Draft: December 17, 2007
JACQUET MODULES OF LOCALLY ANALYTIC
REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS
I. CONSTRUCTION AND FIRST PROPERTIES

Matthew Emerton
Northwestern University
Abstract. Let G be a reductive group defined over a p-adic local field L, let P bea parabolic subgroup of G with Levi quotient M, and write G := G(L), P := P(L),and M := M(L). In this paper we construct a functor JP from the category ofessentially admissible locally analytic G-representations to the category of essentiallyadmissible locally analytic M -representations, which we call the Jacquet module
functor attached to P , and which coincides with the Jacquet module functor of [5]on the subcategory of admissible smooth G-representations. We establish several
important properties of this functor.

Contents
1. Lie-theoretic preliminaries
2. Functional-analytic preliminaries
3. Construction of the Jacquet module functor
4. Some properties of the Jacquet module functor
References

7
12
21
40
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The goal of this paper is to introduce in the theory of locally analytic representations of p-adic reductive groups an analogue of the Jacquet module functors thatare defined in [5] for smooth representations of p-adic reductive groups.
We refer the reader to the papers [16, 17, 18, 20] of Schneider and Teitelbaum
for the foundations of locally analytic representation theory. Some additional contributions to the theory have been supplied by [8], [12], and [14]. (The short article[10] provides a summary of some of these results, as well as a brief discussion of
the Jacquet module functors that are the subject of this paper and its sequel [11].)
We will rely in particular on the concept of essentially admissible locally analytic
representation introduced in [8], which generalizes the concept of admissible locallyanalytic representation introduced in [20].
We now introduce some notation that will be in force throughout this introduction. Let L be a finite extension of Qp for some prime p, let G be a connected
reductive linear algebraic group over L, let P denote a parabolic subgroup of G,
let N denote the unipotent radical of P, and write M := P/N (so that M is T
theLevi quotient of P). Choose an opposite parabolic P to P; the intersection P P
then provides a lifting of M to a Levi factor of P (and indeed making a choiceof P is equivalent to making a choice of such a lifting of M). Write G := G(L),The author would like to acknowledge the support of the National Science Foundation (award
number DMS-0070711)

1

2

MATTHEW EMERTON

M := M(L), N := N(L), P := P(L), and P := P(L). Fix an extension K of L,
complete with respect to a discrete valuation extending the discrete valuation on
L. All representations will be on K-vector spaces (whether or not this is explicitly
stated).
We briefly recall the theory of Jacquet module functors for smooth representations [5], before turning to a discussion of the theory for locally analytic representations that is the subject of this paper.
Jacquet module functors for smooth representations. The Jacquet module of asmooth G-representation V with respect to P is defined to be the space VN of
N -coinvariants of V . The formation of Jacquet modules yields a functor fromsmooth G-representations to smooth M -representations, satisfying the following
three properties: (i) it takes admissible smooth G-representations to admissible
smooth M -representations [5, thm. 3.3.1]; (ii) it is exact [5, prop. 3.2.3]; (iii) if V
is non-zero and irreducible, then its Jacquet module is non-zero if and only if V
appears as a subrepresentation of a representation parabolically induced from a
representation of M [5, thm. 3.2.4].
Property (iii) follows from the obvious Frobenius reciprocity formula
(0.1)

∼

HomG (V, (IndG
P W )sm ) −→ HomM (VN , W ),where V is an arbitrary smooth G-representation, W is an arbitrary smooth M representation W , and (IndG
P W )sm denotes the smooth induction of W from P to
G. The proofs of properties (i) and (ii) are more subtle. They follow from a key
observation of [5], which is that, although the Jacquet module of an admissible
smooth representation V is at first defined as a space of coinvariants, and hence asa quotient of V , it admits an alternative description as a subspace of V . (This is thetheory of the “canonical lifting” developed in [5, §4].) This alternative description
is given in terms of the action of certain Hecke operators. It allows one to prove
properties (i) and (ii), and also forms the basis of the Casselman duality theorem[5, cor. 4.2.5]. One way to phrase this latter result is as follows: for any admissible
smooth M -representation U and admissible smooth G-representation V there is a
canonical isomorphism
(0.2)

∼

G
U )sm , V ) −→ HomM (U (δ), VN ),HomG ((IndP

where U (δ) denotes the twist of U by the modulus character of P (which is trivial
on N , and so may be thought of as a character of M ). Note that in the source of
this isomorphism, the parabolic induction appears “on the wrong side” (from thenaive point of view of Frobenius reciprocity).
Jacquet module functors for locally analytic representations. For any locallyL-analytic group H, denote by Repla.c (H) the category of locally analytic Hrepresentations on compact type locally convex topological K-vector spaces. (See[17] for the precise definition of this category.) For any object V of Repla.c (G), one
may certainly consider the space of Hausdorff N -coinvariants of V (i.e. the Hausdorff completion of the space of N -coinvariants of V – the quotient topology on
this latter space might be non-Hausdorff). This construction gives rise to a functorfrom Repla.c (G) to Repla.c (M ) which satisfies an obvious analogue of (0.1), withsmooth parabolic induction being replaced by locally analytic parabolic induction,and the Hom-spaces being replaced by spaces of continuous homomorphisms. (SeeJACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

3[12, §4] for a discussion of Frobenius reciprocity in the locally analytic context.)
Unfortunately, it is not known (to this author at least) whether this functor preserves any of the natural admissibility conditions that one may impose on objects ofRepla.c (G) and Repla.c (M ). (such as strong admissibility, admissibility, or essential
admissibility, as defined in [17], [20], and [8] respectively).
In this paper we define a functor JP on Repla.c (G) which agrees with the usual
Jacquet module functor on admissible smooth representations, and which takes essentially admissible G-representations to essentially admissible M -representations.
The construction of this functor is modelled on the approach to smooth Jacquet
functors via Hecke operators that was discussed above. Just as in the smooth case,for any G-representation V , the construction of JP (V ) will depend only on the
structure of V as a P -representation, and so we initially define the functor JP in
the more general context of locally analytic P -representations.
In subsection 3.1 we define a certain full subcategory Repzla.c (M ) of Repla.c (M ).
For U an object of Repzla.c (M ), let Ccsm (N, U ) denote the space of compactly supported locally constant U -valued functions on N . The formation of this space yields
a functor Ccsm (N, – ) : Repzla.c (M ) → Repla.c (P ). (See subsection 3.5.)
The following result follows from theorem 3.5.6 below.
Theorem 0.3. The functor Ccsm (N, – ) admits a right-adjoint.
We define the Jacquet module functor JP : Repla.c (P ) → Repzla.c (M ) to be theright adjoint of Ccsm (N, – ), twisted by the modulus character δ of P . Thus for U in
Repzla.c (M ) and V in Repla.c (P ), there is a natural isomorphism
(0.4)

∼

sm
LP (Cc (N, U ), V ) −→ LM (U (δ), JP (V )).

If we compose JP with the forgetful functor Repla.c (G) → Repla.c (P ) then we
obtain a functor Repla.c (G) → Repzla.c (M ) (which we again denote by JP ). Notethat for any object U of Repzla.c (M ), the space Ccsm (N, U ) embeds as a closed P invariant subspace of the parabolic induction IndG
U . (It can be identified with thePGsubspace of locally constant functions in IndP U whose support lies in the open cellof G/P .) Thus the adjointness formula (0.4) is a weak analogue of formula (0.2).
We have already mentioned the following fundamental result (proved as theorem 4.2.32 below).
Theorem 0.5. The functor JP takes essentially admissible locally analytic Grepresentations to essentially admissible locally analytic M -representations.
It is natural to ask whether the other basic properties of Jacquet modules ofsmooth representations extend to the locally analytic setting. It follows directlyfrom its construction that the functor JP is left exact and additive. On theother hand, simple examples show that it is not right exact in general. (Thisshows, incidentally, that JP does not coincide with the functor of Hausdorff N coinvariants in general.) As for the relation with parabolic induction, one mighthope to strengthen (0.4) so as to obtain an analogue of (0.2). In fact, this is possible; however, the details are a little involved, and so we will postpone them to thesequel [11]. Here is one of the results that we will establish there.
Theorem 0.6. Suppose that G is quasi-split, and that P is a Borel subgroup of
G (so that the Levi factor M is a maximal torus of G). If V is a topologically4

MATTHEW EMERTON

irreducible admissible locally analytic G-representation that admits a continuous
injection into an admissible continuous representation of G on a K-Banach space,and if JP (V ) is non-zero, then V is isomorphic to a subquotient of the locallyanalytic parabolic induction from P to G of a locally analytic character of M .
Global motivations. Our original motivation for defining the functor JP was not
local, but global. In the paper [9], it is applied to the problem of p-adic interpolationof automorphic forms, and yields a generalization of the theory of the eigencurve
developed in [7]. (If the reader recalls the important role played by the Hecke
operator at p in the construction of [7], then they will get a hint of the relation
between the problem of p-adic interpolation and the construction of this paper, inwhich Hecke operators also play a key role.)
Some properties of the functor JP . We establish a number of additional properties of the functor JP in this paper. Before describing them, we introduce some morenotation. Let Repes (G) (respectively Repes (M )) denote the category of essentiallyadmissible locally analytic G-representations (respectively M -representations). Let
g denote the Lie algebra of G, let ZG denote the centre of G, and let z(g) denotethe centre of the universal enveloping algebra of g. Similarly, let m denote the Liealgebra of M, let ZM denote the centre of M , and let z(m) denote the centre of
the universal enveloping algebra of m. Let ẐM denote the rigid analytic space thatparameterizes the locally L-analytic characters of ZM , and for any χ ∈ ẐM (K),and object V of Repes (G), let JPχ (V ) denote the χ-eigenspace for the ZM -action on
JP (V ) (which is a closed admissible subrepresentation of JP (V ) [8, cor. 6.4.14]).
We now summarize the main properties of JP that are proved in this paper.
c(N, – ) : Repzla.c (M ) → Repla.c (P ) depends(0.7) The definition of the functor Csm
on the choice of P (or equivalently, on the lifting of M to a Levi factor of P).
However, we show that the functor JP is independent, up to natural isomorphism,of the choice of P.

(0.8) Note that there are natural actions of ZG × z(g) on Repla.c (G), and of
ZM × z(m) on Repzla.c (M ). Also, since M is the Levi quotient of P , there arenatural injections ZG → ZM and z(g) → z(m) (the latter is the “unnormalized
Harish-Chandra homomorphism”, recalled in subsection 1.3 below; it depends onthe choice of P ). These maps induce the upper horizontal arrow in the diagram
ZG × z(g)/ ZM × z(m)Aut(Repzla.c (M )) × End(Repzla.c (M ))Aut(Repla.c (G)) × End(Repla.c (G))

/ Aut(JP ) × End(JP ),in which the left-hand vertical arrow and upper right-hand vertical arrow arise fromthe natural actions referred to above, and in which the lower horizontal arrow andthe lower right-hand vertical arrow are induced by the functorial nature of JP . We
prove that this diagram commutes.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

5(0.9) If V is an object of Repla.c (P ), so that JP (V ) is an object of Repzla.c (M ),then regard JP (V ) as a P -module, by having P act through its quotient M . For
any choice of a compact open subgroup P0 of P and of a lifting of M to a Levifactor of P , we construct an M0 N0 -equivariant map(0.10)JP (V ) → V,T
Tfunctorial in V . (Here M0 = M P0 , N0 = N P0 .) Extending the terminologyof [5, p. 40], we refer to this map as the “canonical lifting” determined by the given
choice of P0 and of a lifting of M .
(0.11) Suppose that G is quasi-split over L, and take P to be a Borel subgroupof G. The Levi quotient M is then a torus, and so M = ZM . Passing to duals induces an anti-equivalence of categories between the category Repes (M ), andthe category of coherent rigid analytic sheaves on M̂ . Thus, for any object V in
Repes (G), its Jacquet module JP (V ) gives rise to a coherent rigid analytic sheafon M̂ . The support of this sheaf is a Zariski closed subset of M̂ , which we denote
by Exp(JP (V )). Passing to the derivative of a character induces a natural map
M̂ → m̌, and hence (by restriction) a map(0.12)Exp(JP (V )) → m̌.

We prove that if V is an admissible locally analytic representation of G, then themap (0.12) has discrete fibres.
(0.13) Let W be a finite dimensional algebraic representation of G, write B =EndG (W ), let X be an admissible smooth representation of G over B, and set
V = X ⊗B W (so that V is an admissible locally W -algebraic representation of G).
We prove that JP (V ) is naturally isomorphic as an M -representation to XN ⊗B W N .
In particular, JP (V ) is an admissible locally W N -algebraic representation of M .
(Taking W to be the trivial representation, we find that JP restricted to admissible
smooth representations coincides with the Jacquet module functor of [5].)(0.14) Assuming that L = Qp and that G is split over K, we give a criterion
for the functor JP to commute with the passage to locally algebraic vectors. Moreprecisely, if V is an essentially admissible locally analytic representation of G, and if
W is a finite dimensional irreducible algebraic representation of G, then the closed
embedding VW −lalg → V induces a closed embedding
JP (VW −lalg ) → JP (V )W N −lalg .
(Here the subscripts indicate subspaces of locally algebraic vectors, transforming
locally according to W and W N respectively.) If χ ∈ ẐM (K), then we may restrictto χ-eigenspaces, and so obtain a closed embedding(0.15)JPχ (VW −lalg ) → JPχ (V )W N −lalg .

Let ψ denote the character through which ZM acts on W N . (There is such a ψ,
since G, and so ZM , is assumed to be split over K.) If χ and ψ do not coincide
locally, and so in particular if χ is not locally algebraic, then both the source andtarget of (0.15) vanish.

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MATTHEW EMERTON

Suppose that χ and ψ do coincide locally. If V admits a G-invariant norm, andif χ is of “non-critical slope” with respect to the irreducible representation W N (inthe sense of definition 4.4.3 below), then we prove that (0.15) is an isomorphism.
Admissibility vs. essential admissibility. The Jacquet module of an admissible
locally analytic G-representation V , while certainly essentially admissible, need not
be an admissible locally analytic M -representation (as the examples of [9] show).
Nevertheless, for admissible locally analytic representation V that are topologicallyof finite length, one might hope that JP (V ) is again admissible, of finite length.
Since JP is a left exact functor, it would suffice to prove this when V is topologicallyirreducible. For those irreducible representations that admit a continuous injection
into an admissible continuous representation of G, this follows from theorem 0.6
The arrangement of the paper. The first two sections are preliminary in nature. Section 1 recalls various standard Lie-theoretic results that we will require.
Section 2 is devoted to the development of a theory of compact operators for topological modules over certain topological K-algebras, generalizing the theory of [6].
The main results are propositions 2.2.6 and 2.3.5. The key technical problem withwhich we must deal is that of analyzing compact operators on Banach modules overa Banach algebra that do not admit an orthonormal basis. The same problem has
also been studied by Ash and Stevens [1]; their approach is quite different to ours.
Section 3 presents the construction of the Jacquet module functor. The order ofdevelopment of the ideas is somewhat different to that given in this introduction:
we define the Jacquet module directly before establishing its characterization it
as an adjoint functor. In the preliminary subsection 3.1 we recall the definitions
and some properties of various categories of locally analytic representations. The
definition of the Jacquet module functor is the subject of subsections 3.2, 3.3,
and 3.4. Properties (0.7), (0.8), and (0.9) follow easily from the construction, andare all established in subsection 3.4. Finally, the adjointness formula (0.4) (and so
also theorem 0.3) is established in subsection 3.5.
Section 4 establishes the deeper properties of the Jacquet module functor. Aftermaking some preliminary constructions in subsection 4.1, in subsection 4.2 we givethe proof of theorem 0.5, which relies heavily on the results of section 2. As a
byproduct of the argument, we obtain a proof of property (0.11) above. Subsection 4.3 establishes property (0.13), while property (0.14) is proved in subsection 4.4
using Verma module techniques.
Notation and conventions. Throughout the paper, we fix a finite extension L of
Qp (for some fixed prime p), an algebraic closure L of L, as well as an extension
K of L, complete with respect to a discrete valuation extending that on L. We let
ordL denote the discrete valuation on L (normalized so as to take the value 1 ona uniformizer of L), ordL denote its extension to L, and ordK its extension to K.
We let OL and OK denote the ring of integers in each of L and K respectively.
We follow closely the notational and terminological conventions introduced in[8], and will adhere to the conventions laid down in section 0 of that paper. In
particular, if G is topological group (or more generally, a topological semigroup),
then we will distinguish between a topological action of G on a topological Kvector space V (that is, an action of G by continuous endomorphisms of V ) and acontinuous action of G on V (that is, an action for which the action map G×V → V
is jointly continuous). Also, we will often write “convex K-vector space” as anabbreviation for “locally convex topological K-vector space”.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

7

Acknowledgments. I would like to thank the following individuals for their many
helpful comments and discussions during the long course of writing this paper:
Christophe Breuil, Stephen DeBacker, Mark Kisin, Raghavan Narasimhan, Mohan
Ramachandran, Peter Schneider, Matthias Strauch, and Jeremy Teitelbaum. Special thanks go to David Ben-Zvi, Roman Bezrukavnikov, and Robert Kottwitz for
their stimulating conversation on the subject of this note in particular, as well as
representation theory in general. I would also like to thank Glenn Stevens for his
encouraging remarks, and especially for showing me a preliminary version of theforthcoming paper [1].
1. Lie-theoretic preliminaries(1.1) Let G be a split connected reductive linear algebraic group defined overthe field K. We will briefly recall the highest weight theory of representations of G,
and also of its Lie algebra g.
Since G is split, we may find a split maximal torus T in G, and a Borel subgroup
B containing T. We let N denote the unipotent radical of B. We let g, t, b, andn denote the Lie algebras of G, T, B, and N respectively. We let X • (T)+ denotethe cone of dominant weights in X • (T). (It consists of all weights that pair nonnegatively with the coroots that are positive with respect to the chosen Borel B.)
Theorem 1.1.1. Let W be a finite dimensional irreducible algebraic representationof G over K.
(i) The subspace W N of W is one dimensional, and T acts on this space throughan element χ ∈ X • (T)+ .
(ii) Associating χ to W induces a bijection between the isomorphism classes ofirreducible algebraic G-representations over K and the elements of X • (T)+ .
Proof. This summarizes the standard highest weight theory for the irreducible representations of the split group G. 
In the situation of the theorem 1.1.1, the dominant weight χ is called the highest
weight of W , and any basis element of W N is called a highest weight vector of W .
If t denotes the Lie algebra of T, then ť (the K-dual to t) is naturally isomorphic
to K ⊗Z X • (T ). We say that an element of ť is infinitesimally integral if it assumes
integral values on all positive coroots. We let ť+ denote the cone of infinitesimally
integral elements of ť that pair non-negatively with all positive coroots.
Theorem 1.1.2. Suppose that we are given an irreducible representation of g ona finite dimensional K-vector space W .
(i) The subspace W n of W is one dimensional, and t acts on this space throughan element χ in ť+ .
(ii) Associating χ to W induces a bijection between the isomorphism classes ofirreducible finite dimensional representations of g over K and the elements of ť+ .
Proof. This summarizes the standard highest weight theory for the reductive Liealgebra g. 
Just as for G-representations, the dominant weight χ is called the highest weightof W , and any basis element of W n is called a highest weight vector of W .
The following results provide the link between the irreducible representations of
G and those of g.

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MATTHEW EMERTON

Theorem 1.1.3. If W is a finite dimensional algebraic representation of G definedover K then the induced representation of g is also irreducible.
Proof. This follows from the fact that G is connected. 
Corollary 1.1.4. If W is a finite dimensional K-vector space equipped with anirreducible representation of g, then we may lift the g-action on W to a finite
dimensional algebraic representation of G if and only if the highest weight of W liesin X • (T). In this case, there is a uniquely determined finite dimensional algebraic
representation of G on W lifting the given g-action.
Proof. This follows from theorem 1.1.3, together with the fact that irreducible
representations of both G and g are determined by their highest weights. (Weremark that since N is connected, the inclusion W N ⊂ W n is an isomorphism forany representation W of G, and so the highest weight spaces of W , thought ofalternatively as a G-representation or as a g-representation, coincide.) 
Related to the highest weight theory of irreducible representations of g is thetheory of Verma modules of g, which we now recall.
Definition 1.1.5. Let χ ∈ ť be a character of t. Regarding χ as a character ofb that is trivial on n, let K(χ) denote the field K regarded as a b-module via thecharacter χ. We define the Verma module of highest weight χ to be the left U (g)module Ver(χ) = U (g) ⊗U (b) K(χ).
We refer to the element 1 ⊗ 1 of Ver(χ) as the highest weight vector of Ver(χ),
and denote it by v(χ). The lie algebra b acts on v(χ) through the character χ.
In addition to the action of g, the Verma module Ver(χ) is equipped with acertain “adjoint” action of the algebraic group B, which we now explain. The
adjoint action of the group G on g induces a natural adjoint action of G on U (g).
Now U (b) is invariant under the restriction of this adjoint action to B, as is thecharacter χ. Thus Ver(χ) is naturally equipped with an adjoint action of B.
We can differentiate the adjoint B-action on Ver(χ) to obtain an adjoint b-action.
Of course b also acts on Ver(χ) via restricting the left g-module structure on Ver(χ).
The adjoint action and the left-module action of b are related in the following way:
if b ∈ b and Y ∈ Ver(χ) then adb (Y ) + χ(b)Y = bY.
We will require the following structure theorem for Verma modules whose highest
weight lies in ť+ . (In the statement of theorem, n denotes the opposite nilpotent
to n determined by t.)
Proposition 1.1.6. Let χ lie in ť+ .
(i) If α is a positive simple root of T over K, if X−α ∈ n is a non-zero element ofthe (one dimensional) root space in n corresponding to the negative simple root −α,mα +1
and if mα = hχ, α̌i (where α̌ is the coroot corresponding to α), then X−αv(χ) isannihilated by n, is fixed by the adjoint action of N, and generates a U (g)-submoduleof Ver(χ) isomorphic to Ver(χ − (mα + 1)α).
(ii) If for each positive simple root α as in (i) we let φα : Ver(χ − (mα + 1)α) →Ver(χ) denote the homomorphism of U (g)-modules obtained by mapping v(χ−(mα +mα +11)α) to X−αv(χ), then the cokernel of the map
M
α

⊕α φα

Ver(χ − (mα + 1)α) −→ Ver(χ)

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

9

is a finite dimensional irreducible representation of g of highest weight χ.
Proof. This is well-known. One reference is [13, thm. 4.37], together with its proof,
in particular lemma 4.40. For ease of comparison of our statements with those of
Knapp, we note that the Knapp uses a different normalization to label the Vermamodules: our Ver(χ) is his V (χ+ρ) (where ρ denotes one-half the sum of the positiveroots). We also remark that if sα denotes the simple reflection corresponding to thepositive simple root α, then χ+ρ−(mα +1)α = χ+ρ−hχ+ρ, α̌iα = sα (χ+ρ). (1.2) We let G be a split connected reductive linear algebraic group over K, let
P be a parabolic subgroup of G, let N denote the the unipotent radical of P, andlet M be a Levi factor of P. Choose a Borel B of G that is contained in P, and amaximal torus T of B that is contained in M (so that T is also a maximal torusof M). We let N0 denote the unipotent radical of B, let ZG denote the centre of G,
and let ZM denote the centre of M. We also let g, p, n, m, b, n0 , and t denoteT the
Lie algebras of G, P, N, M, B, N0 , and T respectively.
The
intersection
M
B is aT
T
Borel subgroup of M, with unipotent radical M N0 . Also N0 = (M N0 )N, and so(1.2.1)n0 = (m

\

n0 )

M

n.

Let ∆(G, T) ⊂ X • (T) denote the set of positive roots of T with respect to G
(that is, the set of characters appearing in the adjoint action of T on n0 ). We let∆(G, T)s denote the subset of ∆(G, T) consisting of simple positive roots. The set∆(G, T)s is a basis for a finite index sublattice of X • (T/ZG ).
Similarly, we let ∆(M, T) denote the set of positive roots of T with respecttoT
M (that is, the set of characters appearing in the adjoint action of T on m n0 ). If
we let ∆(M, T)s denote the subset of ∆(M, T) consisting of simple positive roots,then ∆(M, T)s is the basis of a finite index sublattice of X • (T/ZM ).
T
Lemma 1.2.2. There is an equality ∆(M, T) ∆(G, T)s = ∆(M, T)s .
Proof. Clearly the left-hand side is contained in the right-hand side. On the other
hand, since n is an ideal of p, and so in particular of n0 , we see that any root whose
decomposition in terms of simple
T roots involves an element of ∆(G, T)s \ ∆(M, T)
appears in n, and so not in m n0 . Consequently any element of ∆(M, T)s is simple
even when regarded as an element of ∆(G, T). This proves the lemma. 
We let ∆(G, ZM ) ⊂ X • (ZM ) denote the set of positive restricted roots of ZM(that is, the set of characters appearing in the adjoint action of ZM on n). We
let ∆(G, ZM )s denote the subset consisting of simple positive restricted roots. The
decomposition (1.2.1) shows that the elements of ∆(G, ZM ) are the restrictions to
ZM of the elements of ∆(G, T) \ ∆(M, T).
Lemma 1.2.3. Restriction of characters from T to ZM induces a bijection∼∆(G, T)s \ ∆(M, T)s −→ ∆(G, ZM )s .
S
Proof. Lemma 1.2.2 shows that ∆(G, T)s = ∆(G, T)s \ ∆(M, T)s ∆(M, T)s . As
already remarked, ∆(G, T)s is a basis for a finite index sublattice of X • (T/ZG ),while ∆(M, T)s is a basis for a finite index sublattice of X • (T/ZM ). It follows that

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MATTHEW EMERTON

the restriction of ∆(G, T)s \ ∆(M, T)s to ZM is a basis for a finite index sublatticeof X • (ZM /ZG ). In particular, restriction to ZM yields an embedding(1.2.4)∆(G, T)s \ ∆(M, T)s ,→ ∆(G, ZM ).

