Locally Analytic Vectors in Representations of Locally P-adic Analytic Groups

168 Pages·2017·1.1 MB·English

by  Matthew J. Emerton

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
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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].