Draft: October 10, 2007
JACQUET MODULES OF LOCALLY ANALYTIC
REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS
II. THE RELATION TO PARABOLIC INDUCTION

Matthew Emerton
Northwestern University
To Nicholas
Contents
1. Preliminaries
2. Parabolically induced representations
3. A variant of the method of Amice-Vélu and Vishik
4. Proof of Theorem 0.13
5. Examples, complements, and applications
Appendix
References

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20
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52
59
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Let L be a finite extension of Qp (for some prime p), let G be (the L-valued
points of) a connected reductive linear algebraic group over L, and let P be aparabolic subgroup of G, with unipotent radical N and Levi factor M . We also
fix an extension K of L, complete with respect to a discrete valuation extendingthat of L. All representations will be on K-vector spaces (whether or not this is
explicitly stated).
In the paper [7] we defined a Jacquet module functor JP that takes locallyanalytic G-representations to locally analytic M -representations, which when restricted to the category of admissible smooth representations of G, coincides with
the Jacquet module functor of [5] (i.e. the functor of N -coinvariants). This functor
plays an important role in the study of certain global questions related to the p-adic
interpolation of automorphic forms [8], and its applications in that context provided
the original motivation for its introduction and study. However, it is natural to ask
whether this functor also has an intrinsic representation theoretic significance. In
considering this question, we are guided by the fact that in the theory of admissible smooth G-representations, the Jacquet module functor is both left, and, formore subtle reasons – namely the Casselman Duality Theorem – right adjoint toparabolic induction. More precisely, one has the two formulas
(0.1)

∼

HomG (V, (IndG
P U )sm ) −→ HomM (JP (V ), U )

and
(0.2)

∼

HomG ((IndG
U )sm , V ) −→ HomM (U (δ), JP (V )),P

The author would like to acknowledge the support of the National Science Foundation (award
number DMS-0070711)

1

2

MATTHEW EMERTON

where U and V are arbitrary admissible smooth representations of M and G respectively, P denotes an opposite parabolic to P , U (δ) denotes U with its M -action
twisted by the modulus character δ of P (thought of as a character of M ), andG(IndG
P U )sm and (IndP U )sm denote the smooth induction of U to G from P and P
respectively. (See [5, §4].)
In this paper we investigate the relation between the locally analytic Jacquet
module functor JP and parabolic induction, with the goal of extending the adjointness formula (0.2) to the locally analytic context. (The formula (0.1) cannot extend
to the locally analytic context, since JP is not right exact, as Example 5.1.9 below
shows.) The formula that we will establish not only serves to establish the representation theoretic significance of the functor JP , but also has important global
applications – for example, to the construction of p-adic L-functions attached toautomorphic forms [8, § 4.5].
The first result that we establish in the direction of our goal is the following
lemma, proved in Subsection 2.8. (We refer to the Notations and conventions
below for the definition of the various categories of locally analytic representations
that appear in the statement of this and subsequent results.)
Lemma 0.3. If U is any object of Repzla.c (M ), then there is a closed M -equivariant
U ). (Here IndG
U denotes the locally analytic parabolic
embedding U (δ) → JP (IndG
P
P
induction of U from P to G.)
For any object U of Repzla.c (M ), this lemma allows us to regard U (with its M action twisted by δ) as a subspace of JP (IndG
U ). We define IPG (U ) to be the closedPG
G-subrepresentation of IndP U generated by the image of U under the canonicallifting of JP (IndG
U ) to IndG
U . (Recall from [7, (0.9)] that the canonical liftPP
depends for its definition on a choice of compact open subgroup N0 of N , as well
as a lift of M to a Levi factor of P . Here andT below we fix such a choice of N0 ,while we take the Levi factor to be M = P P . In fact IPG (U ) is well-definedindependent of the choice of N0 ; see Lemma 2.8.3 below.) The subrepresentation
U that can be “detected” by its Jacquet module. (SeeIPG (U ) is the part of IndG
P
Corollary 5.1.4 below for a precise statement in the case when G is quasi-split and P
is a Borel subgroup.) For example, if U is an admissible smooth M -representation,Gthen IPG (U ) = (IndP
U )sm . Passing to Jacquet modules thus yields, for any U ∈zRepla.c (M ) and V ∈ Repla.c (G), an injection
(0.4)

G

LG (IP (U ), V ) → LM (U (δ), JP (V )).

It is not the case in general that (0.4) is surjective, for the following reason.
If V is an object of Repla.c (P ), then the canonical lifting JP (V ) → V is a U(p)equivariant map (where U(p) acts on JP (V ) through its quotient U(m)), and so
induces a U(g)-equivariant map(0.5)

U(g) ⊗U(p) JP (V ) → V.

(Here we are using gothic letters to denote Lie algebras, and U(–) to denote uniG
(U ) in (0.5) (for any object U ofversal enveloping algebras.) Taking V = IndP
Repzla.c (M )), and taking into account Lemma 0.3, we obtain in particular a U(g)equivariant map(0.6)

U(g) ⊗U(p) U → IPG (U ).

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

3(We have dropped the twist by δ, since δ is a smooth character, and so induces
the trivial character of p.) Any map in the source of (0.4) induces a commutative
diagram(0.7)

U(g) ⊗U(p) U(0.6)


IPG (U )/ U(g) ⊗U(p) JP (V )
(0.5)


/ V.

Definition 0.8. We say that a map in the target of (0.4) is balanced if the kernelof the induced map(0.5)

U(g) ⊗U(p) U −→ U(g) ⊗U(p) JP (V ) −→ V
contains the kernel of (0.6). We let LM (U (δ), JP (V ))bal denote the subspace ofLM (U (δ), JP (V )) consisting of balanced maps. (Note that this space depends notjust on JP (V ) as an M -representation, but on its realization as the Jacquet moduleof V . On the other hand, the kernel of (0.6), and hence the condition that a map
be balanced, is independent of the choice of N0 used to determine the canonicallifting; see Lemma 0.18 below.)
The commutativity of (0.7) shows that LM (U (δ), JP (V ))bal contains the imageof (0.4).
The present paper aims to study the following question.
Question 0.9. Is the injection(0.10)Gbal

LG (IP (U ), V ) → LM (U (δ), JP (V ))induced by (0.4) an isomorphism?
A positive answer (perhaps with some hypotheses on U and V ) would yield ageneralization of (0.2) to the locally analytic context. (Note that the condition ofmaps being balanced does not arise in the context of smooth representations, since
the Lie algebra actions on smooth representations are trivial.)
Before stating our main result, we make two further definitions.
Definition 0.11. Let U be an object of Repla.c (M ), and let A denote the spaceof linear M -equivariant endomorphisms of U . We say that U is allowable if for any
pair of finite dimensional algebraic M -representations W1 and W2 , each element ofLA[M ] (U ⊗K W1 , U ⊗K W2 ) is strict (i.e. has closed image). (Here each U ⊗K Wi
is regarded as an A[M ]-module via the A action on the left-hand factor along withthe diagonal M -action.)
Any essentially admissible locally analytic representation of M is allowable, inthe sense of this definition [6, Prop. 6.4.11, Prop. 6.4.16].
Definition 0.12. We say that an admissible locally analytic G-representation V
is very strongly admissible if there is a continuous G-equivariant K-linear injection
V → W , where W is an admissible continuous representation of G on a K-Banach
space.

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MATTHEW EMERTON

Any very strongly admissible locally analytic G-representation is strongly admissible (see [6, Prop. 6.2.4] and [16, Thm. 7.1]), and conversely when G is compact[6, Prop. 6.5.9]. The author does not know an example of a strongly admissible
locally analytic representation that is not very strongly admissible.
We may now state our main result (proved in Subsection 4.3).
Theorem 0.13. If U is an allowable object of Repzla.c (M ), and if V is a verystrongly admissible G-representation, then the map (0.10) is an isomorphism.
When combined with an explicit computation of Jacquet modules of the locallyanalytic principal series (see Proposition 5.1.5 and Remark 5.1.8 below), Theorem 0.13 has the following corollaries (proved in Subsection 5.3).
Corollary 0.14. Suppose that G is quasi-split, and that P is a Borel subgroupof G. If V is a topologically irreducible very strongly admissible locally analytic
representation of G for which JP (V ) 6= 0 then V is a quotient of IPG (U ) for somefinite dimensional locally analytic representation U of the maximal torus M of G.
Corollary 0.15. Suppose that G is quasi-split, and that P is a Borel subgroup of
G. If V is an admissible locally analytic representation of G of finite length, whose
composition factors are very strongly admissible, then JP (V ) is a finite dimensional
M -representation.
As we remarked above, in addition to these purely representation theoretic applications, Theorem 0.13 has applications in the global context. For example, inthe case when G = GL2 (Qp ), it can be applied to the construction of two-variable
p-adic L-functions attached to overconvergent p-adic modular forms [8, §4.5].
A sketch of the proof of Theorem 0.13. For any object U of Repla.c (M ), denoteby Ccsm (N, U ) the space of locally constant compactly supported U -valued functions
on N ; see Subsection 2.2 below for a description of the topology of Ccsm (N, U ), andof the natural P -action on this space. Let IPG (U )(N ) denote the (g, P )-invariant
closed subspace of IPG (U ) consisting of elements supported on N . (Here we regard N
as an open subset of P \G; we refer to Subsection 2.3 for a definition of the supportof an element of IndG
U as a subset of P \G. Note that Lemma 2.3.5 implies thatPU .) The open immersion N → P \G induces a closed
IPG (U )(N ) is closed in IndGP
sm
U which factors through IPG (U )(N ),P -equivariant embedding Cc (N, U ) → IndG
Pand hence induces a (g, P )-equivariant map U(g) ⊗U(p) Ccsm (N, U ) → IndG
U. If
P
lp
sm
G
IP (U ) (N ) denotes the coimage of this map (i.e. the quotient of U(g)⊗p Cc (N, U )by the kernel of this map), then there is induced a continuous (g, P )-equivariant
injection(0.16)IPG (U )lp (N ) → IPG (U )(N ).

If U is an allowable object of Repzla.c (M ), then IPG (U ) is a local closed GG
subrepresentation of IndP
U , polynomially generated by degree zero (using terminology to be introduced in Section 2), and IPG (U )lp (N ) consists of the locallypolynomial elements of IPG (U )(N ); see Subsection 2.8. The results of Section 2
imply that IPG (U ) is generated as a G-representation by IPG (U )(N ), and that (0.16)
has dense image.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II5

We can now outline the proof of Theorem 0.13. Let U be an allowable object ofRepzla.c (M ), and V be a very strongly admissible locally analytic G-representation.
The adjointness result of [7, Thm. 3.5.6] yields an isomorphism(0.17)∼sm
LM (U (δ), JP (V )) −→ L(g,P ) (U(g) ⊗U(p) Cc (N, U ), V ).

The following result follows essentially by definition.
Lemma 0.18. The isomorphism (0.17) restricts to an isomorphismbal

LM (U (δ), JP (V ))∼−→ L(g,P ) (IPG (U )lp (N ), V ).

Now restriction from IPG (U ) to IPG (U )(N ) induces a map(0.19)GG

LG (IP (U ), V ) → L(g,P ) (IP (U )(N ), V ),while (0.16) induces a map(0.20)GG
lp
L(g,P ) (IP (U )(N ), V ) → L(g,P ) (IP (U ) (N ), V ).

Taking into account Lemma 0.18, we see that Theorem 0.13 will follow if we canshow that each of these maps is an isomorphism.
The proof that (0.19) is an isomorphism is the subject of Subsection 4.1. It
relies crucially on the fact that IPG (U ) is generated by IPG (U )(N ) (and in fact,
on the slightly stronger result of Lemma 2.4.13, which also addresses the issue of
topologies), and on the assumption that the G-action on V is locally analytic. The
proof that (0.20) is an isomorphism is the subject of Subsection 4.2. Since themap (0.16) is injective, with dense image (U being allowable), the key point is toshow that any element of the source of (0.20) remains continuous when IPG (U )lp (N )is equipped with the topology obtained by regarding it as a subspace of IPG (U )via (0.16). This is done using a variant of the method of Amice-Vélu and Vishik[1, 18]; it is here that we use the assumption that V is very strongly admissible.
The arrangement of the paper. Section 1 develops some necessary preliminary
results. In Subsection 1.1 we establish a useful structural result for objects ofRepzla.c (G), for any locally L-analytic group G. In Subsection 1.2 we prove somebasic results on what we call locally integrable g-representations; these are continuous representations of the Lie algebra g of a locally L-analytic group G for which
the action on any vector may be integrated to an action of some compact open
subgroup of G. In Subsection 1.3 we prove a certain technical lemma. In Subsection 1.4 we describe some simple functional analysis related to spaces of compactly
supported functions, and also to spaces of germs of locally analytic functions, on
locally L-analytic spaces. In Subsection 1.5 we recall a result from highest weight
theory.
In Section 2 we return to the setting of the introduction. After an initial discussion of locally analytic parabolic induction in Subsection 2.1, and after establishingsome notation in Subsection 2.2, in Subsections 2.3 through 2.7 we develop theGstructure theory of what we call local closed G-subrepresentations of IndP
U. In
GSubsection 2.8 we apply these results to the representation IP (U ).

6

MATTHEW EMERTONAs above, let N0 denote a compact open subgroup of N . Let ZM denote the+centre of M , and define ZM= {z ∈ ZM | zN0 z −1 ⊂ N0 }. In Section 3 we fix
a K-Banach space U equipped with a continuous ZM -action (note that there is+then a natural action of the semidirect product N0 ZM
on C la (N0 , U )), a pair of+la
N0 ZM -invariant closed subspaces S ⊂ T ⊂ C (N0 , U ), and a K-Banach space+
W equipped with a continuous N0 ZM-action. We then give a criterion for the
restriction map LN0 Z + (T, W ) → LN0 Z + (S, W ) to be an isomorphism. (Actually,MM
we replace S and T by their ultrabornologicalizations; this is harmless from thepoint of view of our intended applications.) This result provides the variant of theresults of Amice-Vélu and Vishik on tempered distributions that was alluded toabove. The criterion that we give is described in terms of certain extra structure
whose existence we assume. In Subsection 3.1 we describe this extra structure, andstate our isomorphism criterion in terms of it, while the proof of the isomorphism
criterion is given in Subsection 3.2.
In Section 4 we present the proof of Theorem 0.13. In fact we prove a moregeneral result, in which IPG (U ) is replaced by any local closed G-subrepresentationG
X of IndP
U that is polynomially generated by bounded degrees (as defined in
Subsection 2.7); this result will be useful in future applications. Subsection 4.1(respectively Subsection 4.2) proves that (the analogue for arbitrary X of) themap (0.19) (respectively the map (0.20)) is an isomorphism. In Subsection 4.3 webring these results together to prove our main result, which then has Theorem 0.13
as a consequence.
In Section 5 we give some complements to and applications of our results, focusing on the case when G is quasi-split and P is a Borel subgroup. In Subsection 5.1
we present some results concerning the Jacquet modules of parabolically induced
representations. In Subsection 5.2 we establish a certain technical result (Theorem 5.2.18). In Subsection 5.3 we prove Corollaries 0.14 and 0.15. In an appendix
we establish some general results about admissible locally analytic representations
that are required in Subsection 5.2.
A special case of the main theorem. A proof of the main theorem of Section 4,
namely Theorem 4.3.2, in the special case when G = GL2 (Qp ), the parabolic P is a
Borel subgroup, the representation U being induced is a locally analytic character,Gand the local closed G-subrepresentation X of IndP
U is taken to be IndG
U itself,Pis given in the appendix to [3]. The argument in this special case follows the samelines as in the general case, but with many simplifications, both in the group theory
and in the functional analysis. The reader interested in following the details of ourargument in the general case may find it helpful to first study the argument in thissimplified setting.
Notation and conventions. As in the introduction, we fix a prime p, a finite
extension L of Qp , and an extension K of L that is complete with respect to adiscrete valuation that extends the discrete valuation on L. All vector spaces that
we consider will have coefficients in K (whether or not this is explicitly mentioned).
Throughout this paper, we will systematically identify linear algebraic groups
over L with their groups of points; this should cause no confusion, since for a linear
algebraic group, the set of L-valued points is Zariski dense in the corresponding
algebraic group.
If V is a topological vector space, then by a topological action of a group (orsemigroup, ring, Lie algebra, etc.) on V we mean an action on V via continuous

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

7

endomorphisms. (We use this term in distinction to a continuous action on V ,meaning an action for which the action map is jointly continuous, with respect tothe topology on V and whatever topology is under consideration on the object thatis acting.)
We refer to [6, §1] for the notion of a BH-space, and of a BH-subspace of aconvex K-space. If V is a BH-space, then we let V denote the latent Banach space
underlying V . (See [6, Def. 1.1.1, Prop. 1.1.2].) More generally, if V is an arbitrary
Hausdorff convex K-vector space, then we let V denote the ultrabornologicalizationof V ; that is, V := lim
W , where W runs over all BH-subspaces of V . (It follows−→
Wfrom [6, Prop. 1.1.2] that the transition maps are continuous, and we endow V withthe locally convex inductive limit topology.) Since V is a locally convex inductive
limit of Banach spaces, it is ultrabornological [2, ex. 20, p. III.46]. The continuous
injections W → V induce a continuous bijection V → V , which is an isomorphism
if and only if V is itself ultrabornological.
We refer to [6, §1] for the notion of a space of LB-type, and of an LB-space, andto [2, Cor., p. II.34] for the version of the Open Mapping Theorem that applies tomaps between LB-spaces.
All locally L-analytic spaces appearing are assumed to be paracompact (andhence strictly paracompact in the sense of [13, p. 446], by [12, Satz 8.6]).
For any locally L-analytic group G, we let Reptop.c (G) denote the category whose
objects are Hausdorff locally convex K-vector spaces of compact type, equippedwith a topological action of G, and whose morphisms are continuous G-equivariant
K-linear maps. We let Repla.c (G) denote the full subcategory of Reptop.c (G) consisting of locally analytic representations of G on convex K-vector spaces of compact
type. (The notion of a locally analytic representation of G is defined in [13, p. 12];see also [6, Def. 3.6.9].)
If the centre ZG of G is topologically finitely generated then we let Repes (G)denote the full subcategory of Repla.c (G) consisting of essentially admissible locally analytic representations of G (as defined in [6, Def. 6.4.9]). We let Repad (G)denote the full subcategory of Repes (G) consisting of admissible locally analytic
representations of G (as defined in [16]; see also [6, Def. 6.1.1]).
There is another full subcategory of Repla.c (G) that we will consider, to be denoted by Repzla.c (G). The objects of this category are convex K-spaces V of compact
type, equipped with a locally analytic G-representation, that may be written as theunion of an increasing sequence of ZG -invariant BH-subspaces. Recall that bydefinition, any object of Repes (G) is an object of Repzla.c (G).
We refer to [7, §3.1] for a summary of the basic results regarding these categories
of G-representations.
We briefly recall the notion of an analytic open subgroup of an L-analytic group,referring to [6, §3.5, §5.2] for a detailed discussion. If G is an L-analytic group, thenan analytic open subgroup of G actually consists of the following data: an open
subgroup H of G, a rigid L-analytic group H that is isomorphic as a rigid analytic∼
space to a closed ball, and a locally L-analytic isomorphism H −→ H(L). Although
all three pieces of data are required to specify an analytic open subgroup, typicallywe will refer simply to an analytic open subgroup H of G. We adopt the convention
that if an analytic open subgroup is denoted by H (possibly decorated with somesubscripts or superscripts), then the underlying rigid analytic group will always bedenoted by H (decorated with the same superscripts or subscripts). Given analytic

8

MATTHEW EMERTON

open subgroups H1 and H2 of G, and an inclusion H2 ⊂ H1 , we say that thisis an inclusion of analytic open subgroups if it arises from a rigid analytic open
immersion H2 → H1 by passing to L-valued points.
If H is an analytic open subgroup of G, and g ∈ G, then we will write gH todenote a copy of the rigid analytic space H, with the understanding that the locallyanalytic space of L-valued points of gH is to be naturally identified with the coset∼∼gH ⊂ G via the locally L-analytic isomorphism (gH)(L) = H(L) −→ H −→ gH(the final isomorphism being provided by left multiplication by g). For `example,if {gi }i∈I is a collection of right coset representatives of H in G, then i∈I gi H
is a rigid` analytic space whose set of L-valued points is naturally isomorphic to
G = i∈I gi H as a locally L-analytic space. We will apply similar notation withregard to the left cosets Hg of H in G.
Finally, we recall from [6] that we call an analytic open subgroup H of G good
if there exists an OL -Lie sublattice h of the Lie algebra g of G (i.e. h is a finitely
generated OL -submodule of g which spans g over L, and which is closed under the
Lie bracket) on which the Lie bracket may be exponentiated to form a rigid analytic
group H, whose underlying group of L-valued points may be identified with H (in amanner compatible with the inclusion h ⊂ g). Good analytic open subgroups of G
exist, and are cofinal among all analytic open subgroups of G [17, LG
T 5.35, Cor. 2].
If H is a good analytic open subgroup H of G, and r ∈ (0, 1) |L× |, then thepolydisk of radius r around the origin of H is an affinoid rigid analytic group, which
we denote by Hr . We let Hr := Hr (L) denote the
S corresponding locally analytic
groups of L-valued points. We also write H◦ := r<1 Hr , and write H ◦ := H◦ (L).
(Note that H ◦ = Hr for r close enough to 1.) See [6, §5.2] for more details of these
constructions.
1. Preliminaries(1.1) Let G be a locally L-analytic group, and let H be a compact open subgroupof G. Note that HZG is then an open subgroup of G.
Lemma 1.1.1. We may find a a cofinal sequence {Hi }i≥0 of analytic open subgroups of H, chosen so that H normalizes each Hi . (Recall that this means that H
normalizes the group of L-rational points Hi of Hi , and that the resulting action of
H on Hi via conjugation is induced by a corresponding action of H on Hi via rigid
analytic automorphisms.)
Proof. Let g denote the Lie algebra of G (and hence also of H), and fix an OL -Liesublattice h of g. We may find an OL -Lie sublattice h0 of h which is invariant under
the adjoint action of H. Replacing h0 by pn h0 for sufficiently large n if necessary, we
may furthermore assume that h0 exponentiates to a good analytic open subgroup
H0 of H. If we let Hi be the good analytic open subgroup attached to pi h0 for eachi ≥ 0, then {Hi }i≥0 satisfies the requirements of the lemma. 
We now prove a useful structural result regarding objects of Repzla.c (G).
Lemma 1.1.2. If {Hi }i≥0 is any cofinal sequence of analytic open subgroups of
H satisfying the conclusion of Lemma 1.1.1, and if U ∈ Repzla.c (G), then we may∼
write U −→ lim
Ui , where each Ui is a K-Banach space equipped with an Hi -analytic−→i

HZG -action, and the transition maps Ui → Ui+1 are compact, injective, and HZG equivariant.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

9

Proof. Since U is a locally analytic representation of G, and hence of H, on acompacttype space (and so in particular on an LB-space), we may write U =S
i≥0 U1,i , where {U1,i }i≥0 is an increasing sequence of H-invariant BH-subspacesof U , such that for each i ≥ 0, the induced H-representation on U 1,i is Hi -analytic.
z(This follows from [6, Prop.
S 3.2.15, Thm. 3.6.12].) Since U is an object of Repla.c (G),
we may also write U = i≥0 U2,i , where {U2,i }i≥0 is an increasing sequence of ZG invariant BH-subspaces of U .
◦For any i ≥ 0, the unit ball U2,iof U 2,i (with respect to some norm defining
the Banach space structure on U 2,i ) is bounded in U , and thus so is the closure◦in U (since H is compact, and so acts equicontinuously on U , by [6,of HU2,i◦Lem. 3.1.4]). The K-span of HU2,iis thus an HZG -invariant BH-subspace of U
(since U is complete; see [2, Cor., p. III.8]) that contains U2,i . Replacing U2,i bythis BH-subspace, we may assume that each U2,i is in fact HZG -invariant.
By passing to appropriate subsequences of the two sequences {U1,i }i≥0 and{U2,i }i≥0 of BH-subspaces of U , and relabelling as necessary, we may furthermore
assume that U1,i ⊂ U2,i for all i ≥ 0 [2, Prop. 1, p. I.20]. Passing to latent Banach
space structures (taking into account [6, Prop. 1.1.2 (ii)]) and then to Hi -analytic
vectors (taking into account [6, Thm. 3.6.3]) this inclusion induces continuous injections
U 1,i = (U 1,i )Hi −an → (U 2,i )Hi −an → U 2,i .
∼We deduce that U −→ lim(U 2,i )Hi −an .
−→iBy [6, Prop. 1.1.2 (ii)] and the functoriality of passing to Hi -analytic vectors,
we see that the HZG -action on U2,i induces a HZG -action on (U 2,n )Hi for eachi ≥ 0. Since U is of compact type, by passing to a subsequence of {(U 2,i )Hi −an }i≥0
and then relabelling we may furthermore assume that each of the transition maps
(U 2,i )Hi −an → (U 2,i+1 )Hi+1 −an is compact. Taking Ui = (U 2,i )Hi −an gives thelemma. (1.2) Let G be a locally L-analytic group with Lie algebra g. In this subsection
we develop some simple definitions and results related to g-representations that are
locally integrable, in the sense that the g-action on any vector may be integrated
to an action of some compact open subgroup of G.
Let H be an analytic open subgroup of G. If V is a K-Banach space, thenC an (H, V ) := C an (H, K) ⊗K V is again a K-Banach space, equipped with mutually commuting actions of H by each of the left and right regular actions. By [6,Prop. 3.3.4], these actions are H-analytic in the sense of [6, Def. 3.6.1]. If V isfurthermore equipped with a topological g-action (which is then necessarily continuous, since V is Banach, and so barrelled), then we obtain a topological actionof g on C an (H, V ), namely the action induced by the action on V (the “pointwise
action”), which commutes with each of the regular H-actions.
We define a g × g-action on C an (H, V ) by having the first factor act throughthe pointwise action, and the second factor act through the derivative of the left
regular H-action; this g × g-action commutes with the right regular H-action. If∆ : g → g × g denotes the diagonal embedding, then composing the g × g-action
with ∆ induces yet another g-action on C an (H, V ); we denote this copy of g acting
on C an (H, V ) by ∆(g), to avoid confusion.

10

MATTHEW EMERTON

Definition 1.2.1. We write VH−int := C an (H, V )∆(g) , the closed subspace of invariants in C an (H, V ) under the action of ∆(g), and refer to VH−int as the space of
H-integrable vectors in V . Note that since the right regular H-action on C an (H, V )commutes with the ∆(g)-action, this action restricts to an H-action on VH−int , andwe regard VH−int as an H-representation in this way.
Since VH−int is a closed subspace of the Banach space C an (H, V ), it is again a
Banach space. The formation of VH−int is evidently functorial in V . Evaluation at
the identity of H induces a natural continuous map(1.2.2)

VH−int → V.