Any element of ∆(G, ZM )sP
is the restriction to ZM of some element α ∈ ∆(G, T)\∆(M, T). We may write α = β∈∆(G,T)s nβ β with nβ ≥ 0. Since any β ∈ ∆(M, T)s
has trivial restriction to ZM , we may omit these terms
P from the sum, and not alter
the restriction to ZM . Thus we assume that α = β∈∆(G,T)s \∆(M,T)s nβ β. As already observed, the simple roots β appearing in this sum have non-zero restrictions
to ZM . Thus, if the restriction of α to ZM is simple, then we see that α itself mustbe simple. Hence any element of ∆(G, ZM )s lies in the image of (1.2.4).
Now consider an arbitrary α ∈ ∆(G, T)s \ ∆(M, T)s . The restriction α|ZM is alinear combination of elements of ∆(G, ZM )s with non-negative
coefficients. By theP
result of the preceding paragraph, we may write α|ZM = β∈∆(G,T)s \∆(M,T)s nβ β|ZM ,for some nβ ≥ 0. Since (1.2.4) induces
an injection of the span of its domainPinto X • (ZM ), we deduce that α = β∈∆(G,T)s \∆(M,T)s nβ β. Since α is simple, we
conclude that nβ = 0 if β 6= α, while nα = 1. Thus α|ZM is also simple, and so theimage of (1.2.4) is contained in ∆(G, ZM )s . SFor any root α of T in G (that is, any element of ∆(G, T) −∆(G, T)), we letsα denote the corresponding simple reflection in the Weyl group W (G, T) of T withrespect to G. Note that the Weyl group W (M, T) of T with respect to M naturally
embeds as a subgroup of the Weyl group W (G, T). Indeed, it is identified with thecentralizer of the subtorus ZM of T in W (G, T).
Lemma 1.2.5. If w ∈ W (M, T) and α is a root of T in G, then for any χ ∈ X • (T),the characters sα (χ) and sw(α) (w(χ)) have the same restriction to ZM .
Proof. Indeed, w(sα (χ)) = sw(α) (w(χ)), and w acts trivially on ZM . 
Let W be an irreducible algebraic G-representation. Highest weight theory showsthat W N is an irreducible algebraic M-representation, and hence ZM acts on W N
through a character ψ ∈ X • (ZM ). This extends to the highest weight characterψ̃ ∈ X • (T) of W . If α is any element of ∆(G, ZM )s , then lemma 1.2.3 shows that
α lifts uniquely to α̃ ∈ ∆(G, T)s . Note that ψ and α are both well-determined,
independent of the choice of B contained in P, while ψ̃ and α̃ in general depend on
this choice.
Let ρ(G, T) denote one-half of the sum of the elements of ∆(G, T).
Lemma 1.2.6. The restriction of sα̃ (ψ̃ + ρ(G, T)) + ρ(G, T) to ZM is independentof the choice of the Borel subgroup contained in P.
Proof. If B0 is any Borel subgroupcontained in P, then we may choose the maximalT
torus T to be contained in B B0 , and then may write B0 = wBw−1 for some elementw ∈ W (M, T). The corresponding lift of α (respectively ψ) is thus equal to w(α̃)(respectively w(ψ̃)). Also ρ(G, T) is replaced by w(ρ(G, T)). The lemma is now
seen to be a consequence of lemma 1.2.5. (1.3) Let G denote a connected reductive linear algebraic group over K, let P
denote a parabolic subgroup of G, let N denote the unipotent radical of P, and write
M = P/N (the Levi quotient of P). Let g, m, and p denote the Lie algebras of G, M

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

11

and P respectively, and in addition let z(g) (respectively z(m)) denote the centre of
the universal enveloping algebra U (g) of g (respectively the centre of the universal
enveloping algebra U (m) of m). In this subsection we recall the construction of theunnormalized Harish-Chandra homomorphism z(g) → z(m).
We use the inclusion p ⊂ g to regard the universal enveloping algebra U (p) as a
subalgebra of U (g).
Proposition 1.3.1. The sum U (p) + U (g)n is a subalgebra of U (g), that contains
U (g)n as an ideal, and that contains the centre z(g) of U (g) as a subalgebra.
Proof. The first two claims follows from the fact that n is an ideal in p. To prove thefinal claim,
T choose an opposite parabolic P to P, with unipotent radical N; the interLie algebra
section P P provides a lift of M to a Levi factor of P. If n denotes
L theL
of N, then there is a direct sum decomposition U (g) ∼= n U (p) U (m) U (g)n. If
A is the maximal split torus in the centre of M, then since the adjoint A-action on
U (g) preserves this decomposition, and since the A-invariant subspace of n U (p) is
trivial, we see that the elements of z(g) (which areLnecessarily invariant under theadjoint A-action) must lie in the direct sum U (m) U (g)n = U (p) + U (g)n. 
We define the “unnormalized Harish-Chandra homomorphism” γ 0 : z(g) → U (m)to be the composite of the inclusion z(g) ⊂ U
L(p)+U (g)n of the preceding proposition
with the projection U (p) + U (g)n = U (m) U (g)n → U (m).
Proposition 1.3.3. The map γ 0 is injective with image lying in z(m) ⊂ U (m).
Proof. Since z(g) lies in the centre of U (p) + U (g)n, its image under γ 0 lies in z(m).
By construction the kernel of γ 0 is contained in U (g)n. If W is any finite dimensional
representation of G, then W N is annihilated by U (g)n. Since W is generated as a
G-representation by W N (apply highest weight theory over an extension of K thatsplits G), and since z(g) commutes with the G-action on W , we see that the kernelof γ 0 annihilates W . An element of U (g) that annihilates every finite-dimensional
representation of G necessarily vanishes, and thus γ 0 is injective. (1.4) We recall some basic structure theory of tori over the local field L. If T
is a torus defined over L, let S denote the maximal split subtorus of T. Let X•(respectively X • ) denote the cocharacter lattice (respectively the character lattice)of T over L, and similarly let Y• (respectively Y • ) denote the cocharacter lattice(respectively the character lattice) of S. Each of X • and X• is equipped with anaction of Gal(L/L), compatible with respect to the canonical pairing between thetwo of them. The natural embedding Y• → X• identifies Y• with the sublattice ofGalois invariants in X• . The natural surjection X • → Y • identifies Y • with thequotient by its torsion subgroup of the group of Galois coinvariants of X • .
Write S := S(L) and T := T(L), and let S 0 (respectively T 0 ) denote the maximal
compact subgroup of S (respectively T ). If t ∈ T , then the element of Hom(X • , Q)
defined by χ 7→ ordL (χ(t)) is Galois invariant, and so defines a map T → Q ⊗Z Y• .
Denote the image by Y•0 . Restricted to S, this yields a surjection S → Y• , withkernel S 0 . Altogether, we obtain a diagram of short exact sequences(1.4.1)

0

/ S0

/S

/ Y•

/0

0


/ T0


/T


/ Y•0

/ 0.

12

MATTHEW EMERTON

Both Y• and Y•0 are free Z-modules of finite rank, and so each of the short exact
sequences appearing in this diagram is split. The cokernel of the inclusion Y• → Y•0
is finite. (All this follows from the discussion of [4, §3.2] applied to T.)
We let T̂ denote the rigid analytic variety of locally L-analytic characters of T (asdiscussed in [8, §6.4]). If χ is an element of T̂ (K), then the function t 7→ ordK (χ(t))
is trivial on T 0 , and so factors to yield a morphism Y•0 → Q.
Definition 1.4.2. If χ ∈ T̂ (K), then we define slope(χ) to be the element of Q ⊗Z
Y • corresponding to the morphism Y•0 → Q described in the preceding paragraph.
We remark that if χ in fact happens to be an element of X • (that is, an algebraic
character of T that can be defined over K), then slope(χ) is equal to the image ofχ in Y • . (This follows immediately from the construction of the diagram (1.4.1).)
2. Functional-analytic preliminaries(2.1) In this subsection we develop some operator theory. If X is a set, thenc0 (X, K) denotes the K-Banach space of functions f : X → K whose values become
arbitrarily small outside sufficiently large finite subsets of X (topologized via the
sup norm). We embed X into c0 (X, K) by identifying a point x ∈ X with thefunction 1x that is equal to 1 at the point x and vanishes on X \ {x}. For any
K-vector space V , we let F(X, V ) denote the K-vector space of V -valued functions
on X.
Proposition 2.1.1. If V is a Hausdorff convex K-vector space, then the natural
map L(c0 (X, K), V ) → F(X, V ), given by restricting maps to the subset X ofc0 (X, K), is injective, and its image contains the set of maps X → V for which theimage of X is contained in a bounded complete subset of V .
Proof. The discussion of [15, pp. 59–60] shows that X topologically spans c0 (X, K),
implying the claimed injectivity. It is also proved there that if V is quasi-complete
then the map under consideration has image equal to the set of maps X → V withbounded image. If V is arbitrary, and if X → V has image contained in a bounded
complete subset of V , then this map factors as a composite X → U → V in which
the second arrow is continuous and K-linear, with source equal to a Banach space(and so in particular quasi-complete). Thus the first arrow extends to a continuous
K-linear map c0 (X, K) → U, which when composed with the second arrow yields
the required extension of the given map X → V . 
Proposition 2.1.2. Let V and W be convex K-vector spaces, with W Hausdorff
and quasi-complete, and let X be a collection of continuous K-linear maps from V
to W . The following are equivalent:(i) The collection of maps X is equicontinuous.
(ii) The map X × V → W that describes the action of the maps in X extends(in a necessarily unique fashion) to a jointly continuous map c0 (X, K) × V → W.
Proof. Suppose that (i) holds. By assumption we are given a map X → Ls (V, W )with equicontinuous image. We infer from [15, lem. 6.8, cor. 6.11, prop. 7.13] thatthe closure of the image is equicontinuous, bounded and complete. By Proposition 2.1.1, this map uniquely determines a continuous K-linear map c0 (X, K) →Ls (V, W ), and thus a separately continuous K-bilinear map(2.1.3)c0 (X, K) × V → W.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

13

Since the closed OK -lattice spanned by X is the unit ball of c0 (X, K) (with respect
to the sup norm on this space), the unit ball of c0 (X, K) maps to an equicontinuous
subset of L(V, W ). It follows directly that (2.1.3) is jointly continuous.
Conversely, if we are given a jointly continuous map c0 (X, K) × V → W, thenwe see that the entire unit ball of c0 (X, K) (and so in particular the set X) induces
an equicontinuous set of mappings V → W. 
We next recall some properties of equicontinuous collections of maps.
Lemma 2.1.4. (i) If X and Y are equicontinuous collections of K-linear maps
from T to V and from U to W respectively, then the collection X ⊗Y := {x⊗y | x ∈X, y ∈ Y } of maps T ⊗K,π U → V ⊗K,π W is again equicontinuous.
(ii) If X and Y areStwo equicontinuous collections of K-linear maps U → V
then so is the union X Y .
(iii) If X is an equicontinuous collection of K-linear maps U → V , and if α :T → U and β : V → W are continuous K-linear maps, then the set β ◦ X ◦ α :={β ◦ x ◦ α | x ∈ X} is an equicontinuous collection of maps T → W.
(iv) If X is a collection of K-linear maps U → V , and α : T → U is a continuous
K-linear strict surjection, then X is an equicontinuous collection of maps if andonly if X ◦ α := {x ◦ α | x ∈ X} is an equicontinuous collection of maps T → V.
Proof. All four parts of the lemma are immediate. 
Definition 2.1.5. Let φ be a continuous K-linear endomorphism of a convex Kvector space V . We say that φ is power-bounded if the set {φn | n ≥ 0} of endomorphisms of V is equicontinuous.
For any element x in K × , denote by K hhxtii the Tate algebra over K of power
series in t that converge on the closed disk |t| ≤ |x|−1 . (If x = 1, then we omit
it from the notation, and write simply K hhtii.) Similarly denote by K hhxt, xt−1 ii
the Tate algebra over K of power series in t and t−1 that converge on the closed
annulus |x| ≤ |t| ≤ |x|−1 . (If |x| > 1 then set K hhxt, xt−1 ii = 0.) For any Kˆ K K hhxtii
Banach algebra A, denote by A hhxtii the completed tensor product A ⊗(and similarly for A hhxt, x−1 tii).
Proposition 2.1.6. A continuous K-linear endomorphism φ of a quasi-complete
Hausdorff convex K-vector space V is power-bounded if and only if V admits the
structure of a topological K hhtii-module with respect to which t acts via φ, and thismodule structure is unique if it exists.
∼

Proof. Since the map 1n 7→ tn induces a topological isomorphism c0 (N, K) −→K hhtii, proposition 2.1.2 shows that φ is power-bounded if and only if there is a
jointly continuous map(2.1.7)

K hhtii × V → Vsuch that (tn , v) 7→ φn (v) for every v ∈ V , and that φ uniquely determines (2.1.7)(assuming that it exists). Since K [t] is dense in K hhtii, we see immediatelythat (2.1.7) (if it exists) gives V the structure of a K hhtii-module. 
Proposition 2.1.8. (i) If φ is a power-bounded endomorphism of a convex Kvector space V , and if U is a second convex K-vector space, then idU ⊗ φ is a
power-bounded endomorphism of U ⊗K,π V.

14

MATTHEW EMERTON(ii) Let α : U → V and β : V → U be a pair of maps between quasi-complete
Hausdorff convex K-vector spaces. Let φ := β ◦α and ψ := α◦β. Then φ is a powerbounded endomorphism of U if and only if ψ is a power-bounded endomorphism of
V . If these equivalent conditions hold, and if U and V are equipped with the K hhtiimodule structure of proposition 2.1.6, then α and β are K hhtii-linear maps.
(iii) Let U → V be a strict surjection of Hausdorff quasi-complete convex Kvector spaces, and let ψ and φ be endomorphisms of U and V respectively that fit
into a commutative diagram
U

V

ψ

φ

/U

/ V.

If ψ is power-bounded, then so is φ, and the surjection U → V is furthermore ahomomorphism of K hhtii-modules.
Proof. Part (i) follows immediately from part (i) of lemma 2.1.4.
As for part (ii), write X := {φn | n ≥ 0} andSY := {ψ n | n ≥ 0}. From thedefinitionof φ and ψ we see that Y = α ◦ X ◦ β {idV }, and that X = β ◦ Y ◦S
α {idU }, and so lemma 2.1.4 (ii) and (iii) show that X is equicontinuous if andonly if Y is so. Assuming that they are equicontinuous, and noting that α◦φ = ψ◦αand that β ◦ψ = φ◦β, we see that α and β are K [t]-linear, and hence K hhtii-linear,by continuity.
Part (iii) follows from lemma 2.1.4 (iii) and (iv) and the definition of the K hhtiimodule structure on U and V in terms of ψ and φ respectively. 
Proposition 2.1.9. Let R be a locally convex topological K-algebra, let U and V
be locally convex topological R-modules, and let α : U → V and β : V → U becontinuous R-linear maps such that the composites φ = β ◦ α and ψ = α ◦ β areeach equal to an element t ∈ R. (In the situation of proposition 2.1.8 (ii), if φand ψ are power-bounded, then these hypotheses are satisfied for any R admitting acontinuous injection into K hhtii whose image contains K [t].) If S is a topologicalR-algebra in which the element t becomes invertible, then the natural S-linear map
S ⊗R,π U → S ⊗R,π V induced by α is a topological isomorphism.
Proof. The endomorphisms idS ⊗ φ of S ⊗R,π U and idS ⊗ ψ of S ⊗R,π V are both
topological isomorphisms, since t is a unit in S. The proposition thus follows fromthe formulas β ◦ α = φ and α ◦ β = ψ. 
Proposition 2.1.10. If φ is a continuous endomorphism of a K-Banach space V ,then xφ is power-bounded if x ∈ K has sufficiently large valuation.
Proof. Fix a norm defining the topology of V ; this determines a corresponding
operator norm on Lb (V, V ). For any x ∈ K for which xφ has operator norm at
most one, all powers of xφ again have operator norm at most one, and so xφ is
power bounded. (2.2) In this subsection we extend the notion of a compact operator on a convex
K-vector space to the context of Banach modules over a Banach algebra A, and we
prove a spectral theorem for such operators (proposition 2.2.6 below) which has the
merit of applying to A-Banach modules that are not necessarily orthonormalizable.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

15(Note that since we are assuming that K is discretely valued, any K-Banach space isisomorphic to c0 (X, K) for some set X [15, prop. 10.1]. Thus the orthonormalizable
ˆ K V, for some K-Banach
A-Banach modules are precisely those of the form A ⊗space V .) We fix the K-Banach algebra A for the duration of this subsection.
Lemma 2.2.1. Let M and N be A-Banach modules, and let V be a K-Banachˆ K V → M and N → M withspace. If we are given A-linear continuous maps A ⊗the image of the first map being contained in the image of the second then we mayˆ K V → N that gives a commutative diagram
find an A-linear map A ⊗
ˆK V
A⊗
HH
HH
HH
HH
H$

/ M.
N
Proof. Let P denote the image of the map N → M , equipped with the topologythat it inherits as a subspace of M , and let P1 denote the same abstract vector space,but equipped with the (Banach space) topology that it inherits as a quotient of N .
By assumption the given map V → M factors through the inclusion P ⊂ M , andthe closed graph theorem shows that the induced map V → P1 is again continuous[8, prop. 1.1.2]. By [15, prop. 10.5(a)] (and its proof) we may find a continuous
K-linear splitting of the surjection N → P1 . Composing such a splitting with thecontinuous map V → P1 induces a map of K-Banach spaces V → N, which in turn
ˆ K V → N. This is the required map. 
induces a map of A-Banach modules A ⊗Lemma 2.2.2. If V is a K-Banach space and M is an A-Banach module, then
ˆ K V → M has its image contained in a finitely
a continuous A-linear map A ⊗generated A-submodule of M if and only if it may be factored as a composite ofˆ K V → Ar → M for some natural number r.
continuous A-linear maps A ⊗Proof. The if direction is immediate. Its converse follows from Lemma 2.2.1. Definition 2.2.3. (i) If V and W are K-Banach spaces, then an A-linear morˆK V → A⊗
ˆ K W is called A-compact if we may write it as the limit
phism φ : A ⊗ˆ K V, A ⊗ˆ K W ) of a sequence {φn } of A-linear maps A ⊗ˆK V → A⊗ˆK W,
in Lb (A ⊗ˆK W.
each of whose image is contained in a finitely generated A-submodule of A ⊗(ii) If M and N are A-Banach modules, then a continuous A-linear map φ : M →N is called A-compact if there exist K-Banach spaces V and W and a commutative
diagram of continuous A-linear maps
/ A⊗
ˆK W

ˆK V
A⊗

M

φ


/ N,

in which the left hand vertical arrow is surjective, and the upper horizontal arrowis A-compact in the sense of (i).
The compatibility of parts (i) and (ii) of the preceding definition, in the case oforthonormalizable A-Banach modules, is a consequence of the following lemma.

16

MATTHEW EMERTON

Lemma 2.2.4. Let φ : M → N be A-compact in the sense of definition 2.2.3 (ii). If
V and W are K-Banach spaces, and if we are given continuous A-linear morphisms
ˆ K V → M and A ⊗ˆ K W → N , the latter being surjective, then we may lift φ to aA⊗ˆ
ˆ K W that is A-compact in the sense of definition 2.2.3 (i).
morphism A ⊗K V → A ⊗Proof. Follow definition 2.2.3 (ii) and form a commutative diagram/ A⊗
ˆ K W1

ˆ K V1
A⊗

M


/N

φ

in which V1 and W1 are K-Banach spaces, the left hand vertical arrow is surjective,
and the upper horizontal arrow is A-compact in the sense of definition 2.2.3 (i). Useˆ K V → M and A ⊗ˆ K W1 → N to A-linear maps
lemma 2.2.1 to lift the maps A ⊗ˆK V → A⊗ˆ K V1 and A ⊗ˆ K W1 → A ⊗ˆ K W respectively. Composing these lifts
A⊗ˆK V →
with the upper horizontal map of the commutative diagram yields a map A ⊗ˆ K W which lifts φ and satisfies the conditions of definition 2.2.3 (i). A⊗In [6, §A.1] the notion of a completely continuous A-homomorphism between
A-Banach modules is introduced, but although its definition is given in the context
of arbitrary Banach modules, most of the results of this reference are proved justfor orthonormalizable Banach modules. (In comparing our discussion with that of[6, §A], the reader should note that in that reference all Banach spaces are considered to being equipped with a particular choice of norm, while in this paper
we follow [15] and hence do not regard our Banach spaces as being so equipped.)
Lemma 2.2.4 shows that any map between orthonormalizable A-Banach modules
that satisfies the condition of part (ii) of definition 2.2.3 in fact satisfies the condition of part (i) of that definition, and so our definition of an A-compact morphism
between orthonormalizable A-Banach modules agrees with the notion of a completely continuous A-linear map introduced in [6]. However, if M and N are notorthonormalizable, the two notions are distinct (as far as we know). The class of
A-compact maps may be characterized as being the smallest class of continuous
A-linear maps between A-Banach modules that contains the completely continuous
A-linear maps (as defined in [6]) between orthonormalizable A-Banach modules,
and that satisfies conditions (i) and (iii) of the following proposition.
Proposition 2.2.5. (i) If M1 → M , M → N and N → N1 are continuous Alinear maps of A-Banach modules, the second being A-compact, then the composite
M1 → M → N → N1 is A-compact.
(ii) If A → B is a continuous homomorphism of K-Banach algebras, and if M →ˆA M →
N is an A-compact map of A-Banach modules, then the induced map B ⊗ˆ A N is B-compact.
B⊗(iii) If we are given a commutative diagram of continuous A-linear maps between
A-Banach modules
/N
M

P


/ Q,

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

17

in which the upper horizontal arrow is A-compact and the left-hand vertical arrowis surjective, then the bottom horizontal arrow is also A-compact.
Proof. For part (i), note that the maps M1 → M and N1 → N induce the top
horizontal arrows of the commutative diagramˆ K M1
A⊗

/ A⊗
ˆK M

ˆK N
A⊗

/ A⊗
ˆ K N1


M1


/M


/N


/ N1 ,

in which the vertical arrows are the natural maps induced by the A-module structures on each of M, M1 , N , and N1 respectively. Lemma 2.2.4 shows that we mayˆK M → A⊗ˆ K N satisfying deflift the A-compact map M → N to a map φ : A ⊗inition 2.2.3 (i). If we fill in the above diagram with φ, then the composite of theˆ K M1 → A ⊗ˆ K N1 that again satisfies defupper horizontal arrows yields a map A ⊗inition 2.2.3 (i). Thus the composite of the lower horizontal arrows of this diagram
satisfies condition (ii) of that definition, and so is A-compact, as claimed.
Part (ii) follows from the definition, once we note that for a K-Banach space V∼
ˆ A (A ⊗ˆ K V ) −→
ˆ K V , and take into accountthere is a natural isomorphism B ⊗B⊗
Lemma 2.2.2, while part (iii) follows directly from the definition. We can now prove the main result of this subsection.
Proposition 2.2.6. Let φ be an A-compact endomorphism of an A-Banach module M . If x ∈ K × has sufficiently large valuation then:(i) The endomorphism xφ of M is power-bounded, in the sense of definition 2.1.5.
(ii) If we regard M as a topological K hhxtii-module (with t acting via φ on M ),
ˆ K hhxtii M is finitely generated as an A hhxt, xt−1 ii-module.
then K hhxt, xt−1 ii ⊗
ˆ K V → M for some K-Banach
Proof. Choose a continuous A-linear surjection A ⊗space V (e.g. take V = M ). Lemma 2.2.4 shows that we may lift φ to an A-linear
ˆ K V that satisfies definition 2.2.3 (i).
endomorphism ψ of A ⊗Proposition 2.1.10 shows that xψ is power-bounded if ordK (x) is large enough.
Proposition 2.1.8 (iii) implies that xφ is then also power-bounded, and that theˆ K V → M is K hhxtii-linear. Since it is also A-linear, it is an A hhxtiisurjection A ⊗linear surjection. It thus induces an A hhxt, xt−1 ii-linear surjection
K hhxt, xt−1 iiˆ⊗K hhxtiiˆ V → K hhxt, xt−1 ii
A⊗
K

ˆ
⊗

M.

