−an of H- analytic vectors in V. (If we ignore questions of topology, then VH−an consists of 1In the main body of the text, in contrast to this introduction, the field K of coefficients is assumed merely to be spherically complete with respect to a valuation extending the discrete valution on L. Nevertheless, several of our results require the additional hypothesis that K be discretely valued, and so we have imposed this hypothesis on K throughout the introducion in ordertosimplifythediscussion. Inthemainbodyofthetext,whenagivenresultrequiresforits validitythatK bediscretelyvalued,wehavealwaysindicatedthisrequirementinthestatement oftheresult(withoneexception: itisassumedthroughoutsection6.5thatKisdiscretelyvalued, andsothisisnotexplictlymentionedinthestatementsoftheresultsappearinginthatsection). 0.1. LOCALLY ANALYTIC VECTORS AND LOCALLY ANALYTIC REPRESENTATIONS 3 those locally analytic vectors with “radius of analyticity” bounded below by H.) We define V to be the locally convex inductive limit over all locally analytic open la subgroups H of G of the spaces VH−an. The representation V of G is said to be locally analytic if V is barrelled, and if the natural map V → V is a bijection. If V is an LF-space (we recall the la meaning of this, and some related, functional analytic terminology in section 1.1 below), then we can show that if this map is a bijection, it is in fact a topological isomorphism. Thus given alocally analytic representation of G on an LF-space V, we may write V −∼→l−i→mVHn−an, where Hn runs over a cofinal sequence of analytic n open subgroups of G. The category of admissible locally analytic G-representations, introduced in [27], admits a useful description from this point of view. We show that a locally analytic G-representation on a Hausdorff convex K-vector space V is admissible if and only if V is an LB-space, such that for each analytic open subgroup H of G, the space VH−an admits a closed H-equivariant embedding into a finite direct sum of copies of the space Can(H,K) of rigid analytic functions on H. Recallthatin[27],alocallyanalyticG-representationV isdefinedtobeadmis- sible if and only if V is of compact type, and if the dual space V(cid:4) is a coadmissible module under the action of the ring Dan(H,K) of locally analytic distributions on H, for some (or equivalently, every) compact open subgroup H of G. For this definitiontomakesense(thatis,forthenotionofacoadmissibleDan(H,K)-module to be defined), the authors must prove that the ring Dan(H,K) is a Fr´echet-Stein algebra, in the sense of [27, def., p. 8]. This result [27, thm. 5.1] is the main theorem of that reference. In order to establish our characterization of admissible locally analytic repre- sentations,weareledtogiveanalternativeproofofthistheorem,andanalternative description of the Fr´echet-Stein structure on Dla(H,K), which is more in keeping with our point of view on locally analytic representations. While the proof of [27] relies on the methods of [16], we rely instead on the methods used in [1] to prove the coherence of the sheaf of rings D†. We also introduce the category of essentially admissible locally analytic G- representations. To define this category, we must assume that the centre Z of G is topologicallyfinitely generated. (Thisis a rathermild condition, which is satisfied, for example, if G is the group of L-valued points of a reductive linear algebraic group over L.) Supposing that this is so, we let Zˆ denote the rigid analytic space parameterizing the locally analytic characters of Z, and let Can(Zˆ,K) denote the Fr´echet-Stein algebra of K-valued rigid analytic functions on Zˆ. Let V be a convex K-vector space of compact type equipped with a locally analytic G-representation, and suppose that V may be written as a union V = limV , where each V is a Z-invariant BH-subspace of V. The Dla(H,K)-action −→ n n n on the dual space V(cid:4) then extends naturally to an action of the completed tensor product algebra Can(Zˆ,K)⊗ˆ Dla(H,K). Our proof of the fact that Dla(H,K) is K Fr´echet-SteingeneralizestoshowthatthiscompletedtensorproductisalsoFr´echet- Stein. We say that V is an essentially admissible locally analytic representation of G if, furthermore, V(cid:4) is a coadmissible module with respect to this Fr´echet-Stein algebra, for some (or equivalently, any) compact open subgroup H of G. 4 INTRODUCTION It iseasy toshow, using the characterizationofadmissible locallyanalytic rep- resentations described above, that any such locally analytic representation of G is essentiallyadmissible. Conversely,ifV isanyessentiallyadmissiblelocallyanalytic representation of G, and if χ is a K-valued point of Zˆ, then the closed subspace Vχ ofV onwhichZ actsthroughχisanadmissible locallyanalyticrepresentation ofG. ThegeneraltheoryofFr´echet-Steinalgebras[27,§3]impliesthatthecategory of essentially admissible locally analytic G-representations is abelian. Since the category of coadmissible Can(Zˆ,K)-modules is (anti)equivalent to the category of coherent rigid analytic sheaves on the rigid analytic space Zˆ, one may think of a coadmissible Can(Zˆ,K)⊗ˆ Dla(H,K)-module as being a “coherent K sheaf” of coadmissible Dla(H,K)-modules on Zˆ (or, better, a “coherent cosheaf”). Thus,roughlyspeaking,onemayregardanessentiallyadmissiblelocallyanalyticG- representation V as being a family of admissible locally analytic G-representations parameterized by the space Zˆ, whose fibre (or, better, “cofibre”) over a point χ∈ Zˆ(K) is equal to Vχ. The category of essentially admissible locally analytic representations provides thesettingfor theJacquetmodule constructionforlocally