Lemma 1.2.3. (i) The H-action on VH−int is H-analytic.
(ii) The map (1.2.2) is injective and g-equivariant, if we endow VH−int with theg-action obtained by differentiating the H-action. (This make sense, by (i).)
Proof. Claim (i) follows from [6, Prop. 3.6.2], since VH−int is a closed subrepresentation of the H-analytic H-representation C an (H, V ).
If we differentiate the H-action on VH−int , we obtain a g-action on VH−int , withrespect to which the map (1.2.2) (which we recall is induced by evaluating elementsof VH−int ⊂ C an (H, V ) at the identity element of H) is immediately checked to be gequivariant. The kernel of (1.2.2) is thus g-invariant, and so an element in the kernelof (1.2.2) is a rigid analytic V -valued function on H whose power series expansion
at the identity vanishes identically. Consequently (1.2.2) has trivial kernel, and (ii)is proved. 
Definition 1.2.4. We say that a topological g-action on a K-Banach space V is
H-integrable if the map (1.2.2) is a bijection (in which case the Open Mapping
Theorem shows that it is a topological isomorphism, since the source and domain
are both Banach spaces).
If the Banach space V is equipped with an H-integrable action of g, then we may
use the isomorphism (1.2.2) to transport the H-action on VH−int to an H-action on
V . We will say that this H-action is obtained by “integrating the g-action”.
Lemma 1.2.5. If V is a K-Banach space equipped with an H-analytic action ofH, then the g-action on V obtained by differentiating the H-action is H-integrable,
and the map (1.2.2) is H-equivariant.
Proof. If we define the ∆1,2 (H)-action on C an (H, V ) as in the discussion of [6, §3.3],
then its derivative gives the ∆(g)-action described above, and so we have continuous
H-equivariant injections(1.2.2)C an (H, V )∆1,2 (H) −→ C an (H, V )∆(g) −→ V.
By definition (see [6, Def. 3.3.1, Def. 3.6.1]), our assumption implies that the composite of these injections is a bijection, and thus so is the second, that is, themap (1.2.2). JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

11

Proposition 1.2.6. If V is a K-Banach space equipped with a topological g-action,and if W is a closed g-invariant subspace of V , then the natural commutative diagram/ VH−int
WH−intW/V(in which the vertical arrows are provided by (1.2.2), the bottom horizontal arrow is
the inclusion, and the top horizontal arrow arises from the bottom one by applying
the functor “pass to H-integrable vectors”) is Cartesian (in the category of KBanach spaces).
Proof. By [6, Prop. 2.1.23], the given closed embedding of W into V induces aclosed embedding C an (H, W ) → C an (H, V ). Passing to ∆(g)-invariants yields the
proposition. 
Corollary 1.2.7. If V is a K-Banach space equipped with an H-integrable actionof g, and if W is a closed g-invariant subspace of V , then the g-action on W isagain H-integrable, and the inclusion W ⊂ V is H-equivariant, if we endow W and
V with the H-actions provided by integrating their respective g-actions.
Proof. This follows directly from Proposition 1.2.6.

Proposition 1.2.8. Let H2 ⊂ H1 be an inclusion of analytic open subgroups ofG. If V is a K-Banach space equipped with a topological g-action, then there is anatural g-equivariant continuous injection VH1 −int → VH2 −int , uniquely determinedby the requirement that the diagram/ VH2 −int
VH1 −int
KK
KK
KK
KK
KK 
%
V,
in which the diagonal and vertical arrows are provided by (1.2.2), commutes.
Proof. Since H2 ⊂ H1 is an inclusion of analytic open subgroups, by definition it
arises from a rigid analytic open map H2 → H1 by passing to L-valued points.
The required map is obtained from the pull-back map C an (H1 , V ) → C an (H2 , V ) bypassing to ∆(g)-invariants. 
Let V be a convex K-space equipped with a topological g-action. If W is ag-invariant BH-subspace of V , then [6, Prop. 1.1.2 (ii)] shows that the g-action
on W lifts to a topological g-action on the latent Banach space W ; and so for any
analytic open subgroup H of G, we may define W H−int following Definition 1.2.1.
If W1 ⊂ W2 is an inclusion of g-invariant BH-subspaces of V , and H2 ⊂ H1 an
inclusion of analytic open subgroups of G, then we obtain a commutative diagramof continuous g-equivariant injections(W 1 )H1 −int

/ (W 2 )H −int
(W 1 )H2 −int
/ (W 2 )H −int ,

1

2

12

MATTHEW EMERTON

in which the bottom (respectively top) arrow is induced by the functoriality of theformation of H1 -integrable (respectively H2 -integrable) vectors, while the two vertical arrows are induced by Proposition 1.2.8. (The commutativity of this diagram
follows from the naturality of the maps given by Proposition 1.2.8.) Composing
either the left-hand vertical and bottom arrows or the top and right-hand verticalarrows of this diagram, we obtain a continuous g-equivariant injection(W 1 )H1 −int → (W 2 )H2 −int .

(1.2.9)As W ranges over the directed set of all BH-subspaces of V , and H ranges over the
directed set of all analytic open subgroups of G, the maps (1.2.9) make a directed
set out of the collection of g-representations W H−int .
Definition 1.2.10. If V is a convex K-space equipped with a topological g-action,then we define the space of locally integrable vectors, Vlint , in V to be the locally
W H−int , where H runs over the directed set of
convex inductive limit Vlint := lim−→
H,Wanalytic open subgroups of G, W runs over the directed set of g-invariant BHsubspaces of V , and the transition maps are defined via (1.2.9).
Since each of the space W H−int appearing in the inductive limit that defines Vlint
is equipped in a natural way with a continuous g-action, and since the transition
maps (1.2.9) are g-equivariant, we see that Vlint is equipped in a natural way with acontinuous g-action. (One sees immediately that the g-action on Vlint is topological;
however it is then necessarily continuous, since Vlint is barrelled, being defined as
the locally convex inductive limit of Banach spaces). The formation of Vlint (withits g-action) is obviously functorial in V , and the maps (1.2.2) (with V replaced by
the spaces W ) induce a natural continuous g-equivariant injection
Vlint → V.

(1.2.11)

Proposition 1.2.12. If V is a Hausdorff convex K-vector space equipped with a
topological g-action, then the natural map (Vlint )lint → Vlint (obtained by substituting
Vlint for V in (1.2.11)) is a topological isomorphism.
Proof. Let W be a g-invariant BH-subspace of V , and let H be an analytic open
subgroup of V . Then by definition of Vlint (and taking into account Lemma 1.2.3)
the space W H−int is identified with a g-invariant BH-subspace of Vlint . It is also
equipped with an H-analytic representation, by Lemma 1.2.3, and so Lemma 1.2.5
shows that the map (W H−int )H−int → W H−int (obtained by taking V to be W H−int
in (1.2.2)) is a topological isomorphism. Passing to the locally convex inductive
limit over all such W and H, we obtain the isomorphism that forms the top row of
the commutative diagram(1.2.13)lim (W H−int )H−int

−→
H,W

∼

W H−int
/ lim
−→
H,W


(Vlint )lint∼/ Vlint ,in which the left vertical arrow is the continuous injection induced by definition of(Vlint )lint , the right vertical arrow is the isomorphism induced by definition of Vlint ,JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

13

and the bottom arrow is induced by replacing V by Vlint in (1.2.11). Since the top
and right-hand arrows of (1.2.13) are topological isomorphisms, while the other two
arrows are continuous injections, we see that these other two arrows (in particular,
the bottom arrow) are also topological isomorphisms, as required. 
Proposition 1.2.14. For any Hausdorff convex K-vector space V of LB-type
equipped with a topological g-action, Vlint is an LB-space.
Proof. Since V is the union of a countable collection of BH-spaces, and since G
admits a countable neighbourhood basis of the identity consisting of analytic open
subgroups, we may replace the directed set appearing in the inductive limit ofDefinition 1.2.10 by a cofinal directed set that is countable. Thus Vlint may be
written as the locally convex inductive limit of a sequence of Banach spaces, andso by definition is an LB-space. 
Proposition 1.2.15. Let V be a Hausdorff convex K-vector space equipped with a
topological g-action, and let W be a g-invariant closed subspace of V . Let W1 denotethe preimage of W in Vlint under (1.2.11), which is a g-invariant closed subspaceof Vlint . Then the map Wlint → Vlint (induced by functoriality of the formation of
locally integrable vectors) induces a continuous bijection of Wlint onto W1 .
Proof. Consider the diagram of continuous g-equivariant injections/ W1
WlintE
EE
EE
EE
E" 
W

!
/ Vlint/ V,

in which the outer part of the diagram (obtained by omitting W1 ) arises from thenatural transformation (1.2.11) and the functoriality of the formation of locallyintegrable vectors, while the square is a Cartesian diagram of topological K-vector
spaces. We must show that any vector w ∈ W1 lies in the image of the map
Wlint → W1 . From the definition of Vlint and the construction of W1 , we see that
we may find a g-invariant BH-subspace X of V , and an analytic open
T subgroup
H of G, so that under the map Vlint → V , the image of T
w lies in W X, while w
lies in X H−int . Since W is closed in V , we see that W X is a BH-subspace ofT
W , and that W X is a closed g-invariant subspace of X. Proposition 1.2.6 thus
Tshows that w lies in (W X)H−int . From the definition of Wlint , we see that w does
lie in the image of the map Wlint → W1 , as required. 
Definition 1.2.16. We say that a topological g-action on a convex K-space V is
locally integrable if V is barrelled, and if the map (1.2.11) is a bijection.
Proposition 1.2.17. If V is an LB-space equipped with a locally integrable gaction, then the continuous bijection (1.2.11) is in fact a topological isomorphism.
Proof. Proposition 1.2.14 implies that Vlint is again an LB-space. Thus (1.2.11) isa continuous bijection between LB-spaces, and so by the Open Mapping Theorem
necessarily is a topological isomorphism. 

14

MATTHEW EMERTON

Proposition 1.2.18. If V is a barrelled convex K-space equipped with a locallyanalytic G-action, then the resulting g-action on V makes V a locally integrable
g-representation.
Proof. By assumption the map Vla := lim
lim W H−an is an isomorphism (the in−→ −→H

W

ductive limit being taken over the directed set of analytic open subgroups H ofG, and (H being fixed), the directed set of H-invariant BH-subspaces W of V ).
Each of the Banach spaces W H−an is an H-analytic representation of H, and so
by Lemma 1.2.5 the continuous injection (W H−an )H−int → W H−an is an isomorphism. Since the images of the spaces (W H−an ) in V (with respect to the maps
W H−an → W → W ⊂ V ) have union equal to all of V by assumption, we see thatindeed the map (1.2.11) is a bijection. 
Proposition 1.2.19. Let V be a Hausdorff convex K-vector space equipped witha locally integrable representation of g, and let W be a closed G-invariant subspaceof V . If W is barrelled, then W is also a locally integrable g-representation.
Proof. Proposition 1.2.15 implies that the natural map Wlint → W is a bijection. 
Proposition 1.2.20. If V is a Hausdorff convex K-vector space equipped with∼Vi −→ V,a topological g-action, and if there is a g-equivariant isomorphism lim−→
i∈I

where {Vi }i∈I is a g-equivariant inductive system of Hausdorff K-vector spaces,
each equipped with a locally integrable action of g, then the g-action on V is again
locally integrable.
Proof. Since V is the locally convex inductive limit of the barrrelled spaces Vi , itis barrelled. Functoriality of the formation of locally integrable vectors yields thecommutative diagramlim(Vi )lint
/V
−→
i∈I


lim
Vi
−→
i∈I

lint


/ V.

The left-hand vertical arrow and lower horizontal arrow are both continuous bijections, by assumption, and thus so is the right-hand vertical arrow (since it is apriori a continuous injection). 
Proposition 1.2.21. Let H be an analytic open subgroup of G, let W be a Banach space equipped with an H-integrable action of g, let Y be a dense g-invariant
subspace of W , and let V be an LB-space equipped with a topological G-action.
Suppose that φ : W → V is a continuous K-linear map, with the property that φ|Yfactors through a g-equivariant map Y → VH−an . Then φ itself factors through acontinuous H-equivariant map W → VH−an .
Proof. Since g acts in an H-integrable fashion on W , Lemma 1.2.3 provides an Hanalytic H-action on W . Consider the continuous map Φ : H × W → V defined by
Φ(h, w) = φ(hw) − hφ(w). If we fix w ∈ Y , then the resulting map is an element of
C an (H, V ) (since φ|Y factors through VH−an ), and all its derivatives vanish (since φ|Y
is g-equivariant). Thus Φ|H×Y vanishes, and since Y is dense in W , we conclude that

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

15

Φ vanishes, and thus that φ is H-equivariant. Since H acts H-analytically on W ,we conclude that φ factors through a continuous H-equivariant map W → VH−an ,as claimed. 
Corollary 1.2.22. Let H be an analytic open subgroup of G, let W be a Banach
space equipped with an H-integrable action of g, and let V be an LB-space equippedwith a locally analytic G-action. If φ : W → V is a g-equivariant continuous Klinear map, then there is an analytic open subgroup H 0 ⊂ H such that φ factors
through a continuous H 0 -equivariant map W → VH0 −an .
Proof. By assumption Vla → V is a bijection. Since V is an LB-space and W is a
Banach space, it follows from the definition of Vla [6, Def. 3.5.3] and [2, Prop. 1,
p. I.20] that we may find an inclusion of analytic open subgroups H 0 ⊂ H so that φfactors through a continuous map W → U , where U is an H 0 -invariant BH-subspaceof V for which the H 0 -action on the underlying latent Banach space U is H0 -analytic.
The corollary thus follows from Proposition 1.2.21 (taking Y = W ). 
Proposition 1.2.23. Let W be an LB-space over K equipped with a locally in∼
W i , where each Wi is a g-invariant BHtegrable g-action, and write W −→ lim−→
i≥0subspace of W on which the g-action is again locally integrable. Fix a g-invariant
subspace Y of W , and for each i ≥ 0 let Yi denote the preimage of Y under thecontinuous injection W i → W . Suppose that:(i) Yi is dense in W i for each i ≥ 0;(ii) Yi contains a Banach subspace of W i that generates Yi as g-module.
In addition, let V be a convex K-space of LB-type equipped with a topological
G-action, and let φ : W → V be a continuous K-linear map with the property thatφ|Y factors through a g-equivariant map Y → Vla . Then φ itself factors through acontinuous g-equivariant map W → Vla .
Proof. Fix a value of i ≥ 0, and let Bi ⊂ Yi be a Banach subspace of Yi thatgenerates Yi as a g-module. (Such a subspace exists, by assumption (ii).) Since
Bi ⊂ Y, we have that φ(Bi ) ⊂ Vla by assumption. Thus φ(Bi ) ⊂ VH−an for someanalytic open subgroup H of V (since Bi is a Banach space; cf. the proof of thepreceding corollary). Since W i is a Banach space on which the g-action is locallyintegrable, it follows from [2, Prop. 1, p. I.20] that by shrinking H if necessary, we
may arrange for the g-action on W i to be H-integrable. Since φ|Y is g equivariant,since U(g)Bi = Yi , and since VH−an is a g-invariant subspace of Vla , we see that φ|Yi
is g-equivariant, with image lying in VH−an . Proposition 1.2.21 and assumption (i)thus imply that φ|W i factors through a continuous g-equivariant map W i → VH−an ,and so in particular through a continuous g-equivariant map W i → Vla . Lettingi → ∞ yields the proposition. (1.3) Let H2 ⊂ H1 be an inclusion of analytic open subgroups of a locally Lanalytic group G, and let U be a K-Banach space. We endow C an (H1 , U ) (respectively C an (H2 , U )) with an H1 -action (respectively an H2 -action) via the right regular representation. As was recalled in Subsection 1.2, these actions are H1 -analyticand H2 -analytic respectively and differentiating them induces a g-action on each of
C an (H1 , U ) and C an (H2 , U ). (As usual g denotes the Lie algebra of G.) Restricting
functions from H1 to H2 yields a g-equivariant map C an (H1 , U ) → C an (H2 , U ).

16

MATTHEW EMERTON

Lemma 1.3.1. Let X be a closed g-invariant subspace of C an (H2 , U ). If f ∈C an (H1 , U ) is such that f|H2 ∈ X, then for each h ∈ H1 the function (hf )|H2 again
lies in X.
Proof. Since the H1 -action on C an (H1 , U ) is H1 -analytic, the map h 7→ hf is a rigid
analytic map from H1 to C an (H1 , U ), and thus
h 7→ (hf )|H2

(1.3.2)is a rigid analytic map from H1 to C an (H2 , U ). Since the map f 7→ f|H2 is gequivariant, since X is g-invariant, and since f|H2 lies in X, one furthermore computes that all derivatives of arbitrary order of (1.3.2) at e lie in X. Thus (since X
is closed) we see that (1.3.2) is in fact an X-valued rigid analytic function on H1 .
This proves the lemma. (1.4) For any locally L-analytic space X and any Hausdorff convex K-vector
space U , we let Ccla (X, U ) denote the vector space of compactly supported locallyanalytic U -valued functions on X. If Ω ⊂ Ω0 is an inclusion of compact open subsetsof X, then extension by zero defines an injective map(1.4.1)

C la (Ω, U ) → C la (Ω0 , U ).

Extension by zero also defines an injective map(1.4.2)C la (Ω, U ) → Ccla (X, U ),for each compact open subset Ω of X. Clearly the maps (1.4.2) are compatible withthe maps (1.4.1), and so we obtain a map(1.4.3)C la (Ω, U ) → Ccla (X, U ),
lim
−→
Ω

in which the inductive limit is taken over all compact open subsets of X.
Lemma 1.4.4. The map (1.4.3) is an isomorphism of abstract K-vector spaces.
Proof. This follows from the very definition of a function having compact support. 
Lemma 1.4.5. If we equip source and target with their usual topology (as recalledin [6, Def. 2.1.25] or [13, p. 5]), then (1.4.1) is a closed embedding.
`
Proof. If we write Ω00 := Ω0 \ Ω, then the equation Ω0 = Ω Ω00 expresses Ω0 as the
disjoint union of two compact open subsets. Thus there is a natural isomorphism∼
C la (Ω0 , U ) −→ C la (Ω, U ) ⊕ C la (Ω00 , U ). The map (1.4.1) corresponds to the inclusionof the first direct summand, and so is indeed a closed embedding. In particular, this lemma shows that the transition maps in the inductive limitof (1.4.3) are continuous, and so we may endow it with its locally convex inductive
limit topology.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

17

Definition 1.4.6. We topologize Ccla (X, U ) by equipping lim
C la (Ω, U ) with its−→Ω

locally convex inductive limit topology, and declaring (1.4.3) to be a topologicalisomorphism.
The formation of Ccla (X, U ) is evidently covariantly functorial in U , and contravariantly functorial with respect to proper maps in X.
The obvious natural map Ccla (X, U ) → C la (X, U ) is a continuous injection. Sincela
C (X, U ) is Hausdorff (see the remark following [6, Prop. 2.1.26]) we see that
Ccla (X, U ) is also Hausdorff. Alternatively, this follows from the fact that the transition maps in (1.4.3) are closed embeddings (by Lemma 1.4.5), since a strict inductive limit of Hausdorff convex spaces is Hausdorff [2, Prop. 9, p. II.32].
Lemma 1.4.7. If X is σ-compact and U is of compact type, then Ccla (X, U ) is ofcompact type.
Proof. Since X is σ-compact, we may find a cofinal subsequence of compact open
subsets of X. By [6, Prop. 2.1.28] the inductive limit (1.4.3) is thus isomorphic to
the inductive limit with injective transition maps of a sequence of compact typespaces, and so is again of compact type. Lemma 1.4.8. If G is a locally L-analytic group and U is a Hausdorff locallyconvex K-space, then the right regular action of G on Ccla (G, U ) is locally analytic.
Proof. We fix a compact open subgroup G0 of G. We may then restrict the compact
open subsets Ω in the inductive limit (1.4.3) to consist of finite unions of right cosetsof G0 in G. The right regular action of G0 on such a coset gG0 induces a locallyanalytic action of G on C la (gG0 , U ), by the discussion following [6, Def. 3.6.9],
and hence on C la (Ω, U ) for any finite union Ω of such cosets. It follows from [6,Prop. 3.6.17] that Ccla (G, U ) is locally analytic as a G0 -representation, and so also
as a G-representation. 
Definition 1.4.9. If x is a point of a locally L-analytic space X, and U is a
Hausdorff locally convex K-vector space, then we let C ω (X, U )x denote the spaceof germs at the identity x of locally analytic U -valued functions on X; that is
C la (Ω, U ),C ω (X, U )x = lim−→Ω3xwhere the locally convex inductive limit is taken over the directed set of open
neighbourhoods Ω of x in X, and the transition maps are given by restricting
functions.
Lemma 1.4.10. If the V is a convex K-space of compact type, then for any point
x on a locally L-analytic space X, the space of germs C ω (X, V )x is again of compact
type.
Proof. Let X0 be a chart around x in X, so that there exists a closed rigid analytic∼ball X0 such that X0 −→ X0 (L). Let {Xi }i≥0 be a cofinal sequence of admissible
open affinoid balls containing x in X0 , chosen so that the inclusion Xi+1 ⊂ Xi is∼
relatively compact. Write V −→ lim
Vi , where each transition map Vi → Vi+1 is a−→icompact injection between K-Banach spaces. There is then a natural topological∼isomorphism lim
C an (Xi , Vi ) −→ C ω (X, K)x , where the transition maps defining the
−→
i

18

MATTHEW EMERTON

inductive limit are obtained by restricting from Xi to Xi+1 , together with mappingfrom Vi into Vi+1 . We may rewrite the transition map C an (Xi , Vi ) → C an (Xi+1 , Vi+1 )in the form
(1.4.11)

ˆ Vi → C an (Xi+1 , K) ⊗
ˆ Vi+1 ,C an (Xi , K) ⊗K

K

where the arrow is obtained as the completed tensor product of the restriction map
C an (Xi , K) → C an (Xi+1 , K) with the transition map Vi → Vi+1 . The former map iscompact since the inclusion Xi+1 ⊂ Xi is relatively compact, while the latter map iscompact by assumption. Thus it follows from [6, Cor. 1.1.27] and [11, Lem. 18.12]
that (1.4.11) is injective and compact, and hence that C ω (X, K)x is of compact
type, as claimed. (1.5) In this subsection we recall a standard lemma from highest weight theory.
Not knowing a reference, we include a proof.
We first establish the necessary notation. Let g be a split reductive Lie algebra
over K, let p be a fixed Borel subalgebra
L of g, with nilpotent radical n, let m be ap, chosen
Cartan subalgebra
of
p
(so
that
p
=
m
n), let p be an opposite Borel toLT
so that p p = m, and let n be the nilpotent radical of p (so that p =`m n). We
let ∆ denote the set of roots of m acting on g. We write ∆ = ∆+ ∆− as the
disjoint union of the positive roots (i.e. the roots appearing in n) and the negativeroots (i.e. the roots appearing in n). If r ∈ ∆+ is a positive root, then we fix a basis
element Xr of the r-root space. We also fix a basis element X−r of the −r-root
space, chosen so that if Hr := [Xr , X−r ], then [Hr , X±r ] = ±2X±r . The elements
Xr (respectively X−r ) for r ∈ ∆+ form a basis for n (respectively n) as a vector
space.
The Weyl group W of g with respect to m acts simply transitively on the setof Borel subalgebras of g containing m, and also acts on ∆, and these actions arecompatible, in the following sense: if w(p) is the transform of p under some w ∈ W ,with nilpotent radical w(n), then w(∆+ ) is the set of roots which appear in w(n).
(Equivalently, the elements Xw(r) , for r ∈ ∆+ , form a basis for w(n).)
T
If w ∈ W , let ∆w := ∆+ w(∆− ), and let Iw := {(kr )r∈∆w | kr ∈ Z≥0 }. Thus
∆w denotes the set of roots that n and w(n) have in common, while Iw denotes theset of non-negative integral multi-indices indexed by the set ∆w . We will denote atypical element of Iw by k, and a typical component of k by kr (r ∈ ∆w ).
If r ∈ ∆+ , we let sr ∈ W denote the reflection associated to r. If w ∈ W , then thelength of w is the minimal number of factors appearing in a factorization of w into a
product of simple reflections (i.e. elements sr attached to simple roots r ∈ ∆+ ). If w
has length l, then the order of ∆w is also equal to l, and for an appropriately chosenordering of the elements r1 , . . . , rl of ∆w , we have the factorization w = sr1 · · · srl .
k
If V is a g-representation, then for any integer k ≥ 0 let V n denote the p∞ksubrepresentation of V consisting of elements annihilated by n . We write V n :=Skn
. (Note that this is a g-subrepresentation of V .) If w ∈ W and k ∈ Iw ,k≥0 V
we write(V n )k := {v ∈ V n | (Hr − kr )v = 0 for all r ∈ ∆w }.
∞Lemma 1.5.1. Let V be a g-representation, and suppose that V = V n . If w ∈ W
has length l, and if r1 , . . . , rl is an ordering of the elements of ∆w , chosen so that

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

19

w = sr1 .P. . srl , then the space V w(n) of w(n)-invariants in V is contained in thesubspace k∈Iw (X−r1 )kr1 · · · (X−rl )krl (V n )k of V .
Proof. We prove the lemma by induction on the length l of the element w ∈ W .
If w is the identity element (i.e. l = 0), then ∆w is empty, and the assertion to be
proved reduces to the tautology V n = V n . Suppose now w is non-trivial (so that∆w is non-empty). Note that r1 lies in ∆w , and that −r1 is a simple root of w(n).
k ≥ 0, let (V w(n) )k := {v ∈ V w(n) | (Xr1 )k v = 0}. Clearly V w(n) =S For eachw(n))k . We first prove that (V w(n) )k = (V w(n) )Hr1 (Hr1 +1)···(Hr1 +k−1) (=:k≥0 (V{v ∈ V w(n) | Hr1 (Hr1 + 1) · · · (Hr1 + k − 1)v = 0}). If k = 0, then there is nothing
to prove, so we may assume that k ≥ 1. Now let v ∈ (V w(n) )k . By assumption
X−r1 v = (Xrk1 )v = 0. Thus we compute that
0 = (Xr1 )k X−r1 v = (X−r1 (Xr1 )k + (Xr1 )k−1 (Hr1 + k − 1))v
= (Xr1 )k−1 (Hr1 + k − 1)v.
Hence (V w(n) )k−1 ⊂ (Hr1 +k −1)(V w(n) )k ⊂ (V w(n) )k−1 (the first inclusion holding
from the fact that (Hr1 + k − 1) is coprime to Hr1 (Hr1 + 1) · · · (Hr1 + k − 2),which annihilates (V w(n) )k−1 , by induction on k), and so in fact (V w(n) )k−1 =(Hr1 + k − 1)(V w(n) )k . Suppose now that v ∈ V w(n) is such that (Hr1 + k − 1)v ∈w(n)(V w(n) )k−1 ; we wish to show that v ∈ Vk
. By what we have just shown, we mayfind v 0 ∈ (V w(n) )k such that (Hr1 + k − 1)v = (Hr1 + k − 1)v, and so replacing v byv − v 0 , we may assume that (Hr1 + k − 1)v = 0. If we choose l minimally so thatv ∈ (V w(n) )l , then, by what we have shown, (V w(n) )l−1 3 (Hr1 + l − 1)v = (l − k)v.
Since l is minimal, we find that l = k, and so v ∈ (V w(n) )k . Thus we have shownthat (V w(n) )k = {v ∈ V w(n) | (Hr1 + k − 1)v ∈ (V w(n) )k−1 }. By induction on k,
we conclude that (V w(n) )k = (V w(n) )Hr1 (Hr1 +1)···(Hr1 +k−1) . Consequently, we may
decompose V w(n) as the direct sum
V w(n) =(1.5.2)M(V w(n) )Hr1 =−k ,k≥0,with the non-zero elements of the kth direct summand being annihilated by Xrk+1
1
but by no smaller power of Xr1 .
If we write w0 := sr1 w = sr2 · · · srl ∈ W , then ∆w0 = ∆w \ {r1 } = {r2 , . . . , rl }.
By induction we may assume that
(1.5.3)0

V w (n) ⊂X
k0 ∈∆(X−r2 )kr2 · · · (X−rl )krl (V n )k0 .

w0

We claim that
(1.5.4)0(Xr1 )k (V w(n) )Hr1 =−k ⊂ (V w (n) )Hr1 =k .