K hhxtiiHence, if the source of this map is finitely generated as an A hhxt, xt−1 ii-module,
then the same will be true of its target. In summary, we see that (i) holds, andˆ K V and φ with ψ.
that in proving (ii), we may replace M with A ⊗We thus suppose that M is an orthonormalizable A-Banach module, equippedwith an A-linear endomorphism φ that may be written as the limit in Lb (M, M ) ofa sequence of A-linear maps φn , each having image contained in a finitely generated
A-submodule of M . We will prove that if x ∈ K × is such that xφ is power-bounded,ˆ K hhxtii M is a finitely generated A hhxt, xt−1 ii-module.
then K hhxt, xt−1 ii ⊗Choose n sufficiently large so that the element x−1 (φ − φn ) of L(M, M ) is topoφ0

φ00

n
n
logically nilpotent. Applying lemma 2.2.2, we factor φn as M −→
Ar −→
M

18

MATTHEW EMERTON

for some r ≥ 0. Define P
a continuous A-linear map ξ : M → A hhxt, xt−1 iir∞
i −i−1 0
via the formula ξ(m) =φn (x−i (φ − φn )i (m)). (This expression is
i=0 x t−1
well-defined, since x (φ − φn ) is topologically nilpotent.) The map ξ induces aˆ K M → A hhxt, xt−1 iir ,continuous A hhxt, xt−1 ii-linear map ξ 0 : K hhxt, xt−1 ii ⊗00
−1while φn induces a continuous A hhxt, xt ii-linear map ξ 00 : A hhxt, xt−1 iir →ˆ K hhxtii M. One computes that ξ 00 ◦ ξ 0 coincides with the quotient
K hhxt, xt−1 ii ⊗
ˆ K M → K hhxt, xt−1 ii ⊗
ˆ K hhxtii M. Thus ξ 00 is a surjection,
map K hhxt, xt−1 ii ⊗−1
ˆ K hhxtii M is finitely generated over A hhxt, xt−1 ii. and so K hhxt, xt ii ⊗To conclude this subsection, let us note the following strengthened version of thepreceding proposition, in the special case where A = K.
Proposition 2.2.7. If φ is a compact endomorphism of a K-Banach space V , thenfor x ∈ K × of sufficiently large valuation we have:(i) The endomorphism xφ of V is power-bounded, in the sense of definition 2.1.5.
(ii) If we regard V as a topological K hhxtii-module (with t acting via φ on V ),
ˆ K hhxtii V is a finite dimensional K-vector space.
then K hhxt, xt−1 ii ⊗Proof. Part (i) is proposition 2.2.6 (i) in the case A = K, while (ii) is a restatementof the usual theory of compact operators on K-Banach spaces [21]. (2.3) This subsection extends the theory of compact operators to topologicalmodules of compact type over a certain class of topological K-algebras.
Definition 2.3.1. A compact type topological algebra over K is a locally convex∼
topological K-algebra that is isomorphic to a locally convex inductive limit A −→An , where {An }n≥0 is a directed sequence of K-Banach algebras, with transition
lim
−→
n

maps that are injective and of compact type as maps of convex K-vector spaces.
In particular, a compact type topological K-algebra is of compact type as a
topological K-vector space.
When we write that a topological module over a topological K-algebra is locallyconvex, or of compact type, we mean simply that it is so when regarded as a
topological K-vector space.
Lemma 2.3.2. If A → B is a continuous homomorphism of compact type topological K-algebras, and if M is a compact type topological A-module, then the completed
ˆ A M is a compact type topological B-module.
tensor product B ⊗ˆ A M is a Hausdorff quotient of the completed tensor prodProof. By definition, B ⊗ˆ
uct B ⊗K M . The latter space is of compact type [8, prop. 1.1.32], and thus so isthe former. 
Fix a compact type topological K-algebra A. The following definition, while alittle ad hoc, will serve our purposes.
Definition 2.3.3. A continuous A-linear morphism φ : M → N between compact
type topological A-modules is called A-compact if it may be factored as a composite
of continuous A-linear maps M → N1 → N, where N1 is a compact type topologicalA-module satisfying the following condition: there is a continuous K-linear map ofconvex K-vector spaces V → N1 , with V a convex K-vector space of compact type,ˆ K V → N1 is surjective, and such that the composite
such that the natural map A ⊗V → N1 → N is a compact map of convex K-vector spaces.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

19

Note that when A = K, the notion introduced in the preceding definition reduces
to the usual notion of a compact map (between compact type K-vector spaces).
We next establish some stability properties of the notion of A-compact maps.
Lemma 2.3.4. (i) If M → N is an A-compact map of compact type topologicalA-modules, and if M1 → M and N → N1 are any two continuous A-linear mapsof compact type topological A-modules, then the composite M1 → M → N → N1 is
A-compact.
(ii) If A → B is a continuous K-linear homomorphism of compact type topological K-algebras, and if M → N is an A-compact map of compact type A-module,
ˆA M → B ⊗ˆ A N is B-compact.
then the induced map B ⊗(iii) If we are given a commutative diagram of continuous A-linear maps between
compact type topological A-modules
M

/N


P


/ Q,

in which the upper horizontal arrow is A-compact and the left-hand vertical arrowis surjective, then the bottom horizontal arrow is also A-compact.
Proof. Property (i) follows immediately from the definition of an A-compact map,as does part (ii), once one takes into account lemma 2.3.2.
To prove (iii), let us first factor the top horizontal map (which is assumed to be
A-compact) as M → N1 → N , as in definition 2.2.8. Thus we assume that thisfactorization is A-linear, that N1 is a topological A-module of compact type, andthat there is a map V → N1 with V of compact type for which the induced map
ˆ K V → N1 is surjective, and for which the composite V → N is compact. LetA⊗
Q1 denote the compact type A-module obtained as the quotient of N1 by the kernelof the composite N1 → N → Q. The kernel of the surjection M → P is containedin the kernel of the composite M → N1 → Q1 . Since P is a strict quotientof M (both M and P being of compact type), we see that this latter composite
factors through P . Thus the given map P → Q factors as P → Q1 → Q. Theˆ K V → N1 induces a surjection A ⊗ˆ K V → Q1 , and the composite
surjection A ⊗V → N1 → Q1 is compact, since the first arrow is. This shows that the map P → Q
satisfies the requirements of definition 2.3.3, and hence is A-compact. 
We now establish the analogue of proposition 2.2.6 in the setting of compact
type algebras and modules.
Fix a sequence {rn }n≥0 of real numbers converging to 1 from above. LettingRn (x) denote the Tate algebra of rigid analytic functions defined over K thatconverge on the disk |t| ≤ rn |x|−1 , define K hhxtii† := lim
Rn (x). Each Rn (x) is−→nnaturally a Banach space, and if we equip K hhxtii† with its locally convex inductive
limit topology, it becomes a compact type topological K-algebra. Letting Sn (x)denote the Tate algebra of rigid analytic functions over A on the annulus rn−1 |x| ≤|t| ≤ rn |x|−1 , we may similarly define the compact type algebra K hhxt, xt−1 ii† =lim
Sn (x). Both K hhxtii† and K hhxt, xt−1 ii† are independent (up to canonical−→nisomorphism) of the choice of sequence {rn }.

20

MATTHEW EMERTON

If A = lim
An is a compact type topological K-algebra (written as the locally con−→nvex inductive limit, with compact and injective transition maps, of the K-Banach∼
ˆ K K hhxtii† −→ˆ K Rn (x) andalgebras An ), then we write A hhxtii† := A ⊗
lim
An ⊗
−→
n

∼
ˆ K K hhxt, xt−1 ii† −→ˆ K Sn (the isomorphisms folA hhxt, xt−1 ii† := A ⊗
lim
An ⊗
−→
n

lowing from [8, prop. 1.1.32 (i)]).
Proposition 2.3.5. Let φ be an A-compact endomorphism of a compact type topological A-module M . If x ∈ K × has sufficiently large valuation then:(i) The endomorphism xφ of M is power-bounded endomorphism, in the senseof definition 2.1.5.
(ii) If we regard M as a topological K hhxtii† -module, with t acting via φ on M ,ˆ K hhxtii† M is finitely generated as an A hhxt, xt−1 ii† -module.
then K hhxt, xt−1 ii† ⊗Proof. By assumption, we may factor the map φ in the form M → M1 → M, whereˆ K V → M1 forM1 is a compact type A-module for which there is a surjection π : A ⊗some compact type K-vector space V , with the additional property that the composite V → M1 → M is a compact K-linear map. Applying propositions 2.1.8 (ii)and 2.1.9 (taking the algebras R and S of proposition 2.1.9 to be K hhxtii† andK hhxt, xt−1 ii† ), we see that it suffices to prove the proposition with M1 in placeof M , and with the composite M1 → M → M1 (which we denote by φ1 ) in place
of φ.
∼
∼
Write A −→ lim
An as in definition 2.3.1, and write V −→ lim
Vn as the lo−→−→nncally convex inductive limit of a sequence of Banach spaces with compact injective
ˆ K Vn → A ⊗ˆK Vtransition maps. By [8, prop. 1.1.32], the natural map lim
An ⊗
−→
n

is a topological isomorphism. For each natural number n, let Wn denote the Kˆ K Vn by the kernel of the composite
Banach space obtained as the quotient of An ⊗ˆˆAn ⊗K Vn → A ⊗K V → M1 . For each value of n there is a continuous injection
ιn : Wn → M1 , and the induced map lim
Wn → M1 is a topological isomorphism.
−→
n

φ1

Since the composite V −→ M1 −→ M1 is compact, it factors through a continuous map αm : V → Wm , for some m [3, prop. 1, p. I.20]. At the expense of replacing
m by m + 1 if necessary, we may assume that αm is a compact map (since the map
Wm → Wm+1 is compact). If U denotes the image of Wm in M1 , then (since it isA-linear, and so in particular Am -linear) the map φ1 restricts to an Am -linear endomorphism of U . The closed graph theorem then implies that this endomorphism of
U lifts to a continuous Am -linear endomorphism ψm of Wm [8, prop. 1.1.2], which
by proposition 2.2.5 is Am -compact (in the sense of definition 2.2.3), since αm iscompact.
Proposition 2.1.10 implies that xψm is power-bounded if x has sufficiently largenvaluation. Thus the sequence of maps {xn ψm}n≥0 is equicontinuous, and thus son n
is the sequence of maps {ιm ◦ x ψm ◦ αm }n≥0 from V to M1 , by lemma 2.1.4 (iii).
ˆK V →
Extending these sequence of maps by A-linearity to a sequence of maps A ⊗M1 , we obtain precisely the sequence of composites {xn φn+1
◦
π}
,
which
is thusn≥01
also equicontinuous (by lemma 2.1.4 (i) and (iii)). Finally, since π is a surjection of
spaces of compact type, it is strict, and so lemma 2.1.4 (iv) implies that the sequenceof map {xn φn+1}n≥0 is equicontinuous, and hence that xφ1 is power bounded.
1
If m0 ≥ m, then φ1 equally well induces a compact map ψm0 : Wm0 → Wm0 .

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

21

ˆ K Vm0 subjects
Since φ1 is A-linear (and so in particular Am0 -linear), since Am0 ⊗onto Wm0 , and since φ1 (Vm0 ) ⊂ Wm , we find that ψm0 admits a factorization intojmhm
ˆ Am Wm −→continuous Am0 -linear maps Wm0 −→
Am0 ⊗
Wm0 , where jm is induced0by the natural map Wm → Wm , and hm is defined so that ψm0 = jm ◦ hm . (Wehave also appealed again to [8, prop. 1.1.2].) The composite hm ◦ jm is then equalˆ ψm . It thus follows from proposition 2.1.8 (i) and (iii) that if xψm is
to idAm0 ⊗power-bounded, then the same is true of xψm0 .
For each natural number n, let rn , Rn (x) and Sn (x) be as defined in thediscussion preceding the statement of the proposition. Proposition 2.2.6 implies
ˆ Rm (x) Wm (respectively Sm0 (x) ⊗ˆ R 0 (x) Wm0 ) is a finitely generated
that Sm (x) ⊗mˆˆAm ⊗K Sm (x) module (respectively Am0 ⊗K Sm0 (x) module). Proposition 2.1.9 implies that the continuous Am0 -linear map
ˆ Sm0 (x))(Am0 ⊗Kˆ⊗ˆ K Sm (x)Am ⊗ˆ Wm → Sm0 (x)Sm (x) ⊗Rm (x)

ˆ⊗

Wm0

Rm0 (x)ˆ K Sm0 (x)-modules).
induced by jm is an isomorphism (of finitely generated Am0 ⊗Passing to the locally convex inductive limit in m now shows that the completed
ˆ K hhxtii† M1 is a finitely generated module overtensor product K hhxt, xt−1 ii† ⊗A hhxt, xt−1 ii, as required. 
It is convenient to note that in the situation of part (ii) of the preceding proposition, the natural map
ˆ M → K hhxt, xt−1 ii†
K hhxt, xt−1 ii† ⊗
K [t]

ˆ
⊗

M

K hhxtii†

is an isomorphism. Indeed, this follows immediately from the fact that K [t] isdense in K hhxtii† .
Proposition 2.3.6. Let V be a compact type K-vector space equipped with a compact endomorphism φ. If x ∈ K × has sufficiently large valuation, then:(i) The endomorphism xφ of V is power-bounded, in the sense of definition 2.1.5.
(ii) If we regard M as a topological K hhxtii† -module (with t acting via φ on M ),
ˆ K hhxtii† V is a finite dimensional K-vector space.
then K hhxt, xt−1 ii† ⊗Proof. This follows from proposition 2.2.7 in the same way that proposition 2.3.5
follows from proposition 2.2.6. 
3. Construction of the Jacquet module functor(3.1) If G is a topological group (or semigroup), we let Reptop.c (G) denote thecategory whose objects are Hausdorff locally convex K-vector spaces of compact
type, equipped with a topological action of G, and whose morphisms are continuous G-equivariant K-linear maps. (Recall that by a “topological action” of G on a
topological vector space V we mean an action of G on V by continuous endomorphisms.)
Lemma 3.1.1. (i) If U and V are two objects of Reptop.c (G), then the direct
product U × V , with the diagonal G-action, is again an object of Reptop.c (G).
(ii) If V is an object of Reptop.c (G), and if U is a closed G-invariant subspaceof V , then U is again an object of the category Reptop.c (G), as is the quotient V /U .

22

MATTHEW EMERTON(iii) If {Vn }n≥0 is an inductive sequence of objects in Reptop.c (G), having injective transition maps, then the locally convex inductive limit lim
Vn again lies in−→nReptop.c (G).
ˆ K V (with the diagonal(iv) If U and V are two objects Reptop.c (G), then U ⊗action of G) is again an object of Reptop.c (G).
Proof. This is immediate, once one recalls the following facts: a finite direct productof spaces of compact type is again of compact type; any closed subspace or Hausdorff
quotient of a convex K-vector space of compact type is again of compact type; thelocally convex inductive limit of any inductive sequence of convex K-vector spacesof compact type having injective transition maps is again of compact type; thecompleted tensor product of two convex K-vector spaces of compact type is againof compact type [8, prop. 1.1.32 (i)]. The preceding lemma implies that Reptop.c (G) is an additive category, admitting
kernels, cokernels, images and coimages. More precisely, if φ : U → V is a morphism
in Reptop.c (G), then the kernel of φ is the usual kernel of φ as a map of vector spaces,
regarded as a closed subspace of U ; the coimage of φ is φ(U ), regarded as a quotientof U ; the image of φ is the closure of φ(U ), regarded as a subspace of V ; and thecokernel of φ is the quotient of V by the closure of φ(U ).
Suppose now that G is a locally L-analytic group. We let Repla.c (G) denotethe 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 [17, p. 12]; see also [8, def. 3.6.9].)
Lemma 3.1.2. Let G be a locally L-analytic group.
(i) If U and V are two objects of Repla.c (G), then the direct product U × V , withthe diagonal G-action, is again an object of Repla.c (G).
(ii) If V is an object of Repla.c (G), and if U is a closed G-invariant subspaceof V , then U is again an object of the category Repla.c (G), as is the quotient V /U .
(iii) If {Vn }n≥0 is an inductive sequence of objects in Repla.c (G), having injecVn again lies intive transition maps, then the locally convex inductive limit lim−→nReptop.c (G).
ˆ K V (with the diagonal(iv) If U and V are two objects Repla.c (G), then U ⊗action of G) is again an object of Repla.c (G).
Proof. This follows from standard properties of convex K-vector spaces of compact
type, together with proposition 3.5.5 and 3.5.15 and theorem 3.6.12 of [8]. In particular, we see that Repla.c (G) is an additive category that admits kernels,
cokernels, images and coimages. Furthermore, these coincide with the corresponding constructions in the larger category Reptop.c (G).
Continuing to suppose that G is locally L-analytic, and also supposing thatthe centre ZG of G is topologically finitely generated, we let Repes (G) denote thefull subcategory of Repla.c (G) consisting of essentially admissible locally analytic
representations of G (as defined in [8, def. 6.4.9]). We let Repad (G) denote the fullsubcategory of Repes (G) consisting of admissible locally analytic representations of
G (as defined in [20]; see also [8, def. 6.1.1]).

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

23

Theorem 3.1.3. Each of Repes (G) and Repad (G) is an abelian category. Furthermore, the construction of direct products, kernels, cokernels, images and coimages
coincide with their construction in Reptop.c (G).
Proof. In the case of admissible locally analytic representations, this is proved in[20] (see also [8, cor. 6.1.23]). In the case of essentially admissible locally analytic
representations, it is proved in [8, prop. 6.4.11]. 
A key point in theorem 3.1.3 is that, since images and coimages agree in an
abelian category, any morphism in Repes (G) is necessarily strict, with closed image.
There is one last 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.
As in [8, §6.4], let ẐG denote the rigid analytic space of locally L-analytic characters on ZG , and let C an (ẐG , K) denote the nuclear Fréchet algebra of K-valued
rigid analytic functions on ẐG . Evaluation at points of ZG induces a naturalmap ZG → C an (ẐG , K), with image lying in the group of invertible elements ofC an (ẐG , K). It follows from [8, prop. 6.4.7] that Repzla.c (G) consists of those objects V in Repla.c (G) for which the ZG -action on V extends to a C an (ẐG , K)-module
structure on V , with the map C an (ẐG , K) × V → V that defines this module structure being separately continuous.
Lemma 3.1.4. Let G be a locally L-analytic group.
(i) If U and V are two objects of Repzla.c (G), then the direct product U × V , withthe diagonal G-action, is again an object of Repzla.c (G).
(ii) If V is an object of Repzla.c (G), and if U is a closed G-invariant subspaceof V , then U is again an object of the category Repzla.c (G), as is the quotient V /U .
(iii) If {Vn }n≥0 is an inductive sequence of objects in Reptop.c (G), having injective transition maps, then the locally convex inductive limit lim
Vn again lies in−→nRepzla.c (G).
ˆ K V (with the diagonal(iv) If U and V are two objects Repzla.c (G), then U ⊗action of G) is again an object of Repzla.c (G).
Proof. This follows from lemma 3.1.2, together with standard properties of BHspaces (such as those recalled in [8, §1.1]). (For part (iv), we also refer to [8,prop. 1.1.32 (i)].) 
By definition any object of Repes (G) is an object of Repzla.c (G).
(3.2) If Z is an abelian group and Z + is a submonoid of Z, then the groupalgebra K[Z] contains the monoid algebra K[Z + ] as a subalgebra, and the functor
HomZ + (K[Z], – ) is right adjoint to the forgetful functor from the category of Zmodules to the category of Z + -modules. More precisely, if M is a Z + -module,
then the natural map HomZ + (K[Z], M ) → M given by evaluation at the identity
element of K[Z] realizes the Z-module HomZ + (K[Z], M ) as the final object in the
category of Z-modules equipped with a Z + -equivariant map to M .
In this subsection we will explain a topological analogue of the preceding construction. We fix a topologically finitely generated abelian locally L-analytic group
Z. (See [8, prop. 6.4.1] for some alternative characterizations of such locally Lanalytic groups.) We also fix a topological submonoid Z + of Z, such that Z +24

MATTHEW EMERTON

generates Z as a group, and contains a compact open subgroup of Z. Recall thecategories Repzla.c (Z) and Reptop.c (Z + ), defined in subsection 3.1. There is an obvious forgetful functor Repzla.c (Z) → Reptop.c (Z + ). We will construct a right adjoint
to this functor.
Definition 3.2.1. If V is any object of Reptop.c (Z + ), then we write
Vfs := Lb,Z + (C an (Ẑ, K), V ).
(The right hand side denotes the subspace of Lb (C an (Ẑ, K), V ) consisting of Z + equivariant maps; the inclusion of Z into the group of units of C an (Ẑ, K) allows us
to regard C an (Ẑ, K) as a Z-module, and so in particular as a Z + -module.) Motivated by [7], we refer to Vfs as the “finite slope part” of V .
If V is an object of Reptop.c (Z + ), then since the elements of Z + act on C an (Ẑ, K)
and V via continuous operators, and since V is Hausdorff, the space Vfs is a closed
subspace of the Hausdorff space Lb (C an (Ẑ, K), V ). The natural action of Z on
C an (Ẑ, K) induces an action of Z on Lb (C an (Ẑ, K), V ), and since Z is abelian, Vfs
is obviously Z-invariant. Thus Vfs is equipped with a natural Z-action.
The formation of Vfs is obviously functorial in V . Evaluation of linear maps at
the identity of C an (Ẑ, K) induces a natural Z + -linear map(3.2.2)

Vfs → V.

The following lemma provides a description of the strong dual of Vfs .
Lemma 3.2.3. If V is an object of Reptop.c (Z + ), then there is a natural isomor∼ˆ K[Z + ] Vb0 .
phism (Vfs )0b −→ C an (Ẑ, K) ⊗Proof. It follows from [15, cor. 18.8] and [8, prop. 1.1.32] that Lb (C an (Ẑ, K), V ) is a
ˆ K Vb0 . Since Vfs iscompact type space whose strong dual is isomorphic to C an (Ẑ, K) ⊗an
defined to the the closed subspace of Lb (C (Ẑ, K), V ) consisting of Z + -equivariant
maps, it follows that its strong dual (Vfs )0b is naturally isomorphic to the Hausdorff
ˆ K Vb0 . ˆ K[Z + ] Vb0 of C an (Ẑ, K) ⊗quotient C an (Ẑ, K) ⊗Proposition 3.2.4. (i) If V is an object of Reptop.c (Z + ), then Vfs is an object ofRepzla.c (Z). Thus V 7→ Vfs yields a functor Reptop.c (Z + ) → Repzla.c (Z).
(ii) For any objects W and V of Repzla.c (Z) and Reptop.c (Z + ) respectively, themap (3.2.2) induces a natural topological isomorphism(3.2.5)∼

Lb,Z (W, Vfs ) −→ Lb,Z + (W, V ).