analyticrepresentations that is the subject of the papers [8] and [10]. These functors are in turnapplied in [9] to construct “eigenvarieties” (generalizing the eigencurve of [7]) that p-adically interpolate systems of eigenvalues attached to automorphic Hecke eigenforms on reductive groups over number fields. Letuspointoutthatfunctionalanalysiscurrentlyprovidesthemostimportant technicaltoolinthetheoryoflocallyanalyticrepresentations. Indeed,sincecontin- uous p-adic valued functions on a p-adic group are typically not locally integrable forHaarmeasure (unlesstheyarelocallyconstant), thereissofarnorealanalogue in this theory of the harmonic analysis which plays such an important role in the theory of smooth representations (although one can see some shades of harmonic analysis in the theory: the irreducibility result of [23, thm. 6.1] depends for its proof on Fourier analysis in the non-compact picture, and Hecke operators make an appearance in the construction of the Jacquet module functor of [8]). Thus one relies on softer functional analytic methods to make progress. This memoir is no exception; it relies almost entirely on such methods. 0.2. The organization of the memoir A more detailed summary of the memoir now follows, preceding chapter by chapter. In chapter 1 we develop the non-archimedean functional analysis that we will require in the rest of the memoir. Section 1.1 is devoted to recalling various pieces of terminology that we will need, and to proving some results for which we could not find references in the literature. None of the results are difficult, and most or all are presumably well-known to experts. In section 1.2 we recall the theory of Fr´echet-Steinalgebrasdevelopedin[23,§3],andprovesomeadditionalresultsthat we will require in our applications of this theory. In chapter 2 we recall the basics of non-archimedean function theory. Sec- tion 2.1 recalls the basic definitions regarding spaces of continuous, rigid analytic, and locally analytic functions with values in locally convex K-vector spaces, and establishes some basic properties of these spaces that we will require. Section 2.2 introduces the corresponding spaces of distributions. In section 2.3 we recall the 0.2. THE ORGANIZATION OF THE MEMOIR 5 definitionandbasicpropertiesoftherestrictionofscalarsfunctor,inboththerigid analytic and the locally analytic setting. Inchapter3wepresentourconstructionofthespaceoflocallyanalyticvectors attached to a representation of a non-archimedean locally L-analytic group G. Aftersome preliminaries insections 3.1and 3.2, in section3.3we suppose that G is the group of points of an affinoid rigid analytic group defined over L, and define the space of analytic vectors. In section 3.4 we extend this construction to certain non-affinoid rigid analytic groups. In section 3.5, we return to the situation in which G is a locally L-analytic group, and construct the space of locally analytic vectors attached to any G- representation. Insection3.6werecallthenotionoflocallyanalyticrepresentation, andalsointroducetherelatednotionofanalyticrepresentation,andestablishsome basic properties of such representations. Chapter 4 begins by recalling, in section 4.1, the notion of smooth and locally finitevectorsinaG-representation. Themainpointofthissectionistoprovesome simple factsaboutrepresentations in which everyvectoris smoothor locallyfinite. In section 4.2 we assume that G is the group of L-valued points of a connected reductive linear algebraic group G over L. For any finite dimensional algebraic representationW ofGoverK,andforanyG-representationV,wedefinethespace VW−alg of locally W-algebraic vectors in V, and study some of its basic properties. Asin[18],therepresentationV issaidtobelocallyalgebraicifeveryvectorofV is locallyW-algebraicforsomerepresentationW ofG. Weprovethatanyirreducible locally algebraic representation is isomorphic to the tensor product of a smooth representation of G and a finite dimensional algebraic representation of G (first proved in [18]). One approach to analyzing representations V of G, the importance of which hasbeenemphasized by Schneider andTeitelbaum, istopasstothe dual spaceV(cid:4), and to regard V(cid:4) as a module over an appropriate ring of distributions on G. The goal of chapter 5 is to recall this approach, and to relate it to the view-point of chapter 3. In section 5.1 we prove some simple forms of Frobenius reciprocity, and apply these to obtain a uniform development of the dual point of view for continuous, analytic, and locally analytic representations. In section 5.2 we recall the descrip- tion of algebras of analytic distributions via appropriate completions of universal envelopingalgebras. Insection5.3weusethisdescription, togetherwiththemeth- ods of [1, §3], to present a new construction of the Fr´echet-Stein structure on the ring Dla(H,K) of locally analytic distributions on any compact open subgroup H of G. In fact, we prove a slightly more general result, which implies not only that Dla(H,K) is a Fr´echet-Stein algebra, but also that the completed tensor product A⊗ˆ Dla(H,K) is Fr´echet-Stein, for a fairly general class of commutative Fr´echet- K Stein K-algebras A, namely, those possessing a Fr´echet-Stein structure each of whose transition maps admits a certain kind of good integral model (as specified in definition 5.3.21). In chapter 6 we study the various admissibility conditions that have arisen so far in locally analytic representation theory. Insection6.1wepresentouralternativedefinitionofthecategoryofadmissible locally analytic G-representations, and prove that it is equivalent to the definition presented in [27].