S
Note that w0 (∆+ ) = (w(∆+ ) \ {−r1 }) {r1 }. Since (Xr1 )k+1 annihilates the space(V w(n) )Hr1 =−k , the left-hand side of (1.5.4) is annihilated by Xr1 . On the other
hand, if r ∈ w(∆+ )\{−r1 }, then any iterated commutator [Xr1 , [Xr1 , . . . , [Xr1 , Xr ]]]20

MATTHEW EMERTON

either lies in w(n) or else vanishes. Thus the left-hand side of (1.5.4) is annihilated
by Xr . This shows that
(1.5.5)0(Xr1 )k (V w(n) )Hr1 =−k ⊂ V w (n) .

On the other hand, it follows immediately from the formula [Xr1 , X−r1 ] = Hr1 that
(1.5.6)(Xr1 )k (V w(n) )Hr1 =−k ⊂ V Hr1 =k .

The inclusion (1.5.4) follows from (1.5.5) and (1.5.6).
One easily deduces from (1.5.3) thatX0
(X−r2 )kr2 · · · (X−rl )krl ((V n )k0 )Hr1 =k(1.5.7) (V w (n) )Hr1 =k ⊂k0 ∈∆w0

X=(X−r2 )kr2 · · · (X−rl )krl (V n )k .

k∈∆w s.t. kr1 =kFinally, a simple commutator calculation shows that the actions of (X−r1 )k (Xr1 )kand (−1)k Hr1 (Hr1 +1) · · · (Hr1 +k −1) on V w(n) coincide. Thus (X−r1 )k (Xr1 )k actson (V w(n) )Hr1 =−k as multiplication by k!, and hence (combining (1.5.4) and (1.5.7))(V w(n) )Hr1 =−k = (X−r1 )k (Xr1 )k (V w(n) )Hr1 =−kX⊂(X−r1 )k · · · (X−rl )krl (V n )k .

k∈∆w s.t. kr1 =k

Taken together with (1.5.2), this proves the lemma. 
2. Parabolically induced representations(2.1) Let G be a connected reductive linear algebraic group over L. Fix aparabolic subgroup P of G, with Levi quotient M . If U is an object of Repla.c (M ),then we definelaIndG
P U := {f ∈ C (G, U ) | f (pg) = pf (g) for all p ∈ P },regarded as a closed subspace of C la (G, U ), and equipped with the right regularaction of G. (Here we regard P as acting on U through its Levi quotient M .)We may find a compact open subgroup G0 of G such that G = P G0 . (Forexample, G0 could be taken to beTa special, good, maximal compact subgroup of G[4, p. 140].) If we write P0 = G0 P, then as a G0 -representation, we may identify
IndG
P U with the closed subspace
la
0
IndG
P0 U = {f ∈ C (G0 , U ) | f (pg) = pf (g) for all p ∈ P0 }of the space C la (G0 , U ). This latter space is of compact type, by [6, Prop. 2.2.28],
and thus IndG
P U is a space of compact type. It follows from [6, Prop. 3.5.11,
Lem. 3.6.14] that IndG
P U is an object of Repla.c (G).
Evaluation at the identity of G induces a continuous P -equivariant map
IndG
P U → U.
It follows from [6, Prop. 5.1.1 (iii)] that if V is an object of Repla.c G, then thismap induces a “Frobenius reciprocity” isomorphismG∼

LG (V, IndP U ) −→ LP (V, U ).
Let us remark that a thorough discussion of the basic properties of locally analytic
induction, including this Frobenius reciprocity isomorphism, appeared in the 1997
dissertation of Feaux de Lacroix [10, §4].

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

21

Proposition 2.1.1. If U is an object of Repzla.c (M ), then IndG
P U is an object ofRepzla.c (G).
Proof. Apply Lemma 1.1.1 to find a cofinal sequence {Hi }i≥1 of analytic open subgroups of G0 , each normalized by G0 . For each i ≥ 1, let Pi denote the intersectionof Hi with P (thinking of P as a closed rigid analytic subspace of G), and let Mi
denote the image of Pi under the projection P → M. Then {Mi }i≥1 forms a cofinal
sequence of analytic open subgroups of M , each of which is normalized by the image
M0 of P0 in M .
∼Apply Lemma 1.1.2 to write U −→ lim
Ui , where each Ui is a K-Banach space−→i≥1equipped with a locally analytic M0 ZM -action that is Mi -analytic. Then we may
write
∼

G0
IndG
P U −→ IndP0 U∼−→ lim{f ∈ C la (G0 , Ui )Hi −an | f (pg) = pf (g) for all p ∈ P0 , g ∈ G0 }.
−→iEach term in this inductive limit is a Banach space whose image in IndG
P U isinvariant under ZG (since ZG ⊂ ZM ), and thus we have exhibited IndG
U
as an
P
object of Repzla.c (G). 
Proposition 2.1.2. If U is an admissible (respectively strongly admissible, respectively very strongly admissible) locally analytic representation of M , then IndG
P U
is an admissible (respectively strongly admissible, respectively very strongly admissible) locally analytic representation of G.
∼

G0
Proof. As above, we exploit the isomorphism IndG
P U −→
T IndP0 U. Fix a choice oflifting of M to a Levi factor of P , and write M0 := G0 M. By definition, U is anadmissible (respectively strongly admissible) locally analytic representation of M
if and only if it is such a representation of M0 .
T
Let P denote an opposite parabolic to P , chosen so that P P = M, ourfixed choice of Levi factor of P . Applying [7, Prop. 4.1.6] (after fixing a minimal
parabolic of G contained in P ) we choose a cofinal sequence {Hi }i≥0 of analytic
open subgroups of G0 , whose underlying rigid analytic subgroups Hi admit a rigid
analytic Iwahori decomposition Ni × Mi × Ni → Hi with respect to P and P . Notethat the subgroups Mi := Mi (L) then form a cofinal sequence of analytic open
subgroups of M0 .
Let U be an admissible locally analytic M0 -representation. Fix i ≥ 0. If g ∈ G0 ,let U g denote U regarded as a locallyTanalytic representation of g −1 P g , via(g −1 pg) · u = pu, and write Pig = Hi g −1 P g. Choose an analytic open subgroup Lgi of the locally analytic group Pig , for which the closed embedding Lgi ⊂ Hi
arises from a rigid analytic closed embedding Lgi ⊂ Hi . Since U is an admissible locally analytic M0 -representation, and so in particular an admissible locallyanalytic P0 -representation, U g is an admissible locally analytic Pig -representation.
In particular, the space (U g )Lgi −an of Lgi -analytic vectors in U g is a Banach space
admitting a Lgi -equivariant closed embedding (U g )Lgi −an → C an (Lgk , K)r , for somer ≥ 0. (Here and below we are using the characterization of admissible locallyanalytic representations provided by [6, Def. 6.1.1], and the discussion followingthat definition.)

22

MATTHEW EMERTON

The decomposition of G0 into right Hi -cosets induces a natural Hi -equivariant
closed embedding0
IndG
P0 U →

M

i
IndH
Ug →
Pg

M

i
IndH
U g,
Lg

i

g∈G0 /Hiig∈G0 /Hi

and hence [6, Prop. 3.3.23] a closed embedding of Hi -representations
0(IndG
P0 U )Hi −an →

M

i
(IndH
U g )Hi −an .
Lgig∈G0 /Hii
We will show that each (IndHU g )Hi −an admits an Hi -equivariant closed emLgi0bedding into C an (Hi , K)r , for some r ≥ 0. and thus conclude that (IndG
P0 U )Hi −anansadmits a closed Hi -equivariant embedding into C (Hi , K) , for some s ≥ 0. Since0
i ≥ 0 was arbitrary, we will have established that IndG
P0 U is an admissible locallyanalytic representation.
iU g )Hi −an is equal to the
It follows from [6, props. 3.3.23, 3.3.24] that (IndHLgii
U g under the natural map C an (Hi , U g ) → C la (Hi , U g ). Any
preimage of IndHLgii
U g must actually lie
element in C an (Hi , U g ) whose image in C la (Hi , U g ) lies in IndH
Lg
i
an
g g
in C (Hi , (U )Li −an ), as follows immediately from the closed embedding Lgi ⊂ Higii
U g as a subspace of C la (Hi , U g ). Thus (IndHand the definition of IndH
Li U )Hi −an isLgii
U g under the natural map C an (Hi , (U g )Lgi −an ) → C la (Hi , U g ),the preimage of IndH
Lgiand so in particular embeds as a closed subspace of C an (Hi , (U g )Lgi −an ). The closed
embedding (U g )Lgi −an → C an (Lgi , K)r thus induces a closed embedding∼iU g )Hi −an → C an (Hi , C an (Lgi , K))r −→ C an (Lgi × Hi , K)r .
(IndH
Lg
i

i
U g , this closed embedding lies in the closed
Again, by the definition of IndH
Lg
i
g
an
r
subspace of C (Li × Hi , K) consisting of functions f satisfying the conditionf (pg , h) = f (1, pg h) for any pg ∈ Lgi , h ∈ Hi . Restricting to the closed subvariety
1 × Hi of Lgi × Hi induces an isomorphism between this subspace and C an (Hi , K)r .
iU g )Hi −an → C an (Hi , K)r .
Thus we obtain the required closed embedding (IndHLgi
If we suppose that U is a strongly admissible locally analytic representation of0M0 , then a similar, but more straightforward, argument shows that IndG
P0 U is astrongly admissible locally analytic representation of G0 . Indeed, since U is strongly
admissible, it admits a closed M0 -equivariant embedding U → C la (M0 , K)r , forsome r ≥ 0. (See the remark following [6, Def. 6.2.1].) Thus C la (G0 , U ) admits aclosed embedding into∼

C la (G0 , C la (M0 , K))r −→ C la (M0 × G0 , K)r .
la0
The closed subspace IndG
P0 U of C (G0 , U ) lands in the subspace of functions f forwhich f (m, g) = f (1, mg) (for m ∈ M0 , g ∈ G0 ). Restricting such functions to
1 × G0 ⊂ M0 × G0 induces an isomorphism of this subspace with C la (G0 , K)r . Thuslar0
we obtain a closed embedding IndG
P0 U → C (G0 , K) , as required.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

23

Finally, suppose that U is very strongly admissible, and choose an admissible Banach space representation W of M for which U admits a continuous M -equivariant
injection U → W . There is then a continuous G-equivariant injection(2.1.3)IndG
P U → {f ∈ C(G, W ) | f (pg) = pf (g) for all p ∈ P }.

(The target of this map is the continuous induction of W from M to G.) The
preceding argument shows that the source of (2.1.3) is a strongly admissible Grepresentation, and a similar argument, using the fact that W is an admissible
Banach space representation of M , shows that the target of (2.1.3) is an admissible
Banach space representation of G. By definition, we conclude that the sourceof (2.1.3) is a very strongly admissible locally analytic G-representation. (2.2) Let G be a connected reductive linear algebraic group over L. Fix aparabolic subgroup P of G, a LeviTfactor M of P , and let P denote the opposite
parabolic to P , chosen so that P P = M. Let N and N denote the unipotent
radicals of P and P respectively, so that P = M N and P = M N . Fix an object U
of Repla.c (M ).
Recall the definitions of the convex spaces Ccla (N, U ) and C ω (N, U )e from Subsection 1.4. These are of compact type, by Lemma 1.4.7 and 1.4.10 respectively.
We endow Ccla (N, U ) with a locally analytic P -action as follows: If f ∈ Ccla (N, U )and p ∈ P , then factor p = mn with m ∈ M and n ∈ N , and define the functionpf ∈ Ccla (N, U ) via the formula (pf )(n0 ) = mf (m−1 n0 mn) for all n0 ∈ N ; here macts on the element f (m−1 n0 mn) of U via the given locally analytic action of M
on U . Lemma 1.4.8 shows that the N -action is locally analytic. It is easily verified
that this is also true of the M -action (and hence of the P -action). We will notgive a direct proof here, however, since the local analyticity of the P -action follows
from Lemma 2.3.6 below and the fact that G, and so P , acts locally analyticallyon IndGU.
P
Let Ccsm (N, U ) denote the space of compactly supported locally constant functions on N . Note that Ccsm (N, U ) coincides with the closed subspace of n-invariantsin Ccla (N, U ). Thus Ccsm (N, U ) is a closed P -invariant subspace of Ccla (N, U ), andso in particular is an object of Repla.c (P ). (By construction this locally analytic
P -action on Ccsm (N, U ) agrees with that defined in [7, §3.5].)
We define a locally analytic M -action on C ω (N, U )e as follows: If f ∈ C ω (N, U )e ,and m ∈ M , then let the germ mf ∈ C ω (N, U )e be defined via the formula(mf )(n) = mf (m−1 nm) for n ∈ N sufficiently close to e (i.e. close enough forf (m−1 nm) to be defined); here m acts on the element f (m−1 nm) of U via thegiven locally analytic action of M on U . Again, we omit the (easy) direct proof
that this action is locally analytic, since local analyticity of the action follows
from Corollary 2.3.4 below and the fact that the P (and hence the M -action) on
U )e is locally analytic (by Lemma 2.3.5 (i)).
(IndG
P
We let ZM denote the centre of M , let A denote the maximal split torus in ZM ,and fix a coweight α : L× → A that pairs strictly negatively with each restrictedroot of A acting on N . Set z0 = α(p)−1 . In the remainder of this subsection we
introduce some auxiliary constructions that will be useful for what follows. We
apply [7, Prop. 4.1.6] (after fixing a minimal parabolic of G contained in P ) to finda cofinal sequence {Hi }i≥0 of good analytic open subgroups of G that admit a rigid
analytic Iwahori decomposition with respect to P and P , such that Hi is normal in

24

MATTHEW EMERTON

H0 for each i ≥ 0, such that each inclusion Hi+1 ⊂ Hi is relatively compact, and∼such that Ni := z0i N0 z0−i for all i ≥ 0. Let Hi −→ Ni Mi Ni be the rigid analytici
Iwahori decomposition of Hi . Also, fix an increasing sequence {NS }i≥0i of compact
0
open subsets of N , chosen so that N ⊃ N0 , and so that N = i≥0 N .
We now define two submonoids of the centre ZM of M that will play an importantrole in what follows.
+−
Definition 2.2.1. Write ZM= {z ∈ ZM | zN0 z −1 ⊂ N0 } (respectively ZM= {z ∈+−−1
ZM | N0 ⊂ zN0 z }). Note that each of ZM and ZM is a submonoid of ZM .
∼Write U −→ lim
Ui , where Ui is a K-Banach space equipped with an Mi -analytic−→naction of M0 , and the transition maps are compact, injective, and M0 -equivariant.
In the case when U is an object of Repzla.c (M ), we furthermore choose the Ui tobe ZM -invariant, and the transition maps to be ZM -equivariant (as we may, byLemma 1.1.2).
For each pair of integers i, j ≥L0, we define the K-Banach spaces of functions
an
Ai,j := C an (Ni , Uj ) and Bi,j :=n∈Ni \N0 C (Ni n, Uj ) (where the direct sum isover a set of left coset representatives of Ni in`N0 ). These are spaces of Uj -valued
rigid analytic functions on the spaces Ni and n∈Ni \N0 Ni n respectively.
If i0 ≥ i ≥ 0 and j 0 ≥ j ≥ 0, then there are continuous injections
Ai,j → Ai0 ,j 0(2.2.2)
andBi,j → Bi0 ,j 0(2.2.3)

induced by the continuous` injection Uj →`Uj 0 , together with restricting functions
from Ni to Ni0 , or from n∈Ni \N0 Ni n to n∈Ni0 \N0 Ni0 n.
Taking the germ of an element of Ai,j at e yields a continuous injection
Ai,j → C ω (N, U )e .

(2.2.4)

Passing to the inductive limit in i and j, we obtain an isomorphism(2.2.5)∼lim
Ai,j −→ C ω (N, U )e .
−→i,j(The transition maps in the inductive limit are provided by the maps (2.2.2).) Also,
extension by zero yields a natural continuous injection(2.2.6)

Ai,j → Ccla (N, U ).

Composing (2.2.6) with the natural map Ccla (N, U ) → C ω (N, U )e given by passing
to the germ of a locally analytic function at the identity yields the map (2.2.4).
Regarding an element of Bi,j as a Uj -valued function on N0 we obtain a continuous injection(2.2.7)

Bi,j → C la (N0 , Uj ).

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

25

Fixing j, and passing to the inductive limit in i, we obtain an isomorphism(2.2.8)∼lim
Bi,j −→ C la (N0 , Uj ).
−→i

The continuous injection Uj → U induces a continuous injection(2.2.9)C la (N0 , Uj ) → C la (N0 , U ),which when composed with (2.2.7) yields a continuous injection
Bi,j → C la (N0 , U ).

(2.2.10)

Passing to the inductive limit in i and j yields an isomorphism(2.2.11)∼lim
Bi,j −→ C la (N0 , U ),−→i,jwhich may also be obtained by taking the inductive limit with respect to j of the isomorphisms (2.2.8). (The transition maps in the inductive limits (2.2.8) and (2.2.11)
are provided by the maps (2.2.3).)(2.3) We maintain the notation of Subsection 2.2. In this subsection, and thosethat follow, we will consider parabolic induction from P to G (rather than from P
to G).
Unless the M -action on our fixed object U of Repla.c (M ) is trivial, the elementsof IndG
U are not U -valued functions on P \G. (Rather, they are sections of a bundleP
on P \G, with fibres isomorphic to U .) Nevertheless, any element f of IndG
U hasPa well-defined support (that is, region over which its germ is non-zero) on P \G,which is a compact open subset of this quotient. (The support of f as a function
on G is an open and closed subset of G, invariant under left multiplication by P ,and so corresponds to a compact open subset of P \G.)U and Ω is a compact open subset of P \G, then we write f|Ω toIf f ∈ IndG
P
denote the functionf (g) if P g ∈ Ω
f|Ω (g) =0 otherwise.
The function f|Ω is clearly again an element of IndG
U.
P
Definition 2.3.1. (i) If Ω is any open subset of P \G, we let (IndG
U )(Ω) denotePthe subspace of elements whose support is contained in Ω.
(ii) If x ∈ P \G, then we define the stalk of IndG
U at x to bePGU )x := lim(IndG
U )(Ω),
(IndP
P
−→
Ω3x

where the locally convex inductive limit is taken over the directed set of compact
open neighbourhoods of x, with the transition map (IndG
U )(Ω) → (IndG
U )(Ω0 )PP
0(for Ω ⊃ Ω an inclusion of neighbourhood of x) being given by f 7→ f|Ω0 . If Ω
is an open subset of P \G containing x, then we denote by stx (for “stalk at x”)
GG
the continuous map (IndP
U )(Ω) → (IndPU )x , arising from the description of thetarget as an inductive limit.
T
Since N P = {e}, the map n 7→ P n is an open immersion(2.3.2)

N → P \G,and we thus regard N as an open subset of P \G.

26

MATTHEW EMERTON

Lemma 2.3.3. If Ω is a compact open subset of N , regarded also as a compact
open subset of P \G via (2.3.2), then there is a natural topological isomorphism∼G
C la (Ω, U ) −→ (IndP
U )(Ω).
Proof. The preimage of Ω (regarded as a subset of P \G) in G is equal to P Ω. We
may write G = P Ω t (G\(P Ω)); this is a partition of G into two open sets invariant
under the left action of P on G. Thus we obtain a topological isomorphism
∼

G
(IndP
U )(Ω) −→ {f ∈ C la (P Ω, U ) | f (pg) = pf (g) for all g ∈ P Ω}.

T
Since N P = {e}, multiplication induces a locally analytic isomorphism P × Ω →∼GP Ω. Thus restricting to Ω induces an isomorphism (IndP
U )(Ω) −→ C la (Ω, U ). That
we do indeed obtain an isomorphism follows from the fact that P acts (through M )in a locally analytic fashion on U , so that we may extend any element of C la (Ω, U )Gin a unique manner to an element of (IndP
U )(Ω) (using the natural orbit mapla
U → C (P , U ) provided by [6, Thm. 3.5.7, Thm. 3.6.12]). 
Corollary 2.3.4. The open immersion (2.3.2) induces an M -equivariant topolog∼
ical isomorphism C ω (N, K)e −→ (IndG
U )e .
P
Proof. This follows immediately from Lemma 2.3.3, by passing to the inductive
limit over the compact open neighbourhoods Ω of e. (The claimed M -equivariancefollows immediately from a consideration of the M -action on C ω (N, K)e as defined at the end of Subsection 2.2, the definition of the induction IndG
U, and thePconstruction of the isomorphism.) As usual, for any x ∈ P \G, we write Gx to denote the stabilizer of x in G (withrespect to the action of G on P \G given by right multiplication). If x = P g, thenof course Gx = g −1 P g.
Lemma 2.3.5. (i) For each x ∈ P \G, the stalk (IndG
U )x is a compact type space,Pequipped in a natural way with a (g, Gx )-action. Furthermore, the g-action is locallyintegrable, and the Gx -action is locally analytic.
G(ii) If Ω is an open subset of P \G, then (IndP
U )(Ω) is a closed g-invariant
G
subspace of IndP U . Furthermore, the g-action is locally integrable.
(iii) For any g ∈ G and x ∈ P \G, the action of g induces a topological isomor∼GU )gx −→ (IndPU )x , compatible with the (g, Ggx )-action on thephism g· : (IndGP
source and the (g, Gx )-action on the target (in the sense that g(Xf ) = Adg (X)gf
and g(g 0 f ) = (gg 0 g −1 )gf for any g ∈ G, X ∈ g, g 0 ∈ Ggx (note that gg 0 g −1 is thenG
an element of Gx ), and f ∈ (IndP
U )gx ).
(iv) For each x ∈ P \G, the map stx : IndG
U → (IndG
U )x is (g, Gx )-equivariant.
P
P
G
Proof. Since g acts on IndP
U through differential operators, it is obvious thatG
G
(IndP U )(Ω) is a g-invariant subspace of IndP
U for any open subset Ω of P \G.
G
Thus for any x ∈ P \G, the stalk (IndP U )x is described as the locally convex
inductive limit of g-representations, with g-equivariant transition maps, and so isnaturally a topological g-representation. Incidentally, we see that the claim of (iv)regarding g-equivariance is true by definition of the g-action on (IndG
U )x .
P
It is clear that for any g ∈ G and x ∈ P \G, the action of g induces a topological∼Gisomorphism (IndP
U )gx −→ (IndG
U )x , such that g(Xf ) = Adg (X)gf for any
PJACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

27

X ∈ g and f ∈ (IndG
U )gx . In particular, if g ∈ Gx then g induces a topologicalP
automorphism of (IndG
U )x , and so we find that (IndG
U )x is a topological (g, Gx )PP
representation. This establishes (iii) (since the remaining claimed compatibility“g(g 0 f ) = (gg 0 g −1 )gf ” follows directly from the definitions of the actions involved)G
and a part of (i). By construction of the Gx -action on (IndPU )x , we see that theclaim of (iv) regarding Gx -equivariance holds (and this completes the proof of (iv)).
U )e is of compact type. Since
Corollary 2.3.4 and Lemma 1.4.10 show that (IndGP∼GU )e −→ (IndPU )x foracting by g −1 ∈ G gives a topological isomorphism (IndGP
G
any x = P g ∈ P \G, we see that every (IndP U )x is of compact type, establishinganother claim of (i).
T
By definition we see that (IndG
U )(Ω) = x∈Ω ker(stx ). The target of eachP
map stx is of compact type, and so Hausdorff, and hence ker(stx ) is closed forG
all x ∈ P \G. Consequently (IndG
U )(Ω) is closed. The g-action on IndP
U is loP
cally integrable, by Proposition 1.2.18, and so Proposition 1.2.19 shows that theU )(Ω). This completes the proof of (ii).
same is true of the the g-action on (IndGP
To complete the proof of (i), and hence of the lemma, we must show that theg-action and Gx -action on (IndG
U )x are locally integrable and locally analyticPU )xrespectively. By the result of the preceding paragraph, we see that (IndGP
is the g-equivariant locally convex inductive limit of a sequence of locally integrable g-representations, and so by Proposition 1.2.20 it is a locally integrable
g-representation. Since the Gx -action on IndG
U is locally analytic, and sincePGstx : IndG
U
→
(Ind
U
)
is
surjectiveandGxx -equivariant, it follows from [6,PP
G
Prop. 3.6.14] that the Gx -action on (IndP U )x is also locally analytic. Lemma 2.3.6. The open immersion (2.3.2) induces a P -equivariant topological∼isomorphism Ccla (N, U ) −→ (IndG
U )(N ).
P
Proof. Since any element of (IndG
U )(N ) has compact support, we see that thePisomorphisms of Lemma 2.3.3 give rise to a continuous bijection(2.3.7)G
Ccla (N, U ) → (IndPU )(N ).