Proof. To prove (i), note that the isomorphism of lemma 3.2.3 shows that (Vfs )0b is
a topological C an (Ẑ, K)-module. It follows from [8, 6.4.7] that Vfs lies in Repzla.c (Z).
Now let W be as in (ii). (Thus W is of compact type, and hence is barrelled.)
As recalled in the subsection 3.1, the Z-action on W endows W with the structureof a C an (Ẑ, K)-module, for which the induced map C an (Ẑ, K) ⊗K,ı W → W is
continuous. Thus we obtain natural mapsanan
Lb (W, V ) −→ Lb (C (Ẑ, K) ⊗K,ı W, V ) → Lb (W, Lb (C (Ẑ, K), V )),JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

25

the second map being the continuous bijection of [8, prop. 1.1.35]. One easily
checks that the composite of these maps takes any Z + -equivariant map in thesource to a Z + -equivariant map in the target. Since W and Vfs are both Z-modules,
the inclusion Lb,Z (W, Vfs ) ⊂ Lb,Z + (W, Vfs ) is an equality, and so we obtain a map
Lb,Z + (W, V ) → Lb,Z (W, Vfs ). It is easily checked to provide an inverse to (3.2.5). The preceding result implies in particular that V 7→ Vfs is a right adjoint to theforgetful functor Repzla.c (Z) → Reptop.c (Z + ).
Proposition 3.2.6. (i) The functor V 7→ Vfs is additive.
(ii) If U → V is an injective map in the category Reptop.c (Z + ), then the morphism Ufs → Vfs induced by functoriality is again injective.
(iii) If U → V is a closed embedding in the category Reptop.c (Z + ), then themorphism Ufs → Vfs induced by functoriality is again a closed embedding. Furthermore, in this situation, the morphism Vfs /Ufs → (V /U )fs induced by functoriality
is a continuous injection.
(iv) If {Vn }n≥0 is an inductive sequence in the category Reptop.c (Z + ) having injective transition maps, and if we form the locally convex inductive limit V = limVn−→n(an object of Reptop.c (Z + ), by lemma 3.1.1), then the natural map lim(Vn )fs → Vfs−→n

is a topological isomorphism.
Proof. Parts (i) and (ii) are immediate from the construction of Ufs and Vfs .
Let U → V be a closed embedding as in (iii). Applying Lb,Z + (C an (Ẑ, K), – )to the short exact sequence 0 → U → V → (V /U ) → 0 yields the sequence ofcontinuous Z-equivariant maps
0 → Lb,Z + (C an (Ẑ, K), U ) → Lb,Z + (C an (Ẑ, K), V ) → Lb,Z + (C an (Ẑ, K), V /U ).
Since U embeds as a closed subspace of V , this is exact as a sequence of abstract
vector spaces and the second arrow is a closed embedding. This proves part (iii).
Since C an (Ẑ, K) is a Fréchet space, if V = lim
Vn as in (iv), then the natural map
−→
n

lim
Lb,Z + (C an (Ẑ, K), Vn ) → Lb,Z + (C an (Ẑ, K), V ) is a continuous bijection. Since−→n

its source and target are of compact type, it is a topological isomorphism. 
Lemma 3.2.7. Let H be a locally L-analytic group H, and let the object V of
Reptop.c (Z + ) be equipped with a locally analytic H-representation that commuteswith the Z + -action on V .
(i) The H-action on Vfs induced by functoriality is also locally analytic.
(ii) If h denotes the Lie algebra of H, then the action of U (h) on Vfs , induced
by functoriality from the U (h)-action on the locally analytic H-representation V ,coincides with the U (h)-action on Vfs , obtained by regarding it as a locally analytic
H-representation via (i).
∼ˆ K[Z + ] Vb0 −→ (Vfs )0b . It folProof. Lemma 3.2.3 yields an isomorphism C an (Ẑ, K) ⊗lows from [17, cor. 3.3] that the H-action induced on the source of this isomorphismby the H-action on V extends to a topological Dla (H, K)-module structure. Since
the isomorphism is natural, it intertwines the H-action on the source with the Haction on the target induced by the H-action on V , and thus this latter action
also extends to a topological Dla (H, K)-module structure. Thus (again by [17,26

MATTHEW EMERTON

cor. 3.3]) the H-action on Vfs induced by the H-action on V is locally analytic.
This proves (i). Part (ii) is clear. Lemma 3.2.8. For any object V of Repzla.c (Z), regarded via the forgetful functor
as an object of Reptop.c (Z + ), the natural map (3.2.2) is a topological isomorphism.
Proof. This follows from the universal property satisfied by Vfs , together with thefact that for any objects V and W of Repzla.c (Z), the natural map LZ + (W, V ) →LZ (W, V ) is an isomorphism. 
Proposition 3.2.9. If V and W are objects of Reptop.c (Z + ) and Repzla.c (Z) re∼ˆ K W −→
ˆ K W )fs .
spectively, then there is a natural isomorphism Vfs ⊗(V ⊗Proof. Since W is of compact type, it follows from [8, prop. 1.1.35] and [15,Cor. 18.8] that for any compact type spaces U1 and U2 , there is an isomorphism∼
ˆ Wb0 , U2 ) −→
ˆ W ).
L(U1 ⊗
L(U1 , U2 ⊗(3.2.10)KKˆ K W is an object of Reptop.c (Z + ), while if U is an object of Repzla.c (Z)Now V ⊗
ˆ K W (as follows from part (iv) of lemmas 3.1.2 and 3.1.4 respecthen so is U ⊗tively). Taking into account the universal property of Vfs and (V ⊗K W )fs , andapplying (3.2.10), we obtain natural isomorphisms∼

00ˆ W ) −→ LZ (U ⊗
ˆ Wb , Vfs ) = LZ + (U ⊗
ˆ Wb , V )LZ (U, Vfs ⊗
K

K

K

∼

∼

ˆ W ) −→ LZ (U, (V ⊗ˆ W )fs ).
−→ LZ + (U, V ⊗KK∼ˆ K W −→
ˆ K W )fs . Thus we obtain the required natural isomorphism Vfs ⊗(V ⊗Proposition 3.2.11. If V is an object of Reptop.c (Z + ) and if X is any collection ofcontinuous Z + -equivariant endomorphisms of V , then the closed embedding V X →∼V induces an isomorphism (V X )fs −→ (Vfs )X . (Here the superscript “X” denotes
the closed subspace consisting of vectors annihilated by each of the elements of X.)
Proof. This is immediate from the definition of the finite slope parts and the evident∼
isomorphism Lb,Z + (C an (Ẑ, K), V X ) −→ Lb,Z + (C an (Ẑ, K), V )X . 
If χ is a locally L-analytic K-valued character of Z (i.e. a point of Ẑ(K)) and if
V is an object of Reptop.c (Z + ), then we defineV χ = {v ∈ V | zv = χ(z)v for all z ∈ Z+}.
We regard V χ as a locally analytic representation of Z, by having Z act throughthe character χ. Note that since V , and hence its closed subspace V χ , is of compact
type, and since Z acts on V χ through a scalar, the representation V χ is certainly
an object of Repzla.c (Z). Note also that the preceding construction and notation
applies in particular to objects of Repzla.c (Z), which may be regarded as objects ofReptop.c (Z + ), by restricting the Z-action to a Z + -action.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

27

Proposition 3.2.12. If χ ∈ Ẑ(K) and if V is an object of Reptop.c (Z + ), then the∼map (3.2.2) induces an isomorphism (Vfs )χ −→ V χ .
Proof. The universal property of Vfs , applied to the map V χ → V , yields a map
V χ → Vfs , and hence a map V χ → (Vfs )χ . On the other hand, the map (3.2.2)
certainly induces a map (Vfs )χ → V χ . The two maps constructed are immediately
checked to be inverse to one another. We may regard the monoid Z + as a directed set, directed under the relationshipof divisibility. (That is, for a pair of elements z1 , z2 ∈ Z + , we have z1 < z2 if there
exists z ∈ Z + such that z2 = zz1 .) If z ∈ Z + and V is an object of Reptop.c (Z + ),then following the notation introduced in the statement of proposition 3.2.11, we let
V z denote the Z + -invariant closed subspace of V consisting of vectors annihilatedby the element z ∈ Z + . If z1 , z2 ∈ Z + such that z1 < z2 , then V z1 ⊂ V z2 .
Definition 3.2.13. If V is an object of Reptop.c (Z + ), then we define Vnull :=lim
V z , the locally convex inductive limit being indexed by the monoid Z + , directed,
−→
z∈Z +

as above, under the relation of divisibility.
Let V be as in the preceding definition. For each z ∈ Z + , we have a closed
embedding V z → V. Since Z + is commutative, each of the closed subspaces V z
is also a Z + -invariant subspace of V . Passing to the inductive limit, we obtain a
topological Z + -action on Vnull , and a continuous Z + -equivariant injection
Vnull → V.

(3.2.14)

Proposition 3.2.15. (i) If V is an object of Reptop.c (Z + ), then Vnull is also an
object of Reptop.c (Z + ).
(ii) Suppose that {Vn }n≥0 is a Z + -equivariant inductive sequence objects ofReptop.c (Z + ), having injective transition maps. If V denotes the locally convex
inductive limit V = lim
Vn (also an object of Reptop.c (Z + ), by lemma 3.1.1), then−→n

the natural map lim(Vn )null → Vnull is a topological isomorphism.
−→nProof. By assumption, Z + contains a compact open subgroup Z0 of Z, and theinductive limit in definition 3.2.13 may be regarded as being indexed by Z + /Z0 ,rather than by Z + . The quotient Z + /Z0 is a subsemigroup of the finitely generated
group Z/Z0 , and hence is countable, and thus the inductive limit appearing in
definition 3.2.13 can be replaced by a countable inductive limit.
If V is an object of Reptop.c (Z + ) and z ∈ Z + , then V z is a closed Z + -invariant
subspace of V , which, by lemma 3.1.1 (ii), is again an object of Reptop.c (Z + ). Thus
part (iii) of the same lemma, when combined with the observation of the preceding
paragraph, shows that Vnull is again an object of Reptop.c (Z + ). This proves (i).
To prove (ii), suppose that V = lim
Vn is the locally convex inductive limit of−→nan inductive sequence of objects of Reptop.c (Z + ) having injective transition maps.
Clearly the natural map lim(Vn )null → Vnull is a continuous bijection. Since source−→nand target are both of compact type, it is necessarily a topological isomorphism. 

28

MATTHEW EMERTON

Definition 3.2.16. If V is an object of Reptop.c (Z + ), we say that V is null if thenatural map Vnull → V is a continuous bijection (or equivalently, since source andtarget are spaces of compact type, a topological isomorphism).
Lemma 3.2.17. If V is a null object of Reptop.c (Z + ), then Vfs = 0.
Proof. Proposition 3.2.6 (iv), combined with the observation in the first paragraphof the proof of proposition 3.2.15, shows that for any object V of Reptop.c (Z + ),the natural map lim(V z )fs → (Vnull )fs is an isomorphism. Thus it suffices to
−→
z∈Z +

show that (V z )fs = 0 for any object V of Reptop.c (Z + ). Proposition 3.2.11 gives∼
an isomorphism (V z )fs −→ (Vfs )z . The target of this isomorphism is clearly trivial,
since the z-action on Vfs arises from an action of the group Z. +
Lemma 3.2.18.
L If V is a finite dimensional object of Reptop.c (Z ), then the natural map Vnull Vfs → V (induced by (3.2.2) and (3.2.14)) is an isomorphism.

Proof. This is an immediate consequence of the elementary spectral theory of commuting linear operators on finite dimensional vector spaces. Lemma 3.2.19. Let Y be a subsemigroup of Z + that generates Z as a group. If
V is an object of Reptop.c (Z + ), then the natural map Vfs = Lb,Z + (C an (Ẑ, K), V ) →Lb,Y (C an (Ẑ, K), V ) is a topological isomorphism.
Proof. The map in question is an inclusion of closed subspaces of Lb (C an (Ẑ, K), V ),and we must show that it is an equality, that is, that any Y -equivariant map
φ : C an (Ẑ, K) → V is necessarily Z + -equivariant. By assumption, if z ∈ Z + , thenwe may find y, y 0 ∈ Y such that zy = y 0 . If f ∈ C an (Ẑ, K), then (taking into accountthat φ is Y -equivariant) we computeφ(zf ) = φ(y 0 y −1 f ) = y 0 φ(y −1 f ) = zyφ(y −1 f ) = zφ(f ).
Thus φ is automatically Z + -equivariant, as required. 
Proposition 3.2.20. Let U and V be two objects of Reptop.c (Z + ), and let φ :U → V , ψ : U → V be a pair of continuous K-linear maps satisfying the following
conditions:(i) There is a subsemigroup Y of Z + that generates Z as a group, such that φand ψ are both Y -equivariant.
(ii) There is an element z ∈ Z + , such that ψ ◦ φ : U → U (respectively φ ◦ ψ :V → V ) is equal to the endomorphism of U (respectively V ) induced by the actionof z.
Then the map Lb (C an (Ẑ, K), U ) → Lb (C an (Ẑ, K), V ) induced by φ restricts to a∼Z-equivariant topological isomorphism Ufs −→ Vfs .
Proof. The map φ induces a map φ̃ : Lb,Y (C an (Ẑ, K), U ) → Lb,Y (C an (Ẑ, K), V )which is evidently Z-equivariant, and whose source and target are equal to Ufs andVfs respectively, by lemma 3.2.19. To show that φ̃ is an isomorphism, it suffices toshow that it is a bijection, since its source and target are of compact type.
Assumption (ii) implies that ker φ ⊂ U z . Thus ker φ̃ ⊂ Lb,Y (C an (Ẑ, K), U z ) =z(U )fs = 0 (the two equalities following from lemmas 3.2.19 and 3.2.17 respectively).
Thus φ̃ is injective. If f : C an (Ẑ, K) → V lies in Vfs , then define f˜ : C an (Ẑ, K) → U

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

29

via f˜(α) = ψ ◦ f (z −1 α). Since ψ is Y -equivariant, one sees that f˜ is Y -equivariant,and hence lies in Ufs . Furthermore, the composite φ ◦ f˜ is equal to f . Thus φ̃ issurjective. ∼

Now let A be a K-Fréchet algebra, written as a projective limit A −→ lim
An ,
←−
n

where each An is a compact type topological K-algebra, each of the transition maps
An+1 → An is compact, and the map A → An has dense image for each n. Suppose
also that the projective system An is cofinal with a projective system of K-Banach
algebras, so that A is a nuclear Fréchet algebra, in the sense of [8, def. 1.2.12].
Choose an exhaustive increasing sequence {Ẑn }n≥0 of admissible open affinoid
subdomains of Ẑ, with the property that each inclusion Zn ⊂ Zn+1 is relatively∼
compact. There is then an isomorphism C an (Ẑ, K) −→ lim
C an (Ẑn , K)† (where←−n

C an (Ẑn , K)† denotes the space of overconvergent (in Ẑ) rigid analytic functions onẐn ). In fact, as already recalled, C an (Ẑ, K) is a nuclear Fréchet algebra. (Theprojective sequence {C an (Ẑn , K)}n≥0 of K-Banach algebras is cofinal with the projective sequence {C an (Ẑn , K)† }n≥0 , and satisfies the conditions of [8, def. 1.2.12].)
ˆ K A is again a nuclear Fréchet
By [8, lem. 1.2.13] the tensor product C an (Ẑ, K) ⊗algebra, and the proof of that lemma yields an isomorphism(3.2.22)

ˆ A → lim (C an (Ẑn , K)† ⊗ˆ An ).
C an (Ẑ, K) ⊗
←−
K

n

K

(Here we are using the fact that the projective sequences {C an (Ẑ, K)† }n≥0 and{An }n≥0 are both cofinal with projective sequences of Banach spaces.)
Lemma 3.2.23. Let V be a convex K-vector space of compact type, equipped with
an A-module structure for which the multiplication map A × V → V is separately
continuous. Suppose also that V is equipped with a topological Z + -action commuting
with the given A-action, so that V is an object of Reptop.c (Z + ). Then the Amodule structure on Vfs induced by functoriality makes Vfs an A-module, and themultiplication map A × Vfs → A is again separately continuous.
Proof. By [8, prop. 1.1.35], the action of A on V induces a continuous map A →Lb,Z + (V, V ), which when composed with (3.2.2) yields a continuous map A →Lb,Z + (Vfs , V ). Composing this map with the inverse of the isomorphism (3.2.5)
gives a continuous map A → Lb,Z (Vfs , Vfs ). The A-module structure induced on Vfs
by this map is precisely the A-module structure on V induced by the functorialityof the construction of finite slope parts, and so the multiplication map A×Vfs → Vfs
is separately continuous, as claimed. 
In the context of the preceding lemma, the A-module structure on Vfs induces a
topological A-module structure on its strong dual (Vfs )0b [8, prop. 1.2.14]. Since Vfs
is an object of Repzla.c (Z), the same reference shows that (Vfs )0b is also a topologicalˆ K A-module.
C an (Ẑ, K)-module. Thus (Vfs )0b is in fact a topological C an (Ẑ, K) ⊗Proposition 3.2.24. In the above situation, suppose given the following data:(i) For each n ≥ 0 a compact type topological An -module Un , equipped with an
An -linear action of Z + , as well as an An+1 [Z + ]-linear transition map Un+1 → Un ,
ˆ An+1 Un+1 → Un is An -compact (insuch that the induced An [Z + ]-linear map An ⊗the sense of definition 2.3.3);

30

MATTHEW EMERTON(ii) An element z ∈ Z + , such that for each n ≥ 0, the map Un → Un induced
ˆ An+1 Un+1 → Un , so as to give aby z factors through the transition map An ⊗commutative diagram/ Unˆ An+1 Un+1
An ⊗
rr
r
r
r
ˆz
z
id ⊗
rrr
r
r

xr

/ Un .
ˆA
An ⊗
Un+1
n+1

∼

(iii) An A[Z + ]-equivariant isomorphism Vb0 −→ lim
Un .
←−
n

ˆ K A-module (inThen we may conclude that (Vfs )0b is a coadmissible C an (Ẑ, K) ⊗the sense of [8, def. 1.2.8]).
∼∼

Proof. The isomorphisms Vb0 −→ lim
Un and C an (Ẑ, K) −→ lim
C an (Ẑn , K)† , to←−←−nngether with [8, prop. 1.1.29] and lemma 3.2.3, yield isomorphisms∼∼
ˆ Vb0 −→
ˆ Un .
(Vfs )0b −→ C an (Ẑ, K) ⊗lim
C an (Ẑn , K)† ⊗←−K[Z + ]

K[Z + ]

nThus to prove the proposition, it suffices to prove that:(a) For each n ≥ 0, the natural map(3.2.25)

ˆ An
C an (Ẑn , K)† ⊗Kˆ Un+1C an (Ẑn+1 , K)† ⊗ˆ⊗C an (ẐK[Z + ]† ˆ
n+1 ,K) ⊗K An+1

ˆ Un→ C an (Ẑn , K)† ⊗K[Z + ]

is an isomorphism;
ˆ K[Z + ] Un is a finitely generated C an (Ẑn , K)† ⊗ˆ K An -module,(b) C an (Ẑn , K)† ⊗for any n.
To prove (a), note that the map (3.2.25) may be written more simply as(3.2.26)

ˆ (An ⊗
ˆ Un+1 ) → C an (Ẑn , K)† ⊗ˆ Un .
C an (Ẑn , K)† ⊗K[Z + ]

K[Z + ]

An+1

Let z be as in hypothesis (ii). Since the image of z in C an (Ẑn , K)† is invertible,
proposition 2.1.9 implies that (3.2.26), and hence (3.2.25), is an isomorphism.
To prove (b), let z ∈ Z + be as in hypothesis (ii). Since each Ẑn is an affinoid domain, the rigid analytic functions on Ẑn induced by each of z and z −1 arebounded, and so for x ∈ K × with ordK (x) sufficiently large, the map K[z, z −1 ] →C an (Ẑn , K)† extends to a map K hhxz, xz −1 ii† → C an (Ẑn , K)† . Hypotheses (i)and (ii) together imply that z acts as an An -compact operator on Un , and thus
proposition 2.3.5 (i) implies that (perhaps after increasing the lower bound on
ordK (x)) the operator xz is a power-bounded endomorphism of Un , and hence that
Un is naturally a K hhxzii† -module. For such values of x we obtain a surjection
C an (Ẑn , K)†ˆ⊗K hhxz, xz −1 ii†

K hhxz,xz −1 ii†

ˆ⊗
K hhxzii†∼−→ C an (Ẑn , K)†Unˆ⊗
K hhxzii†Un∼ˆ Un → C an (Ẑn , K)† ⊗ˆ Un .
−→ C an (Ẑn , K)† ⊗K[z]

K[Z + ]

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

31(The second isomorphism follows from the remark following the proof of proposition 2.3.5.) By Proposition 2.3.5 (ii), together with [8, prop. 1.2.5], the source
ˆ K An -module (again afterof this surjection is a finitely generated C an (Ẑn , K)† ⊗increasing the lower bound on ordK (x), if necessary), and thus so is the target. 
We close this subsection by establishing a variant of lemma 3.2.19. We first
define a certain subcategory of Reptop.c (Z + ). By assumption, Z + contains an open
subgroup Z0 of Z. We let Repzla.c (Z + ) denote the full subcategory of Reptop.c (Z + )consisting of objects which may be written as a union of an increasing sequenceof Z + -invariant BH-subspaces, and on which Z0 acts through a locally analytic
representation (for one, or equivalently any, choice of such open subgroup Z0T).
Let us now fix a closed subgroup Y of Z, as well as a submonoid Y + of Y Z + ,such that Y + contains an open subgroup of Y and generates Y as a group, andsuch that Y Z + = Z. If V is an object of Reptop.c (Z + ), then by restricting theZ + -action to Y + , we may also regard V as an object of Reptop.c (Y + ). We write
VZ−fs to denote the finite slope part of V as a Z + -representation, and write VY −fs
to denote the finite slope part of V as a Y + -representation. There is a natural
Y -equivariant map
VZ−fs → VY −fs(3.2.27)(constructed either by a consideration of the universal property of VY −fs , or by considering the definitions directly, and restricting maps from C an (Ẑ, K) to C an (Ŷ , K)).
By functoriality, the Z + -action on V induces a Z + -action on VY −fs , and so VY −fs
is in fact naturally a Y Z + = Z-representation. The map (3.2.27) is clearly Zequivariant.
Proposition 3.2.28. If V is an object of Repzla.c (Z + ), then the map (3.2.27) is a
Z-equivariant topological isomorphism.
Proof. Let W be an object of Repzla.c (Z). The universal property of VY −fs yields an∼
isomorphism LY (W, VY −fs ) −→ LY + (W, V ), and hence (recalling that Z = Y Z + )∼an isomorphism LZ (W, VY −fs ) −→ LZ + (W, V ). The proposition will follow provided
that we prove that VY −fs lies in Repzla.c (Z).
Proposition 3.2.7 (i) shows that Z0 acts locally analytically on VY −fs . By assumption (and [8, prop. 1.1.2]), we may write V ∼Vn , where each Vn is a K-Banach
= lim
−→
n
S
space equipped with a topological Z + -action. Also, we may write Ŷ = n Ŷn ,where {Ŷn } is an increasing sequence of affinoid subdomains of Ŷ , and hence∼
write C an (Ŷ , K) −→ lim
C an (Ŷn , K); each C an (Ŷn , K) is a K-Banach space. From←−n[3, prop. 1, p. I.20] we conclude that the natural map lim
Lb (C an (Ŷn , K), Vn ) →−→nLb (C an (Ẑ, K), V ) is a Y -equivariant continuous bijection, and so we obtain a continuous bijection lim
Lb,Y (C an (Ŷn , K), Vn ) → VY −fs . Since each of C an (Ŷn , K) and−→n

Vn is a Banach space, the same is true of the spaces Lb,Y (C an (Ŷn , K), Vn ). The action of Z + on Vn and of Y on C an (Ŷn , K) induces a topological action of Z = Y Z +on Lb,Y (C an (Ŷn , K), Vn ). Thus VY −fs is indeed an object of Repzla.c (Z). 
The following lemma is helpful for verifying the hypothesis of the preceding
proposition, in the context of proposition 3.2.24.