The source of (2.3.7) is of compact type, by Lemma 1.4.7, and so is the target,GU.
since by Lemma 2.3.5 (ii) it is a closed subspace of the compact type space IndPThe Open Mapping Theorem thus implies that (2.3.7) is an isomorphism. As in
the proof of Corollary 2.3.4, the P -equivariance of this isomorphism is immediately
checked. We transport the (g, P )-action on (IndG) given by Lemma 2.3.5 (iii) to a
P e(g, P )-action on C ω (N, U )e via the isomorphism of Corollary 2.3.4, and hence regard C ω (N, U )e as a (g, P )-representation. Similarly, we transport the g-action on(IndG
U )(N ) to a g-action on Ccla (N, U ) via the isomorphism of Lemma 2.3.6, andP
hence regard Ccla (N, U ) as a (g, P )-representation.
GU )(N ) → IndG
U with the isomorphism
Composing the closed embedding (IndPP
of Lemma 2.3.6 and the continuous injection (2.2.6) yields for each i ≥ 0 a continuous injection(2.3.8)G
Ai,i → IndP
U,

28

MATTHEW EMERTON

which we use to regard Ai,i as the latent Banach space underlying a BH-subspaceGof IndP
U . Similarly, composing the isomorphism of Corollary 2.3.4 with (2.2.4)
yields for each i ≥ 0 a continuous injection
Ai,i → (IndG
U )e ,
P(2.3.9)via which we regard Ai,i as the latent Banach space underlying a BH-subspace ofG(IndPU )e .
Proposition 2.3.10. (i) For any i ≥ 0 the image of Ai,i in IndG
U under (2.3.8)
P
is Hi -invariant. The resulting Hi -action on Ai,i (given by [6, Prop. 1.1.2 (ii)]) is
furthermore Hi -analytic.
GU )e under (2.3.9) is g-invariant.
(ii) For any i ≥ 0, the image of Ai,i in (IndPz−
If U is an object of Repla.c (M ), then it is furthermore ZM-invariant.
Proof. Since Hi is a rigid analytic group, the group operation on Hi induces a rigid
analytic map∼

Ni × Hi → Hi −→ Ni × Mi × Ni

(2.3.11)(where theT second arrow is the rigid analytic Iwahori decomposition of Hi ). Set
P i := Hi P = N i Mi , and writei(IndHUi )Hi −an := {f ∈ C an (Hi , Ui ) | f (pg) = pf (g) for all p ∈ P i , g ∈ Gi }P
i(where P i acts on Ui through its quotient Mi ). Taking into account (2.3.11) andthe fact that the Mi -action on Ui is Mi -analytic, we see that restricting functions
to Ni induces an isomorphism(2.3.12)

∼

i
(IndH
Ui )Hi −an −→ C an (Ni , Ui ) = Ai,i .
P
iOn the one hand, right translation by elements of Hi equips the source of (2.3.12)
with an Hi -analytic Hi -action. On the other hand, since Hi admits an Iwahori deHicomposition, any element of (IndPUi )Hi −an evidently admits a unique extension byizero to an element of IndG
U , and we obtain a continuous Hi -equivariant injectionP(2.3.13)HiGUi )Hi −an → IndP
U,
(IndP
i

which fits into the commutative diagram(2.3.12)i/ Ai,i(IndHU)
P i i Hi −anNNNNNN(2.3.8)
NNN
N&


IndG
U.
P

Part (i) follows from the fact that the diagonal arrow is Hi -equivariant, while thehorizontal arrow is an isomorphism.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

29

Consider the commutative diagram
Ai,i(2.3.8)/ IndG U
P
II
II
II
ste
I
(2.3.9) II
$ 
(IndG
U )e .
P

Part (i) shows that the image of the horizontal arrow is g-invariant, and since thevertical arrow is g-equivariant, by Lemma 2.3.5 (iv), we see that the image of the
diagonal arrow is also g-invariant. To complete the proof of (ii), we must show that−when U is an object of Repzla.c (M ) this image is also ZM-invariant.
Since we have chosen Ui to be ZM -invariant when U is an object of Repzla.c (M ),we find that for any z ∈ ZM and f in the image of (2.3.9), the translate zf is a−
Ui -valued rigid analytic function on zNi z −1 . If z ∈ ZM
, then Ni ⊂ zNi z −1 , and−1(since Ni and zNi z are good analytic open subgroups of G) this inclusion extends
to a rigid analytic embedding Ni ⊂ zNi z −1 . Thus zf in particular restricts to a−
Ui -valued rigid analytic function on Ni . Hence the image of (2.3.9) is ZM-invariant,as claimed. −
We use [6, Prop. 1.1.2 (ii)] to lift the (g, ZM)-action on the image of (2.3.9) given−by part (ii) of the preceding lemma to a continuous (g, ZM)-action on Ai,i . Part (i)of the proposition (together with Lemma 1.2.5) shows that the g-action on Ai,i isthen Hi -integrable, and that the resulting Hi -action on Ai,i is compatible with themap (2.3.8).
GU )(N0 ) is a closed g-invariant subspace of IndG
U,
Lemma 2.3.5 shows that (IndPP
laand so C (N0 , U ) (which the isomorphism of Lemma 2.3.6 maps isomorphically to
U )(N0 )) is a closed g-invariant subspace of Ccla (N, U ).
(IndG
PLemma 2.3.14. For each i ≥ 0, the image of Bi,i under the continuous injection (2.2.10) is a g-invariant subspace of C la (N0 , U ). The induced g-action on Bi,i
is locally integrable.
Proof. There is a natural isomorphism(2.3.15)∼

Bi,i −→

M

nAi,i .

n∈Ni \N0

Proposition 2.3.10 (i) shows that the image of Ai,i in C la (N0 , U ) is g-invariant, and(together with Lemma 1.2.5) that the g-action on Ai,i is locally integrable. Sincetranslation by n intertwines the action of X ∈ g on Ai,i with the action of Adn (X)on nAi,i , and since Adn is an automorphism of g, we see that the image of nAi,i isalso g-invariant, and that the induced g-action on nAi,i is again locally integrable,
for each n ∈ Ni \N0 . Taking into account the isomorphism (2.3.15), the lemma isproved. 
We close this subsection with a result that strengthens part (i) of Lemma 2.3.5.
For the remainder of this subsection, let us break with our conventions and write N
to denote the algebraic group underlying N . Since N is unipotent, it is σ-affinoidS∞j
in the sense of [6, §3.4], i.e. it may be written as an increasing union N = j=1 N ,30

MATTHEW EMERTONjjjwhere N is an admissible affinoid open subgroup of N. We write N := N (L)jto denote the locally analytic group of L-valued points of N (which is a compactjjopen subgroup of N ). We may and do choose the affinoid groups N so that N isj
Zariski dense in N for each j ≥ 1.
G
Lemma 2.3.16. The stalk (IndPU )e , which by Lemma 2.3.5 (i) is a locally analytic P -representation, is N-analytic as an N -representation, in the sense of [6,Def. 3.6.1].

Proof. By [6, Thm. 3.3.16], it suffices to show that for each j ≥ 1, and for eachGf ∈ (IndP
U )e , the orbit map of : N j → (IndG
U )e is induced by a rigid analyticPjfunction on N .
We may choose i ≥ 0 such that f lies in the image of (2.3.9), and hence (via theiUi )Hi −an , and hence
isomorphism (2.3.12)) may regard f as an element of (IndHPialso (via the embedding (2.3.13)) as an element of IndG
U (extended by zero fromP
P Hi to G). Again breaking with our conventions, write G to denote the algebraic
group underlying G. There is an algebraic, and hence rigid analytic, action of N
on G via conjugation. Explicitly, we consider the action
N×G→G(2.3.17)

given by the formula (n, g) 7→ n−1 gn on the level of L-valued points. Evidentlyje ∈ G is fixed by this action. Since N is affinoid, and since the collection of affinoid
open subgroups {Hi0 } forms a cofinal sequence of open affinoid neighbourhoods ofe in G, we may find i0 ≥ 0 such that (2.3.17) restricts to a rigid analytic mapj

N × Hi0 → Hi ,and so in particular to a rigid analytic mapj

N × Ni0 → Hi .
This map induces a corresponding map on spaces of Ui -valued rigid analytic functions:jj∼
C an (Hi , Ui ) → C an (N × Ni0 , Ui ) −→ C an (N , C an (Ni0 , Ui )).
Hi
Precomposing this with the closed embedding (IndPUi )Hi −an → C an (Hi , Ui ), andijjpostcomposing it with the map C an (N , C an (Ni0 , Ui )) → C an (N , (IndG
U )e ) inducedPby (2.2.4) and the isomorphism of Corollary 2.3.4, we obtain a continuous map
j

Hi
(IndP
Ui )Hi −an → C an (N , (IndG
U )e ).
Pi

Evaluating this map on the element f of its source, we obtain a rigid analytic map(2.3.18)jU )e .
N → (IndG
Pj

A consideration of the formula f (nn) = f (n−1 nn) for n ∈ Ni0 and n ∈ N showsjjthat the restriction of (2.3.18) to the group N of L-valued points of N coincides
with of . Thus of is indeed a rigid analytic map, and the lemma is proved. (2.4) We maintain the notation of the preceding subsections. In this subsecU (namely, the local closed
tion we introduce certain G-subrepresentations of IndG
P
subrepresentations) that play a crucial role in our later argument.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

31

Definition 2.4.1. We say that a vector subspace X of IndG
U is local if for any
Pf ∈ X and compact open subset Ω of P \G the function f|Ω again lies in X.
G
Definition 2.4.2. Let X be a local subspace of IndPU.
TG(i) For any open subset Ω of P \G, we write X(Ω) = X (IndP
U )(Ω).
(ii) If x ∈ P \G, then we define the stalk of X at x to be

Xx := lim
X(Ω),
−→
Ω3x

where the locally convex inductive limit is taken over the directed set of compact
open neighbourhoods of x, with the transition map X(Ω) → X(Ω0 ) (for Ω ⊃ Ω0 an
inclusion of neighbourhood of x) being given by f 7→ f|Ω0 .
GU.
We begin by establishing some basic lemmas regarding local subspaces of IndPG
Lemma 2.4.3. Let X be a local subspace of IndP
U. We have the following characterization of X, namely

X = {f ∈ IndG
U | stx (f ) ∈ Xx for all x ∈ P \G}.
P
G
Proof. Let us write Y = {f ∈ IndP
U | stx (f ) ∈ Xx for all x ∈ P \G}. It is clearthat X ⊂ Y, and we must show that this is an equality. Let f be an element of Y .
For each element x of P \G, choose an element Fx of X whose stalk at x coincides
with that of f . We may find an analytic chart Yx containing x such that both fand Fx are rigid analytic on the underlying rigid analytic ball Yx , and thus coincide
on Yx . Since P \G is compact, we may partition it into a disjoint union of open
subsets
P Ωx ⊂ Yx , as x ranges over some finite set S of points of P \G. If we set
F := x∈S Fx|Ωx , then F lies in X (since X is local), and coincides with f . Lemma 2.4.4. If X is a local subspace of IndG
U, then so is its closure Y inP
U.
Furthermore,
for
any
open
subsetΩofP\G,the closure of X(Ω) is equal
IndG
P
to Y (Ω).
Proof. Since the map(2.4.5)

f 7→ f|Ω(for any compact open subset Ω of P \G) is a continuous endomorphism of IndG
U,P
we see that if X is local, then so its closure Y . If we fix an open subset Ω of P \G,then (taking into account Lemma 2.3.5 (ii) and the fact that Y is closed) we seethat Y (Ω) is closed, and so certainly contains the closure of X(Ω). Conversely, if
F ∈ Y (Ω), we will show that F lies in the closure of X(Ω). Replacing Ω by the
support of F , we may furthermore assume that Ω is compact. Since F = F|Ω , andF does lie in the closure of X by assumption, this again follows from the continuityof (2.4.5). 32

MATTHEW EMERTON

Lemma 2.4.6. If X is a local closed N -invariant subspace of IndG
U , then thePstalk Xe is a closed subspace of (IndG
U
)
.
e
P
Proof. We will use the notation introduced in Subsection 2.2. Via Corollary 2.3.4
we identify (IndG
U )(N )e with C ω (N, U )e , and so regard Xe as a subspace ofPωC (N, U )e , and for each i ≥ 0 we let Xi denote the closed subspace of Ai,i obtained as the preimage of X under the continuous map (2.3.8).
The natural map lim
Xi → Xe is evidently an isomorphism of abstract K-vector−→ispaces. We claim that for each i ≥ 0, the preimage of Xi+1 in Ai,i under thetransition map Ai,i → Ai+1,i+1 provided by (2.2.2) is equal to Xi . Given this,
it follows from [6, Prop. 1.1.41], Lemma 1.4.10, and the isomorphism (2.2.5) thatlim
Xi embeds as a closed subspace of C ω (N, U )e . This will complete the proof of−→ipart (i).
To prove the claim, suppose that f ∈ Ai,i is such that f|Ni+1 lies in Xi+1 .
Lemma 1.3.1 implies that for each n ∈ Ni , the function (nf )|Ni+1 = n(f|Ni+1 n ) liesin Xi+1 . Since X is N -invariant, we find that f|Ni+1 n lies in X for each n ∈ Ni .
Letting
P n run over a set of left coset representatives of Ni+1 in Ni , we find thatf = n f|Ni+1 n lies in X, and so in Xi . 
The next set of results is aimed at providing a useful description of local closed
G
U in terms of closed (g, P )-invariant subspaces of
G-invariant subspaces of IndP
G
(IndP U )e .
G
Proposition 2.4.7. Let X be a local closed G-invariant subspace of IndP
U.
(i) Xe is a closed (g, P )-invariant subspace of (IndG
U
)
.
e
P
(ii) If x = P g ∈ P \G, then the isomorphism of Lemma 2.3.5 (iii) induced by∼multiplication by g induces an isomorphism Xx −→ Xe .
(iii) We have the following characterization of X, namely

X = {f ∈ IndG
U | gstx (f ) ∈ Xe for all x = P g ∈ P \G}.
P
G
Proof. It is clear that Xe is a (g, P )-invariant subspace of (IndPU )e . The remainderof part (i) follows from Lemma 2.4.6, part (ii) is immediate, and part (iii) is a
restatement of Lemma 2.4.3 in the situation of the proposition (taking into accountthe statement of part (ii)). We will establish a converse to the preceding Proposition. We begin by introducing some notation. Suppose that X is a subspace of (IndG
U )e . For each i ≥ 0, let Xi
Pdenote the preimage of X under the map (2.3.9). Recall from Proposition 2.3.10 (i)that for each i ≥ 0, the space Ai,i is equipped with an Hi -analytic Hi -action.
Lemma 2.4.8. If X is a closed g-invariant subspace of (IndG
U )e , then for eachPi ≥ 0 the subspace Xi of Ai,i is closed and Hi -invariant.
Proof. Since (2.3.9) is continuous and g-invariant, we see that Xi is a closed ginvariant subspace of Ai,i . The lemma then follows from Lemma 1.2.5 and Corollary 1.2.7. 
We now prove the converse to Proposition 2.4.7.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

33

Proposition 2.4.9. If X is a closed (g, P )-invariant subspace of (IndG
U )e , andP
if we defineG
X = {f ∈ IndP
U | gstx (f ) ∈ X for all x = P g ∈ P \G}(where g ∈ G acts on stx (f ) via the isomorphism of Lemma 2.3.5 (iii)), then X is aG
closed local G-invariant subspace of IndP
U, and the natural map Xe → X induced
by ste is an isomorphism.
Proof. Note that X is well-defined, since we have assumed that X is P -invariant,and X is G-invariant, local, and closed in IndG
U by construction. Furthermore, theP
map Xe → X is certainly an injection, and we must show that it is also surjective.
G
U be an element for which ste (f ) is equal to some given element of
Let f ∈ IndP
X . We will find a neighbourhood Ω of e such that f|Ω lies in X, and thus showthat ste (f ) lies in Xe .
Choose a value of i ≥ 0 so that f|Ni lies in the image of Ai,i under the continuous injection (2.3.8). Since f|Ni lies in Xi (using the notation introduced above),it follows from Lemma 2.4.8 that nf|Ni lies in Xi for all n ∈ Ni . We conclude
that ste (nf ) = stn (f ) lies in X for all n ∈ Ni , and thus that f|Ni lies in X, asrequired. 
Propositions 2.4.7 and 2.4.9 together show that closed local G-invariant subspaces of IndG
U are determined by their stalks at the base point e of P \G, andP
that the association X 7→ Xe provides a bijection between the lattice of local
closed G-invariant subspaces of IndG
U , and the closed (g, P )-invariant subspaces
P
U
)
.
The
following
resultwill
be useful for detecting such subspacesof (IndGeP
Gof (IndP U )e .
U )e is P -invariant if andLemma 2.4.10. A closed g-invariant subspace of (IndG
P
only if it is M -invariant.
Proof. The only if direction is obvious. The if direction follows from Lemma 2.3.16,
which shows that an n-invariant closed subspace of (IndG
U )e is automatically N P
invariant. GU and Ω is an open subset of P \G,
If X is a local G-invariant subspace of IndP
then the inclusion X(Ω) ⊂ X together with the G-action on X induces a map(2.4.12)

K[G] ⊗K X(Ω) → X,

which is continuous if we endow K[G] with its finest convex topology and the tensor
product with the inductive tensor product topology. (This tensor product may bedescribed less canonically as a direct sum of copies of X(Ω), indexed by the elementsof G.)
U and Ω is
Lemma 2.4.13. If X is a local closed G-invariant subspace of IndG
P
an open subset of P \G, then (2.4.12) is a strict surjection.
Proof. Since P \G S
is compact, we may find a finite set of elements {g1 , . . . , gn } ⊂ Gnsuch that P \G = i=1 Ωgi , and thus (since X is local) such that the map(2.4.14)nM
i=1

X(Ω) → X

34

MATTHEW EMERTON

P
defined by (fi ) 7→
i gi fi is surjective. Since the source and target of (2.4.14)
are of compact type, it is furthermore a strictL
surjection. Since it factors as thencomposite of(2.4.12)andthecontinuous
map
i=1 X(Ω) → K[G] ⊗K X definedPby (fi ) 7→ i gi ⊗ fi , it is immediate that (2.4.12) is also a strict surjection, as
claimed. (2.5) We maintain the notation of the preceding subsections. We let C pol (N, K)
denote the ring of algebraic K-valued functions on the affine algebraic group N ;i.e. if N denotes the linear algebraic group underlying N , then C pol (N, K) is thesubspace of C la (N, K) obtained by restricting the global sections of the structure
sheaf of N over K to the set N of L-valued points of N. Since N is Zariski dense
in N, the ring of functions C pol (N, K) is naturally identified with the affine ringof N over K.
Since N is unipotent, we may naturally identify it as an algebraic variety with its
Lie algebra n (regarded as an affine space over L); we denote this identification as
usual via X 7→ exp(X). The group law on N is then determined by the Lie bracket
on n together with the Baker-Cambpell-Hausdorff formula:1
exp(X) exp(Y ) = exp(X + Y + [X, Y ] + · · · ).
2This identification induces a natural isomorphism between C pol (N, K) and the symmetric algebra over K of the dual to n:(2.5.1)∼

C pol (N, K) −→ Sym•K ň,

where ň := HomL (n, K) is the K-dual space to n.
Since the group structure on N is induced by the algebraic group law of N,
the space of functions C pol (N, K) is invariant under the right regular action of N .
We may extend the right regular N -action on C pol (N, K) to a P -action as follows.
If m ∈ M , define (mf )(n) = f (m−1 nm) for any m ∈ M, f ∈ C pol (N, K), andn ∈ N . One immediately checks that mf again lies in C pol (N, K). Using theequality P = M N , we combine the M and N -action on C pol (N, K) into a P -action.
The resulting P -action on C pol (N, K) is algebraic, in the sense that C pol (N, K)
may be written as the union of an increasing sequence of finite dimensional P invariant subspaces, on each of which P acts through an algebraic representation.
Differentiating this P -action we make C pol (N, K) a p-module (where p denotes the
Lie algebra of P ). Note that since P is a connected algebraic group, this p-action
uniquely determines the P -action that gives rise to it.
We define a bilinear pairing C pol (N, K) × U(n) → K via(2.5.2)(f, X) 7→ (Xf )(e)(where as usual e denotes the identify element of N ). We may use this pairing togive another useful description of C pol (N, K).
Lemma 2.5.3. The pairing (2.5.2) induces an n-equivariant isomorphism∼∞

C pol (N, K) −→ HomK (U(n), K)n .

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

35(Here HomK (U(n), K) is equipped with its right regular U(n)-action, and the target
is the U(n)-submodule of HomK (U(g), K) consisting of elements that are annihilated
by nk for some k ≥ 0.)
Proof. This is well-known and straightforward. ∞

Note that since the n-action on HomK (U(n), K)n is locally finite by definition, we may integrate the n-action to obtain an N -action. The isomorphism ofLemma 2.5.3 is then N -equivariant (since we observed above that the N -action on
C pol (N, K) is also obtained by integrating the corresponding n-action).
We endow C pol (N, K) with its finest convex topology; since C pol (N, K) is ofcountable dimension, this topology makes it a compact type space. The preceding description of the P -action on C pol (N, K) shows that it is P0 -analytic for any
analytic open subgroup P0 of P , and so in particular is locally analytic.
If W is any Hausdorff convex K-space, then we write C pol (N, W ) := W ⊗Kpol
C (N, K) to denote the space of polynomial functions on N with coefficients in W .
We endow C pol (N, W ) with the inductive tensor product topology. Since C pol (N, K)
is endowed with its finest convex topology, we see that C pol (N, W ) is topologicallyisomorphic to a direct sum of copies of W (indexed by the elements of a basis of
C pol (N, K)). We will apply this notation with W taken either to be U (our fixed
object of Repla.c (M )), or one of the BH-subspaces Ui of U (in the notation of
Subsection 2.2).
We let P act on U through its quotient M , and define a P -action on C pol (N, U )by taking the tensor product of the P -action on U and the P -action discussed above
on C pol (N, K). If we regard C pol (N, U ) as a space of U -valued functions on N , thenthis action is given by the formula(2.5.4)(mnf )(n0 ) = mf (m−1 n0 mn)

for m ∈ M, n, n0 ∈ N , and f ∈ C pol (N, U ). Since the M -action on U and the
P -action on C pol (N, K) are both locally analytic, the same is true of the P -action
on C pol (N, U ). Differentiating this action makes C pol (N, U ) a U(p)-module.
We will now explain how the p-action on C pol (N, U ) may be extended in a naturalmanner to a g-action, making C pol (N, U ) a (g, P )-module. We will do this by giving
another description of C pol (N, U ), which will naturally exhibit the (g, P )-action on
this space.
We first note that the isomorphism of Lemma 2.5.3 induces an isomorphism(2.5.5)∼∞

C pol (N, U ) −→ HomK (U(n), U )n .

If we regard U as a p-representation, via the quotient map p → m, then the iso∼morphism U(p) ⊗K U(n) −→ U(g) induced by multiplication in U(g) gives rise to anatural isomorphism(2.5.6)∞

HomU(p) (U(g), U )n∼−→ HomK (U(n), U )n∞(where the source is defined by regarding U(g) as a U(p)-module via left multiplication). Combining the isomorphisms (2.5.5) and (2.5.6) yields an isomorphism(2.5.7)∼∞

C pol (N, U ) −→ HomU(p) (U(g), U )n .

We may use this isomorphism to transport the U(g)-action on the target inducedby right multiplication to a U(g)-action on the source.

36

MATTHEW EMERTON

Lemma 2.5.8. The g-action on C pol (N, U ) arising from the isomorphism (2.5.7)
is continuous, and makes C pol (N, U ) a (g, P )-representation.
Proof. This is easily checked by the reader.



We will need to consider certain gradings on the spaces C pol (N, K). Recall that
we have fixed a coweight α of the maximal split torus in ZM that pairs strictly
negatively with all the restricted roots of N . The adjoint action of α on g andthe coadjoint action of α on ň induce gradings on g and on Sym•K ň. (Note thatthe grading on the latter space typically does not coincide with the grading bysymmetric powers).
We use the isomorphism (2.5.1) to transport the grading on Sym•K ň to a gradingof C pol (N, K). We let C pol,d (N, K) denote the dth graded piece of C pol (N, K) withrespect to this grading. Our assumption on α implies that C pol,d (N, K) is finite
dimensional for each d, and vanishes if d < 0. Since the grading on C pol (N, K) is
defined using a coweight of the centre of M , we see that each of the graded pieces
C pol,d (N, K) is an M -invariant subspace of C pol (N, K).
We writeM
0
C pol,≤d (N, K) :=C pol,d (N, K)
d0 ≤d

and
C pol,>d (N, K) :=M

0C pol,d (N, K).

d0 >dSince our choice of α ensures that n is graded in negative degrees, we see that eachof the spaces C pol,≤d (N, K) is n-invariant, and hence forms a P -invariant subspaceof C pol (N, K).
For any Hausdorff convex K-vector space W we induce a grading on the tensor
product C pol (N, W ) = W ⊗K C pol (N, K) by defining
C pol,d (N, W ) := W ⊗K C pol,d (N, K)
for all d ≥ 0. We also write
C pol,≤d (N, W ) = W ⊗K C pol,≤d (N, K) =M

0C pol,d (N, W )

d0 ≤d

and
C pol,>d (N, W ) = W ⊗K C pol,>d (N, K) =M

0C pol,d (N, W )d0 >dfor all d ≥ 0.
Taking W = U , each of the spaces C pol,d (N, U ) (respectively C pol,≤d (N, U )) is a
closed M -invariant (respectively closed P -invariant) subspace of C pol (N, U ).
Lemma 2.5.9. The g-action on C pol (N, U ) arising from the isomorphism (2.5.7)
is a graded action, if we endow each of g and C pol (N, U ) with the grading inducedby α described above.
Proof. This is clear. JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

37

Suppose now that Y is an open affinoid rigid analytic polydisk contained in N.
Since the space of polynomial functions on N is dense in the space of rigid analytic
functions on Y, we obtain a continuous injection(2.5.10)M∼

C pol,d (N, Uj ) −→ C pol (N, Uj ) → C an (Y, Uj )d≥0with dense image. Thus we may regard C an (Y, Uj ) as a completion of C pol (N, Uj )with respect to a certain topology. We now describe this completion more explicitly.
We begin by introducing some notation.
Definition 2.5.11. Let W d (for d ≥ 0) denote a sequence of Banach spaces,
each equipped with a fixed norm || – ||d that defines the topology on W d . Let
L
L
d
d
ˆ
to thed≥0 W denote the completion of the direct sumd≥0 W with respect
L
d
d d
d
norm || (w ) || := maxd≥0 || w || (where (w ) denotes an element of d≥0 W d ).
L
Note that the construction of ˆ d≥0 W d depends on the particular choice ofnorms || – ||d , and not just on the topological vector space structures of the Banach
spaces W d .
Let us now fix a norm on Uj that defines the topology on Uj ; the induced supnorm on elements of C an (Y, Uj ) then defines the topology on C an (Y, Uj ). Notethat for any fixed degree d, the continuous injection (2.5.10) restricts to a closed
embeddingC pol,d (N, Uj ) → C an (Y, Uj ).

(2.5.12)

If we write || – ||dY,j to denote the pull-back of the sup norm on C an (Y, Uj ) to
the Banach space C pol,d (N, Uj ) via (2.5.12), then || – ||dY,j defines the topology on
C pol,d (N, Uj ).
Lemma 2.5.13. The map (2.5.10) induces a topological isomorphismMˆ∼

C pol,d (N, Uj ) −→ C an (Y, Uj ),d≥0

the source of the isomorphism being defined via Definition 2.5.11, using the norms|| – ||dY,j .
Proof. This follows directly from definition of the space of rigid analytic functions
C an (Y, Uj ). 
Recall from Subsection 2.2 that Ai,j := C an (Ni , Uj ), for any i, j ≥ 0. The
previous discussion applies in particular when we take Y := Ni . In this case (2.5.10)
induces a continuous injection(2.5.14)

M∼

C pol,d (N, Uj ) −→ C pol (N, Uj ) → Ai,jd≥0

with dense image. Writing || – ||di,j := || – ||dNi ,j , we obtain the following special caseof Lemma 2.5.13.