32

MATTHEW EMERTON

Lemma 3.2.29. If V ∈ Reptop.c (Z + ) may be written as the union of an increasing
sequence of Z + -invariant F H-subspaces, then it may be written as the union of anincreasing sequence of Z + -invariant BH-subspaces.
∼

Proof. Appealing to [8, prop. 1.1.2], we may write V −→ lim
Wn as the limit of an−→ninductive sequence of Fréchet spaces, each equipped with a topological Z + -action,such that the transition maps Wn → Wn+1 and the maps Wn → V are continuous,
injective and Z + -equivariant. Since V is of compact type, it is in particular an
LB-space, and so we may choose the sequence {Wn } so that each transition map
factors as Wn → Xn → Wn+1 , where Xn is a Banach space. (See [8, props. 1.1.2
and 1.1.10].) Let Z0 be a compact open subgroup of Z contained in Z + . The
quotient Z + /Z0 is a submonoid of the finitely generated abelian group Z/Z0 , andso is itself a finitely generated monoid. Fix a subset S of Z + that is the union ofZ0 and a finite set whose image in Z + /Z0 is a generating set. Since Z0 is compact,
the set S acts equicontinuously on each of the spaces Wn .
Let An+1 be an open neighbourhood of the origin of Wn+1 , and choose an open
neighbourhood Bn+1 of the origin of Wn+1 such that SBn+1 ⊂ An+1 . If we let
An (respectively Bn ) denote the preimage in Wn of An+1 (respectively Bn+1 ),then since the map Wn → Wn+1 factors through the Banach space Xn , and is
Z + -equivariant, the open subsets An and Bn of Wn are commensurable with one
another (i.e. there exists x ∈ K × such that xAn ⊂ Bn ⊂ x−1 An ), and SBn ⊂An . Thus if Yn denote the Banach space obtained by completing Wn with respect
to the gauge of An (or equivalently, of Bn ), then the S-action on Wn extends
to an equicontinuous S-action on Yn , and hence to a topological Z + -action. By
construction, the map Wn → Xn factors through the Z + -equivariant map Wn →∼Yn . Thus we obtain an isomorphism V −→ lim
Yn , where each Yn is a Banach space−→nequipped with a topological Z + -action. This proves the lemma. 
In the context of proposition 3.2.24, dualizing condition (iii) yields an isomor∼
phism V −→ lim(Un )0b . Since each Un is of compact type, each of the strong duals−→n(Un )0b is a Fréchet space. Thus the hypothesis of lemma 3.2.29 is satisfied.
(3.3) We now return to the situation considered in the introduction. Fix aconnected reductive linear algebraic group G defined over L, as well as a parabolic
subgroup P of G and a Levi factor M ⊂ P. Let N denote the unipotent radical ofP, so that P = MN. Write G := G(L), P := P(L), M := M(L), and N := N(L).
Let ZG (respectively ZM ) denote the centre of G (respectively M), and write ZG :=ZG (L) (respectively ZM := ZM (L)), so that ZG (respectively ZM ) is the centre of0
G (respectively M ). Taking T to be ZM in the discussion of subsection 1.4, let ZM
denote the maximal compact subgroup of ZM , and choose a sublattice Λ of ZM
which splits the bottom row of (1.4.1).
Lemma 3.3.1. If N0 and N00 are two compact open subgroups of N , and if wewrite Λ0 := {λ ∈ Λ | λN00 λ−1 ⊂ N0 }, then Λ0 generates Λ as a group.
Proof. This is a standard consequence of the theory of roots. Indeed, there exists
a positive constant C such that if ordL (α(λ)) > C for every positive restricted root
α of ZM , then λ ∈ Λ0 . (See for example [5, prop. 1.4.3].) The subsemigroup ofelements satisfying this inequality generates Λ as a group. JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

33

Proposition 3.3.2. Let {(Ni , Ni0 )}1≤i≤n be any finite sequence of pairs of compact
open subgroups of N .
(i) If Y = {z ∈ ZM | zNi z −1 ⊂ Ni0 for all 1 ≤ i ≤ r}, then Y generates ZM as agroup.
(ii) If Y 0 = {m ∈ M | mNi m−1 ⊂ Ni0 for all 1 ≤ i ≤ r}, then Y 0 and ZM
together generate M as a semigroup. In particular, Y 0 generates M as a group.
Proof. If N0 denotes the compact open subgroup of N generated by all the Ni , andN00 denotes the intersection of all the Ni0 , then it suffices to prove the proposition
with the sequence of pairs {(Ni , Ni0 )} replaced by the single pair (N0 , N00 ). Let N000
denote the compact open subgroup of N generated by the elements z0 n0 z0−1 , as z00
and n0 range over all elements of ZM
and N0 respectively. Lemma 3.3.1 gives a0
subsemigroup Λ of Λ that generates Λ as a group, such that λN000 λ−1 ⊂ N0 for00λ ∈ Λ0 . Then Λ0 ZM⊂ Y , and Λ0 ZM
generates ZM as a group. This proves (i).
If m ∈ M , then lemma 3.3.1 applied to the pair mN0 m−1 and N00 of compact
open subgroups of N gives an element z ∈ ZM such that z(mN0 m−1 )z −1 ⊂ N00 .
Thus zm ∈ Y 0 , and so m ∈ z −1T
Y 0 ⊂ ZM Y 0 . This proves (ii).
0ZM ⊂ Y 0 , it follows from (i) and (ii) together that
Since, by definition, Y = Y
0
Y generates M as a group. TTFix a compact open subgroup P0 of P , and write M0T:= M P0 , N0 := N P0 ,+M + := {m ∈ M | mN0 m−1 ⊂ N0 }, and ZM:= M + ZM . There are inclusions++
M0 ⊂ M and ZG ⊂ ZM .
+Corollary 3.3.3. The abelian group ZM is generated by its submonoid ZM.

Proof. This is a special case of proposition 3.3.2. Definition 3.3.4. Let M + ×Z + ZM denote the quotient of the monoid M + × ZM
M
by the equivalence relation generated by the relations (mz + , z) ∼ (m, z + z), for any+m ∈ M + , z + ∈ ZM
and z ∈ ZM .
The morphism of monoids M + × ZM → M defined by (m, z) 7→ mz clearlyfactors through M + ×Z + ZM , yielding a morphismM(3.3.5)

M + ×Z + ZM → M.
M

Proposition 3.3.6. The morphism (3.3.5) is an isomorphism.
Proof. Suppose that (m1 , z1 ) and (m2 , z2 ) are two elements of the product M + ×
ZM , such that m1 z1 and m2 z2 are equal as elements of M . Corollary 3.3.3 implies++
that we may find an element z ∈ ZM
such that z1 z2−1 z also lies in ZM, yielding the−1−1−1
relations (m1 , z1 ) ∼ (m1 z1 z2 z, z z2 ) = (m2 z, z z2 ) ∼ (m2 , z2 ). Thus (m1 , z1 )and (m2 , z2 ) are related by the kernel of the canonical map M + × ZM → M + ×Z +M
ZM , and (3.3.5) is injective. Proposition 3.3.2 (ii) shows that M + and ZM together
generate M , and thus that (3.3.5) is surjective. (3.4) We are now ready to define our Jacquet module functors. We retain thenotation of the previous subsection.
Let n denote the Lie algebra of N . If V is a locally analytic representation of P ,then let V n denote the closed subspace of n-invariant elements of V . The subspace
V n is P -invariant (since n is invariant under the adjoint action of P ). In particular,
the subgroup N of P acts smoothly on V n [8, cor. 4.1.7].

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MATTHEW EMERTON

Definition 3.4.1. Let V be a locally analytic representation of P . If N0 is any
compact open subgroup of N
operator πN0 : V n →R , then we define the projection
N0nV
as follows: πN0 v = N0 nv dn, for all v ∈ V . (The measure dn is Haar
measure on N , normalized so that N0 has measure one.)More concretely, P
if v ∈ V n , then v is fixed by some subgroup N00 of N0 , and so
0 −1
πN0 v = [N0 : N0 ]n∈N0 /N00 nv.
We now fix a compact open subgroup P0 of P , and define M0 , N0 , M + and++
ZM as they were defined preceding corollary 3.3.3. That corollary shows that ZM
T+generates ZM as a group. Also, ZM contains the compact open subgroup ZM P0
of ZM .
Definition 3.4.2. If m ∈ M + , define πN0 ,m : V n → V N0 as follows: πN0 ,m (v) =πN0 (mv), for all v ∈ V n .
The alternative formula πN0 ,m (v) = mπm−1 N0 m (v) is also useful. Note that the
restriction of πN0 ,m to V N0 induces an endomorphism of V N0 .
Lemma 3.4.3. If m ∈ M + , then the endomorphism πN0 ,m of V N0 is continuous.
Proof. Since m ∈ M + , we have an inclusion mN0 m−1 ⊂ N0 . The operator πN0 ,mP
1
is defined by the formula πN0 ,m (v) =−1 nmv, for any[N0 : mN0 m−1 ] n∈N0 /mN0 m
N0v ∈ V . Since N acts via continuous automorphisms of V , the lemma follows. 
Lemma 3.4.4. If m, m0 ∈ M + , then πN0 ,m πN0 ,m0 = πN0 ,mm0 .
Proof. Since m ∈ M + there is an inclusion N0 ⊂ m−1 N0 m. Thus πm−1 N0 m πN0 =πm−1 N0 m , and so we compute, for any v ∈ V ,πN0 ,m πN0 ,m0 (v) = mπm−1 N0 m πN0 (m0 v)= mπm−1 N0 m (m0 v) = πN0 (mm0 v) = πN0 ,mm0 (v),as claimed. 
The preceding two lemmas show that the operators πN0 ,m define a topologicalaction of the monoid M + on V N0 . In particular, we obtain a topological action of+
ZM
on V N0 . Suppose now that V is of compact type (so that V lies in the category
Repla.c (P ) defined in subsection 3.1). The closed subspace V N0 of V is then also of+compact type, and is equipped with a topological ZM-action. Thus it is an object+of the category Reptop.c (ZM ), and following definition 3.2.1 (with Z = ZM and+
Z + = ZM), we may take its finite slope part.
Definition 3.4.5. For any object V of Repla.c (P ), we define JP (V ) := (V N0 )fs .
Proposition 3.2.4 shows that V N0 is an object of Repzla.c (ZM ), and so in particular is a convex K-vector space of compact type, equipped with a locally analytic
representation of ZM .
Proposition 3.4.6. If V is an object of Repla.c (P ), then the locally analytic
ZM -representation on JP (V ) extends in a natural way to a locally analytic M representation.
+
Proof. Since ZM
lies in the centre of M + , the M + -action on V N0 induces an M + action on (V N0 )fs . Proposition 3.3.6 then implies that the M + -action and ZM

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

35

action on (V N0 )fs -action together determine an M -action on (V N0 )fs . It remains toshow that this action is locally analytic. Since M0 is compact open in M , it suffices
to show that the M0 -action on (V N0 )fs is locally analytic.
The group M0 normalizes N0 . This implies that M0 ⊂ M + , and that for m ∈ M0 ,the operator πN0 ,m on V N0 coincides with multiplication by m. Since V N0 is a
closed M0 -invariant subspace of V , which is locally analytic as a P -representation,and so in particular as an M0 -representation, we see that the M0 -action on V N0+
is locally analytic. The operators πN0 ,m commute with the action of ZM
on V N0 ,N0and so lemma 3.2.7 (i) shows that the M0 -action on (V )fs is locally analytic. Thus JP defines a functor Repla.c (P ) → Repzla.c (M ), which in subsection 3.5 will
be characterized as an adjoint functor. Since any object of Repla.c (G) gives rise toan object of Repla.c (P ) by restricting the group action from G to P , we also obtaina functor JP : Repla.c (G) → Repzla.c (M ), which in subsection 4.2 will be shown to
restrict to a functor Repes (G) → Repes (M ). The remainder of this subsection isdevoted to proving some more elementary properties of JP .
Lemma 3.4.7. (i) The functor JP is additive.
(ii) If U → V is a continuous injection in Repla.c (P ), then the morphism induced
by functoriality, JP (U ) → JP (V ), is again injective.
(iii) If U → V is a closed embedding in Repla.c (P ), then the morphism induced
by functoriality, JP (U ) → JP (V ), is again a closed embedding. Furthermore, inthis situation, the morphism JP (V )/JP (U ) → JP (V /U ) induced by functoriality isa continuous injection.
(iv) If {Vn }n≥0 is an inductive sequence in the category Repla.c (P ), having injective transition maps, and if V denotes the locally convex inductive limit V = limVn−→n(an object of Repla.c (P ), by lemma 3.1.2), then the natural map lim
JP (Vn ) →−→n

JP (V ) is a topological isomorphism.
Proof. The additivity of the formation of N0 -invariants is clear. If U → V is
injective (respectively a closed embedding), then the same is true of the induced
map U N0 → V N0 . Also, if V = lim
Vn , then V N0 = lim
VnN0 (taking into account the
−→
−→
n

n

fact that a continuous bijection between spaces of compact type is an isomorphism).
The lemma now follows from proposition 3.2.6. 
If we compose the map JP (V ) = (V N0 )fs → V N0 , provided by (3.2.2), with theinclusion V N0 → V, we obtain a natural map
JP (V ) → V.

(3.4.8)It is immediate from the construction that this map is M0 N0 -equivariant (where
the compact open subgroup M0 N0 of P is regarded as acting on JP (V ) through its
quotient M0 ). As will become clear in subsection 4.3, this is a generalization of theso-called “canonical lifting” of [5, p. 40]. It also provides the map of property (0.9)
stated in the introduction.
If χ is a locally analytic K-valued character of ZM , and V is an object ofRepla.c (P ), then we write++
V N0 ,ZM =χ = {v ∈ V N0 | πN0 ,z v = χ(z)v for all z ∈ ZM}.

As in the introduction, we let JPχ (V ) denote the closed subrepresentation of JP (V )on which ZM acts through the character χ.

36

MATTHEW EMERTON

Proposition 3.4.9. If V is an object of Repla.c (P ), then (3.4.8) induces an iso+∼morphism JPχ (V ) −→ V N0 ,ZM =χ .
Proof. This follows from proposition 3.2.12. 
We now determine the extent to which the functor JP depends on the choice of
the Levi factor M and the open subgroup P0 of P .
If M 0 is another lift of M to a Levi subgroup of P , then we may write M 0 =nM n−1 for some (uniquely determined) n ∈ N . If we set N00 = nN0 n−1 and(M 0 )+ = nM + n−1 , then (M 0 )+ = {m0 ∈ M 0 | m0 N00 (m0 )−1 ⊂ N00 }. Thus for any
00
object V of Repla.c (P ), we may define Hecke operators πN00 ,m0 : V N0 → V N0 for0m0 ∈ (M 0 )+ , and so regard V N0 as an (M 0 )+ -module.
Let us use the isomorphism given by conjugation by n to identify M and M 0 (andhence M + and (M 0 )+ ). (This is the same identification that arises by identifying
each of M and M 0 with the Levi quotient of P .)Proposition 3.4.10. For any object V of Repla.c (P ), multiplication by n in0∼
duces an isomorphism of M + -modules V N0 −→ V N0 , and hence an isomorphism∼
N0
N00(V )fs −→ (V )fs of objects in Repla.c (M ).
Proof. This is immediate. Proposition 3.4.10 shows that the functor JP is independent (up to a natural
isomorphism) of the choice of Levi factor of P used in its construction. We next
show that it is similarly independent of the choice of compact open subgroup P0 .
T
Proposition 3.4.11. If P00 is an open subgroup of P0 , and if N00 = N P00 , then0
the projection πN0 : V N0 → V N0 induces an M -equivariant topological isomorphism0∼(V N0 )fs −→ (V N0 )fs .
0+
Proof. Choose z ∈ ZM
such that zN0 z −1 ⊂ N00 , and write φ := πN0 : V N0 →0
V N0 , ψ := πN00 ,z : V N0 → V N0 . Let (M + )0 denote the submonoid of M thatT
conjugates N00 into itself, and write Y 0 = M + (M + )0 . One checks easily that φ0and ψ intertwine the Y 0 -action on each of V N0 and V N0 . (Here m ∈ Y 0 acts on thesespaces through the operators πN00 ,m and πN0 ,m respectively.) T
Also, ψ ◦ φ = πN00 ,z ,while φ ◦ ψ = πN0 ,z . Proposition 3.3.2 (i) shows that Y := Y 0 ZM generates ZM
as a group, and thus, by proposition 3.2.20, we see that φ induces a ZM -equivariant
0∼
isomorphism (V N0 )fs −→ (V N0 )fs that intertwines the Y 0 -action on its source andtarget. Since proposition 3.3.2 (ii) shows that Y 0 generates M as a group, thisisomorphism is in fact M -equivariant. 

Although the preceding results show that the functor JP is determined, up toa natural isomorphism, independently of any particular choice of Levi factor andcompact open subgroup of P , the “canonical lifting” (3.4.8) does in general dependon these choices. (In light of this, the term “canonical lifting” is a little misleading.)
Of course, there is one obvious example in which the construction of JP , andhence also the canonical lifting, does not require any choices.
Proposition 3.4.12. Suppose that V is an object of Repzla.c (G). (For example,
V could be an object of Repes (G).) Taking P to be G, the morphism JG (V ) → V
provided by (3.4.8) is a topological isomorphism.
+
Proof. In this case N0 = N = 1, ZG= ZG , and lemma 3.2.8 gives the result. We finish this subsection by establishing property (0.8) of the introduction.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

37

Proposition 3.4.13. The diagram of (0.8) is commutative.
+
Proof.
T Although M is not a group,+ it contains the compact open subgroup M0 =M P0 . Thus we write Repla.c (M ) to denote the category of convex K-vector
spaces of compact type, equipped with a topological action of M + , which are locally+analytic as representations of M0 . Note that ZM
× z(m) acts as symmetries of the+category Repla.c (M ).
The functor JP is the composite of two functors: the functor V 7→ V N0 , mapping
Repla.c (G) to Repla.c (M + ), and the functor W 7→ Wfs , mapping Repla.c (M + ) to
Repla.c (M ). Thus to verify that the diagram of (0.8) commutes, it suffices to verify
the commutativity of the diagrams(3.4.14)

ZG × z(g)/ Z + × z(m)Aut(V ) × End(V )/ Aut(V N0 ) × End(V N0 ),M(where V is any object of Repla.c (G), and the bottom horizontal arrow is inducedby the functoriality of the formation of N0 -invariants), and(3.4.15)+ZM× z(m)/ ZM × z(m)Aut(W ) × End(W )/ Aut(Wfs ) × End(Wfs ),(where W is any object of Repla.c (M + ), and the bottom arrow is induced by thefunctoriality of the formation of finite slope parts). Here we are taking into account+the fact that the map ZG → ZM factors through the inclusion ZM→ ZM .
Let us begin by verifying the commutativity of (3.4.14). This diagram is the+
direct product of two diagrams, the first involving ZG and ZM, and the second
involving z(g) and z(m). We verify the commutativity of each factor separately.
If V is an object of Repla.c (G), then the action of ZG on V preserves V N0 . Thus
for any z ∈ ZG , the operator πN0 ,z coincides with the action of z on V N0 . Recalling+that the action of ZM
on V N0 is defined via the operators πN0 ,z , the commutativityof the first factor of (3.4.14) follows.
Let γ 0 : z(g) → z(m) denote the the map of proposition 1.3.3. If X ∈ z(g), thenwe may write X = γ 0 (X) + X 0 , with X 0 ∈ U (g)n. If v ∈ V N0 ⊂ V n , then we seethat Xv = γ 0 (X)v. Recalling that the U (m)-action, and hence the z(m)-action, on
V N0 is defined by differentiating the M0 -action on this space, the commutativityof the second factor of (3.4.14) follows.
We now turn to establishing the commutativity of (3.4.15). Again, we regard thisdiagram as the product of two factors, and treat each factor separately. We must++first show that the ZM-action on Wfs , induced by functoriality from the ZM-action++on W , coincides with the ZM -action on Wfs obtained from regarding ZM as a submonoid of ZM . This follows from the definition of Wfs as Lb,Z + (C an (ẐM , K), W ),M
or alternatively, from the universal property of Wfs .
Secondly, we must show that the z(m)-action on Wfs , induced by functoriality
from the z(m)-action on W , coincides with the z(m)-action on Wfs , obtained by

38

MATTHEW EMERTON

differentiating the M0 -action on Wfs . This follows from lemma 3.2.7 (ii), and so weare done. 
The preceding result implies in particular that when V is an object of Repla.c (G),the canonical lifting (3.4.8) is both ZG and z(g)-equivariant (if ZG and z(g) act on
JP (V ) through the maps ZG → ZM and z(g) → z(m) considered above).
(3.5) In this subsection we characterize JP as an adjoint functor. We begin bydescribing the functor to which it is adjoint. As in the preceding section we fix aparabolic subgroup P of G, and a Levi factor M of P , so that P = M N .
For U an object of Repla.c M , let Ccsm (N, U ) denote the space of locally constant,compactly supported functions on N , with values in U . Since N is the union of its
compact open subgroups, we may regard Ccsm (N, U ) as the inductive limit(3.5.1)∼

Ccsm (N, U ) ←− limF(N1 /N0 , U ),−→N0 ,N1

where N0 is an open subgroup of the compact open subgroup N1 of N , and theinductive limit is taken as N0 shrinks to the identity while N1 grows without bound.
The quotient N1 /N0 is finite, and so F(N1 /N0 , U ) (the space of U -valued functions
on N1 /N0 ) is isomorphic to a finite direct sum of copies of U . In this way, Ccsm (N, U )is written as an inductive limit of finite direct sums of copies of U . The transition
maps are obviously continuous and injective, and thus Ccsm (N, U ) may be regarded
as an inductive limit with injective transition maps of spaces of compact type. In
particular, it is naturally a space of compact type.
The right regular action of N on Ccsm (N, U ) equips this space with a smooth(and so locally analytic) action of N . We extend this to an action of P = M N bydefining the following action of an element mn (with m ∈ M and n ∈ N ) on anelement f of Ccsm (N, U ):(mnf )(n0 ) = mf (m−1 n0 mn)(the right-hand side denoting the action of m on the element f (m−1 n0 mn) of U ).
One easily checks that this does yield an action of P on Ccsm (N, U ).
If N0 ⊂ N1 are compact open subgroups of N , then a sufficiently small open
subgroup of M will leave both N0 and N1 invariant under conjugation. The description (3.5.1) of Ccsm (N, U ) and the fact that M acts locally analytically on U
then implies that Ccsm (N, U ) is an object of Repla.c P . Its construction is clearlyfunctorial in U , and so defines a functor Repla.c M → Repla.c P . We will show thatthe functor JP is right adjoint (up to a twist) to (the restriction to Repzla.c M of)
the functor Ccsm (N, – ).
As usual, we denote by δ the modulus character of P , which describes the effectof right multiplication by elements of P on a left-invariant Haar measure of P .
This is a smooth character that factors through the natural surjection P → M ;concretely, if m ∈ M , then δ(m) is equal to [N0 : mN0 m−1 ]−1 , for any compact
open subgroup N0 of N . For an object U of Repla.c M , we let U (δ) denote theobject of Repla.c M obtained by twisting the M -action on U by the character δ.
Lemma 3.5.2. If U is an object of Repzla.c M , then there is a natural isomorphism∼
JP (Ccsm (N, U )) −→ U (δ).

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

39

Proof. Fix a compact open subgroup P0 of P , with respect to which we will com+pute JP , and let M0 , M + , N0 , ZM
and so on have the same meanings as in subsection 3.4. Composing the map (3.4.8) (with V taken to be Ccsm (N, U )) with the map
induced by evaluating at the identity of N yields a continuous map(3.5.3)

JP (Ccsm (N, U )) → U.

We will show that this map is a topological isomorphism of vector spaces, thatintertwines the action of M up to a twist by δ.
If u is an object of U , let fu denote the element of Ccsm (N, U ) that is constanton N0 , with value equal to u, and that vanishes elsewhere. Clearly fu is invariant
under N0 , and one easily computes that πN0 ,m (fu ) = δ(m)fmu = fδ(m)mu foreach m ∈ M + . Thus the association of fu to u induces an M + -equivariant closed
embedding(3.5.4)

U (δ) → (Ccsm (N, U ))N0 ,and hence (taking into account proposition 3.2.6 (iii) and lemma 3.2.8) an M equivariant closed embedding(3.5.5)

U (δ) → JP (Ccsm (N, U )).

It is easily checked that (3.5.5) provides an inverse to (3.5.3), and hence we haveobtained our required isomorphism. (It is easily seen that this isomorphism isindependent of the choice of the compact open subgroup P0 of P , up to the natural
isomorphism provided by proposition 3.4.11.) We can now prove the main result of this subsection.
Theorem 3.5.6. If V is an object of Repla.c P and U is an object of Repzla.c M ,then passing to Jacquet modules yields a natural isomorphism(3.5.7)∼sm
LP (Cc (N, U ), V ) −→ LM (U (δ), JP (V )).

Proof. Lemma 3.5.2, and functoriality of the formation of Jacquet modules, yields
the map (3.5.7). We first prove that (3.5.7) is surjective. As in the proof oflemma 3.5.2, we fix P0 , N0 and so on, and consider the corresponding canonical lifting JP (V ) → V N0 . If we are given φ ∈ LM (U (δ), JP (V )), we may compose
it with the canonical lifting to obtain a map φ̃ : U (δ) → V N0 , which intertwines
the action of M + on the source with the action of the Hecke operators πN0 ,m (for+m ∈ M + ) on the target. For any z ∈ ZM
and f ∈ C sm (N0 , U ) define Iz (f ) ∈ V via
Iz (f ) = δ(z)−1Z

n−1 z φ̃(z −1 f (n))dn.

N(Here z −1 acts on the values f (n) via the action of ZM ⊂ M on U , and dn isHaar measure on N , normalized to give N0 measure one.) One checks (using the+equivariance property of φ̃, and the fact that it takes values in V N0 ) that if z ∈ ZM
is chosen such that f is locally constant on the left cosets of zN0 z −1 in N , then+Iz (f ) = Izz0 (f ) for any z 0 ∈ ZM
. Thus the net Iz (f ) (indexed by the elements of

40

MATTHEW EMERTON+
ZM, directed by the relation of divisibility) is eventually constant, and in particular
the limit I(f ) := lim
Iz (f ) exists. The map f 7→ I(f ) is easily seen to yield an
−→
+
z∈Z
M

element of LP (Ccsm (N, U ), V ), which yields the given map φ upon passage to Jacquet
modules. Thus (3.5.7) is surjective.
We turn to proving that (3.5.7) is injective. Let P + denote the product N0 M + ⊂P , and let C sm (N0 , U ) denote the subspace of Ccsm (N, U ) consisting of locally constant U -valued functions on N0 . One sees that P + is a subsemigroup of P , that
the P -action on Ccsm (N, U ) restricts to a P + -action on C sm (N0 , U ), and that theembedding C sm (N0 , U ) → Ccsm (N, U ) induces an isomorphism∼

K[N ] ⊗K[N0 ] C sm (N0 , U ) −→ Ccsm (N, U ),and so also (since M + generates M as a group) an isomorphism(3.5.8)∼smsm
LP (Cc (N, U ), V ) −→ LP + (C (N0 , U ), V ).

The map (3.5.4) induces a map(3.5.9)LP + (Csm(N0 , U ), V ) → LM + (U (δ), V N0 ).