38

MATTHEW EMERTON

Lemma 2.5.15. The map (2.5.14) induces a topological isomorphismMˆ pol,d∼C(N, Uj ) −→ Ai,j ,d≥0

the source of the isomorphism being defined via Definition 2.5.11, using the norms|| – ||di,j .
Note that the Banach spaces appearing as direct summands in the source of theisomorphism of Lemma 2.5.15 are independent of i; however, the norms || – ||di,j dodepend on the choice of i (as they must, since Ai,j certainly depends on i as wellas j). The next result quantifies the dependence of these norms on i.
Lemma 2.5.16. For any i ≥ 0, the norms || – ||d0,j and || – ||di,j on C pol,d (N, Uj ) are
related by the formula|| – ||di,j = |p|di || – ||d0,j .
Proof. Since z0 = α(p)−1 and Ni = z0i N0 z0−i , this follows from the fact that thegrading on C pol (N, Uj ) is defined via the coweight α. ∼

The isomorphism lim
Uj −→ U of Subsection 2.2 induces an isomorphism−→i(2.5.17)∼lim
C pol (N, Uj ) −→ C pol (N, U ).
−→j

Passing to the inductive limit of the maps (2.5.14), and taking into account theisomorphisms (2.5.17) and (2.2.5), we obtain a map(2.5.18)

C pol (N, U ) → C ω (N, U )e .

Lemma 2.5.19. The maps (2.5.18) is a continuous (g, M )-equivariant injection
with dense image.
Proof. Since for each i, j ≥ 0 the morphism (2.5.14) is a continuous injection, with
dense image, the same is true of (2.5.18). The claimed equivariance is easily checkedby the reader. Composing (2.5.18) with the isomorphism of Corollary 2.3.4 then yields a continuous (g, M )-equivariant injection(2.5.20)GU )eC pol (N, U ) → (IndP

with dense image.
In the remainder of this subsection we introduce some terminology and basic
results related to spaces of locally polynomial functions.
Definition 2.5.21. Let Cclp (N, U ) denote the tensor productC pol (N, U ) ⊗K Ccsm (N, K)∼(−→ U ⊗K C pol (N, K) ⊗K Ccsm (N, K)∼−→ C pol (N, K) ⊗K Ccsm (N, U )).
equipped with the inductive (or equivalently, by [6, Prop. 1.1.31], projective) tensor product topology. (The superscript “lp” stands for “locally polynomial”.) It
is a compact type convex K-space, which we endow with a continuous (g, P )representation by having U(g) act via its action on the first factor C pol (N, U ),and having P act diagonally.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

39

Lemma 2.5.22. The P -action on Cclp (N, U ) is locally analytic.
Proof. This follows from [6, 3.6.18] and the fact that the P -action on each of
C pol (N, U ) and Ccsm (N, K) is locally analytic. 
Multiplication of polynomials by smooth functions induces a map(2.5.23)

Cclp (N, U ) → Ccla (N, U ).

Lemma 2.5.24. The morphism (2.5.23) is a continuous (g, P )-equivariant injection, whose image is equal to the subspace of Ccla (N, U ) consisting of functions fthat are locally polynomial, in the sense that in a neighbourhood of each point of N ,the function f is defined by an element of C pol (N, U ).
Proof. This is easily confirmed by the reader. Definition 2.5.25. For any open subset Ω of N , define Cclp (Ω, U ) := C pol (N, U )⊗K
Ccsm (Ω, K). For d ≥ 0, let Cclp,≤d (Ω, U ) denote the closed subspace C pol,≤d (N, U )⊗K
Ccsm (Ω, K) of C lp (Ω, U ).
The closed embedding Ccsm (Ω, K) → Ccsm (N, K) given by extending functions
by zero induces a closed embedding Cclp (Ω, U ) → Cclp (N, U ), via which we regard
Cclp (Ω, U ) as a g-invariant closed subspace of Cclp (N, U ).
Lemma 2.5.26. The closed subspace Cclp (Ω, U ) of Cclp (N, U ) is equal to the preimage of the closed subspace Ccla (Ω, U ) of Ccla (N, U ) under the map (2.5.23).
Proof. This is immediate. 
The composite of the map (2.5.23) with the isomorphism of Lemma 2.3.6 restricts
to a continuous injection(2.5.27)G
U )(Ω).
Cclp (Ω, U ) → (IndP

for any open subset Ω of N . The description of the image of (2.5.23) provided by
Lemma 2.5.24 shows that the image of (2.5.27) is a local subspace of IndG
U (inP
the sense of Definition 2.4.1), whose stalk at e is equal to the image of (2.5.20).
(2.6) In this section we develop some further results related to representations
on spaces of polynomial functions. Although these results are not required for the
proof of the main theorem in Subsection 4.3, they will be useful in applications.
(See for example Propositions 2.7.16 and 5.2.11 below.)
Fix a chart Y around e ∈ N and integers d, j ≥ 0. Let Πdj : C an (Y, Uj ) →pol,dC(N, Uj ) denote the projection onto the dth direct summand in the isomorphismof Lemma 2.5.13.
Lemma 2.6.1. If I denotes
C an (Y, Uj ) to itself, thenP∞ thed identityanoperator fromthere is an equality I = d=1 Πj in L(C (Y, Uj ), C an (Y, Uj ))s (i.e. the series on
the right converges in weak topology to the identity operator).
Proof. This is clear from the isomorphism of Lemma 2.5.13. 
Let d/dt denote the standard non-zero tangent vector at the identity of the torus
L× , and write ∂α to denote the pushforward of d/dt with respect to α (so ∂α is a
non-zero element in the Lie algebra of ZM ). The action of ZM on C ω (N, U ) being

40

MATTHEW EMERTON

locally analytic, it induces an action of ∂α on C ω (N, U )e , as well as on C an (Y, Uj )e
for each open affinoid polydisk Y contained in N and each j ≥ 0. These actions
induce maps
(2.6.2)K[∂α ] → L(C ω (N, U )e , C ω (N, U )e )sand(2.6.3)K[∂α ] → L(C an (Y, Uj ), C an (Y, Uj ))s .

(Here K[∂α ] denotes the polynomial ring over K in ∂α , or equivalently, the universal
enveloping algebra of the lie algebra of the one dimensional torus L× . Note alsothat the targets of these two maps are equipped with their weak topologies.)
The following result is a special case of the theory of diagonalizable modules
developed in [10, §1.3.1]. We give a complete proof, since [10] remains unpublished,
and the argument is in any case straightforward.
Proposition 2.6.4. If ∂α acts on the locally analytic M -representation U via a
scalar, then for each d, j ≥ 0, the projection Πdj lies in the closure of the imageof (2.6.3).
ˆ K Uj , and that the ∂α -action on this
Proof. Recall that C an (Y, Uj ) = C an (Y, K) ⊗tensor product is simply the tensor product action of the ∂α -action on each factor.
By assumption ∂α acts on U , and hence Uj , through a scalar λ.
Fix d ≥ 0. For any r ≥ 0 such that 0 ≤ d < pr , let φd,r : Zp → {0, 1} denote thecharacteristic function of the coset d+pr Zp ⊂P
Zp . It is a continuousfunction on Zp ,∞and so admits a Mahler expansion φd,r (x) := k=0 ad,r,k xk , with ad,r,k ∈ Zp . Let
Ppr −1
Pd,r := k=0 ad,r,k xk ∈ Zp [x]. Note that then Pd,r (x) = φd,r (x) if 0 ≤ x < pr ;more concretely,
1 if x = d(2.6.5)
Pd,r (x) =0 if 0 ≤ x < pr and x 6= d.
Let f ∈ C an (Y, Uj ), and via the isomorphism of Lemma 2.5.13 write f =P∞0pol,d0(N, Uj ) = C pol,d (N, K) ⊗K Ui for each d0 ≥ 0, andd0 =1 fd0 , where fd0 ∈ C
0||fd0 ||dY,j → 0 as d0 → ∞. Taking into account (2.6.5), one computes that
(2.6.6)

Pd,r (∂α − λ)f =∞X

X

Pd,r (d0 )fd0 = fd +d0 =0

Pd,r (d0 )fd0 .

d0 ≥prSince Pd,r (d0 ) ∈ Zp for all d0 ≥ 0, we find that00lim ||Pd,r (d0 )fd0 ||di,j ≤ 0lim ||fd0 ||di,j = 0,

d0 →∞

d →∞

which when combined with (2.6.6) shows thatlim Pd,r (∂α − λ)f = fd .

r→∞Thus Pd,r (∂α − λ) converges weakly to Πdj as r → ∞, and so Πdj does lie in theclosure of the image of (2.6.3). JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

41

The preceding discussion applies in particular taking Y := Ni for some i ≥ 0.
We then write
Πdi,j : Ai,j = C an (Ni , Uj ) → C pol,d (N, Uj )for the projection denoted above by Πdj . The projections Πdi,j are compatible withthe transition maps of (2.2.5) and (2.5.17), and so passing to the corresponding
inductive limits, they yield maps
Πd : C ω (N, U )e → C pol,d (N, U ).
Corollary 2.6.7. (i) If I denotes
identity operator from C ω (N, U )e to itself,
P∞ thedthen there is an equality I = d=1 Π in L(C ω (N, U )e , C ω (N, U )e )s (i.e. the series
on the right converges in weak topology to the identity operator).
(ii) If ∂α acts on the locally analytic M -representation U via a scalar, then foreach d, j ≥ 0, the projection Πdj lies in the closure of the image of (2.6.2).
Proof. The claims follow from Lemma 2.6.1 and Proposition 2.6.4, together with
the isomorphism (2.2.5). (2.7) We maintain the notation of the preceding subsections.
GU )e , then we let X pol denote theDefinition 2.7.1. If X is a subspace of (IndP
preimage of X under the map (2.5.20).

The map (2.5.20) induces a continuous injection(2.7.2)

X pol → X .

GLemma 2.7.3. If X is a closed (g, M )-invariant subspace of (IndPU )e , then X polpol
is closed (g, P )-invariant subspace of C (N, U ), and (2.7.2) is a (g, M )-equivariant
map.

Proof. Evidently X pol is a closed (g, M )-invariant subspace of C pol (N, U ), since themap (2.5.20) is continuous and (g, M )-equivariant. Furthermore (2.7.2) is (g, M )equivariant, being a restriction of (2.5.20). Since the N -action on C pol (N, U ) is obtained by integrating the n-action, we see that X pol , being both M and n-invariant,
is in fact P = M N -invariant. The following lemma gives a result in the converse direction to that just proved.
Lemma 2.7.4. If X is a closed (g, M )-invariant subspace of C pol (N, U ), and if X̂
Gdenotes the closure in (IndPU )e of the image of X under (2.5.20), then X̂ is aG(g, P )-invariant subspace of (IndPU )e .
Proof. Since X is assumed to be (g, M )-invariant, the same is true of its image
under (2.5.20), and thus of the closure X̂ of this image. Lemma 2.4.10 shows that
if U is an object of Repzla.c (G) then X is in fact (g, P )-invariant. GDefinition 2.7.5. If X is a local closed G-invariant subspace of IndP
U, thenfor any open subset Ω of N we let X lp (Ω) denote the preimage of X(Ω) under
the continuous injection (2.5.27). For any d ≥ 0, we let X lp,≤d (Ω) denote theintersection of X lp (Ω) and Cclp,≤d (Ω, U ).

42

MATTHEW EMERTON

For any open subset Ω of N , the map (2.5.27) evidently restricts to a continuous
injection(2.7.6)

X lp (Ω) → X(Ω).

In the case when Ω = N , Lemma 2.5.24 implies that (2.5.27) is (g, P )-equivariant.
Since X(N ) is a closed (g, P )-invariant subspace of (IndG
U )(N ), we thus see thatP
X lp (N ) is a closed (g, P )-invariant subspace of Cclp (N, U ).
Since Xe is a closed subspace of (IndG
U )e (by Proposition 2.4.7 (i)), Lemma 2.7.3
Pshows that (Xe )pol is a closed (g, P )-invariant subspace of C pol (N, U ). By defini∼tion we have an isomorphism C pol (N, U ) ⊗K C sm (N, K) −→ Cclp (N, U ) (see Definition 2.5.21), which restricts to a closed embedding(2.7.7)(Xe )pol ⊗K C sm (N, K) → Cclp (N, U ).

Lemma 2.7.8. The map (2.7.7) induces an isomorphism∼(Xe )pol ⊗K C sm (N, K) −→ X lp (N ).
Proof. Let Y denote the image of (2.7.7). To show that Y ⊂ X lp (N ), it suffices(since X is local) to show that for every element F ∈ (Xe )pol , and for every sufficiently small compact open subset Ω of N , the function F|Ω lies in X. (Here F|Ωdenotes the locally polynomial function defined by F on Ω, and by zero elsewhere.)GU,
If n ∈ Ω, then F|Ω = n−1 (nF )|Ωn−1 . Since X is an N -invariant subspace of IndPpolpolwhile (Xe )is an N -invariant subspace of C (N, U ) (by Lemma 2.7.3), we seethat it suffices to treat the case when Ω is a sufficiently small neighbourhood of e.
It then follows from the very definition of (Xe )pol that F|Ω lies in X for sufficiently
small neighbourhoods Ω of e.
To prove that X lp (N ) ⊂ Y, it suffices to show that if a locally polynomial functionf ∈ Ccla (N, U ) lies in X, then for any sufficiently small compact open subset Ω of N
we have f|Ω = F|Ω for some F ∈ (Xe )pol . If n ∈ Ω, then f|Ω = n−1 (nf )|Ωn−1 . Again
using the N -invariance of X and of (Xe )pol , we see that it suffices to treat the casewhen Ω is a sufficiently small neighbourhood of e. Since f is locally polynomial,
we see that F := ste (f ) is a polynomial and that f|Ω = F|Ω for any sufficiently
small neighbourhood Ω of e. Since f ∈ X, it follows that F lies in (Xe )pol . This
completes the proof of the lemma. GProposition 2.7.9. If X is a local closed G-invariant subspace of IndP
U , thenthe following are equivalent.
(i) The image of (Xe )pol in Xe under the map (2.7.2) (with X taken to be Xe )is dense in Xe .
(ii) The image of (2.7.6) is dense in X(Ω) for some (or equivalently, for every)
non-empty open subset Ω of N .
GFurthermore, X is then equal to the closed G-subrepresentation of IndP
U generated by the image of (2.7.6) for any non-empty open subset Ω of N .

Proof. Suppose that (i) holds. The image of (2.7.6) for Ω = N is a local (g, P )invariant subspace of X(N ). If we let Y denote the closure of this image, then Y
is (g, P )-invariant, and Lemma 2.4.4 shows that Y is again local. We claim that

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

43

there is an equality of stalks Yn = Xn for all n ∈ N . Granting this, it follows from
Lemma 2.4.3 that Y = X(N ). Since, by Lemma 2.4.4, Y (Ω) coincides with theclosure of the image of (2.7.6) for any open subset Ω of N , we conclude that (ii)holds.
If we consider stalks at e, then it is clear that Ye contains the image of (2.7.2)
GU )e , by(applied to (Xe )pol ), and is contained in Xe . Since Ye is closed in (IndP
Lemma 2.4.6, our hypothesis implies that ste (Y ) = Xe . Since both Y and X(N )are N -invariant, it follows that indeed Yn = Xn for all n ∈ N , as claimed.
Conversely, suppose that (ii) holds, i.e. that (2.7.6) has dense image for someopen subset Ω of N . Since the map ste is continuous, the image of the composite(2.7.10)(2.7.6)

st

e
G
(IndP
U )e
X lp (Ω) −→ X(Ω) −→is dense in Xe = ste (X). The isomorphism of Lemma 2.7.8 restricts to an isomorphism∼(Xe )pol ⊗K C sm (Ω, K) −→ X lp (Ω).
Thus we see that the image of (2.7.10) coincides with the image of (2.7.2), andso (i) holds.
Finally, if condition (ii) holds then X(Ω) is by assumption the closure of theimage of (2.7.6). Since (2.4.12) is a surjection (X being local) we see that X is indeed
U generated by the image of (2.7.6). the closed G-subrepresentation of IndG
PDefinition 2.7.11. We say that a subspace of C pol (N, U ) is graded if it is gradedwith respect to the grading induced by α on C pol (N, U ) that was discussed in
Subsection 2.5.
Suppose now that X is a closed graded (g, M )-invariant subspace of C pol (N, U ).
The isomorphism (2.5.17) then induces isomorphisms
M∼∼Xj −→ lim
Xdj ,(2.7.12)X −→ lim
−→
−→
j

j

d≥0

where Xj denotes the preimageof X under the natural injection C pol (N, Uj ) →T pol,dpoldC (N, U ), and Xj := Xj C
(N, Uj ) (where the intersection takes place inC pol (N, Uj )). If we let X̂i,j denote the closure in Ai,j of the image of Xj under (2.5.14), then the isomorphism of Lemma 2.5.15 induces an isomorphismMˆ∼X̂i,j −→
Xdj ,d≥0where the target is defined via Definition 2.5.11, using the restriction of the norms|| – ||di,j to Xdi,j .
From this description of X̂i,j it is clear that X̂i,j coincides with the preimageof X̂i0 ,j 0 under the transition map (2.2.2), for any i0 ≥ i, j 0 ≥ j. It follows from[6, Prop. 1.1.41] together with the isomorphism (2.2.5) and the isomorphism ofCorollary 2.3.4 that we have an isomorphism(2.7.13)∼lim
X̂i,j −→ X̂
−→i,j(where as in the statement of Lemma 2.7.4, we let X̂ denote the closure of inGU )e of the image of X under (2.5.20)).
(IndP

44

MATTHEW EMERTON

Corollary 2.7.14. The evident inclusion X ⊂ X̂pol is in fact an equality.
Proof. If follows directly from the explicit description of X̂ provided by (2.7.12)
and (2.7.13) that the preimage of X̂ under the injection(2.5.20)U )e
C pol (N, Uj ) −→ C pol (N, U ) −→ (IndG
P
coincides with Xj , for each j ≥ 0, and thus that the preimage of X̂ under (2.5.20)
coincides with X, as claimed. 
Definition 2.7.15. If X is a local closed G-invariant subspace of IndG
U, thenP
we say that X is polynomially generated if (Xe )pol is a graded (in the sense of
Definition 2.7.11) subspace of C pol (N, U ), and if the image of (Xe )pol under (2.5.20)
is dense in Xe .
We say that X is polynomially generated by bounded degrees
if furthermoreT(Xe )pol is generated as a g-module by (Xe )pol,≤d := (Xe )pol C pol,≤d (N, U ), forsome d ≥ 0.
If U is an object of Repzla.c (M ), then the results of this Subsection and of Subsection 2.4 show that the polynomially generated local closed G-invariant subspaces ofG
U are in natural bijection with the closed graded (g, M )-invariant subspaces
IndP
of C pol (N, U ), the bijection being provided by X 7→ (Xe )pol .
We close this subsection with a result which helps illustrate the scope of Definition 2.7.15.
Proposition 2.7.16. If the M -representation U admits a central character, thenGevery local closed G-subrepresentation of IndP
U is polynomially generated. If U
is furthermore finite dimensional, then any such G-subrepresentation is in fact
generated by bounded degrees.
Proof. If X is a local closed G-subrepresentation of IndG
U , then it follows fromP
Proposition 2.4.7 (i) that Xe is a (g, P )-invariant, and so in particular ∂α -invariant,GU )e . Corollary 2.6.7 then implies that (Xe )pol is graded,closed subspace of (IndPand that its image under (2.5.20) is dense in Xe . Thus X is polynomially generated,
as claimed.
If U is finite dimensional, then C pol (N, U ) is a finitely generated U(g)-module.
Since U(g) is Noetherian, (Xe )pol is then also finitely generated, and so is generated
over g by (Xe )pol,≤d for some sufficiently large value of d. This shows that X isgenerated by bounded degrees. (2.8) We maintain the notation of the previous subsections. We begin by proving
Lemma 0.3.
Proof of Lemma 0.3. Composing the closed embedding(2.8.1)

Ccsm (N, U ) → Ccla (N, U )with the isomorphism of Lemma 2.3.6 yields a closed embedding(2.8.2)

Ccsm (N, U ) → IndG
U.
P

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

45

Passing to Jacquet modules, and taking into account [7, Lem. 3.5.2], we obtain aclosed embedding U (δ) → JP (IndG
U ), as required. PRecall from the introduction that if U is an object of Repzla.c (M ), then IPG (U ) isdefined to be the closed G-subrepresentation of IndG
U generated by the image ofPGGU under the canonical lifting of JP (IndP U ) to IndP U . Recall from the discussionof [7, §3.4] that the canonical lifting depends on a choice of compact open subgroup
N0 of N (and also a choice of Levi factor of P ; however, we have already fixed sucha choice). The following lemma gives another description of IPG (U ), which showsthat it is well-defined independently of the choice of N0 .
Lemma 2.8.3. If U is an object of Repzla.c (M ), then IPG (U ) coincides with theclosed G-subrepresentation of IndG
U generated by the image of the closed embedP
ding (2.8.2).
Proof. Fix a compact open subgroup N0 of N , and for u ∈ U , let u|N0 ∈ Ccsm (N, U )denote the function that is identically equal to u on N0 , and that is zero on the
U is defined by sending
complement of N0 in N . The canonical lifting U → IndGP
an element u ∈ U to the image of u|N0 under (2.8.1). (See the discussion following
the proof of [7, Thm. 3.5.6].) Since {u|N0 | u ∈ U } generates Ccsm (N, U ) as a P representation, we see that the image of U under the canonical lift generates theimage of (2.8.2) as a P -representation. Consequently, the closed G-representation
generated by the image of U under the canonical lift coincides with the closed
G-representation generated by the image of (2.8.2). Regarding an element of U as a constant U -valued function on N induces aclosed P -equivariant embedding(2.8.4)

U → C pol (N, U ).

(Here P acts on U through its quotient M .) Tensoring (2.8.4) with Ccsm (N, K) over
K induces a closed P -equivariant embedding(2.8.5)

Ccsm (N, U ) → Cclp (N, U ).

Note that the composite of (2.8.5) and (2.5.23) coincides with (2.8.1).
The P -equivariant maps (2.8.4) and (2.8.5) induce (g, P )-equivariant maps
(2.8.6)

U(g) ⊗U(p) U → C pol (N, U )and(2.8.7)

U(g) ⊗U(p) Ccsm (N, U ) → Cclp (N, U ).

respectively. We may regard (2.8.7) as being obtained from (2.8.6) by tensoringwith Ccsm (N, K) over K.
Lemma 2.8.8. If U is an allowable object of Repla.c (M ), then the image of (2.8.6)
is a closed (g, P )-invariant subspace of C pol (N, U ).
∼Proof. Multiplication in U(g) induces an isomorphism U(n) ⊗K U(p) −→ U(g).
Thus we may rewrite (2.8.6) as a map U(n) ⊗K U → C pol (N, U ). We may further
write this map as the direct sum of maps
(2.8.9)Ud (n) ⊗K U → C pol,d (N, U ) = C pol,d (N, K) ⊗K U,

46

MATTHEW EMERTON

as d ranges over the non-negative integers. (Here Ud (n) denotes the degree d gradedpiece of U(n) with respect to the grading induced by α.)
Since the formation of the maps (2.8.9) is functorial in U , they are EndM (U )equivariant, and hence have closed image (since each of Ud (n) and C pol,d (N, K) isa finite dimensional algebraic M -representation, and U is assumed allowable). It
follows from [2, Cor. 1, p. II.31] that (2.8.6) has closed image, as asserted. 
Proposition 2.8.10. If U is an allowable object of Repzla.c (M ), then IPG (U ) is a local closed polynomially generated G-invariant subspace of IndG
U , and IPG (U )lp (N )P
(as given by Definition 2.7.5) coincides with the image of (2.8.7).
Proof. Let X denote the image of (2.8.6), which by Lemma 2.8.8 is a closed subspaceof C pol (N, U ). Since (2.8.4) identifies U with C pol,0 (N, U ), Lemma 2.5.9 showsthat X is furthermore graded, in the sense of Definition 2.7.11. If X̂ denotes
the closure of the image of X under the map (2.5.20), then Lemma 2.7.4 showsthat X̂ is a (g, P )-invariant subspace of (IndG
U )e , which corresponds to a local
PG
closed subrepresentation X of IndP U via the construction of Proposition 2.4.9.
Corollary 2.7.14 implies that X = X̂pol , and thus that X is polynomially generated
in the sense of Definition 2.7.15.
Proposition 2.7.9 implies that X coincides with the closed G-representation generated by X lp (N ), which Lemma 2.7.8 shows to be isomorphic to X ⊗K Ccsm (N, K).
The definition of X then shows that X lp (N ) is equal to the image of (2.8.7). Since
this image maps under (2.5.27) (taking Ω to be N ) to the g-subrepresentation ofG
IndP
U generated by the image of (2.8.2), we conclude from Lemma 2.8.3 that X
coincides with IPG (U ). This establishes the proposition. In the context of the preceding proposition, note that by construction IPG (U )is in fact polynomially generated in degree zero. Note also that the proof of this
proposition shows that IPG (U )lp (N ) coincides with the space denoted in the samemanner in the introduction.
3. A variant of the method of Amice-Vélu and Vishik(3.1) We maintain the notation introduced in Subsection 2.2, and fix in addition a K-Banach space V equipped with a continuous action of ZM . We recall from
Subsection 1.4 the definition of Ccla (N, V ) as a convex K-vector space. We endow
Ccla (N, V ) with the right regular action of N , and with the ZM -action defined by(zf )(n) = zf (z −1 nz). These actions together give a locally analytic action of N ZM
on Ccla (N, V ). If we embed C la (N0 , V ) as the closed subspace of Ccla (N, V ) consist+ing of functions supported on N0 , then C la (N0 , V ) is an N0 ZM-invariant subspace
laof Cc (N, V ).
L
For each i ≥ 0, consider the space Bi := n∈Ni \N0 C an (Ni n, V ) (the direct sum
ranging over a set of left coset representatives of Ni in N0 ). This may be naturally
identified with the space of Ni -analytic vectors in C la (N0 , V ) with respect to theleft regular action of N0 , and so is equipped with a natural map(3.1.1)

Bi → C la (N0 , V ).

Since any Ni -analytic vector is also Ni+1 -analytic, there is a natural map(3.1.2)Bi → Bi+1 ,JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

47

compatible with (3.1.1) for i and i + 1. We may form the inductive limit lim
Bi−→iwith respect to the maps (3.1.2), and the maps (3.1.1) induce an isomorphism(3.1.3)∼lim
Bi −→ C la (N0 , V );−→ithis gives a concrete realization of C la (N0 , V ) as an inductive limit of Banach spaces.
The right regular N0 -action on C la (N0 , V ) induces an N0 -action on each Bi , andthe maps (3.1.1), (3.1.2), and (3.1.3) are all N0 -equivariant. The action of z0 on∼
C la (N0 , V ) induces an isomorphism z0i B0 −→ C an (Ni , V ). Since the action of any∼anelement n ∈ N0 induces an isomorphism C (Ni , V ) −→ C an (Ni n−1 , V ), we obtainan isomorphismM∼(3.1.4)nz0i B0 −→ Bi .
n∈Ni \N0+
Suppose that S ⊂ T are closed N0 ZM-invariant subspaces of C la (N0 , V ). For
each i ≥ 0, let Ti (respectively Si ) denote the preimage of T (respectively S) underthe map (3.1.1). The closed embedding S ⊂ T induces a closed embedding

Si ⊂ T i(3.1.5)for each i ≥ 0, while the isomorphism (3.1.4) induces closed embeddings(3.1.6)M

nz0i T0 → Tin∈Ni \N0and(3.1.7)M

nz0i S0 → Si .

n∈Ni \N0

As discussed in the Notations and conventions, let T and S denote the ultrabornologicalizations of S and T respectively. It follows from [2, Prop. 1, p. I.20]∼∼
Ti −→ T and limthat there are natural isomorphisms lim
Si −→ S (the transition
−→
−→
i

i

maps for these locally convex inductive limits being induced by the maps (3.1.2)).
Recall that we have continuous bijections(3.1.8)T →T

and

S → S.

The closed embeddings (3.1.5) induce a continuous injection(3.1.9)S→T

in a manner compatible with the bijections (3.1.8).
The reason for considering T and S is that T and S, while being of LB-type –that is, they are a countable union of BH-subspaces, being closed subspaces of the
LB-space C la (N0 , V ) – may not be LB-spaces. The space T (respectively S) may be
regarded as being the same underlying abstract vector space as T (respectively S),

48

MATTHEW EMERTON

endowed with the coarsest locally convex topology that both makes it an LB-space
and is finer than the given topology on T (respectively S).
+
Since the formation of ultrabornologicalizations is functorial, the N0 ZM-action+on T and S induces an N0 ZM -action on each of the space T and S, uniquely+determined by the requirement that the bijections (3.1.8) be N0 ZM-equivariant.
(These actions are a priori separately continuous, but since T and S are LB-, andso barrelled, spaces, they are in fact continuous.) The map (3.1.9) is then also+
necessarily N0 ZM-equivariant.
+
Let W be a K-Banach space equipped with a continuous action of N0 ZM. Our
goal in the remainder of this subsection is to state a theorem giving conditions
under which the map(3.1.10)+ (S, W )+ (T , W ) → L
LN0 ZM
N0 ZM

induced by (3.1.9) is an isomorphism. The proof of this theorem will be given in
the following subsection.
In order to state our theorem, we must introduce further notation. We fix
anorm|| – ||V on V that determines
its Banach space structure. Since Bi :=
L
`
an
an
C
(N
n,
V
)
=
C
(
n, V ) is equal to the space of rigid ani
n∈Ni \N0n∈Ni \N0 Ni`
alytic V -valued functions on the affinoid n∈Ni \N0 Ni n, the choice of the norm|| – ||V on V determines a norm on Bi , namely the usual rigid analytic sup norm.
We denote this norm by || – ||i ; it determines the Banach space structure on Bi . We
also fix a norm || – ||W on W that determines its Banach space structure.
Since N0 is compact and acts continuously on W , we may and do assume that|| – ||W is N0 -invariant (by [6, Lem. 6.5.3] for example). Since z0 acts as a continuous
endomorphism of W , we may find C1 > 0 such that || z0 w ||W ≤ C1 || w ||W for allw ∈ W . Since z0 induces a continuous automorphism of V , we may find C2 > 0such that || z0 (u) ||V > C2 || u || for all u ∈ V. Write C = C1 /C2 .
For each i ≥ 0, fix a topological complement Si⊥ to Si in Ti (as we may, by [11,Prop. 10.5]), and let πi (respectively πi⊥ ) denote the projection Ti → Si with kernelSi⊥ (respectively the projection Ti → Si⊥ with kernel Si ).
Theorem 3.1.11. Suppose that the closed embeddings (3.1.6) and (3.1.7) are isomorphisms, and that there exists  > 0 such that C < 1, and such that for alli0 ≥ i ≥ 0, one has(3.1.12)0|| πi⊥0 (f ) ||i0 ≤ i −i || f ||i

for all f ∈ Ti . Then (3.1.10) is an isomorphism.
As remarked above, we present the proof of this result in Subsection 3.2. The
following lemma gives a criterion for verifying the first hypothesis of Theorem 3.1.11.
Lemma 3.1.13. Let T be a closed subspace of C la (N0 , V ) such that:(i) For all f ∈ T , the element z0−1 (f|N1 ) of C la (N0 , V ) again lies in T .
(ii) For any f ∈ T and any compact open subspace Ω ⊂ N0 , the restriction f|Ω(regarded as an element of C la (N0 , V ) via extending by zero) again lies in T .
Then the closed embedding (3.1.6) is an isomorphism.
T
Proof. Fix i ≥ 0 and n ∈ N0 ,Tand let f ∈ Ti C an (Ni n, V ). Since Ti is N0 invariant, we see that nf ∈ Ti C an (Ni , V ), and so by (i) (applied i times) weJACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

49

Tifind that z0−i nf ∈ T0 . Thus Ti C an (Ni n, V ) ⊂ n−1 zT
0 T0 . The reverse inclusion+
is clear (T being N0 ZM -invariant),and sofact Ti C an (Ni n, V ) = n−1 z0i T0 .
L
T in
an
Since (ii) implies that Ti = n∈Ni \N0 Ti C (Ni n, V ), we see that (3.1.6) is an
isomorphism, as claimed. (3.2) In this subsection we prove Theorem 3.1.11. Thus we assume that theclosed embeddings (3.1.6) and (3.1.7) are in fact isomorphisms, and that (3.1.12)
holds for all i0 ≥ i ≥ 0 and some  (which we fix) satisfying 0 <  < C −1 . We begin
by establishing some estimates.
Lemma 3.2.1. If i ≥ 0 and f ∈ Ti , then || f ||i+1 ≤ || f ||i .
Proof. `Recall that || f ||i+1 is computed as the sup norm of f over the rigid analytic
space n∈Ni+1 \N0 Ni+1 n, while || f ||i is computed as the sup norm of f over the`
rigid analytic space n∈Ni \N0 Ni n. Since the former rigid analytic space is an open
subset of the latter, the claimed inequality obviously holds. 
Lemma 3.2.2. If i ≥ 0 and f ∈ Ti , so that z0 f ∈ Ti+1 , then || z0 f ||i+1 ≥ C2 || f ||i .
`Proof. By assumption f is a rigid analytic function on n∈Ni \N0 Ni n, and so z0 f`
is the rigid analytic function on n∈Ni+1 \N1 Ni+1 n defined by(z0 f )(x) = z0 (f (z0−1 xz0 ))(and extended by zero to`n∈Ni+1 \N0 Ni+1 n). We compute that|| z0 f ||i+1 = sup{|| (z0 f )(x) ||V | x ∈aNi+1 n}n∈Ni+1 \N1a= sup{|| z0 (f (z0−1 xz0 )) ||V | x ∈Ni+1 n}n∈Ni+1 \N1a≥ sup{C2 || f (z0−1 xz0 ) ||V | x ∈Ni+1 n}n∈Ni+1 \N1= C2 sup{|| f (x) ||V | x ∈aNi n} = C2 || f ||i ,n∈Ni \N0

as claimed. 
Lemma 3.2.3. Let t be a continuous semi-norm on T that satisfies the following
conditions:(i) The semi-norm t is N0 -invariant.
(ii) We have t(z0 f ) ≤ C1 t(f ) for all f ∈ T .
Then there exists A > 0 such that t(f ) ≤ AC i || f ||i for all i ≥ 0 and f ∈ Ti .
Proof. Since t is continuous, there is a constant A > 0 such that t(f ) ≤ A|| f ||0
for allPf ∈ T0 . Now let f ∈ Ti . Since (3.1.6) is an isomorphism, we may write
f = n∈Ni \N0 nz0i gn for some gn ∈ T0 . We then compute (taking into accountconditions (i) and (ii)) thatt(f ) ≤ max{t(z0i gn )} ≤ C1i max{t(gn )} ≤ AC1i max{|| gn ||0 }.
On the other hand, Lemma 3.2.2 shows that|| f ||i = max{|| z0i gn ||i } ≥ C2i max || gn ||0 .
Combining these two inequalities (and recalling that C = C1 /C2 ) yields the inequality of the lemma. 

50

MATTHEW EMERTON

Lemma 3.2.4. If t is any continuous semi-norm on T that satisfies conditions (i)and (ii) of Lemma 3.2.3, then for all i0 ≥ i ≥ 0 and f ∈ Ti we have t(πi⊥0 (f )) ≤0
AC i (C)i −i || f ||i .
Proof. This follows from Lemma 3.2.3 (applied to πi⊥0 (f ) ∈ Ti0 ) and (3.1.12). Lemma 3.2.5. If t is any continuous semi-norm on T that satisfies conditions (i)and (ii) of Lemma 3.2.3, then the image of (3.1.9) is dense in T with respect to the
topology induced by t.
Proof. If f ∈ Ti , then it follows from Lemma 3.2.4 and the assumption that C < 1
that limi≤i0 →∞ t(f − πi0 (f )) = limi≤i0 →∞ t(πi⊥0 (f )) = 0. +
We can now prove that (3.1.10) is injective. Let F : T → W be an N0 ZM
equivariant continuous K-linear map T → W, and let t denote the continuoussemi-norm on T defined by t(f ) = || F (f ) ||W for all f ∈ T . It follows directlyfrom the corresponding properties of || – ||W that t satisfies conditions (i) and (ii)of Lemma 3.2.3. If the restriction of F to the image of (3.1.9) vanishes, then the
restriction of t to the image of (3.1.9) also vanishes. Lemma 3.2.5 then implies
that t vanishes identically on T , and thus that F vanishes. Consequently (3.1.10)
is injective.
The proof that (3.1.10) is surjective is similar. We begin by proving the following
result.

Proposition 3.2.6. Let s be a continuous semi-norm on S that satisfies conditions (i) and (ii) of Lemma 3.2.3, with t and T replaced by s and S. Then there
is a uniquely determined continuous semi-norm t on T that satisfies conditions (i)and (ii) of Lemma 3.2.3, and such that s is equal to the composite of t with (3.1.9).
Proof. The uniqueness is a direct consequence of Lemma 3.2.5. As for the existence, we have the following explicit definition of t: If f ∈ Ti , then t(f ) =limi≤i0 →∞ s(πi0 (f )). We will show that this gives a well-defined continuous seminorm on T that satisfies conditions (i) and (ii) of Lemma 3.2.3. Note that if f ∈ Si
then πi0 (f ) = f for all i0 ≥ i, and so it is clear that s is equal to the composite of tand (3.1.9).
To show that t(f ) is well-defined, we must show that the limit exists. For this,it suffices to show that limi≤i0 →∞ s(πi0 +1 (f ) − πi0 (f )) exists and equals zero. We
compute that (for some A > 0)s(πi0 +1 (f ) − πi0 (f ))0≤ AC i +1 || πi0 +1 (f ) − πi0 (f ) ||i0 +10= AC i +1 || πi⊥0 +1 (f ) − πi⊥0 (f ) ||i0 +1(3.2.7)0≤ AC i +1 max{|| πi⊥0 +1 (f ) ||i0 +1 , || πi⊥0 (f ) ||i0 +1 }0

≤ AC i +1 max{|| πi⊥0 +1 (f ) ||i0 +1 , || πi⊥0 (f ) ||i0 }00≤ AC i +1 i −i max{, 1}|| f ||i0= AC i+1 max{, 1}(C)i −i || f ||i(where the first inequality follows from Lemma 3.2.3, applied with s and Si0 +1 in
place of t and Ti , the subsequent equality follows from the equation πi0 +1 (f ) −JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II51

πi0 (f ) = f − πi⊥0 +1 (f ) − (f − πi⊥0 (f )) = πi⊥0 (f ) − πi⊥0 +1 (f ), the second inequality is anapplication of the non-archimedean triangle inequality, the third inequality follows
from Lemma 3.2.1, and the fourth inequality follows from (3.1.12).) Since C < 1
by assumption, we conclude that indeed limi≤i0 →∞ s(πi0 +1 (f ) − πi0 (f )) = 0, and sot is well-defined. It is clear that t is a semi-norm, since s is a semi-norm and the
projections πi0 are linear.
The inequality (3.2.7) immediately generalizes: If i ≤ i0 ≤ i00 , and f ∈ Ti then
(3.2.8)s(πi00 (f ) − πi0 (f )) ≤ max{s(πi0 +1 (f ) − πi0 (f )), . . . , s(πi00 (f ) − πi00 −1 (f ))}0≤ AC i+1 max{, 1}(C)i −i || f ||i ,using (3.2.7) and the fact that C < 1.
Now || πi (f ) ||i ≤ max{|| πi⊥ (f ) ||i , || f ||i } = || f ||i (applying (3.1.12) with i0 = i),which together with Lemma 3.2.3 (putting πi (f ), s and Si in place of f , t and Ti )implies that s(πi (f )) ≤ AC i || f ||i . Combining this and (3.2.8), we find thats(πi0 (f )) ≤ max{s(πi (f )), s(πi0 (f ) − πi (f ))} ≤ AC i max{1, C max{, 1}}|| f ||i .
Passing to the limit as i0 → ∞, we deduce thatt(f ) ≤ AC i max{1, C max{, 1}}|| f ||i .
∼Ti , we conclude that t is
This implies that t is continuous on Ti . Since T −→ lim−→icontinuous on T .
If i ≥ 0 and f ∈ Ti , then for any i0 ≥ i, we compute thatt(f − πi0 (f )) = 0 lim
00i ≤i →∞= 0 lim00s(πi00 (f − πi0 (f )))i ≤i →∞0s(πi00 (f ) − πi0 (f )) ≤ AC i+1 max{, 1}(C)i −i || f ||i(where the inequality follows from (3.2.8)). Passing to the limit as i0 → ∞, andremembering that C < 1, we find that limi≤i0 →∞ t(f − πi0 (f )) = 0, and thus that
πi0 (f ) → f in the t-topology on T as i0 → ∞. In particular, we conclude that theimage of (3.1.9) is dense in T with respect to the t-topology. Since conditions (i)and (ii) of Lemma 3.2.3 hold for s on S by assumption, we conclude that they also
hold for t on T . +
We can now prove that (3.1.10) is surjective. Let F1 : S → W be an N0 ZM
equivariant continuous K-linear map, and let s denote the continuous semi-norm on
S defined by s(f ) = || F1 (f ) ||W . The semi-norm s satisfies conditions (i) and (ii) of
Lemma 3.2.3, and so we may extend s to a semi-norm t on T as in Proposition 3.2.6.
Since t induces the s-topology on the image of S in T under the injection (3.1.9),
since this image is dense in the t-topology on T , since F1 is continuous with respect
to the s-topology on S (by the very definition of s), and since W is complete,
we conclude that there is a t-continuous map F : T → W whose composition+
with (3.1.9) coincides with F1 . Furthermore, since the action of N0 ZM
is continuous+with respect to the t-topology on T , and since F1 is N0 ZM -equivariant, we conclude+
that F is N0 ZM-equivariant. Since t is continuous with respect to the inductive
limit topology on T , we see that F is continuous with respect to the inductive limit
topology on T . Thus F is the required element of LN0 Z + (T , W ) that maps to F1
M
under (3.1.10).

52

MATTHEW EMERTON

4. Proof of Theorem 0.13
(4.1) We maintain the notation introduced in Subsection 2.2. In particular,
we fix an object U of Repla.c (M ), which in fact we assume to lie in Repzla.c (M ).
We also fix a local closed G-invariant subspace X of IndG
U . If V is a locallyP
analytic G-representation, then restriction of morphisms from X to X(N ) induces
a morphism
LG (X, V ) → L(g,P ) (X(N ), V ).

(4.1.1)

The following result show that an element in the target of (4.1.1) is as G-equivariant
as it can be.
Lemma 4.1.2. Suppose that V is an LB-space equipped with a locally analytic
G-representation, and let φ be an element in the target of (4.1.1). If f ∈ X(N )and g ∈ G are chosen so that gf also lies in X(N ), then φ(gf ) = gφ(f ).
Proof. Let Ω ⊂ N be the support of f . Since Ω is compact, we may choose i
sufficiently large so that for any n ∈ Ω we have Ni n ⊂ Ω, and (nf )|Ni is Ni analytic, taking values in Ui . If we fix this value of i, and let Xi denote thepreimage under (2.3.8) of X(N ) (or equivalently, the preimage under (2.3.9) of
Xe ), then we see (taking into account Proposition 2.3.10 and the fact that X is aG−U ) that Xi is an (Hi , ZM)-invariant closed subspace ofG-invariant subspace of IndP
Ai,i , which by assumption contains (nf )|Ni for any n ∈ Ω. Since φ is a continuous
g-equivariant map between the LB-spaces X(N ) and V , and since the G-action
on V is locally analytic, we deduce from [2, Prop. 1, p. I.20] that we may find a
BH-subspace W of V that is Hj -invariant for some j ≥ i, such that the Hj action
on W obtained by [6, Prop. 1.1.1 (ii)] is Hj -analytic, and such that φ induces ag-equivariant continuous map Xi → W .
By assumption the compact open subset Ωg −1 of P \G (which is the support ofgf ) is contained in N (regarded as an open subset of P \G via the open immersion (2.3.2)). Equivalently, the product Ωg −1 , now computed in G (by regardingΩ as a compact subset of G via the closed embedding N ⊂ G), is contained in
P N = N P . If we let Ω denote the projection of Ωg −1 onto N , then Ω is a compact−
subset of N . Thus we may find z ∈ ZM
such that zΩz −1 ⊂ N j .
Fix n ∈ Ω. Since Ωg −1 ⊂ ΩP, we may write
gn−1 = pn(4.1.3)for some n ∈ Ω(4.1.4)−1

and p ∈ P . We thus compute thatφ(g(f|z−1 Ni zn )) = φ(gn−1 z −1 (znf )|Ni ) = φ(pnz −1 (znf )|Ni )= pz −1 φ(znz −1 (znf )|Ni ) = pz −1 (znz −1 )φ((znf )|Ni )= pnnφ(n−1 z −1 (znf )|Ni ) = gφ(n−1 z −1 (znf )|Ni ) = gφ(f|z−1 Ni zn ).

Here the first and last equalities follow from the formulaf|z−1 Ni zn = n−1 z −1 (znf )|Ni ,JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

53

the second and second-last equalities follow from (4.1.3), and the third and thirdlast equalities follow from the P -equivariance of φ. The middle equality of (4.1.4)
follows from the g-equivariance of φ, the fact that (znf )|Ni (which lies in Xi , since−(nf )|Ni does, and since Xi is ZM-invariant) is Hi -analytic, and so also Hj -analytic,
the fact that all the elements of φ(Xi ) are Hj -analytic, and the containment znz −1 ∈−1
N j ⊂ Hj`
(since n ∈ Ω ). We now write Ω as a disjoint union of left z −1 Ni z-cosets,msay Ω = l=1 z −1 Ni znl , and compute thatXX
φ(gf ) = φ(gf|z−1 Ni znl ) =φ(gf|z−1 Ni znl )XX=
gφ(f|z−1 Ni znl ) = gφ(f|z−1 Ni znl ) = gφ(f )(the third equality following from (4.1.4)), as required. 
Theorem 4.1.5. If V is an LB-space equipped with a locally analytic action of G,
then (4.1.1) is an isomorphism.
Proof. Since X(N ) generates X as a G-representation, we see that the map (4.1.1)
is an injection. We must show that is is also surjective.
Let φ be an element of the target of (4.1.1). Since V is endowed with a G-action,
we see that φ induces a natural map
K[G] ⊗K X(N ) → V.

(4.1.6)

We will show that (4.1.6) factors through the map (2.4.12) (with Ω taken to be N ).
Since (2.4.12) is a strict surjection, by Lemma 2.4.13, this will imply that (4.1.6)
factors through a continuous G-equivariant map X → V , which will then be anelement in the source of (4.1.1) that maps under (4.1.1) to φ.
Thus we must prove the following claim: if g1 , . . . , gl is a finite sequence ofelements of G, and f1 , . . . , fl a finite sequence of elements of X(N ), such thatlX

(4.1.7)gi fi = 0 in X,

i=1

then
l
X

(4.1.8)?

gi φ(fi ) = 0 in V.

i=1

Since X is local, it suffices to show that each point x of P \G has a compact open
neighbourhood Ωx such that for any neighbourhood Ω0x ⊂ Ωx of x we havelX

(4.1.9)?

gi φ(fi|Ω0x gi ) = 0.

i=1Indeed, we can then`spartition P \G into a finite disjoint union of such neighbourhoods, say P \G = j=1 Ω0xj , and writing
gi fi =sX

(gi fi )|Ω0x =

s
X

j

j=1

gi fi|Ω0x gi ,
j

j=1

54

MATTHEW EMERTON

so thatfi =sXfi|Ω0x gi ,j

j=1

we conclude that if (4.1.9) holds, then so does (4.1.8).
If x = P g for some g ∈ G, then replacing (g1 , . . . , gl ) by (gg1 , . . . , ggl ), we seethat it suffices to treat the case when x equals the identity coset, which we identify
with the identity e of N under the open immersion (2.3.2). Let Ωe be a compact
open neighbourhood of e in N , chosen so that Ωe gi ⊂ N (as open subsets of P \G)for all i for which gi ∈ P N , and so that Ωe gi (regarded as an open subset of P \G)is disjoint from the support of fi for all other i. It follows from (4.1.7) that
(4.1.10)lX(gi fi )|Ωe =

i=1

l
X

gi fi|Ωe gi = 0 in X(N ).

i=1

Applying the map φ to the second equality of (4.1.10) (which is an equation involving elements of X(N ), since f|Ωe gi = 0 if gi 6∈ N , by virtue of our choice of Ωe ), andPltaking into account Lemma 4.1.2, we deduce that indeed i=1 gi φ(fi|Ωe gi ) = 0,
and we are done. (4.2) We maintain the notation and assumptions of Subsection 4.1. We furG
U isthermore assume that the given local closed G-invariant subspace X of IndP
polynomially generated, in the sense of Definition 2.7.15.
Taking Ω to be N and N0 in turn in (2.7.6) yields continuous injections(4.2.1)

X lp (N ) → X(N )and(4.2.2)

X lp (N0 ) → X(N0 )which sit in the Cartesian diagram(4.2.3)X lp (N0 )/ X(N0 )X lp (N )/ X(N ),

whose vertical arrows are given by extension by zero. The injection (4.2.1) is (g, P )equivariant. If P + denotes the submonoid of P generated by N0 and M + , thenone immediately checks that X(N0 ) is a (g, P + )-invariant subspace of X(N ). Thus
X lp (N0 ) is a (g, P + )-invariant subspace of X lp (N ), and the map (4.2.2) is (g, P + )equivariant.
Note that the N -actions on X(N ) and on X lp (N ) induce isomorphisms(4.2.4)∼

K[N ] ⊗K[N0 ] X(N0 ) −→ X(N )and(4.2.5)∼

K[N ] ⊗K[N0 ] X lp (N0 ) −→ X lp (N ).

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

55

Fix i, j ≥ 0. Let Tj denote the preimage of X(N0 ) under (2.2.9), and let Ti,jdenote the preimage of Tj under (2.2.7) (or equivalently, the preimage of X(N0 )under (2.2.10)). The isomorphisms (2.2.8) and (2.2.11) give rise to a continuous
bijection∼lim
Ti,j −→ Tj−→(4.2.6)iand an isomorphism∼lim
Ti,j −→ X(N0 )−→(4.2.7)i,jrespectively. If T j is the ultrabornologicalization of Tj , then the continuous bijection (4.2.6) induces an isomorphism∼lim
Ti,j −→ T j .
−→i

For any d, i, j ≥ 0, let Sjd denote the preimage of X lp,≤d (N0 ) under (2.2.9), anddlet Si,jdenote the preimage of Sjd under (2.2.7) (or equivalently, the preimage oflp,≤dX(N0 ) under (2.2.10). The isomorphisms (2.2.8) and (2.2.11) give rise to acontinuous bijection
∼

d
lim
Si,j
−→ Sjd
−→(4.2.8)iand an isomorphism(4.2.9)

∼

d
lim
Si,j
−→ X lp,≤d (N0 )
−→
i,j

d
respectively. If S j is the ultrabornologicalization of Sjd , then the continuous bijec-

tion (4.2.8) induces an isomorphism
∼

d

lim
Si,j −→ S j .
−→i+Since the ZM -action on U induces a ZM -action on each Uj , the N0 ZM-action on+lala
C (N0 , U ) induces an N0 ZM -action on C (N0 , Uj ), and each of the subspaces Tj+
and Sjd is N0 ZM-invariant.
lp,≤dSince X
(N0 ) is a closed subspace of X(N0 ), we find that Sjd is a closed
subspace of Tj for each d, j ≥ 0. As in Subsection 3.1, the closed embedding of Sjd+
in Tj induces a N0 ZM-equivariant continuous injectiond

Sj → T j .

(4.2.10)+The isomorphisms (4.2.7) and (4.2.9) give rise to N0 ZM-equivariant isomorphisms∼lim
T j −→ X(N0 ),−→(4.2.11)j(4.2.12)d∼lim
S j −→ X lp,≤d (N0 ),−→jand (as a consequence of (4.2.12))(4.2.13)d∼lim
S j −→ X lp (N0 ).
−→
j,d

56

MATTHEW EMERTON+
Lemma 4.2.14. If W is a K-Banach space equipped with a continuous N0 ZM
action, and if we fix j ≥ 0, then the map(4.2.15)d∼+ (S , W )+ (T j , W ) −→ L
LN0 ZMj
N0 ZM

given by pulling back via (4.2.10) is a natural isomorphism provided d is sufficiently
large (in a manner possibly depending on j).
Proof. We will prove the lemma via an application of Theorem 3.1.11. We fix j, andtake the Banach V of Subsection 3.1 to be Uj ; thus the spaces Bi of Subsection 3.1
correspond to the spaces Bi,j of Subsection 2.2. As in Subsection 3.1, fix norms|| – ||Uj and || – ||W defining the topologies of Uj and W respectively.
We take the spaces S and T of Subsection 3.1 to be Sjd and Tj . It is clear fromthe definition of Sjd and Tj that the conditions of Lemma 3.1.13 hold for both ofthem, and thus that the closed embeddings (3.1.6) and (3.1.7) are isomorphisms.
We write X := (Xe )pol , and use the notation introduced in Subsection 2.7, following Definition 2.7.11. Note that our assumption that X is polynomially generated
implies in particular that Xe = X̂. In terms of that notation, we may writeMTi,j =nX̂i,jn∈Ni \N0

and
nX≤d
i,j =

M

d
Si,j
=

M

M

n∈Ni \N0 d0 ≤dn∈Ni \N0

We define
d,⊥
Si,j
=

0

nXdi,j .