Composing this on the left with (3.5.8), and on the right with the isomorphism
LM + (U (δ), V

N0∼) −→ LM (U (δ), JP (V ))induced by the adjointness property of (V N0 )fs , we obtain (3.5.7). Thus to prove
the injectivity of (3.5.7), it suffices to show that (3.5.9) is injective. This follows
from the fact that the constant functions on N0 obtained as the image of (3.5.4)
clearly generate C sm (N0 , U ) as a P + -representation. 
The adjointness isomorphism of the preceding theorem does not depend on the
choice of the compact open subgroup P0 of P that is used to compute the functor
JP , since this is true of the isomorphism of lemma 3.5.2. However, it does dependon the choice of the Levi factor M of P (or equivalently, on the particular choiceof P ), since the very definition of the functor Ccsm (N, – ) depends on this choice.
If we take U = JP (V ) in theorem 3.5.6 then the identity map of JP (V ) to itself
gives rise to a P -equivariant morphism(3.5.10)

Ccsm (N, JP (V )) → V.

If we fix a compact open subgroup P0 of P , then the constant JP (V )-valued functions on N0 form an M0 N0 -invariant subspace of the source of (3.5.10), which isisomorphic to JP (V ) itself (regarded as an M0 N0 -representation by having thisproduct act through its quotient M0 ). Thus (3.5.10) induces an M0 N0 -equivariant
map JP (V ) → V, which is immediately seen to be the canonical lifting (3.4.8).
4. Some properties of the Jacquet module functor(4.1) This subsection is preliminary to the proof of theorem 0.5. We first recall
some constructions (and the attendant terminology and notation) introduced in [8,§§3.5, 5.2], and then apply them in the context of our reductive group G.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

41

We begin with a discussion that applies to an arbitrary locally L-analytic group,and so for the moment we let G denote any such group. Suppose that (φ, H, H) is alocally analytic chart of G – thus H is a compact open subset of G, H is an affinoid
rigid analytic space over L isomorphic to a closed ball, and φ is a locally analytic∼
isomorphism φ : H −→ H(L) – with the additional property that H is a subgroupof G. Since H is Zariski dense in H, there is at most one rigid analytic groupstructure on H giving rise to the group structure on H. If such a rigid analytic
group structure exists on H, we will refer to the chart (φ, H, H) as an analytic open
subgroup of G. Usually, we will suppress the isomorphism φ, and simply refer toan analytic open subgroup H of G, and write H to denote the corresponding rigid
analytic group determined by H.
The analytic open subgroups of G form a directed set in an obvious fashion: if
H 0 ⊂ H is an inclusion of open subgroups of G each of which is equipped with thestructure of an analytic open subgroup of G, then we say that it is an inclusion ofanalytic open subgroups if it lifts to a rigid analytic map H0 → H on the associatedrigid analytic groups. (Since H 0 and H are Zariski dense in H0 and H respectively,
such a lift is uniquely determined, and is automatically a homomorphism of rigid
analytic groups.) Forgetting the chart structure yields an order-preserving map
from the directed set of analytic open subgroups of G to the set of all open subgroupsof G. Since the group structure on G is locally analytic, the image of this map iscofinal in the directed set of all open subgroups of G.
We now introduce a particular class of analytic open subgroups of G (the “goodanalytic open subgroups”), obtained by applying the Campbell-Baker-Hausdorff
formula to certain OL -Lie subalgebras of g. We begin with some terminology. Byan OL -lattice in g we mean an OL -submodule h of g which is free of finite rank as
an OL -module, and which spans g over L. The gauge of h defines a norm on g, andwe denote by H the rigid analytic closed unit ball in the affine space underlying
g defined by this norm (so by definition we have h = H(L)). By a Lie sublatticeof g, we mean an OL -lattice h in g that is also an OL -Lie subalgebra of g. Notethat if h is any OL -lattice in g, then ah is a Lie sublattice of g for any a ∈ K × of
sufficiently large valuation. It follows from [22, LG Ch. V, § 4] (or more precisely,from the proof of that result) that even more is true: if a ∈ K × has sufficiently
large valuation, then not only is ah a Lie subalgebra of g, but the Baker-CampbellHausdorff formula converges on h and so defines a rigid analytic group structureon H. In particular, g admits a basis of neighbourhoods of the origin consisting of
Lie sublattices h for which the Baker-Campbell-Hausdorff formula defines a rigid
analytic group structure on H.
If h is such a Lie sublattice, giving rise to the rigid analytic group H, we write H
to denote the locally L-analytic group underlying H (so H = H(L) = h as sets).
Assuming that h is sufficiently small, it is proved in [22, LG 5.35, cor. 2] that we
may construct an embedding of locally L-analytic groups exp : H → G, and thus
realize H as an analytic open subgroup of G. We refer to analytic open subgroupsof G constructed in this manner as good analytic open subgroups. Our discussion
shows that the set of good analytic open subgroups is cofinal in the directed set of
all analytic open subgroups of G. Furthermore, if H 0 and H are two good analytic
open subgroups of G, corresponding to Lie sublattices h0 and h of g respectively,
then H 0 ⊂ H if and only if h0 ⊂ h, and if these equivalent conditions hold, then the
inclusion H 0 ⊂ H is necessarily an inclusion of analytic open subgroups.
We continue to let h denote a Lie sublattice of g corresponding to a good analytic

42

MATTHEW EMERTON

open subgroup H of G. For any real number 0 < r < 1, we let Hr denote the rigid
analytic closed ball of radius r in g with respect to the gauge of h. The open
subdomain Hr of h is in fact a rigid analytic subgroup of H, and if r equals theabsolute value of an element of the algebraic closure L of L, then Hr is furthermore
affinoid [2, thm. 6.1.5/4]. (Note that such elements r are dense in the interval(0, 1).) We let
S Hr denote the locally analytic group of L-valued points of Hr . If wewrite H◦ = 0<r<1 Hr , then H◦ is a strictly σ-affinoid rigid analytic open subgroupof H (i.e. it is the union of an increasing sequence of open affinoid subdomains, eachrelatively compact in the next).
For any element g ∈ G, the conjugate gHg −1 is naturally a good analytic open
subgroup of G, corresponding to the Lie sublattice Adg (h). We denote the rigid
analytic group that underlies gHg −1 by gHg −1 . Conjugation by g induces an∼isomorphism of rigid analytic groups H −→ gHg −1 . In particular, we see thatgHg −1 ⊂ H if and only if Adg (h) ⊂ h, and that if these equivalent conditions hold,then the action of g on H by conjugation extends to a rigid analytic endomorphismof H, and also of the rigid analytic groups Hr , for each r ∈ (0, 1), and of H◦ .
We now introduce some notation similar to that introduced in subsections 5.3
and 6.1 of [8]. Let us fix for the moment a compact open subgroup G0 of G. The
space C la (G0 , K) is a compact type space equipped with a locally analytic actionof G0 . (In fact, C la (G0 , K) is the most fundamental example of a strongly admissible
locally analytic G0 -representation.) Let H be a good analytic open subgroup of G,
contained in G0 , such that G0 normalizes H. The discussion of the preceding
paragraph implies that the conjugation action of G on H extends to an action by
rigid analytic automorphisms on Hr , for each r ∈ (0, 1), and of H◦ . (We say thatG0 “normalizes” these rigid analytic groups.) Following [8, def. 3.4.1], we define
M∼
C an (gHr , K).
(4.1.1) C la (G0 , K)H◦ −an = lim
C la (G0 , K)Hr −an −→ lim
←−
←−
r

r

g∈G0 /Hr(Here r ranges through those elements of the interval (0, 1) that are absolute valuesof elements of L. If g ∈ G0 , then we let gHr denote the coset gHr , endowed with the∼structure of a rigid analytic space via the locally L-analytic isomorphism Hr −→gHr obtained via multiplication by g, together with the equality Hr = Hr (L). The∼
second isomorphism then follows from the natural isomorphism C la (G0 , K) −→L
lag∈G0 /Hr C (gHr , K) together with [8, cor. 3.3.26].) Since the inclusion Hr ⊂ Hr 0
is relatively compact if r < r0 , we see that C la (G0 , K)H◦ −an is the projective limitof K-Banach spaces under compact transition maps, and so is a nuclear Fréchetspace. (See [8, prop. 2.1.16], for example.) Since G0 normalizes each Hr , we seethat C la (G0 , K)H◦ −an is naturally equipped with a continuous G0 -action.
If D(H◦ , G0 ) denotes the strong dual to C la (G0 , K)H◦ −an , then D(H◦ , G0 ) is alocally convex topological K-vector space of compact type. In fact the definitionof C la (G0 , K)H◦an induces an isomorphismM∼(4.1.2) D(H◦ , G0 ) := (C la (H, K)H◦ −an )0b −→ lim(
C an (gHr , K))0b−→r

g∈G0 /Hr∼−→ lim−→rMδg ∗ Dan (Hr , K),g∈G0 /Hr

where Dan (Hr , K) denotes the strong dual to C an (Hr , K), and δg denotes the δfunction supported at g. Each of the spaces Dan (Hr , K) is naturally a K-Banach

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

43

algebra [8, cor. 5.1.8]. Since G0 normalizes each Hr , each of the direct sumsLang∈G0 /Hr δg ∗ D (Hr , K) also has a natural structure of K-Banach algebra. Finally, the space D(H◦ , G0 ) is realized as the inductive limit of these topological Kalgebras under continuous K-algebra maps, and so is again a topological K-algebra.
(The multiplication map on D(H◦ , G0 ), while a priori separately continuous, is infact jointly continuous, since D(H◦ , G0 ) is of compact type [8, prop. 1.1.31].)
We now return to the setting of the introduction, in which G is supposed tobe the group of L-valued points of a connected reductive linear algebraic group
G. We will exploit the reductive group structure of G to show that G admits aneighbourhood basis of the identity consisting of a descending sequence {Hn }n≥0of good analytic open subgroups with certain additional nice properties. Amongthese is the existence of certain rigid analytic Iwahori decompositions for each of
the groups Hn . We begin by explaining what we mean by this.
Let Hn be a good analytic open subgroup belonging to our sort-for neighbourhood basis, let P be a parabolic
T subgroup of G, let P be a parabolic subgroup
opposite to P, and let M = P P be the common Levi factor of each of P and P.
Let N (respectively N) denote the unipotent radical of P (respectively P). Write
P, P , M , N
T groups of L-valued points. Set
T, and N to denoteT the corresponding
Mn := Hn M, Nn := Hn N , and N n := Hn N , and let Mn , Nn , and Nn denote the rigid analytic Zariski closure of Mn , Nn , and N n in Hn (the rigid analytic
group underlying the good analytic subgroup H
Tn ). Let hn denoteT the Lie sublatticeTof g corresponding to Hn , and write mn := hn m, nn := hn n, and nn := hn n.
Definition 4.1.3. We will say that the good analytic open subgroup Hn admits arigid analytic Iwahori decomposition with respect to P and P if:(i) Under the identification Hn = hn , the groups Mn , Nn , and N n are identified
with mn , nn , and nn respectively. Thus Mn , Nn , and N n coincide with the goodanalytic open subgroups of M, N , and N obtained by exponentiating mn , nn , and nn ,respectively, and Mn , Nn , and Nn coincide with the rigid analytic groups underlying
Mn , Nn , and N n , respectively.
(ii) The rigid analytic map(4.1.4)

Nn × Mn × Nn → Hn

induced by multiplication in Hn is in fact a rigid analytic isomorphism.
Note that if (4.1.4) is an isomorphism, then the corresponding map(4.1.5)

N n × Mn × N n → H nobtained by passing to L-valued points is also an isomorphism.
We fix a minimal parabolic subgroup P∅ of G, as well as a maximal split torus A∅
in P∅ . The centralizer M∅ of A∅ is a LeviTfactor of P∅ . Denote by P∅ the opposite
parabolic to P∅ , chosen so that M∅ := P∅ P∅ .
The following result refines [5, prop. 1.4.4].
Proposition 4.1.6. We may find a decreasing sequence {Hn }n≥0 of good analytic
open subgroups of G, cofinal in the directed set of all analytic open subgroups of G,
and satisfying the following conditions:(i) For each n ≥ 0, the inclusion Hn+1 ⊂ Hn extends to a relatively compact
rigid analytic map Hn+1 ⊂ Hn .

44

MATTHEW EMERTON(ii) For each n ≥ 0, the subgroup Hn of H0 is normal.
The remaining properties refer to any pair P and P of opposite parabolic subgroups of G, chosen so that P contains P∅ and P contains P∅ . We employ thenotation introduced in the discussion preceding definition 4.1.3.
(iii) Each Hn admits a rigid analytic Iwahori decomposition with respect to P
and P.
(iv) If z ∈ ZM is such that z −1 N 0 z ⊂ N 0 , then z −1 N n z ⊂ N n for each n ≥ 0.
(v) If z ∈ ZM is such that zN0 z −1 ⊂ N0 , then zNn z −1 ⊂ Nn for each n ≥ 0.
(vi) We may find z ∈ ZM such that z −1 N 0 z ⊂ N 0 and zN0 z −1 ⊂ N0 , andsuch that, for each n ≥ 0, the embedding of part (iv) factors through the inclusion
N n+1 ⊂ N n .
Proof. Let a∅ denote
L the Lie algebra of A∅ , let m∅ denote Lie algebra of M∅ , andwrite g = m∅ ⊕ α∈±∆(G,A∅ ) gα , where ∆(G, A∅ ) denotes the set of positive restricted roots of A∅ acting on g, and for any positive or negative restricted root α,
we denote by gα the corresponding restricted root space. Fix an OL -lattice h0 in g,
fix a ∈ K × of sufficiently large valuation, and set
 \ 
 \ 
M
h := ah0 m∅ ⊕ah0 gα .
α∈±∆(G,A∅ )More generally, if π is a uniformizer of L, set hn := π n h. Then, provided that thevaluation of a was chosen large enough, we see that each hn is a Lie sublattice of gwhich exponentiates to a good analytic open subgroup Hn of G. Since the sequence{hn } forms a neighbourhood basis of zero in the Lie algebra g, we see that {Hn }forms a neighbourhood basis of the identity in G. As above, for each n ≥ 0, let
Hn denote the rigid analytic group underlying Hn . By construction, the inclusion
Hn+1 → Hn extends to a rigid analytic embedding Hn+1 → Hn , for each value ofn, and (again by construction) this embedding is relatively compact. Thus we haveestablished (i). Property (ii) is clear from our construction.
By construction, the natural map
 \ 
 \ 
M
∼
(4.1.7)hn m∅ ⊕
hn gα −→ hnα∈±∆(G,A∅ )is an isomorphism, for each n. Suppose now that P is a parabolic subgroup of G
that contains P∅ , that P is the corresponding opposite parabolic, chosen to containP∅ , and that M is the common Levi factor of these two parabolics. As above, we let
N denote the unipotent radical of P, and let N denote the unipotent radical of P.
We write m, n, and n for the Lie algebras of M, N, and N. The structure theory ofparabolic subgroups L
of reductive groups
L shows that there is a subset
L Φ of ∆(G, A∅ )such that m = m∅ ⊕ α∈±Φ gα , n = α∈∆(G,A∅ )\Φ gα , and n = α∈∆(G,A∅ )\Φ g−α .
It thus follows from the the fact that (4.1.7) is an isomorphism that the naturalmap(4.1.8)

nn

M

mn

M

nn → hn

is an isomorphism, for each n ≥ 0. If we fix a value of n, then each of nn , mn , andnn exponentiates to a good analytic open subgroup N n of N , Mn of M , and Nn

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

45

of N , respectively. Following the convention introduced above, we write Nn , Mn ,and Nn to denote the rigid analytic groups underlying the good analytic subgroups
N n , Mn , and Nn . These groups are also equal to the Zariski closures in Hn of N n ,Mn , and Nn , respectively.
Increasing theTvaluation of a if necessary,
we furthermoreTTfind that N n = Hn N , that Mn = Hn M, that Nn = Hn N , and (taking into
account the fact that (4.1.8) is an isomorphism) that the natural rigid analyticmap (4.1.4) is an isomorphism. Altogether, we have established (iii).
−1Note
T that N 0 is closed under conjugation by z , for some z ∈ ZM , if and only
if h T n is closed under the action of Adz−1 . This in turn is true if and only if
π n h n is closed under the action of Adz−1 , which holds if and only if N n is closed
under conjugation by z −1 . Thus (iv) is proved, and (v) is proved similarly.
The
T
theory Tof roots shows that we may find z ∈ ZM such that Adz (h n) ⊂ n andAdz (h n) ⊂ πn. This establishes (vi). 
In the context of the preceding proposition, we may apply the constructionof (4.1.2). Namely, taking the pair (G0 , H) to be the pair (H0 , Hn ) for each n ≥ 0
in turn, we obtain for each n ≥ 0 a compact type topological K-algebra D(H◦n , H0 )as the strong dual to the nuclear Fréchet space C la (H0 , K)H◦n −an . The natural∼
isomorphism C la (H0 , K) −→ lim
C la (H0 , K)H◦n −an induces a natural isomorphism−→n(4.1.9)∼Dla (H0 , K) −→ lim
D(H◦n , H0 ).
←−n(Compare the proof of [8, prop. 5.3.1].) We may similarly apply the constructionof (4.1.2) to the pair (M0 , Mn ), for each n ≥ 0. Thus we obtain for each n acompact type topological K-algebra D(M◦n , M0 ), and a natural isomorphism(4.1.10)∼D(M◦n , M0 ).
Dla (M0 , K) −→ lim←−n

The closed embedding M0 → H0 gives rise to a closed embedding of K-Fréchet
algebras(4.1.11)Dla (M0 , K) → Dla (H0 , K).

This same closed embedding, together with the compatible rigid analytic closed
immersion M◦n → H◦n , gives rise to a closed embedding of topological K-algebras(4.1.12)D(M◦n , M0 ) → D(H◦n , H0 ),for each n ≥ 0. The closed embeddings (4.1.11) and (4.1.12) are compatible with
the isomorphisms (4.1.9) and (4.1.10), and passage to the projective limit in n.
We continue to suppose that we are in the context of proposition 4.1.6, anddefine certain additional good analytic open subgroups of G. Let Z + denote thesubmonoid of ZM consisting of those z such that zN0 z ⊂ N0 and z −1 N 0 z ⊂ N 0 .
An evident variant of proposition 3.3.2 (i) shows that Z + generates
ZM as a group.
T
If n ≥ 0, and if z ∈ Z + , then write H(z)n := (z −1 Hn z) Hn . Since Hn admits
the Iwahori decomposition (4.1.5), we find that z −1 Hn z admits the corresponding∼
Iwahori decomposition z −1 N n z × Mn × z −1 Nn z −→ z −1 Hn z, and thus (taking into

46

MATTHEW EMERTON

account the inclusions of proposition 4.1.6 (iv) and (v), which hold, since z ∈ Z + ),that H(z)n admits the Iwahori decomposition(4.1.13)∼z −1 N n z × Mn × Nn −→ H(z)n .

If hn is the OL -Lie subalgebra of g corresponding
T to the good analytic open
subgroup Hn , then we see that h(z)n := Adz−1 hn hn is an OL -Lie subalgebraof hn , and that the identification of hn with Hn yields an identification of h(z)nwith H(z)n . This latter identification gives H(z)n the structure of a good analytic open subgroup of Hn ; we let H(z)n denote the corresponding rigid analytic
group, of which H(z)n is the group of L-valued points. One sees that the Iwahori
decomposition (4.1.13) extends to a rigid analytic Iwahori decomposition(4.1.14)∼z −1 Nn z × Mn × Nn −→ H(z)n .

(Recall that z −1 Nn z denotes the image of Nn in itself under the map induced by
conjugation by z −1 .)The group H(z)n is a normal open subgroup of H(z)0 , and hence we may form
the compact type algebra D(H(z)◦n , H(z)0 ). There is an isomorphism of topologicalK-algebras∼
Dla (H(z)0 , K) −→ lim
D(H(z)◦n , H(z)0 ).
←−n

The inclusion H(z)0 ⊂ H0 induces a continuous homomorphism of K-Fréchet algebras
Dla (H(z)0 , K) → Dla (H0 , K),
and a consequent isomorphism
Dla (H0 , K) ∼=M

Dla (H(z)0 , K) ∗ δh .

h∈H(z)0 \H0

The same inclusion H(z)0 ⊂ H0 , when coupled with the rigid analytic open immersion H(z)◦n → H◦n , also induces a continuous homomorphism(4.1.15)

D(H(z)◦n , H(z)0 ) → D(H◦n , H0 ).

We define
D(H(z)◦n , H0 ) := D(H(z)◦n , H(z)0 ) ⊗Dla (H(z)0 ,K) Dla (H0 , K)M∼−→
D(H(z)◦n , H0 ) ∗ δh .
h∈H(z)0 \H0

The compact type convex K-space D(H(z)◦n , H0 ) may be identified with the strong
dual of C la (H0 , K)H(z)◦n −an (where the space of H(z)◦n -analytic vectors in C la (H0 , K)
is defined as in [8, def. 3.4.1], with respect to the right-multiplication action of H(z)non H0 ). It is naturally a topological (D(H(z)◦n , H(z)0 ), Dla (H0 , K))-bimodule. The
homomorphism (4.1.15) extends to a continuous homomorphism of topological(D(H(z)◦n , H(z)0 ), Dla (H0 , K))-bimodules(4.1.16)

D(H(z)◦n , H0 ) → D(H◦n , H0 ).

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

47(The left D(H(z)◦n , H(z)0 )-module structure on the target is defined via (4.1.15),
and the right Dla (H0 , K)-module structure via the homomorphism Dla (H0 , K) →D(H◦n , H0 ) arising from the isomorphism (4.1.9).) Also, analogously to (4.1.12),
there is a closed embedding of topological K-algebras(4.1.17)D(M◦n , M0 ) → D(H(z)◦n , H(z)0 ),whose composite with (4.1.15) is equal to the map (4.1.12). In particular, we
obtain a left action of D(M◦n , M0 ) on D(H(z)◦n , H(z)0 ), and also on D(H(z)◦n , H0 ).
Since (4.1.16) is compatible with the left D(H(z)◦n , H(z)0 )-module structure on its
source and target, it is in particular a map of left D(M◦n , M0 )-modules.
Finally, suppose that zT∈ Z + is chosen so that z −1 N n z ⊂ N n+1 . We then define
H(z)n,n+1 := (z −1 Hn z) Hn+1 . Just as with the open subgroup H(z)n , we seethat H(z)n,n+1 admits the structure
T of a good analytic open subgroup associatedto the OL -Lie subalgebra Adz−1 hn hn+1 of g. Our assumption on z shows that
H(z)n,n+1 furthermore admits the rigid analytic Iwahori decomposition(4.1.18)∼z −1 Nn z × Mn+1 × Nn+1 −→ H(z)n,n+1 .

Since H(z)n,n+1 is normal in H(z)0 , we may define the compact type algebra
D(H(z)◦n,n+1 , H(z)0 ), as well as the (D(H(z)◦n,n+1 , H(z)0 ), Dla (H0 , K))-bimodule
D(H(z)◦n,n+1 , H0 ) := D(H(z)◦n,n+1 , H(z)0 ) ⊗Dla (H(z)0 ,K) Dla (H0 , K)M∼−→
D(H(z)◦n,n+1 , H0 ) ∗ δh .
h∈H(z)0 \H0Analogously to (4.1.17), there is a closed embedding of topological K-algebrasD(M◦n+1 , M0 ) → D(H(z)◦n , H(z)0 ),and thus D(H(z)◦n,n+1 , H0 ) may also be regarded as a (D(M◦n+1 , M0 ), Dla (H0 , K))bimodule. The diagram of inclusions of good analytic open subgroups
H(z)n,n+1/ H(z)n


Hn+1


/ Hn

induces a diagram of continuous maps of topological (D(M◦n+1 , M0 ), Dla (H0 , K))bimodules(4.1.19)

ˆ D(M◦ ,M0 ) D(H(z)◦n,n+1 , H0 )D(M◦n , M0 ) ⊗n+1/ D(H(z)◦n , H0 )
ˆ D(M◦ ,M0 ) D(H◦n+1 , H0 )D(M◦n , M0 ) ⊗n+1/ D(H◦n , H0 ) .

(4.2) In this subsection we prove theorem 0.5, that is, that JP restricts to afunctor Repes (G) → Repes (M ). Equivalently, we will show that if V is an object of

48

MATTHEW EMERTONˆ K Dla (M0 , K)-module. Our
Repes (G), then JP (V )0b is a coadmissible C an (ẐM , K) ⊗strategy will be to apply proposition 3.2.24. In order to explain how that result
will be applied, we must first establish some notation.
Choose a Levi factor
T M of P, let P denote the opposite parabolic to P with respect
to M (so that M = P P), and let N denote the unipotent radical of P. Also let P :=P(L) and N := N(L). We apply proposition 4.1.6 to an opposite pair of minimal
parabolic subgroups of G that are compatible with P and P, so as to find a decreasing
sequence {Hn }n≥0 of good analytic open subgroups of G satisfying conditions (i)to (vi) of that proposition. (We will apply conditions (iii), (iv), (v), and (vi) with P
and P taken to be the particular parabolic subgroups just introduced.) We will use
the compact type algebras D(H◦n , H0 ) and D(M◦n , M0 ) introduced in subsection 4.1,
as well as the (D(M◦n , M0 ), Dla (H0 , K))-bimodules D(H(z)◦n , H0 ) (for n ≥ 0, andz ∈ Z + ) and the (D(M◦n+1 , M0 ), Dla (H0 , K))-bimodules D(H(z)◦n,n+1 , H0 ) (for n ≥0 and z ∈ Z + such that z −1 N n z ⊂ N n+1 ).
Choose an exhaustive increasing sequence {ẐG,n }n≥0 of admissible open affinoid
subdomains of ẐG , such that each inclusion ẐG,n ⊂ ẐG,n+1 is relatively compact. If
C an (ẐG,n , K)† denotes the space of overconvergent (in ẐG ) rigid analytic functions
on ẐG,n , then C an (ẐG,n , K)† is a locally convex topological K-algebra of compact
type, and there is a natural isomorphism∼

C an (ẐG,n , K)† .
C an (ẐG , K) −→ lim←−(4.2.1)n

Combining this with (4.1.10) yields (as a special case of (3.2.22)) an isomorphism∼

ˆ Dla (M0 , K) −→ lim C an (ẐG,n , K)† ⊗ˆ D(M◦n , M0 ).
C an (ẐG , K) ⊗
←−
K

K

n

This isomorphism describes a weak Fréchet-Stein structure (in the sense of [8,ˆ K Dla (M0 , K).
def. 1.2.6]) on the nuclear Fréchet algebra C an (ẐG , K) ⊗+Recall that ZM denotes the submonoid of ZM consisting of elements z for which++zN0 z −1 ⊂ N0 , and that the operators πN0 ,z (for z ∈ ZM
) define an action of ZM+
N0+
on V . We let Z denote the submonoid of ZM consisting of elements z suchthat z −1 N 0 z ⊂ N 0 . As mentioned in the preceding section, Z + generates ZM as agroup, and so lemma 3.2.19 shows that we may compute JP (V ) as the finite slope
part of V N0 , regarded as a Z + -module. We will apply proposition 3.2.24 to V N0 ,
ˆ K Dla (M0 , K), and An totaking A to be the nuclear Fréchet algebra C an (ẐG , K) ⊗an† ˆ◦be C (ẐG,N , K) ⊗K D(Mn , M0 ), for each n ≥ 0.
For proposition 3.2.24 to apply, we must introduce the following additional data:
ˆ K D(M◦n , M0 )-module Un as well
(4.2.2) For each n ≥ 0, a C an (ẐG,n , K)† ⊗an† ˆ◦as a C (ẐG,n+1 , K) ⊗K D(Mn+1 , M0 )-linear transition map Un+1 → Un , and a∼
ˆ K Dla (M, K)-linear topological isomorphism (V N0 )0b −→C an (ẐG , K) ⊗
lim
Un .
−→
n

ˆ K D(M◦n , M0 )-linear action of Z + on each Un , such that
(4.2.3) A C an (ẐG,n , K)† ⊗the transition maps Un+1 → Un and the maps (V N0 )0b → Un are Z + -equivariant.
We must then establish the following hypotheses:(4.2.4) For each n ≥ 0, the induced map
ˆ D(M◦n , M0 )C an (ẐG,n , K)† ⊗Kˆ⊗ˆ K D(M◦ ,M0 )C an (ẐG,n+1 ,K)† ⊗n+1

Un+1 → Un

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

49

ˆ K D(M◦n , M0 )-compact (in the sense of definition 2.3.3).
is C an (ẐG,n , K)† ⊗(4.2.5) There exists z ∈ Z + such that the endomorphism of each Un induced byz factors through the map of (4.2.4), so as to render commutative the diagramˆ D(M◦n , M0 )C an (ẐG,n , K)† ⊗
K

ˆ
⊗

ˆz
id ⊗

ˆ D(M◦n , M0 )C an (ẐG,n , K)† ⊗
K

Un+1
/
iii Un
i
i
i
iiii
iiii
i
i
i
z
iiii
tiiii

ˆ
⊗
Un+1
/ Un .
ˆ D(M◦ ,M ),K)† ⊗

ˆ K D(M◦ ,M0 )C an (ẐG,n+1 ,K)† ⊗n+1

C an (ẐG,n+1

K

0

n+1

Let us begin by specifying the modules Un . Combining (4.2.1) with (4.1.9), we
obtain as a special case of (3.2.22) an isomorphism∼

ˆ Dla (H0 , K) −→ lim C an (ẐG,n , K)† ⊗ˆ D(H◦n , H0 ).
C an (ẐG , K) ⊗
←−
K

K

n

This isomorphism describes a weak Fréchet-Stein structure (in the sense of [8,ˆ K Dla (H0 , K).
def. 1.2.6]) on the nuclear Fréchet algebra C an (ẐG , K) ⊗0an
ˆ K Dla (H0 , K)-module (since
The strong dual Vb is a coadmissible C (ẐG , K) ⊗V is an object of Repes (G)) and so [8, thm. 1.2.11 (i)] yields a natural isomorphism∼ˆ D(H◦n , H0 )Vb0 −→ lim
C an (ẐG,n , K)† ⊗
←−
K

n

ˆ
⊗
ˆ K D la (H0 ,K)
C an (ẐG ,K) ⊗Vb0 ,where each of the tensor products
ˆ D(H◦n , H0 )C an (ẐG,n , K)† ⊗Kˆ⊗ˆ K D la (H0 ,K)
C an (ẐG ,K) ⊗Vb0

ˆ K D(H◦n , H0 )-module. Since theis a finitely generated topological C an (ẐG,n , K)† ⊗N0strong dual to V
is naturally identified with the Hausdorff N0 -coinvariants (Vb0 )N0of the strong dual to V , we obtain an isomorphism(4.2.6)∼(V N0 )0b −→
lim
←−

†

an

ˆ
⊗

C (ẐG,n , K)

C an (ẐG ,K)n



D(H◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0

!



.
N0

ˆ Dla (H0 ,K) Vb0 )N0 denotes the Hausdorff N0 -coinvariants of the(Here (D(H◦n , H0 ) ⊗ˆ Dla (H0 ,K) Vb0 .)completed tensor product D(H◦n , H0 ) ⊗For each n ≥ 0, define

 !
Un :=

C an (ẐG,n , K)†ˆ⊗C an (ẐG ,K)

D(H◦n , H0 )

ˆ⊗D la (H,K)

Vb0

.
N0

ˆ Dla (H0 ,K) Vb0 is naturally a topological
The completed tensor product D(H◦n , H0 ) ⊗D(H◦n , H0 )-module, and hence is also (taking into account (4.1.12)) a topologicalˆ Dla (H0 ,K) Vb0 )N0D(M◦n , M0 )-module. As M0 normalizes N0 , the space (D(H◦n , H0 ) ⊗50

MATTHEW EMERTON

inherits a natural quotient D(M◦n , M0 )-module structure (this is “dual” to the factthat V N0 is an M0 -invariant subspace of V ), and hence Un is naturally a compact
ˆ K D(M◦n , M0 )-module for each value of n. Furtype topological C an (ẐG,n , K)† ⊗thermore, the isomorphism (4.2.6) may be rewritten as an isomorphism∼

Un .
(V N0 )0b −→ lim−→n

We next turn to explaining the action of the monoid Z + on the modules Un .
The discussion is somewhat involved, since it will incorporate some constructions
necessary for the verification of (4.2.4) and (4.2.5).
For each z ∈ Z + , the map (4.1.16) induces a map(4.2.7)D(H(z)◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0 → D(H◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0

of left D(H(z)◦n , H(z)0 )-modules. The preceding discussion generalizes to show thatˆ Dla (H0 ,K) Vb0 induced by (4.1.17) descends
the D(M◦n , M0 )-action on D(H(z)◦n , H0 ) ⊗◦ˆ Dla (H0 ,K) Vb0 )N0 , and that (4.2.7)
to an action of D(Mn , M0 ) on (D(H(z)◦n , H0 ) ⊗◦induces a map of D(Mn , M0 )-modules(4.2.8)(D(H(z)◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0 )N0 → (D(H◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0 )N0 .

For each z ∈ Z + , we are going to define a continuous D(M◦n , M0 )-linear map(4.2.9)(D(H◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0 )N0 → (D(H(z)◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0 )N0 ,in such a way that the diagram(4.2.10)(V N0 )0b∼

/ (V 0 )N0b(πN0 ,z )0(V N0 )0b∼

/ (V 0 )N0bv 0 7→1⊗v 0◦/ (D(Hn , H0 )

ˆ⊗D la (H0 ,K)Vb0 )N0(4.2.9)◦v 0 7→1⊗v 0ˆVb0 )N0/ (D(H(z)n , H0 ) la ⊗TTTT
D (H0 ,K)
TTTTv0 7→1⊗v0
TTTT
TTTT
(4.2.8)TTT)ˆ(D(H◦n , H0 )⊗Vb0 )N0
D la (H0 ,K)commutes. (Here, as indicated on the diagram, the left-hand vertical arrow is givenby the action of z ∈ Z + on (Vb0 )N0 , obtained as the transpose of the operator
πN0 ,z on V N0 .) Since the horizontal arrows have dense image (because the maps
Dla (H0 , K) → D(H◦n , H0 ) and Dla (H0 , K) → D(H(z)◦n , H0 ) have dense image, as
one sees by noting that their strong duals are the injections C(H0 , K)H◦n −an →C la (H0 , K) and C(H0 , K)H(z)◦n −an → C la (H0 , K) respectively), the map (4.2.9) is
uniquely determined, if it exists.

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

51

Lemma 4.2.11. The morphism (4.2.9) can be constructed so as to make (4.2.10)
commute.
Proof. Let λ : H0 → N 0 × M0 =: P 0 denote the inverse of the isomorphism (4.1.5)(for n = 0), followed by projection onto the first two factors. If x is an element of
N0 (we use the symbol x rather than n so as to avoid confusion with the index nof the analytic open subgroup Hn ), then
(4.2.12)h 7→ z −1 λ(x−1 λ(h−1 ))−1 z

describes a locally analytic map H0 → z −1 P 0 z ⊂ H0 . This map induces a continuous linear endomorphism sx,z of C la (H0 , K), defined for f ∈ C la (H0 , K) via(sx,z f )(h) = f (z −1 λ(x−1 λ(h−1 ))−1 z).
We regard H0 as the locally analytic group underlying the rigid analytic groupa(4.2.13)hH◦n .

h∈H0 /H◦n (L)(Note that proposition 4.1.6 (ii) implies that we obtain the same decomposition of
H0 if we consider instead left cosets, and so this disjoint union does indeed describe
a rigid analytic group.) The space of rigid analytic K-valued functions on this rigid
analytic group is precisely C la (H0 , K)H◦n −an . We simultaneously regard H0 as the
locally analytic space underlying the rigid analytic spacea(4.2.14)hH(z)◦n .

h∈H0 /H(z)◦n (L)The space of rigid analytic K-valued functions on this rigid analytic space is precisely C la (H0 , K)H(z)◦n −an .
Since Hn admits a rigid analytic Iwahori decomposition, the restriction to Hnof λ induces a rigid analytic map Hn → Nn × Mn , and hence also a rigid analytic◦
map H◦n → Nn × M◦n . Thus (taking into account proposition 4.1.6 (iv) as well) themap (4.2.12) is in fact a rigid analytic map from the rigid analytic group (4.2.13)
to the rigid analytic space (4.2.14), and so sx,z induces a continuous linear map
C la (H0 , K)H(z)◦n −an → C la (H0 , K)H◦n −an . Passing to the transpose, we obtain a continuous linear map s0x,z : D(H◦n , H0 ) → D(H(z)◦n , H0 ). Since (4.2.12) is left M0 equivariant, this map is in fact D(M◦n , M0 )-linear.
As in subsection 3.5, let δ denote the modulus character of P . Let x run over a
set of coset representatives for zN0 z −1 in N0 , and define a map(4.2.15)

ˆ Vb0 → D(H(z)◦n , H0 ) ⊗ˆ Vb0D(H◦n , H0 ) ⊗
K

K

via
(4.2.16)µ ⊗ v 0 7→ δ(z)Xx

s0x,k (µ) ⊗ z −1 x−1 v 0 .

52

MATTHEW EMERTON

We claim that the map (4.2.15) descends to the required morphism (4.2.9). To see
this, we note that formula (4.2.16) also defines a map
K[H0 ] ⊗K Vb0 → K[H0 ] ⊗K Vb0 .

(4.2.17)

In lemma 4.2.19 below we will establish the commutativity of the diagram/ (V 0 )N0

K[H0 ] ⊗K Vb0(4.2.18)

/ (V N0 )0

∼

b

b

(4.2.17)/ (V 0 )N0

K[H0 ] ⊗K Vb0

(πN0 ,z )0/ (V N0 )0

∼

b

b

(in which the left horizontal arrows are induced by the action of H0 on Vb0 , together with the quotient map Vb0 → (Vb0 )N0 ). In particular, the map (4.2.17)
descends to a map (Vb0 )N0 → (Vb0 )N0 . Since K[H0 ] ⊗K Vb0 is dense in each ofˆ K Vb0 and D(H(z)◦n , H0 ) ⊗ˆ K Vb0 , we infer that (4.2.15) similarly deD(H◦n , H0 ) ⊗◦ˆ Dla (H0 ,K) Vb0 )N0 → (D(H(z)◦n , H0 ) ⊗ˆ Dla (H0 ,K) Vb0 )N0 .
scends to a map (D(Hn , H0 ) ⊗If we take this map to be our sought-after morphism (4.2.9), then the commutativityof (4.2.10) follows from the commutativity of (4.2.18). 
Lemma 4.2.19. The diagram (4.2.18) commutes.
Proof. It suffices to show that for any v ∈ V N0 , v 0 ∈ Vb0 , and h ∈ H0 , there is anequality(4.2.20)

δ(z)X

hs0x,z (δh ) ⊗ z −1 x−1 v 0 , vi = hhv 0 , πN0 ,z (v)i.

x(Here we have denoted by δh the element h ∈ H0 regarded as an element of thegroup ring K[H0 ], since this element is identified with the delta function supported
at h under the natural embedding K[H0 ] → Dla (H0 , K); it has nothing to dowith the modulus δ.) It is an immediate consequence of the definition of sx,z thats0x,z (δh ) = δz−1 λ(x−1 λ(h−1 ))−1 z . Thus we computeδ(z)X

hs0x,z (δh ) ⊗ z −1 x−1 v 0 , vi = δ(z)x

= δ(z)

Xhδz−1 λ(x−1 λ(h−1 ))−1 z ⊗ z −1 x−1 v 0 , vi

x

X

0

hv , xzz

−1

−1

λ(x

λ(h

−1

))zvi = δ(z)Xx

hv 0 , xλ(x−1 λ(h−1 ))zvix= δ(z)

X

hv 0 , h−1 hxλ(x−1 λ(h−1 ))zvi.

x

Lemma 4.2.21 below shows that as x ranges over a set of right coset representativesof zN0 z −1 in N0 , the same is true of the elements hxλ(x−1 λ(h−1 )). Thus if we take
into account the definition of πN0 ,z , we find that
δ(z)X

hs0x,z (δh ) ⊗ x−1 z −1 v 0 , vi = hv 0 , h−1 πN0 ,z vi.

xThis establishes (4.2.20).

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

53

Lemma 4.2.21. If x ranges over a set of right coset representatives of zN0 z −1
in N0 , then for any h ∈ H0 , the elements hxλ(x−1 λ(h−1 )) also range through a setof right coset representatives of zN0 z −1 in N0 .
Proof. Define ρ : H0 → N0 via the equation h = λ(h)ρ(h) for any h ∈ H0 . (Thusthe product λ × ρ yields the isomorphism (4.1.5) for n = 0.) Note that hλ(h−1 ) =ρ(h−1 )−1 ∈ N0 , and thus that the lemma is true if and only if the elementsλ(h−1 )−1 xλ(x−1 λ(h−1 )) = ρ(x−1 λ(h−1 ))−1
run over a set of right coset representatives of zN0 z −1 in N0 . Write p = λ(h−1 ) ∈P 0 . Since the inverses of a system of right coset representatives form a system ofleft coset representatives, it is equivalent to show that, as x ranges over a set of left
coset representatives of zN0 z −1 in N0 , the elements ρ(xp) range over such a set ofleft coset representatives.
Let x and x0 be two elements of N0 that lie in the given set of left coset representatives of zN0 z −1 in N0 . Then xp and x0 p are both elements of H0 , and so wemay write xp = λ(xp)ρ(xp) and x0 p = λ(x0 p)ρ(x0 p). Thusx(x0 )−1 = λ(xp)ρ(xp)ρ(x0 p)−1 λ(x0 p)−1 ,and so (zN0 z −1 )x = (zN0 z −1 )x0 if and only if x(x0 )−1 ∈ zN0 z −1 , which holds
if and only if λ(xp)ρ(xp)ρ(x0 p)−1 λ(x0 p)−1 ∈ zN0 z −1 , which holds if and only if
ρ(xp)ρ(x0 p)−1 lies in λ(xp)−1 zN0 z −1 λ(x0 p) ∩ N0 .
Now H0 ∩ zH0 z −1 = P 0 zN0 z −1 . Thus p0 zN0 z −1 p00 ∩ N0 ⊂ H0 ∩ zH0 z −1 ∩ N0 =zN0 z −1 for any pair of elements p0 and p00 of P 0 . Hence (zN0 z −1 )x = (zN0 z −1 )x0
implies that ρ(xp)ρ(x0 p)−1 ∈ zN0 z −1 . Conversely, if ρ(xp)ρ(x0 p)−1 ∈ zN0 z −1 , thenx(x0 )−1 ∈ λ(xp)zN0 z −1 λ(x0 p) ∩ N0 ⊂ zN0 z −1 . This proves the lemma. Let φz denote the composite of (4.2.8) with (4.2.9). (Thus φz is the compositeof the two right-hand vertical arrows in the diagram (4.2.10).) As observed, φz isuniquely determined by the fact that it makes the diagram (4.2.10) commute. Thus,
since the operators (πN0 ,z )0 define a topological action of Z + on (Vb0 )N0 , we see thatˆ Dla (H0 ,K) Vb0 )N0 .
the operators φz define a topological action of Z + on (D(H◦n , H0 ) ⊗an†
Taking the completed tensor product with C (ẐG,n , K) over C an (ẐG , K), weˆ φz ofˆ K D(M◦n , M0 )-linear endomorphisms id ⊗obtain continuous C an (ẐG,n , K)† ⊗
0
an
† ˆ
◦
ˆ
Un = C (ẐG,n , K) ⊗C an (ẐG ,K) (D(Hn , H0 ) ⊗Dla (H0 ,K) Vb )N0 , defining the required
topological action of Z + on Un .
This completes our specification of the data (4.2.2) and (4.2.3). Since V isa locally analytic G-representation, the closed subspace V N0 is certainly a locally analytic
T M0 -representation, and so also a locally analytic representation of
Z0 := M0 Z + . Together with the remarks following the proof of lemma 3.2.29,
this shows that V N0 lies in the category Repzla.c (Z + ) (as defined in the discussion
preceding the statement of proposition 3.2.28).
We now turn to verifying the hypotheses (4.2.4) and (4.2.5). Since M0 normalizes
N0 , we see that the space of Hausdorff N0 -coinvariants D(H◦n , H0 )N0 is naturally acompact type D(M◦n , M0 )-module, for each n ≥ 0.
Proposition 4.2.22. For any n ≥ 0, the continuous D(M◦n , M0 )-linear map(4.2.23)D(M◦n , M0 )

ˆ
⊗

D(M◦
n+1 ,M0 )D(H◦n+1 , H0 )N0 → D(H◦n , H0 )N0

54

MATTHEW EMERTON

induced by the natural map D(H◦n+1 , H0 )N0 → D(H◦n , H0 )N0 is a D(M◦n , M0 )compact map (in the sense of definition 2.3.3).
Proof. For each n ≥ 0 the Iwahori decomposition of H0 , and the rigid analytic
Iwahori decomposition of H◦n , give rise to a topological isomorphism◦∼ˆ D(M◦n , M0 ) ⊗ˆ D(N◦n , N0 )D(H◦n , H0 ) −→ DNn , N 0 ) ⊗KK(where each of the factors in the target is defined via (4.1.2)), and hence an isomorphism of D(M◦n , M0 )-modules(4.2.24)◦∼
ˆ D(M◦n , M0 ),D(H◦n , H0 )N0 −→ D(Nn , N 0 ) ⊗K

provided that the D(M◦n , M0 )-module structure on the target of this map is definedby having M0 act on the first factor via the action of M0 on N0 given by conjugation,and on the second factor by regarding D(M◦n , M0 ) as a left module over itself.
The untwisting lemma [8, lem. 3.6.4] yields (after passing to the appropriate limits, and then taking strong duals) a natural isomorphism of the completed tensor◦
ˆ K D(M◦n , M0 ), equipped with the D(M◦n , M0 )-module strucproduct D(Nn , N 0 ) ⊗ture that appears in (4.2.24), and the same completed tensor product, equippedwith the evident D(M◦n , M0 )-structure defined by the D(M◦n , M0 )-structure on thesecond factor. Composing this isomorphism with (4.2.24) yields an isomorphism ofD(M◦n , M0 )-modules(4.2.25)◦∼

ˆ D(M◦n , M0 ),D(H◦n , H0 )N0 −→ D(Nn , N 0 ) ⊗K

in which the D(M◦n , M0 )-module structure on the target is defined simply by theD(M◦n , M0 )-module structure on the second factor of the completed tensor product.
The naturality of this isomorphism (for the index n + 1 as well as for the index n)shows that we may embed (4.2.23) into the commutative diagram
D(M◦n , M0 )

ˆ (4.2.25)
id ⊗

ˆ
⊗

D(M◦
n+1 ,M0 )D(H◦n+1 , H0 )N0∼K(4.2.23)D(H◦n , H0 )N0◦ˆ D(M◦n , M0 )/ D(Nn+1 , N 0 ) ⊗◦ˆ D(M◦n , M0 ).
/ D(Nn , N 0 ) ⊗(4.2.25)∼

K

To see that (4.2.23) is D(M◦n , M0 )-compact, it now suffices (by lemma 2.3.4 (ii)and the remark following definition 2.3.3) to note that the map Nn+1 → Nn is◦relatively compact (by proposition 4.1.6 (i)), and thus that the map D(Nn+1 , N 0 ) →◦D(Nn , N 0 ) is a compact map of convex K-vector spaces. Corollary 4.2.26. For each n ≥ 0, the mapˆ D(M◦n , M0 )C an (ẐG,n , K)† ⊗Kˆ
⊗

Un+1 → Unˆ K D(M◦ ,M0 )C an (ẐG,n+1 ,K)† ⊗n+1

ˆ K D(M◦n , M0 )-compact map (in the sense of definiof (4.2.4) is a C an (ẐG,n , K)† ⊗tion 2.3.3).