M

M
ˆ

M

nX>d
i,j =

0

nXdi,jn∈Ni \N0 d0 >dn∈Ni \N00(where the completed direct sum is taken with respect to the norms || – ||di,j on the0d ,⊥Banach spaces Xdi,j ). Let πi,jdenote the projection onto the second factor of theisomorphism
M ⊥,d
∼
d
Ti,j −→ Si,j
Si,j .
PP∞
If f ∈ Ti,j , write f =n∈Ni \N0 nfn , where fn =d0 =0 fn,d0 ∈ X̂i,j with
fn,d0 ∈ Xdi,j . Then|| πid0 ,j,⊥ (f ) ||i0 ,j =
=
≤

0

max

sup || fn,d0 ||di0 ,jmax

sup |p|d (i −i) || fn,d0 ||di,jmax

sup |p|(d+1)(i −i) || fn,d0 ||di,jn∈Ni \N0 d0 >d

0

0

0

n∈Ni \N0 d0 >d00n∈Ni \N0 d0 ≥00= |p|(d+1)(i −i) || f ||i,j .
(The second equality follows from the formula of Lemma 2.5.16.) Since |p|d+1 → 0
as d → ∞, we see that if we choose d large enough, then condition (3.1.12) will
be satisfied if we take  = |p|d+1 . It now follows from Theorem 3.1.11 that pullingdback along the map S j → T j induces an isomorphism∼d+ (T j , W ) −→ L
+ (S , W ),LN0 ZMj
N0 ZM

as claimed. JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

57

Theorem 4.2.16. If W is any K-Banach space equipped with a continuous P + action, then pulling back via (4.2.2) induces a natural isomorphism∼lp
LP + (X(N0 ), W ) −→ LP + (X (N0 ), W ).

Proof. Passing to the projective limit over d and j of the maps (4.2.15), and taking into account the statement of Lemma 4.2.14 and the isomorphisms (4.2.11)
and (4.2.13), we find that the map(4.2.17)lp+ (X(N0 ), W ) → L+ (X(N0 ), W )LN0 ZM
N0 ZM

given by pulling back via (4.2.2) is also an isomorphism. Since (4.2.2) has dense
image, we find that a map in the source of (4.2.17) is M + -equivariant if and only if
this is true of its pull-back via (4.2.2). Thus (4.2.17) restricts to the isomorphismin the statement of the theorem. Corollary 4.2.18. If X is polynomially generated by bounded degrees, then for any
K-Banach space W equipped with a continuous G-action, pulling back via (4.2.1)
induces a natural isomorphism∼

lp
L(g,P ) (X(N ), Wla ) −→ L(g,P ) (X (N ), Wla ).

Proof. Since P is generated by P + as a group, and since we have the isomorphisms (4.2.4) and (4.2.5), it suffices to prove that the map(4.2.19)∼

lp
L(g,P + ) (X(N0 ), Wla ) −→ L(g,P + ) (X (N0 ), Wla )given by pulling back via (4.2.2) is an isomorphism. It follows from Theorem 4.2.16
that (4.2.19) is an injection, and that any map in the target may be extended toa continuous P + -equivariant map X(N0 ) → W . Thus to prove the corollary, it
suffices to show that any continuous map X(N0 ) → W with the property that its
composite with (4.2.2) factors through a continuous g-equivariant map X lp (N0 ) →Wla itself necessarily factors through a continuous g-equivariant map X(N0 ) → Wla .
The isomorphism (2.2.11) induces an isomorphism∼

lim
Ti,i −→ X(N0 ).
−→i

If Y denotes the image of (4.2.2), then the preimage Yi of Y in Ti,i is dense indTi,i . Furthermore, this preimage contains the closed Banach subspace Si,iof Ti,ifor all d ≥ 0, and by assumption, if d is sufficiently large this subspace generates
Yi as a g-representation. The corollary now follows from Proposition 1.2.23, once
we note that the g-action on Ti,i is locally integrable (by Proposition 1.2.19 andLemma 2.3.14, since Ti,i is a closed g-invariant subspace of Bi,i ). 
(4.3) We maintain the notation introduced in Subsection 4.1. The closed embedding of X(N ) into X, when composed with (4.2.1), yields a map(4.3.1)

X lp (N ) → X.

58

MATTHEW EMERTON

Theorem 4.3.2. Suppose that X is polynomially generated by bounded degrees. If
W is a K-Banach space equipped with a continuous G-representation, then (4.3.1)
induces an isomorphism∼lp
LG (X, Wla ) −→ L(g,P ) (X (N ), Wla ).

Proof. Theorem 4.1.5 and Corollary 4.2.18 show that we have isomorphisms∼∼lp
LG (X, Wla ) −→ L(g,P ) (X(N ), Wla ) −→ L(g,P ) (X (N ), Wla ),proving the theorem. Corollary 4.3.3. Suppose that X is polynomially generated by bounded degrees.
If V is a very strongly admissible locally analytic G-representation (in the sense of
Definition 0.12), then (4.3.1) induces an isomorphism∼lp
LG (X, V ) −→ L(g,P ) (X (N ), V ).

Proof. By assumption we may find a continuous G-equivariant injection V → W ,where W is a Banach space equipped with a continuous admissible G-action. Since
the G-action on V is locally analytic, this continuous injection factors through acontinuous map V → Wla . Since V and Wla are both admissible locally analytic
G-representations (the latter by [6, Prop. 6.2.4] or [16, Thm. 7.1 (ii)]), this map isfurthermore a closed embedding [16, Prop. 6.4 (ii)].
Theorem 4.3.2 yields an isomorphism( 4.3.4)∼

lp
LG (X, Wla ) −→ L(g,P ) (X (N ), Wla ).

Since X is topologically generated as a G-representation by the image of (4.3.1),
and since V is closed in Wla , we see that a map in the source of (4.3.4) takes values
in V if and only the same is true of the corresponding map in the target. 
We are now ready to prove Theorem 0.13.
Proof of Theorem 0.13. Let U be an allowable object of Repzla.c (M ). Proposition 2.8.10 shows that IPG (U ) is a local closed G-subrepresentation that is polynomially graded and generated by degree zero (and so in particular by bounded
degrees). Corollary 4.3.3 thus yields an isomorphism
∼

G
G
lp
LG (IP (U ), V ) −→ L(g,P ) (IP (U ) (N ), V ),while by Lemma 0.18, passing to Jacquet modules induces an isomorphism
∼

lp
bal
G
L(g,P ) (IP (U ) (N ), V ) −→ LM (U (δ), JP (V )) .

This proves Theorem 0.13.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

59

5. Examples, complements, and applications(5.1) We maintain the notation introduced in Subsection 2.2. We furthermore
suppose that G is quasi-split, and we take P to be a Borel
subgroup of G. Thus P
T
is an opposite Borel to P , and the Levi factor M = P P is a maximal torus in G.
Proposition 5.1.1. If U is an object of Repzla.c (M ), then the inclusion IPG (U ) ⊂G
IndG
U induces an equality of n-invariants IPG (U )n = (IndP
U )n .
P`U )n . We may find a partition P \G = i Xi of
Proof. Fix an element f ∈ (IndGP
P \G into a finite disjoint union of charts, where each chart Xi is of the form Yi gi ,for some chart Yi of N and some gi ∈ G, and whereP f is rigid analytic on the rigid
analytic polydisk Xi underlying Xi . Since f = i f|Xi , it suffices to prove that
(5.1.2)?

f|Xi ∈ IPG (U )nfor each value of i.
Let W be the Weyl group of M in G. We recall the Bruhat decomposition(5.1.3)

P \G =

a

wN.

w∈W

(Here, for each w ∈ W , we use the same letter to denote the image of w in P \G.)Each chart Xi must have non-empty intersection with at least one of the strataof the Bruhat decomposition, and thus we may choose ni ∈ N and wi ∈ W suchthat wi ni ∈ Xi . We may also write wi ni = n0i gi for some n0i ∈ Yi . Translating
the putative inclusion (5.1.2) by wi ni , and noting that Xi (wi ni )−1 = Yi (n0i )−1 , andalso that Adwi ni (n) = Adwi (n), we find that it suffices to prove the containment
wi ni f|Xi ∈ IPG (U )(Yi (n0i )−1 )Adwi (n)for each value of i. Note that Yi (n0i )−1 is a chart of N containing e, and thatwi ni f|Xi is rigid analytic on the polydisk underlying Yi (n0i )−1 . Thus, to simply
notation, we now let Y denote an arbitrary chart of N containing e, and let fGdenote an element of (IndPU )(Y )Adw (n) which is rigid analytic on the polydisk Y
underlying Y . We will prove that f ∈ IPG (U )(Y )Adw (n) (which by the preceding
discussion suffices to establish the proposition).
Choose j ≥ 0 such that f takes values in the BH-subspace Uj of U . We may regard f as anelement of C an (Y, Uj ), and then via the isomorphism of Lemma 2.5.13,
P∞
write f = d=0 fd , where fd ∈ C pol,d (N, Uj ). The action of Adw (n) on C pol (N, Uj )is a graded action, and thus if f is annihilated by Adw (n), so are each of its components fd . Since IPG (U )(Y ) is closed, it suffices to show that fd ∈ IPG (U )(Y )Adw (n)for each d. In other words, we may assume that f , when thought of as an elementof C an (Y, Uj ), in fact lies in C pol (N, Uj ). The isomorphism of (2.5.7), together with
Lemma 1.5.1 (which we may apply after making an extension of scalars so as tosplit G), then shows that f lies in the g-subrepresentation of C pol (N, U ) generated
by C pol (N, U )n , and thus in IPG (U ), as required. 

60

MATTHEW EMERTON

Corollary 5.1.4. If U is an object of Repzla.c (M ), then the inclusion IPG (U ) ⊂∼G
IndP
U includes an isomorphism JP (IPG (U )) −→ JP (IndG
U ).
P
Proof. This follows from the very definition of the functor JP [7, Def. 3.4.5], together with Proposition 5.1.1. 
Proposition 5.1.5. If U is a finite dimensional locally analytic representation ofG
M , then JP (IndP
U ) is a finite dimensional M -representation.
Proof. Extending scalars, we may assume that U is filtered by one dimensional
U is exact, while JP is left
representations of M . Since the formation of IndG
P
exact, we may thus suppose that U is equal to some locally analytic character ψof M . Extending scalars again, if necessary, we may and do assume that G splits
over K.
Let M̂ denote the rigid analytic space over K that parameterizes the locallyanalytic characters of M (as constructed in [6, §6.4]). Proposition 2.1.2 showsψ is a strongly admissible locally analyticthat the locally analytic induction IndG
P
G-representation, and so by [7, Thm. 0.5], the Jacquet module JP (IndGψ) is anPessentially admissible M -representation. Hence its dual space is identified with thespace of sections of a rigid analytic coherent sheaf F on M̂ . The claim of the
proposition is equivalent to showing that F is supported on a finite subset of M̂ .
If χ ∈ M̂ (K) (where K denotes an algebraic closure of K), with field of definitionEχ (a finite extension of K), then the fibre of F at χ is dual to the χ-eigenspaceJPχ (Eχ ⊗K IndGψ) of JP (Eχ ⊗K IndGψ) (= Eχ ⊗K JP (IndGψ)). Thus to prove
P
P
P
χ
G
the proposition, it suffices to prove that JP (Eχ ⊗K IndP ψ) is non-zero for only afinite number of χ ∈ M̂ (K).
Thus we fix an element χ ∈ M̂ (K), and suppose that JPχ (Eχ ⊗K IndGψ) 6= 0. To
Psimplify our notation, we extend scalars if necessary so as to assume that Eχ = K.
We will show that χ belongs to a certain finite set of characters that depends onlyon ψ.
As in the proof of Proposition 5.1.1, we let W denote the Weyl group of M in
G. Choose a labelling {wi | 1 ≤ i ≤ |W |} of the elements of W , with the property
that if 1 ≤ i ≤ j ≤ |W | then the length of wi is less than or equal to that of wj .
S
For any 1 ≤ i ≤ |W |, we write Bi := j≤i wj N ⊂ P \G. From the theory of the
Bruhat decomposition (5.1.3), it is known that each Bi is an open subset of P \G,and that B|W | = P \G. If we write Vi = (IndG
ψ)(Bi ), thenPG
0 = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ V|W | = IndPψis an increasing and exhaustive filtration of IndG
ψ by closed P -invariant subspaces,
P
inducing a corresponding filtration
0 = JPχ (V0 ) ⊂ JPχ (V1 ) ⊂ JPχ (V2 ) ⊂ · · · ⊂ JPχ (V|W | ) = JPχ (IndGψ)
P
Gof JPχ (IndPψ). By [7, Prop. 3.4.9], we may (and do) identify JPχ (Vi ) (for eachi ∈ {1, . . . , |W |}) with the subspace of ViN0 on which the Hecke operators πN0 ,m(for m ∈ M + ) act via the eigenvalues χ(m).
GClaim 5.1.6: For each i ∈ {1, . . . , |W |}, the map f 7→ stwi f ∈ (IndPψ)wi induces
an injection(5.1.7)G
JPχ (Vi )/JPχ (Vi−1 ) → ((IndPψ)wi )n .

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

61

To see this, let f ∈ JP (Vi ), and let Ω ⊂ Bi denote the support of f . Since
πN0 ,m f = χ(m)f for all m ∈ M + , one easily deduces (taking
T into account the factthat wiTN = Bi \ Bi−1 is a closed subset of Bi , so that Ω wi N is compact) either
that Ω wi N T= ∅ (in which case f is supported in Bi−1 , and so lies in JP (Vi−1 ))or else that Ω wi N = wi N0 . In any case, the germ of f at any point of wi N0 is
evidently determined by the germ of f at wi (since f is N0 -invariant), establishing
the claim.
Before drawing conclusions from the claim, we require some computations. We
first compute the action of M on the target of (5.1.7). Translation by wi induces
the first isomorphism in the sequence of isomorphisms∼∼

Gψ)wi −→ (IndGψ)e −→ C ω (N, ψ)e ,(IndPPthe second being provided by Corollary 2.3.4. Let dψ : m → K denote the derivativeof the character ψ. Note that M acts on (C ω (N, ψ)e )n via ψ, and thus that m actson this space via dψ. Lemma 1.5.1 shows that (C ω (N, ψ)e )wi (n) vanishes unlessdψ(Hr ) is a non-negative integer for each r ∈ ∆wi . If this condition holds, write
kr := dψ(Hr ). Lemma 1.5.1 then implies thatkrk(C ω (N, ψ)e )wi (n) ⊂ X−rr11 · · · X−rll (C ω (N, ψ)e )nfor an appropriately chosen ordering r1 , . . . , rl of the elements of ∆wi . Thus M acts1lon (C ω (N, ψ)e )wi (n) via the character r−k· · · r−k
ψ. (In this last expression we are1lregarding the roots as characters of M rather than m, and hence use multiplicative
rather than additive notation.) Applying the isomorphism of Corollary 2.3.4 andGψ)wi )n via the characterthen translating by wi−1 , we find that M acts on ((IndPψi := rk11 · · · rkl l ψ wi
(where ψ wi (m) := ψ(wi mwi−1 )).
We next compute stwi (πN0 ,m f ) in terms of stwi (f ), for f ∈ JPχ (Vi ) and m ∈ M + .
Recall thatX
πN0 ,m f = mπm−1 N0 m f = mδ(m)n0 f,n0 ∈m−1 N0 m/N0

where δ(m) = [N0 : mN0 m
for m ∈ M , namely
δi (m) :=

−1 −1

]

[N0

T

. We also define a variant of the modulus functionT
wi−1 N wi : mN0 m−1 wi−1 N wi ].
[N0 : mN0 m−1 ]We now compute:stwi (πN0 ,m f ) = stwi (mδ(m)X

n0 f )

n0 ∈m−1 N0 m/N0

= ψi (m)δ(m)X

stwi (n0 f )n0 ∈m−1 N0 m/N0

= ψi (m)δ(m)X

n0 stwi (n0 )−1 (f )n0 ∈m−1 N0 m/N0X

= ψi (m)δ(m)n0 ∈(m−1 N0 m

= ψi (m)δi (m)stwi (f )TTwi−1 N wi )/(N0 wi−1 N wi )stwi (f )62

MATTHEW EMERTON(where the second equality follows from the result of the preceding paragraph, while
the second to last equality follows from the following observations: the stabilizer
in G of (the P -coset of) wi is wi−1 P wi , the intersection of the support of f and
wi N is contained in wi N0 , and stwi (f ) is fixed by n, and hence by the elements ofTG
N wi−1 N wi , since the action of wi−1 N wi on (IndPψ)wi is wi−1 Nwi -analytic, byLemmas 2.3.5 (iii) and 2.3.16).
Combining this computation with Claim 5.1.6, we find that JPχ (Vi )/JPχ (Vi−1 )ψ) vanishes unless χ = ψi δi forvanishes unless χ = ψi δi . Consequently JPχ (IndGP
some i ∈ {1, . . . , |W |}. This proves the proposition. 
Remark 5.1.8. Suppose that ψ is a locally dominant algebraic character of M ,i.e. that ψ = θλ where θ is smooth (i.e. locally constant) and λ is an algebraic
character of M that is dominant (with respect to P ). Let W be the irreducible
algebraic representation of G of highest (with respect to P ) weight λ. The reader
can easily check that there is an isomorphism∼Gθ)sm ⊗K W
IPG (ψ) −→ (IndP(where the first factor denotes the smooth parabolic induction of θ). Corollary 5.1.4
and [7, Prop. 4.3.6] taken together then show that∼∼θ)sm )N ⊗K λ.
JP (IndP ψ) −→ JP (IPG (ψ)) −→ ((IndG
P
Combining this with [5, Thm. 6.3.5] (applied with PΘ = PΩ = P ) yields a precise
description of the semi-simplification of the finite dimensional M -representation
JP (IndP ψ).
One can obtain an analogous description of (the semi-simplification of) the finite dimensional M -representation JP (IndP ψ) in general. To see this, note firstthat Claim 5.1.6 can be strengthened slightly: it evidently holds true if we replace JPχ (Vi )/JPχ (Vi−1 ) by JPgen−χ (Vi )/JPgen−χ (Vi−1 ), where the notation “gen − χ”
denotes the generalized χ-eigenspace. The argument in the proof of the proposition then shows that JPgen−χ (Vi )/JPgen−χ (Vi−1 ) vanishes except under the followingTconditions: kr := dψ(Hr ) ∈ Z≥0 for each root r that appears in Adwi (n) n,
and χ = rk11 · · · rkl l ψ wi δi , in which case its dimension is at most one. In fact, in
this latter case, one can show that JPgen−χ (Vi )/JPgen−χ (Vi−1 ) has dimension exactly
one. Thus (for any given choice of character ψ), we can compute the dimension
Gof JPgen−χ (IndPψ) precisely for each of the finitely many characters χ which may
Gappear in JP (IndPψ), and hence determine precisely the semi-simplification ofG
JP (IndP ψ). (We leave it to the reader to check that this general description of
JP (IndGψ) is consistent with the description already given in the case when ψ isP
locally dominant algebraic.)
Example 5.1.9. Suppose that L = Qp and that G = SL2 (Qp ). Take P (respectively P ) to be the Borel subgroupof upper triangular (respectively lower triangularTmatrices),
so
that
M
=
P
P
is
the subgroup of diagonal matrices. The coroot


a
0
∼
α : a 7→yields an isomorphism Q×
p −→ M , and hence yields an isomor0 a−1
phism(5.1.10)∼K −→ mJACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

63

via k 7→ k∂α (where as in Subsection 2.6, we denote by ∂α the pushforward under
α of the element dt/t in the Lie algebra of Q×p ). We let w : m → K denote theinverse of the isomorphism (5.1.10); thus w generates the weight lattice of M . If r
denotes the (unique) root of M acting on n, then r = w2 , while α coincides with thecoroot rˇ associated to r (and so also ∂αand Hrcoincide). The Weyl group W =0 1{1, w} has order two (one can take w =), and the Bruhat decomposition−1 0
decomposes P \G (which is isomorphic to the Qp -points of the projective line) into
the union of N (the affine line) and a point (the P -coset of w, or the point at
infinity).
Let ψ : M → K × be a locally analytic character, and write dψ = kw. Using∼Gthe notation of the proof of Proposition 5.1.5, we see that V1 = (IndPψ)(N ) −→ψ, andCcla (N, ψ) (the isomorphism being given by Lemma 2.3.6), that V2 = IndG
P
that taking stalks at w induces an isomorphism∼ψ)e .
V2 /V1 −→ (IndGP
The proof of Proposition 5.1.5 shows that JP (V2 )/JP (V1 ) = 0 unless k ∈ Z≥0 , thatis, unless ψ is locally dominant algebraic.
Thus if ψ is not locally dominant algebraic, we find that
JP (IndGψ) = JP (Ccla (N, ψ)) = ψδ.
P
Note also that in this case IPG (ψ) = IndG
ψ, as follows from Proposition 2.8.10 andPpolthe fact that C (N, ψ) is an irreducible g-representation when dψ is not dominantG
integral. (Alternatively, one might note IndP
ψ itself is topologically irreducible[13, Thm. 6.1].)
Suppose on the other hand that ψ is locally dominant algebraic, and write ψ =θwk , with θ smooth. As was noted in the preceding remark,∼θ)sm ⊗K (Symk K 2 )IPG (ψ) −→ (IndG
P(recall that Symk K 2 has highest weight equal to wk ) and so
JP (IndGψ) = JP (IPG (ψ)) = JP ((IndGθ)sm ) ⊗ wk .
PP
Applying [5, Thm. 6.3.5], we find that JP ((IndGθ)sm ) is a two dimensional M P
Grepresentation, whose Jordan-Hölder constituents are θδ and θw . Thus JP (IndPψ)
is a two dimensional M -representation, with Jordan-Hölder factors ψδ and ψ w rk .
Finally, we note that there is an is a short exact sequence
0 → IPG (ψ) → IndG
ψ → IndGψr−(k+1) → 0.
PP(This is the short exact sequence (∗) on p. 123 of [14].) Taking Jacquet modules
yields the exact sequence0/ JP (I G (ψ))
P∼

/ JP (IndG ψ)P/ JP (IndG ψr−(k+1) ).
P

64

MATTHEW EMERTON

Since the final object in this sequence is non-zero, we see that the sequence is notexact on the right, and thus that JP is not an exact functor.
(5.2) As in the preceding section we suppose that G is quasi-split, and we takeP to be a Borel subgroup of G. In general, if V → W is a G-equivariant continuous
surjection of admissible locally analytic G-representations, then the induced map
JP (V ) → JP (W ) need not be surjective. This is the case even when V is of the form
ψ for some locally analytic character ψ of M , as Example 5.1.9 shows. The
IndG
P
main result of this section is Theorem 5.2.18, which shows by contrast that whenV = IPG (ψ), any surjection V → W does induce a surjection on the corresponding
Jacquet modules.
Definition 5.2.1. We say that a g-equivariant surjection of g-modules V → W
is locally split if for any finitely generated g-subrepresentation W1 of W , withpreimage V1 in V , the induced surjection V1 → W1 is g-equivariantly split.
Lemma 5.2.2. If V → W is a continuous G-equivariant surjection of admissible
locally analytic G-representations that is locally split as a surjection of g-modules,
then the induced map JP (V ) → JP (W ) is surjective.
Proof. As in the proof of Proposition 5.1.5, let M̂ denote the rigid analytic spaceover K that parameterizes the locally analytic characters of M . For any χ ∈ M̂ (K),
let Idχ denote the kernel of the composite homomorphism U(p) → U(m) → K, inwhich the first arrow is induced by the projection p → m, and the second arrowis induced by the character dχ : m → K (the derivative of χ). Similarly, for eachi ≥ 0 we let Iχ,i denote the kernel of the composite K[Pi ] → K[Mi ] → K, in which
K[Pi ] and K[Mi ] denote the group rings over K of Pi and Mi respectively, the firstarrow is induced by the projection Pi → Mi , and the second arrow is induced byr
χ|Mi . For any i, r ≥ 0, we let V (Idχ ) ,n denote the P -invariant closed subspaceof V annihilated by both the ideal (Idχ )r ⊂ U(p) and the lie algebra n, and letr
V (Iχ,i ) ,Ni denote the Pi -invariant closed subspace of V annihilated by both theideal (Iχ,i )r ⊂ K[Pi ] and the group Ni . We employ similar notation with W inplace of V .
The assumption that the surjection V → W is locally split as a map of g-modules
implies that the induced map(5.2.3)rr

V (Idχ ) ,n → W (Idχ ) ,n

is also surjective. SinceSthe P -actions on V and W are locally analytic, we haverr
an equality V (Idχ ) ,n = i≥0 V (Iχ,i ) ,Ni , and similarly with W in place of V . Avrr
eraging over right Pi -cosets in P0 induces projections V (Iχ,i ) ,Ni → V (Iχ,0 ) ,N0 and(Iχ,i )r ,Ni(Iχ,0 )r ,N0W→W
for any i, r ≥ 0. Applying these projection to the source
and target of (5.2.3), we find that the induced map(5.2.4)rr

V (Iχ,0 ) ,N0 → W (Iχ,0 ) ,N0

is surjective.
+
Since M = ZM , we write M + := ZM. Any element m ∈ M + induces a Hecke
N0
N0
operator πN0 ,m on V
(respectively W ) [7, Def. 3.4.2]. These operators inducean action of the monoid ring K[M + ] on V N0 and W N0 [7, Lem. 3.4.4]. The space

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II
r

65

r

V (Iχ,0 ) ,N0 (respectively W (Iχ,0 ) ,N0 ) is a K[M + ]-submodule of V N0 (respectively
W N0 ), and the map (5.2.4) is M + -equivariant. We let Iχ denote the kernel in∞K[M + ] of the homomorphism K[M + ] → K induced by χ|M + , and let V N0 ,Iχ (rer∞r∞spectively V (Iχ,0 ) ,N0 ,Iχ , respectively W (Iχ,0 ) ,N0 ,Iχ ) denote the subspace of V N0rr(respectively V (Iχ,0 ) ,N0 , respectively W (Iχ,0 ) ,N0 ) consisting of vectors annihilated
by some power of Iχ .
It was shown in the proof of [7, Prop. 4.2.33] that the Hecke operator πN0 ,z0 actsr
as a compact operator on each of the πN0 ,z0 -invariant subspaces V (Iχ,0 ) ,N0 andr
W (Iχ,0 ) ,N0 . It follows from the theory of compact operators that the generalizedrr
χ(z0 )-eigenspace of πN0 ,z0 on each of V (Iχ,0 ) ,N0 and W (Iχ,0 ) ,N0 is finite dimensional, and that the surjection (5.2.4) induces a surjection on generalized χ(z0 )r∞r∞
eigenspaces. Consequently each of the spaces V (Iχ,0 ) ,N0 ,Iχ and W (Iχ,0 ) ,N0 ,Iχ arer∞
finite dimensional, and the surjection (5.2.4) induces a surjection V (Iχ,0 ) ,N0 ,Iχ →r∞
W (Iχ,0 ) ,N0 ,Iχ . Letting r tend to infinity, we obtain a surjection(5.2.5)∞∞

V N0 ,Iχ → W N0 ,Iχ .

The Jacquet module JP (V ) is an essentially admissible locally analytic M representation [7, Thm. 0.5]. Thus the dual space JP (V )0 is naturally isomorphic
to the space of global sections of a coherent sheaf E on M̂ . Similarly, JP (W )0 is
naturally isomorphic to the space of global sections of a coherent sheaf F on M̂ ,and the induced map JP (V ) → JP (W ) corresponds to a map of coherent sheaves(5.2.6)

F → E.

If we complete this morphism at the point χ of M̂ , we obtain a morphism(5.2.7)F̂χ → Êχ

which is naturally identified with the topological dual to (5.2.5) (as follows fromthe definition of the Jacquet module [7, Def. 3.2.5]; compare [7, Prop. 3.4.9]).
Since (5.2.5) is surjective, we see that (5.2.7) is injective.
Applying the above reasoning to the base-change of the map V → W over arbitrary finite extensions of K, we find that the morphism (5.2.6) becomes injective
after being completed at arbitrary K-valued points of M̂ (where K denotes an algebraic closure of K). Thus (5.2.6) is itself injective. Passing to global sections anddualizing again, we find that the map JP (V ) → JP (W ) is surjective, as claimed. 
The following result allows one to verify the hypothesis of the preceding lemma.
Lemma 5.2.8. Let V → W be a G-equivariant continuous surjection of admissible locally analytic G-representations. If there is a cofinal sequence {Hi } of goodanalytic open subgroups of G such that, for each value of i, the space VH◦i −an is
topologically isomorphic, as an Hi◦ -representation, to a finite direct sum of topologically irreducible Hi◦ -representations, then V → W is locally split as a surjection of
g-modules.
Proof. Since any finitely generated g-subrepresentation of W embeds into WH◦i −anfor sufficiently large i, it suffices to show that the induced map VH◦i −an → WH◦i −an
is a split surjection of Hi◦ -representations for all i. That this map is a surjection

66

MATTHEW EMERTON

follows from Corollary A.13 of the appendix. Dualizing, we find that the induced
map(5.2.9)(WH◦i −an )0b → (VH◦i −an )0b

is a closed embedding of Dan (H◦i , K)-modules, these modules being finitely presented, by Corollary A.14 of the appendix. It suffices to show that the embedding (5.2.9) is split.
Our assumption on VH◦i −an implies that (VH◦i −an )0b is topologically semi-simple
as a Dan (H◦i , K)-module (i.e. is a direct sum of topologically irreducible submodules). That the embedding (5.2.9) is split now follows from standard propertiesof semi-simple objects in abelian categories, together with Proposition A.10 of theappendix. As usual, we regard N as an open subset of P \G via the open immersion (2.3.2).
In particular, we regard the identity e ∈ N as a point of P \G.
Lemma 5.2.10. Let H be a good analytic open subgroup of G for which the locallyanalytic open immersion H → G underlies a rigid analytic open immersion H → G.
T
T ×
Let r ∈ (0, 1) |L |, and write P r := Hr P . Let Pr denote the rigid analytic
Zariski closure of P r in Hr . If the orbit of H on e (via the right action of H on
P \G) is contained in N , then Pr \Hr is an affinoid polydisk, and the locally analytic∼
open immersion P r \Hr −→ e · Hr ⊂ N underlies a rigid analytic open immersion
Pr \Hr → N.
Proof. In the course of the proof it will be necessary to distinguish between various
rigid analytic spaces, and their underlying locally analytic manifolds of L-valued
points. Thus we begin by introducing some additional notation for this purpose.
Let g (respectively n) denote the rigid analytic affine space whose underlying
locally analytic set of L-valued points is equal to the Lie algebra g (respectivelyn). As usual, let h denote the OL -Lie sublattice of g from which H is obtained byexponentiation. This sublattice defines a norm on g, the gauge of h, and we let
hr denote the rigid analytic closed polydisk in g of radius r with respect to thisgauge. We let hr denote the set of L-valued points of hr . The exponential map∼
defines a rigid analytic isomorphism exp : hr −→ Hr . Also, since n is a nilpotent Lie∼algebra, the exponential map defines a rigid analytic isomorphism exp : n −→ N. Of
course, these isomorphisms induce isomorphisms of the underlying locally analytic
manifolds of L-valued points. Finally, we note that since the H-orbit of e in P \Gis contained in N , we have the inclusion H ⊂ P N as open subsets of G, and hence
a rigid analytic open immersion H ⊂ PN.
Consider the diagram
hr

/g

∼ exp


Hr

/n
∼ exp

/ PN


/N

of maps of rigid analytic spaces (in which the upper right horizontal arrow is induced∼by the projection g → p\g −→ n, and the lower right horizontal arrow is inducedJACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

67

by the projection PN → N), as well as the corresponding diagram
/n

/g

hr
∼ exp


Hr

∼ exp

/ PN


/ N,

obtained from it by passing to L-valued points. One sees that the first diagram iscommutative (by rigid analytic continuation, since it is evidently so if we replace hr
by a sufficiently small OL -Lie sublattice), and thus so is the second diagram. Thisestablishes the lemma. 
Proposition 5.2.11. Let H be a good analytic open subgroup of G, and let Ω
denote the H ◦ -orbit of e ∈ P \G. If Ω is contained in N , then for any locally analytic
K-valued character ψ of M , the H ◦ -representation IPG (ψ)(Ω)H◦ −an is topologicallyirreducible.
T ×
Proof. For each r ∈ (0, 1) |L |, let Ωr denote the Hr -orbit of e, so that in thenotation of the preceding lemma, Ωr := P r \Hr . We have an isomorphism∼

IPG (ψ)(Ω)H◦ −an −→ lim
IPG (ψ)(Ωr )Hr −an .
←−1>rThus if V is a closed subspace of IPG (ψ)(Ω)H◦ −an , and if Vr denotes the closureof V in IPG (ψ)(Ωr )Hr −an for each 1 > r > 0, then the induced map V → lim
Vr
←−
1>r

is an isomorphism. Hence it suffices to show that IPG (ψ)(Ωr )Hr −an is a topologically irreducible Hr -representation, or equivalently, a topologically irreducible grepresentation, for each 1 > r > 0.
The preceding lemma yields a closed embedding IPG (ψ)(Ωr )Hr −an → C an (Y, ψ)
for some open affinoid polydisk Y in N. If (IPG (ψ)(Ωr )Hr −an )pol denotes the preimage under this isomorphism of C pol (N, ψ) ⊂ C an (Y, ψ), then (IPG (ψ)(Ωr )Hr −an )polis a g-invariant subspace of IPG (ψ)(Ωr )Hr −an . In order to show that this latter
space is topologically irreducible as a g-representation, it suffices, by Lemma 2.6.1
and Proposition 2.6.4, to show that (IPG (ψ)(Ωr )Hr −an )pol is irreducible as a grepresentation. From the construction of IPG (ψ), one sees that this space coincides
with the image of the natural g-equivariant map
U(g) ⊗U(p) K(ψ) → HomU(p) (U(g), K(ψ)).
(Here K(ψ) denotes the one dimensional M -representation given by the character ψ,and the map is obtained by composing (2.8.6) with the isomorphism (2.5.7), taking
U to be K(ψ).) This image is well-known from the theory of Verma modules to be
an irreducible g-representation. 
Lemma 5.2.12. Let H be a subgroup of G, let x be a point of P \G, and letΩ = xH denote the H-orbit of x. If there exists g ∈ G such that x = P g and suchthat Ω ⊂ N g (where N g denotes the translate of N ⊂ P \G by g), then for every
point x0 ∈ Ω, we may find g 0 ∈ G such that x0 = P g 0 and such that Ω ⊂ N g 0 .
Proof. We may write x0 = P gh for some h ∈ H, and take g 0 = gh. Since Ω ⊂ N g,translating on the right by h, and using the fact that Ω = xH is an H-orbit, wefind that also Ω ⊂ N gh = N g 0 , as required. 

68

MATTHEW EMERTON

Lemma 5.2.13. If {Hi } is a cofinal sequence of open subgroups of G, then we mayfind a natural number i0 , such that for any fixed i ≥ i0 there exists a finite subset{g1 , . . . , gm } ∈ G with the property that, if we write Ωj to denote the Hi -orbit of
P gj in P \G, then P \G is the disjoint union of the Hi -orbits Ωj , and there is an
inclusion Ωj ⊂ N gj for each j = 1, . . . , m.
Proof. Since N is an open subset of P \G, and since P g lies in N g for each g ∈ G,
we may choose for each g ∈ G an index ig such that the Hig -orbit of P g lies in
N g. Since P \G is compact, we may cover P \G by a finite number of these orbits,
associated to the elements g1 , . . . , gl say. Let i0 be the maximum of the indices igj
for 1 ≤ j ≤ l.
Fix i ≥ i0 , and choose {x1 , . . . , xm } ⊂ P \G such that P \G is the disjoint unionof the Hi -orbits Ωj 0 := xj 0 Hi for 1 ≤ j 0 ≤ m. (This is possible, since P \G iscompact.) For each 1 ≤ j 0 ≤ m, we may find 1 ≤ j ≤ l such that xj 0 lies in the
Higj -orbit of P gj . By the choice of igj , this orbit is contained in N gj . Lemma 5.2.12
allows us to find gj0 0 such that xj 0 = P gj0 0 , and such that this Higj -orbit is containedin N 0 gj 0 . Since i ≥ i0 ≥ igj , we see that Hi ⊂ Higj , and thus that Ωj 0 ⊂ N gj 0 .
0} as {g1 , . . . , gm } completes the proof of the lemma. 
Relabelling {g10 , . . . , gm
Corollary 5.2.14. We may find a cofinal sequence of good analytic open subgroups{Hi } of G so that for any locally analytic character ψ of M , and any value of i,
the Hi◦ -representation IPG (ψ)H◦i −an is topologically isomorphic to a finite direct sumof topologically irreducible Hi◦ -representations.
Proof. Fix a cofinal sequence {Hi } of good analytic open subgroups of G. Apply
Lemma 5.2.13 to the cofinal sequence {Hi◦ }. If i ≥ i0 , then we may find a finite
subset {g1 , . . . , gm } ⊂ G such that P \G is the disjoint union of the Hi◦ -orbits Ωjof the points P gj , and such thatΩj ⊂ N gj(5.2.15)for each 1 ≤ j ≤ m. We then have a topological isomorphism of Hi◦ -representations∼

IPG (ψ) −→m
M

IPG (ψ)(Ωj ),j=1

which induces an isomorphism(5.2.16)∼

IPG (ψ)H◦i −an −→mM

IPG (ψ)(Ωj )H◦i −an

j=1

upon passing to H◦i -analytic vectors. Translating by gj induces a topological isomorphism(5.2.17)∼

IPG (ψ)(Ωj )H◦i −an −→ IPG (ψ)(Ωj gj−1 )gj H◦ g−1 −anijfor each 1 ≤ j ≤ m, which is compatible with the Hi◦ -action on the source, the∼gj Hi◦ gj−1 -action on the target, and the isomorphism Hi◦ −→ gj Hi◦ gj−1 induced by
conjugation by gj .

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS II

69

Now Ωj gj−1 is the gj H ◦ gj−1 -orbit of e ∈ N , and (5.2.15) implies that Ωj gj−1 ⊂ N.
It follows from Proposition 5.2.11 (applied to the good analytic open subgroup
gHi g −1 ) and from the isomorphism (5.2.17) that IPG (ψ)(Ωj )H◦i −an is a topologicallyirreducible Hi◦ -representation. Taking into account the isomorphism (5.2.16), we
conclude that IPG (ψ)H◦i −an is topologically isomorphic as an Hi◦ -representation to a
finite direct sum of topologically irreducible representations. The proposition follows upon taking the sequence of good analytic open subgroups to be {Hi }i≥i0 . 
Theorem 5.2.18. If U is a finite dimensional locally analytic representation of
M , then any G-equivariant surjection IPG (U ) → W of admissible locally analytic
G-representations induces a surjection of Jacquet modules JP (IPG (U )) → JP (W ).
Proof. We may verify the assertion of the theorem after making a finite extension ofscalars, and thus may assume that U is a direct sum of locally analytic characters.
The theorem thus follows from Lemmas 5.2.2 and 5.2.8 and Corollary 5.2.14. (5.3) In this subsection we prove Corollaries 0.14 and 0.15.
Proof of Corollary 0.14. Since JP (V ) is a non-zero object of Repes (M ), we may finda character χ ∈ M̂ (E) for some finite extension E of K for which the χ-eigenspaceof JP (V ⊗K E) is non-zero. Taking U to be χδ −1 , we let W denote the image ofthe map U(g) ⊗U(p) Ccsm (N, U ) → V ⊗K E corresponding via (0.17) to the inclusionof U (δ) into JP (V ⊗K E). If dχ denotes the derivative of χ (regarded as a weightof the Lie algebra m of M ) then Ccsm (N, U ) is isomorphic to a direct sum of copiesof dχ as a U(p)-module, and so W is a direct sum of copies of a quotient of theVerma module U(g) ⊗U(p) dχ.
The space of n-invariants W n decomposes as a direct sum of weights of m. Furthermore, for every weight λ of m that appears, there is a corresponding character χ̃appearing in JP (V ⊗K E) for which dχ̃ = λ. (Compare the proof of [7, Prop. 4.4.4].)
The theory of Verma modules shows that we may find a weight λ of m appearing in
W n such that λ − µ does not appear in W n for any element µ in the positive cone of
the root lattice of m. Let χ̃ be a character of M appearing in JP (V ⊗K E) for which
λ = dχ̃, and set Ũ = χ̃δ −1 . Our choice of λ ensures that the resulting inclusionŨ (δ) → JP (V ⊗K E) is balanced, and so Theorem 0.13 yields a non-zero E-linear
and G-equivariant map IPG (Ũ ) → V ⊗K E. Forgetting the E-linear structure, andregarding the source and target of this maps simply as K-linear representationsof G, we see that the target is isomorphic to a finite direct sum of copies of the
topologically irreducible representation V . Since the map is non-zero, the projection onto at least one of these direct summands must again be non-zero, and so we
obtain a K-linear and G-equivariant map IPG (Ũ ) → V, as required. 
Proof of Corollary 0.15. The functor JP is left exact (see [7, Thm. 4.2.32]), and so
it suffices to prove the result when V is a topologically irreducible very strongly admissible locally analytic G-representation. If JP (V ) is non-zero then Corollary 0.14
yields a surjection IPG (U ) → V for some finite dimensional locally analytic representation of M . Theorem 5.2.18 shows that the induced map JP (IPG (U )) → JP (V ) issurjective. Proposition 5.1.5 shows that the source of this map is finite dimensional,
and thus so is its target. 

70

MATTHEW EMERTON

Appendix
In this appendix we derive some results concerning admissible locally analytic
representations, based on the results of [6, §5].
Proposition A.1. If V is an admissible locally analytic representation of G, andif H is a good analytic open subgroup of G, then there is a natural isomorphism
Dan (H◦ , K)b∼ˆ⊗
D la (H ◦ ,K)b

Vb0 −→ (VH◦ −an )0b .

Proof. We will use the notation introduced in the proof of [6, Prop. 5.3.1]. Choose×a real number 0 < r < 1 such that r ∈ |L |. There are isomorphisms∼

Dla (H ◦ , K)b −→ limD(H◦rn , H ◦ )
←−

(A.2)

n

and
∼

Vb0 −→ lim(VH◦rn −an )0b .
←−

(A.3)

n

Since furthermore the natural map
Dla (H ◦ , K)b → D(H◦rn , H ◦ )(A.4)

induced by (A.2) has dense image, tensoring (A.2) and (A.3) induces the first of
the sequence of isomorphisms
Dan (H◦ , K)b∼ˆ⊗
D la (H ◦ ,K)b

Vb0 −→ lim
Dan (H◦ , K)b
←−
n

ˆ
⊗

D(H◦
,H ◦ )
rn(VH◦rn −an )0b∼∼−→ lim(VH◦ −an )0b −→ (VH◦ −an )0b .
←−n

The second isomorphism is provided by [6, (6.1.15)] (which [6, Cor. 6.1.19] shows
to be indeed an isomorphism), while the third isomorphism is tautological, since
its source is the projective limit of a constant projective system. Composing these
isomorphisms gives the isomorphism of the proposition. In order to apply the preceding result, it will be useful to establish some facts
concerning topological modules over Dan (H◦ , K)b . The key property of Dan (H◦ , K)bis the isomorphism of topological algebras
(A.5)

∼

(m)

Dan (H◦ , K)b −→ lim
Dan (H◦ , K)b−→mprovided by [6, Prop. 5.2.3], which shows that Dan (H◦ , K)b is coherent, and ofcompact type.

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71

Lemma A.6. Suppose that {Am } is an inductive sequence of Noetherian Banach
algebras, for which the transition maps Am → Am+1 are injective, compact (asmaps between Banach spaces), and flat homomorphisms of K-algebras, and let A :=lim
Am be the inductive limit algebra (which is thus of compact type). If Mm is a−→mfinitely generated Am -Banach module, for some value of m, then A ⊗Am Mm is ofcompact type (in particular it is Hausdorff and complete), and is a finitely presented
A-module.
Proof. For each m0 ≥ m, write Mm0 := Am0 ⊗Am Mm , so that Mm0 is a finitely
generated Am0 -Banach module (by [6, Prop. 1.2.5] and the remark following its
proof). The natural map Am0 → A induces a map(A.7)

Mm0 → A ⊗Am Mm

for each m0 ≥ m.
Our convention is that A ⊗Am Mm is equipped with the quotient topology obtained by regarding it as a quotient of A ⊗K,π Mm . Since A is of compact type, and
Mm is a Banach space, [6, Prop. 1.1.31] shows that this topology coincides with the
topology obtained by regarding A ⊗Am Mm as a quotient of A ⊗K,ı Mm . A similar
remark applies to each of the tensor products Mm0 = Am0 ⊗Am Mm (for m0 ≥ m),since Am0 and Mm are both Banach spaces. From these remarks, together with [6,Lem. 1.1.30], one easily deduces that the natural map(A.8)

lim Mm0 → A ⊗Am M

−→
m0 ≥m

induced by the maps (A.7) is an isomorphism.
For m0 ≥ m, let Nm0 be the quotient of Mm0 by the kernel of the map (A.7). We
may then rewrite the isomorphism (A.8) as an isomorphism
(A.9)

∼

Nm0 −→ A ⊗Am M,
lim
−→

m0 ≥m

where the transition maps in this inductive limit are injective. Since Am0 is Noetherian and Mm0 is finitely generated over Am0 , [6, Prop. 1.2.4] implies that thekernel of (A.7) is a closed subspace of Mm0 , and thus that each Nm0 is a Banach
space. Since each Nm0 is finitely generated over Am0 , and since the transition maps
between the Am0 are compact, we see that the transition maps in the inductive limitof (A.9) are furthermore compact. Thus (A.9) presents A ⊗Am M as an inductive
limit of a sequence of Banach spaces with compact and injective transition maps.
This proves the first claim of the lemma. The claim of finite presentability follows
directly from the fact that Mm is finitely generated over the Noetherian Banach
algebra Am , and hence in fact finitely presented. 
In the context of the preceding lemma, the ring A is coherent (being the inductive limit of a sequence of Noetherian rings via flat transition maps). Thus thecategory of finitely presented A-modules (i.e the full subcategory of the categoryof A-modules whose objects are the finitely presented A-modules) is abelian (andthe formation of kernels, cokernels, and images in this category coincide with thecorresponding constructions in the category of all A-modules). The following result
provides some useful functional analytic information about the category of finitely
presented A-modules.

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MATTHEW EMERTON

Proposition A.10. Let {Am } and A be as in the statement of Lemma A.6.
(i) Every finitely presented A-module has a unique structure of compact typespace making it a topological A-module.
(ii) If f : M → N is an A-linear morphism between finitely presented A-modules,
then f is continuous and strict with respect to the topologies on M and N givenby (i).
Proof. If M is a finitely generated A-module, then we may find a surjection of
A-modules Ar → M for some r ≥ 0. If M is furthermore equipped with a locallyconvex topology with respect to which it becomes a topological A-module, then thismap is necessarily continuous. If the topology on M is of compact type, then the
open mapping theorem shows that this surjection is a strict map of convex spaces,
and the so the topology on M coincides with the topology it inherits as a quotientof Ar . This proves the uniqueness statement of (i).
Suppose now that M is finitely presented. It is a standard (and easily verified)
fact that we may write M ∼= A ⊗Am Mm for some value of m, and some finitely generated Am -module Mm . From [6, Prop. 1.2.4 (i)], we see that Mm may be equippedwith a canonical Am -Banach module structure, and Lemma A.6 then shows that thetensor product topology on M (which certainly makes M a topological A-module)
endows M with a compact type space structure. This establishes the existenceclaim of (i).
Suppose now that M → N is an A-linear morphism of finitely presented Amodules. We may fit this morphism into a commutative diagram of A-linear morphisms
/ As
Ar

M


/ N,

for some r, s ≥ 0. If we endow each of M and N with the topological A-module
structure given by part (i), then we have already observed that the vertical arrows
must be strict, while the upper horizontal arrow is certainly continuous (since A isa topological algebra). Thus the lower horizontal arrow must be continuous. This
proves the continuity claim of (ii).
Now consider a short exact sequence of finitely presented A-modules
0 → M → N → P → 0.
If we endow each of these modules with the topological A-module structure givenby part (i), then the result of the preceding paragraph shows that the morphisms
in this sequence are necessarily continuous. Since the topologies on M , N , andP are of compact type, the open mapping theorem shows that the morphisms arefurthermore necessarily strict.
As was noted above, the kernel or cokernel of a morphism of finitely presented
A-modules is again finitely presented (since A is coherent), and so any injection or
surjection of finitely presented A-modules may be fitted into a short exact sequenceof finitely presented A-modules. The result of the preceding paragraph thus implies
the strictness claim of (ii) for injections and surjections of finitely presented Amodules.

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Any morphism of finitely presented A-modules may be factored as the composite
of a surjection of its source onto its image, and an injection of its image into its
target. Furthermore, as was noted above, this image is again finitely presented(since A is coherent). The strictness claim of (ii) (in full generality) thus follows
from the result for injections and surjections already established. 
The isomorphism (A.5) and [6, Prop. 5.3.11] allow us to apply the preceding(m)results in the particular context when A = Dan (H◦ , K)b and Am = Dan (H◦ , K)b
for m ≥ 0.
Lemma A.11. If H is a good analytic open subgroup of G, then Dan (H◦ , K)b isflat over Dla (H ◦ , K)b , and the natural map
Dan (H◦ , K)b ⊗Dla (H ◦ ,K)b M → Dan (H◦ , K)b(A.12)

ˆ⊗M

D la (H ◦ ,K)b

is an isomorphism for any coadmissible Dla (H ◦ , K)b -module M . Furthermore, thetopological tensor product Dan (H◦ , K)b ⊗Dla (H ◦ ,K)b M is of compact type, and isfinitely presented as a Dan (H◦ , K)b -module.
Proof. We employ the same notation as in the proof of Proposition A.1. From [6,∼(m)Prop. 5.2.3] we obtain an isomorphism Dan (H◦ , K)b −→ lim
Dan (H◦ , K)b , and−→m[6, Prop. 5.3.18] implies that the map (A.4) (for n = 1) factors as(m)

Dla (H ◦ , K)b → D(H◦r , H ◦ ) → Dan (H◦ , K)b→ Dan (H◦ , K)b

for m sufficiently large. The second and third of these maps are flat, by [6,Prop. 5.3.18] and [6, Prop. 5.3.11 (ii)] respectively, and the first is flat, by [16,Rem. 3.2]. Thus Dan (H◦ , K)b is indeed a flat Dla (H ◦ , K)b -module.
(m)
It follows from [6, Prop. 5.3.11 (i)] that Dan (H◦ , K)b is a Noetherian Banach
algebra for any value of m. As was observed in the proof of [6, Thm. 1.2.11], the(m)(m)
tensor product Dan (H◦ , K)b ⊗Dla (H ◦ ,K)b M is a finitely generated Dan (H◦ , K)b Banach module, and hence the natural map(m)

Dan (H◦ , K)b(m)⊗Dla (H ◦ ,K)b M → Dan (H◦ , K)bˆ⊗M

D la (H ◦ ,K)b

is an isomorphism for any m. Since the natural maps
Dan (H◦ , K)b ⊗Dan (H◦ ,K)(m) (Dan (H◦ , K)(m) ⊗Dla (H ◦ ,K)b M )→ Dan (H◦ , K)b ⊗Dla (H ◦ ,K)b M
andDan (H◦ , K)bˆ⊗
D an (H◦ ,K)(m)(Dan (H◦ , K)(m)

ˆ
⊗

M)

D la (H ◦ ,K)b

→ Dan (H◦ , K)bˆ⊗M

D la (H ◦ ,K)b

are also both isomorphisms, it thus follows from Lemma A.6 that (A.12) is indeed
an isomorphism, and also that Dan (H◦ , K)b ⊗Dla (H ◦ ,K)b M is finitely presented asa Dan (H◦ , K)-module. 

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MATTHEW EMERTON

Corollary A.13. If H is a good analytic open subgroup of G, then the functor
V 7→ VH◦ −an on the category of admissible locally analytic G-representations is exact(in the strong sense that it takes an exact sequence of admissible locally analytic
representations to a strict exact sequence of nuclear Fréchet spaces).
Proof. By Proposition A.1 (and the fact that passing to duals is exact, by the
Hahn-Banach theorem), to prove the corollary it suffices to show that the functor
M 7→ Dan (H◦ , K)bˆ⊗M

D la (H ◦ ,K)b

is an exact functor on the category of coadmissible Dla (H ◦ , K)b -modules.
If 0 → M → N → P → 0 is a short exact sequence of coadmissible Dla (H ◦ , K)b modules, then it follows from Lemma A.11 that
0 → Dan (H◦ , K)b ⊗Dla (H ◦ ,K)b M → Dan (H◦ , K)b ⊗Dla (H ◦ ,K)b N→ Dan (H◦ , K)b ⊗Dla (H ◦ ,K)b P → 0
is a sequence of continuous morphisms of finitely presented compact type topologicalDan (H◦ , K)b -modules, which is short exact in the algebraic sense (i.e. ignoring
topologies). Proposition A.10 then implies that the maps are strict. Corollary A.14. If V is an admissible locally analytic representation of G, andif H is a good analytic open subgroup of G, then (VH◦ −an )0 is finitely presented asa Dan (H◦ , K)-module.
Proof. This was noted in the proof of the preceding corollary. 
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Northwestern University
Department of Mathematics
2033 Sheridan Rd.
Evanston, IL 60208-2730, USA
E-mail address: emerton@math.northwestern.edu