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

55

ˆ K Dla (H0 , K)-module, andProof. By assumption Vb0 is a coadmissible C an (ẐG , K) ⊗so we may find a surjection(4.2.27)

ˆ D(H◦n+1 , H0 )rC an (ẐG,n+1 , K)† ⊗Kˆ D(H◦n+1 , H0 )→ C (ẐG,n+1 , K)† ⊗
an

K

ˆ
⊗
ˆ K D la (H0 ,K)
C an (ẐG ,K) ⊗Vb0 ,for some natural number r, and hence also a surjection
ˆ D(H◦n+1 , H0 )rN0 → Un+1 .
C an (ẐG,n+1 , K)† ⊗Kˆ K D(H◦n , H0 )Taking the completed tensor product of (4.2.27) with C an (ẐG,n , K)† ⊗an† ˆ◦over C (ẐG,n+1 , K) ⊗K D(Hn+1 , H0 ) yields a surjection
ˆ D(H◦n , H0 )rC an (ẐG,n , K)† ⊗Kˆ D(H◦n , H0 )→ C an (ẐG,n , K)† ⊗Kˆ⊗ˆ K D la (H0 ,K)
C an (ẐG ,K) ⊗Vb0 ,and hence also a surjection
ˆ D(H◦n , H0 )rN0 → Un .
C an (ẐG,n , K)† ⊗K

The morphism of (4.2.4) may thus be covered by a morphismˆ D(M◦n , M0 )C an (ẐG,n , K) ⊗
K

ˆ
⊗

D(M◦
n+1 ,M0 )D(H◦n+1 , H0 )rN0
ˆ D(H◦n , H0 )rN→ C an (ẐG,n , K) ⊗0K

ˆ K D(H(z)◦n , H0 )-modules, and so the corollary follows from (ii)of C an (ẐG,n , K)† ⊗and (iii) of lemma 2.3.4 together with proposition 4.2.22. 
Proposition 4.2.28. If n ≥ 0, and if z ∈ Z + is chosen so that z −1 N n z ⊂ N n+1 ,then the endomorphism of Un induced by z factors through the map of (4.2.4), insuch a way that the diagram of (4.2.5) commutes.
ˆ φz , where φz is the endomorphism
Proof. The action of z on Un is defined as id ⊗◦0
ˆof D(Hn , H0 ) ⊗Dla (H0 ,K) Vb N defined as the composite of the D(M◦n , M0 )-linear0ˆ (4.2.8)
maps (4.2.8) and (4.2.9). We will prove the proposition by showing that id ⊗factors through the map of (4.2.4). To this end we consider the diagram(4.2.29)
D(M◦n , M0 )

ˆ D(M◦n , M0 )C an (ẐG,n , K)† ⊗
K

ˆ
⊗

D(M◦
n+1 ,M0 )/ U (z)n

U (z)n,n+1

ˆ (4.2.8)
id ⊗


ˆ
⊗
C an (Ẑ◦† ˆG,n+1 ,K) ⊗K D(Mn+1 ,M0 )Un+1(4.2.4)


/ Un

56

MATTHEW EMERTON

induced by the diagram (4.1.19), where to ease typesetting we have writtenˆˆU (z)n,n+1 := C an (ẐG,n , K)†⊗
D(H(z)◦n,n+1 , H0 )⊗Vb0
D la (H0 ,K)

C an (ẐG ,K)

N0

and similarly
†

an



ˆ
⊗

U (z)n := C (ẐG,n , K)

C an (ẐG ,K)

D(H(z)◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0


.
N0

We will show below that the natural map(4.2.30)
D(M◦n , M0 )

ˆ
⊗

D(M◦
n+1 ,M0 )C an (ẐG,n , K)†ˆ⊗C an (ẐG ,K)−→ C an (ẐG,n , K)†

D(H(z)◦n,n+1 , H0 )ˆ⊗C an (ẐG ,K)

D(H(z)◦n , H0 )



ˆ
⊗

S

D la (H0 ,K)

N0



ˆ
⊗

S

D la (H0 ,K)N0ˆ K Dla (H0 , K)-module S. In
is an isomorphism for any coadmissible C an (ẐG , K) ⊗particular, the upper horizontal arrow of (4.2.29) is an isomorphism, and invertingˆ (4.2.8).
this isomorphism then yields the required factorization of id ⊗We turn to proving that (4.2.30) is an isomorphism. To ease notation, weˆ K Dla (H0 , K))denote by M (z)n,n+1 the topological (D(M◦n+1 , M0 ), C an (ẐG , K) ⊗an◦
ˆ K D(H(z)n,n+1 , H0 ), and denote by M (z)n the topologicalbimodule C (ẐG,n , K) ⊗an◦
ˆ K Dla (H0 , K))-bimodule C an (ẐG,n , K)⊗ˆ K D(H(z)◦n , H0 ).
(D(Mn , M0 ), C (ẐG , K)⊗anlaˆ K D (H0 , K) is a Fréchet-Stein algebra, we may find a conSince C (ẐG , K) ⊗
ˆ K Dla (H0 , K) → B for some Noetinuous K-algebra homomorphism C an (ẐG , K) ⊗
ˆ K Dla (H0 , K) on
therian Banach algebra B such that the action of C an (ẐG , K) ⊗each of the topological modules M (z)n,n+1 and M (z)n factors through a correˆ C an (Ẑ ,K) ⊗sponding continuous action of B, and such that S1 := B ⊗ˆ K D la (H0 ,K) S
G
is a finitely presented B-Banach module. It thus suffices to show that the natural
map




◦
ˆ
ˆ S1
ˆ S1
(4.2.31)
D(Mn , M0 )⊗M (z)n,n1 ⊗→ M (z)n ⊗D(M◦n+1 ,M0 )

B

N0

B

N0

is an isomorphism for any finitely presented B-Banach module S1 .
The construction of (4.2.31) is evidently functorial in S1 . If we apply this construction to the members of a finite presentation B r → B s → S1 → 0 of S1 , weobtain the diagram/ (M (z)rn )Nˆ D(M◦ ,M0 ) M (z)rn,n+1
D(M◦n , M0 ) ⊗
n+1
0
N
0



ˆ D(M◦ ,M0 ) M (z)sn,n+1
D(M◦n , M0 ) ⊗n+1N/ (M (z)sn )Nˆ D(M◦ ,M0 ) M (z)n,n+1 ⊗ˆ B S1D(M◦n , M0 ) ⊗
N
n+1



/ M (z)n ⊗ˆ B S1
N


0


0

0

0

0

0

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

57(the horizontal maps being induced by (4.2.31)). The columns of this diagram
may not be exact as sequences of maps of abstract K-vector spaces (since we haveformed completed tensor products, and then passed to Hausdorff N0 -coinvariants),
but they are exact in the category of Hausdorff locally convex K-spaces (i.e. thethird member of each column is isomorphic to the quotient of the second memberof that column by the closure of the image of the first member). Thus to showthat (4.2.31) is an isomorphism, it suffices to show that the natural map
D(M◦n , M0 )

ˆ
⊗

D(M◦
n+1 ,M0 )(M (z)n,n+1 )N0 → (M (z)n )N0

is an isomorphism. For this, it suffices in turn to note that the natural map
D(M◦n , M0 )

ˆ
⊗

D(M◦
n+1 ,M0 )D(H(z)◦n,n+1 , H0 )N0 → D(H(z)◦n , H0 )N0is an isomorphism, as follows from a consideration of the rigid analytic Iwahori
decompositions (4.1.14) and (4.1.18) of each of the good analytic open subgroups
H(z)n,n+1 and H(z)n . 
We may now prove the main theorem of this subsection.
Theorem 4.2.32. If V is an object of Repes (G), then the Jacquet module JP (V ) isan object of Repes (M ). Thus JP induces an additive, left exact functor Repes (G) →Repes (M ).
Proof. Proposition 4.1.6 (vi) yields z ∈ Z + such that z −1 N n z ⊂ N n+1 for eachn ≥ 0. Corollary 4.2.26 and proposition 4.2.28 establish (4.2.4) and (4.2.5), and so
proposition 3.2.24 implies that JP (V )0b is a coadmissible module over the nuclear
ˆ K C an (ẐG , K) ⊗
ˆ K Dla (M0 , K). Since the two actions
Fréchet algebra C an (ẐM , K) ⊗an
0of C (ẐG , K) on JP (V )b (one through the embedding C an (ẐG , K) → C an (ẐM , K)
ˆ K Dla (M0 , K)) coincide, theand the other through the first factor of C an (ẐG , K) ⊗
ˆ K C an (ẐG , K) ⊗
ˆ K Dla (M0 , K) on JP (V )0b facaction of the algebra C an (ẐM , K) ⊗anlaˆ
tors through its quotient C (ẐM , K) ⊗K D (M0 , K) (in which the two copies
of C an (ẐG , K) described above are identified). Thus JP (V )0b is a coadmissible
ˆ K Dla (M0 , K)-module, and so JP restricts to a functor Repes (G) →C an (ẐM , K) ⊗Repes (M ), as required. That this functor is additive and left exact follows fromlemma 3.4.7. 
We now suppose that G is quasi-split, and that P is a Borel subgroup of G, andturn to proving (0.11). The Levi factor M is then a torus, and so M = ZM .
Proposition 4.2.33. If V is an object of Repad (G), then the map Exp(JP (V )) →m̌ has discrete fibres.
S
Proof. Write M̂ = M̂n as the a union of affinoid subdomains. We have to showthat the intersection of Exp(JP (V )) with any of the affinoid subdomains M̂n has afinite fibre over any point of m̌. For this, it suffices to show that for any character χ ∈ˆ C an (M̂ ,K) JP (V )0b )m=χ is finite dimensional form̌, the tensor product (C an (M̂n , K)† ⊗each n, or equivalently, for each sufficiently large value of n. (Here the subscript
indicates the maximal Hausdorff quotient where m acts through χ.) The proof will
be a variant of that used to prove the preceding theorem.

58

MATTHEW EMERTON

We begin by normalizing our various choices of indices. For each n ≥ 0, the composite Dla (M0 , K) → C an (M̂ , K) → C an (M̂n , K)† factors through the natural map
Dla (M0 , K) → D(M◦m , M0 ), for some sufficiently large value of m. Replacing the sequence Hn of analytic open subgroups of G by a cofinal subsequence, and relabelling
if necessary, we may assume that in fact the map Dla (M0 , K) → C an (M̂n , K)† factors through D(M◦n , M0 ). Thus for each n ≥ 0, the continuous homomorphism
Dla (M0 , K) → C an (M̂ , K) determines a continuous homomorphism
D(M◦n , M0 ) → C an (M̂n , K)† .

(4.2.34)

Since V is admissible as a locally analytic G-representation (rather than justessentially admissible) we may strengthen the isomorphism (4.2.6) as follows:∼(V N0 )0b −→ lim(D(H◦n , H0 )
←−
n

ˆ
⊗
D la (H0 ,K)

Vb0 )N0 .

ˆ Dla (H0 ,K) Vb0 )N0 , for n ≥ 0;
Take An to be D(M◦n , M0 ) and Un to be (D(H◦n , H0 ) ⊗variants of corollary 4.2.26 and proposition 4.2.28 then show that the hypothesesof proposition 3.2.24 are satisfied. The proof of that proposition thus yields an
isomorphism∼∼ˆ (V N0 )0b −→JP (V )0b = ((V N0 )fs )0b −→ C an (M̂ , K) ⊗K[Z + ]

ˆ lim (D(H◦n , H0 )C an (M̂ , K) ⊗←−K[Z + ]nˆ⊗
D la (H0 ,K)∼

Vb0 )N0 −→ˆ (D(H◦n , H0 )C an (M̂n , K)† ⊗lim←−
K[Z + ]nˆ⊗
D la (H0 ,K)

Vb0 )N0 ,ˆ K[Z + ] (D(H◦n , H0 ) ⊗ˆ Dla (H0 ,K) Vb0 )N0 is finitely genwhere the space C an (M̂n , K)† ⊗an† ˆ◦erated as a C (M̂n , K) ⊗K D(Mn , M0 )-module, for each n ≥ 0. The continuous
homomorphism (4.2.34) induces a continuous surjection
ˆ D(M◦n , M0 ) → C an (M̂n , K)† ,C an (M̂n , K)† ⊗Kˆ K D(M◦n , M0 )-action on
and by construction the C an (M̂n , K)† ⊗ˆ (D(H◦n , H0 )C an (M̂n , K)† ⊗K[Z + ]

ˆ⊗D la (H0 ,K)

Vb0 )N0

factors through this surjection. Thus we see that for each n ≥ 0, the spaceˆ (D(H◦n , H0 )C an (M̂n , K)† ⊗K[Z + ]

ˆ⊗D la (H0 ,K)

Vb0 )N0

is finitely generated as a C an (M̂n , K)† -module. (In short, we have just repeated theargument of theorem 4.2.32, proving that JP (V ) lies in Repes (M ), in the particularsituation of the current proposition.)
We deduce from [8, thm. 1.2.11 (i)] that there is a topological isomorphism
C an (M̂n , K)†ˆ⊗C an (M̂ ,K)∼ˆ (D(H◦n , H0 )JP (V )0b −→ C an (M̂n , K)† ⊗K[Z + ]

ˆ⊗D la (H0 ,K)

Vb0 )N0 ,JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

59

and hence a topological isomorphism(C an (M̂n , K)†ˆ⊗C an (M̂ ,K)

JP (V )0b )m=χ∼
ˆ (D(H◦n , H0 )−→ (C an (M̂n , K)† ⊗K[Z + ]

ˆ⊗D la (H0 ,K)

Vb0 )N0 )m=χ ,for each n ≥ 0.
If n is chosen large enough, then χ integrates to a rigid analytic character χ̃ ofM◦n , and the natural map (C an (M̂n , K)† )m=χ → (C an (M̂n , K)† )Mn◦ =χ̃ is an isomorphism (as follows from the fact that the natural map Dla (M0 , K) → C an (M̂ , K)
factors through the map (4.2.34)). Thus there is an isomorphism(C an (M̂n , K)†∼ˆ⊗C an (M̂ ,K)

JP (V )0b )m=χ −→ (C an (M̂n , K)†ˆ⊗C an (M̂ ,K)

JP (V )0b )Mn◦ =χ̃ ,and hence an isomorphism(4.2.35)(C an (M̂n , K)†∼ˆ⊗C an (M̂ ,K)

JP (V )0b )m=χ −→ˆ (D(H◦n , H0 )(C (M̂n , K)† ⊗an

K[Z + ]

ˆ (D(M◦n , M )M ◦ =χ̃C an (M̂n , K)† ⊗n
K[Z + ]∼

ˆ⊗D la (H0 ,K)

ˆ
⊗
D(M◦
n ,M )

Vb0 )N0 )Mn◦ =χ̃ −→(D(H◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0 )N0 ).

If z ∈ Z + is such that φz induces a D(M◦n , M0 )-compact endomorphism ofˆ Dla (H0 ,K) Vb0 )N0 , then by lemma 2.3.4 (ii), we see that φz induces(D(H◦n , H0 ) ⊗◦a D(Mn , M )Mn◦ =χ̃ -compact endomorphism ofD(M◦n , M )Mn◦ =χ̃

ˆ
⊗
D(M◦
n ,M )(D(H◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0 )N0 .

Since Mn◦ has finite index in M0 , we see that D(M◦n , M )Mn◦ =χ̃ is in fact a finite
dimensional K-algebra, and thus that φz induces a compact endomorphism ofˆ D(M◦ ,M ) (D(H◦n , H0 ) ⊗ˆ Dla (H0 ,K) Vb0 )N0 . Proposition 2.3.6 implies
D(M◦n , M )Mn◦ =χ̃ ⊗nthat
ˆ (D(M◦n , M )M ◦ =χ̃C an (M̂n , K)† ⊗n
K[Z + ]ˆ⊗D(M◦n ,M )(D(H◦n , H0 )

ˆ⊗D la (H0 ,K)

Vb0 )N0 )is finite dimensional over K. Taking into account (4.2.35), this completes the proofof the proposition. We close this subsection by strengthening the preceding proposition, under suitable hypotheses. We continue to assume that P is a Borel subgroup of G.
Proposition 4.2.36. Let V be an object of Repad (G). If there is a compact open∼
subgroup H of G such that V −→ C la (H, K)r as an H representation, for somenatural number r (so that in particular V is a strongly admissible locally analytic
representation of G), then Exp(JP (V )) is equidimensional of dimension d, where dis the dimension of the maximal torus M of G.

60

MATTHEW EMERTON

Proof. The key point is that our assumption on V will allow us to reduce to asituation in which we may apply the relative theory of compact operators developedin [6], for compact operators on orthonormalizable Banach modules. A careful
explanation of how we make this reduction is somewhat involved, and accounts forthe length of the following argument.
If H 0 is an open subgroup of H, then there is an H 0 -equivariant isomorphism
0∼laC (H, K)r −→ C la (H 0 , K)r[H:H ] , and so in the statement of the proposition, we
may replace H by any such open subgroup H 0 . In particular, after relabelling themembers of the sequence {Hn } if necessary, we may assume that the open subgroup
H appearing in the statement of the proposition is equal to H0 .
We choose z ∈ Z + such that z −1 Nn z ⊂ Nn+1 for each n ≥ 0, and such that−1z
and Z + together generate the group ZM . (A consideration of the proof of
proposition 4.1.6 (vi) shows that we may do this.) Let Y + (respectively Y ) denotethe submonoid (respectively subgroup) of ZM generated by z. The discussion preceding the statement of proposition 4.2.22 shows that V N0 lies in Repzla.c (Z + ), andso by proposition 3.2.28 there is a natural isomorphism∼

JP (V ) −→ Lb,Y + (C an (Ŷ , K), V N0 ).
Dualizing, we obtain an isomorphism∼ˆ (V N0 )0b ,JP (V )0b −→ K {{z, z −1 }} ⊗(4.2.37)

K[z]where K {{z, z −1 }} denotes the ring of entire functions on Ŷ (which is isomorphic
to Gm , thought of as a rigid analytic space).
ˆ Dla (H0 ,K) Vb0
As in the proof of proposition 4.2.33, we set Un := D(H◦n , H0 ) ⊗,N0ˆ Dla (H0 ,K) Vb0
for
each
value
for each n ≥ 0. Also, set U (z)n = D(H(z)◦n , H0 ) ⊗N0of n. There is an isomorphism∼(V N0 )0b −→ lim
Un .
←−(4.2.38)

n

∼

la

r

By assumption, V −→ C (H0 , K) . The untwisting isomorphism (4.2.25) thus
yields isomorphisms of D(M◦n , M0 )-modules◦∼
ˆ D(M◦n , M0 )Un −→ D(Nn , N 0 )r ⊗(4.2.39)

K

and
◦

∼

ˆ D(M◦n , M0 ),U (z)n −→ D(z −1 Nn z, N 0 )r ⊗(4.2.40)

K

where D(M◦n , M0 ) acts on the targets of these isomorphisms via its action on theright hand factor in the completed tensor product. We may fit these isomorphisms
into the following commutative diagram:(4.2.39)

Un∼/ D(N◦ , N )r ⊗
ˆ K D(M◦n , M0 )0nαn(4.2.9)φz
(4.2.40)
∼ /
−1 ◦
r ˆ
U (z)nD(z Nn z, N 0 ) ⊗K D(M◦n , M0 )(4.2.8)( 
Un(4.2.39)∼βn/ D(N◦ , N )r ⊗
ˆ K D(M◦n , M0 ),0nJACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

61

where the vertical arrow labelled βn is obtained by taking the completed tensor
product of the natural map
D(z −1 N◦n z, N 0 )r → D(N◦n , N 0 )r

(4.2.41)(induced by the embedding z −1 N◦n z → N◦n ) with D(M◦n , M0 ) over K. By assumption, the embedding z −1 N◦n z → N◦n factors through the embedding N◦n+1 → N◦n ,and so is relatively compact (by proposition 4.1.6 (i)). Hence (4.2.41) is a compact
map, and so can be factored as
ın,1

ın,2

D(z −1 N◦n z, N 0 )r −→ Wn −→ D(N◦n , N 0 )r ,where Wn is a Banach space. Consequently, the map βn factors asβn,1ˆ D(M◦n , M0 ) −→
ˆ D(M◦n , M0 )Wn ⊗D(z −1 N◦n z, N 0 )r ⊗

K

K

βn,2

ˆ D(M◦n , M0 ),−→ D(N◦n , N 0 )r ⊗K

ˆ id. The composite βn ◦ αn defines a D(M◦n , M0 )-linear action
where βi,n := ıi,n ⊗◦ˆ K D(M◦n , M0 ), compatible with the isomorof z (and thus Y + ) on D(Nn , N 0 )r ⊗phism (4.2.39) and the action of z (and thus Y + ) on Un defined by φz . The
composite βn,1 ◦ α ◦ βn,2 defines a D(M◦n , M0 )-linear action of z (and thus Y + )ˆ K D(M◦n , M0 ). By construction, the maps βn,1 ◦ αn and βn,2 are Y + on Wn ⊗equivariant and D(M◦n , M0 )-linear.
The isomorphisms (4.2.37), (4.2.38), and (4.2.39), [8, prop. 1.1.29], and proposition 2.1.9, together yield isomorphisms(4.2.42)∼

ˆ Un
JP (V )0b −→ lim
K {{z, z −1 }} ⊗
←−
K[z]

n

∼

−→ lim
K {{z, z←−−1◦

ˆ D(Nn , N 0 )r ⊗
ˆ D(Mn◦ , M0 )
}} ⊗
K

K[z]
∼

n

ˆ Wn ⊗
ˆ D(Mn◦ , M0 )K {{z, z −1 }} ⊗−→ lim
←−
K[z]

n

K

(the third isomorphism being induced by the maps βn,1 ◦ αn ).
∼The isomorphism Dla (M0 , K) −→ C an (M̂0 , K) of [8, prop. 6.4.6] shows thatla
D (M0 , K) is equal to the Fréchet algebra of rigid analytic functions on the rigid
analytic space M̂0 . Writing M̂0 as the union of an increasing sequence of ad∼
missible open affinoid subdomains, we may write Dla (M0 , K) −→ lim
An , where←−n{An } is a projective sequence of Tate algebras over K. We also have the isomor∼
phism Dla (M0 , K) −→ limD(M◦n , M0 ). The two projective sequences {An } and←−n{D(M◦n , M0 )} are cofinal [8, prop. 1.2.7]. Passing to subsequences, and relabelling if
necessary, we may assume that the map Dla (M0 , K) → An factors through the map
Dla (M0 , K) → D(M◦n , M0 ), for each n. Tensoring the nth term of the projective
sequence with An over D(M◦n , M0 ), the isomorphism (4.2.42) yields an isomorphism(4.2.43)∼
ˆ Wn ⊗
ˆ An .
JP (V )0b −→ lim
K {{z, z −1 }} ⊗
←−
n

K[z]

K

62

MATTHEW EMERTON

Since ın,2 is compact (being a map from a Banach space to the compact type spaceˆ K An through an An -compact map.
D(M◦n , M0 )), we find that z acts on Wn ⊗In order to show that the support of JP (V )0b is equidimensional of dimensiond, it follows from (4.2.43) that it suffices to show that this is true for each of theAn -modules
ˆ Wn ⊗
ˆ An .
K {{z, z −1 }} ⊗K[z]K

ˆ K An is an orthonormalizable An -Banach module on which z acts via
Since Wn ⊗an An -compact map, this follows from the theory of [6, §A] and [7, §1]. (4.3) In this subsection we establish property (0.13). We maintain the notationof the preceding subsections.
Let V be an admissible smooth representation of G defined over K. If weequip V with its finest convex topology, then it becomes an object of Repad (G)[8, prop. 6.3.2]. Our first goal is to show that JP (V ) is isomorphic as an M representation to the N -coinvariants VN of V . In order to compute JP (V ), we fix alift of the Levi quotient M to a subgroup of P , as well as a compact openTsubgroup
P0 of P . Let P denote the opposite parabolic to P (chosen so that P P = M )and let N denote the unipotent radical of P .
+Lemma 4.3.1. The ZM-representation V N0 is the locally convex inductive limit+of an increasing sequence of finite dimensional ZM-invariant subspaces.

Proof. Let v ∈ V N0 , and choose an open subgroup H1 of G that leaves v invariant.
Shrinking H1 if necessary, we may assume that H1 admits an Iwahori decomposition+
H1 = N 1 M1 N1 with N1 ⊂ N0 . Set Y := {y ∈ ZM
| yN1 y −1 ⊂ N1 , y −1 N 1 y ⊂ N 1 }.
Since V is an admissible smooth G-representation, the space V H1 is finite dimensional. Furthermore, the Hecke operators πN1 ,y for y ∈ Y leave V H1 invariant. (If we let πH1 denote averaging over H1 , then a consideration of the
Iwahori decomposition of H1 shows that πH1 (yv1 ) = πN1 (yv1 ) =: πN1 ,y (v1 ) for
any y ∈ Y and v1 ∈ V H1 .) Hence the span of the set {πN1 ,y (v) | y ∈ Y } is
finite dimensional. A variant of proposition 3.3.2 (i) shows that Y generates+
ZM as a group, and so we may find a finite subset {z1 , . . . , zn } of ZM
so thatSn+
ZM= i=1 zi Y. Since πN0 ,y = πN0 πN1 ,y for y ∈ Y , we conclude that the span of+
the set {πN0 ,z (v) | z ∈ ZM
} is also finite dimensional, and hence that V N0 is the+union of finite dimensional ZM-invariant subspaces. Since V , and hence V N0 , isequipped with its finest convex topology, we see that V N0 is even isomorphic to the
locally convex inductive limit of these invariant subspaces. LProposition 4.3.2. (i) The natural map (V N0 )null JP (V ) → V N0 is an isomorphism.
(ii) Each of (V N0 )null and JP (V ) is an M + -invariant subspace of V N0 .
Proof. Part (i) follows from lemmas 4.3.1 and 3.2.18 and propositions 3.2.6 (iv)and 3.2.15 (ii) (once we recall that, by definition, JP (V ) = (V N0 )fs ). Part (ii)follows by functoriality of the formation of (V N0 )null and JP (V ). 
We now recall some ideas from [5, §3]. Since V is a smooth representation ofG, it is in particular smooth as a representation of N , and consequently, for any
compact open subgroup N00 of N the map πN00 is defined on all of V . As in [5,§3] we let V (N ) denote the subspace of V consisting of those vectors v for which

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

63

πN00 v = 0 for some compact open subgroup N00 of N ; it is easily checked that V (N )is equal to the kernel of the natural map(4.3.3)

V → VN[5, prop. 3.2.1]. (This description of the kernel shows that the functor V 7→ VN ,which is right exact by construction, is in fact an exact functor on smooth N representations [5, prop. 3.2.3].)
By proposition 3.3.2, any compact open subgroup of N is contained in z −1 N0 z+
for some z ∈ ZM
. The equation πN0 ,z = zπz−1 N0 z thus shows that V (N ) may also
be described as the subspace of V consisting of those vectorsthat are annihilatedT+by πN0 ,z for some element z ∈ ZM
. In particular, V (N ) V N0 = (V N0 )null .
Proposition 4.3.4. (i) The restriction of the morphism (4.3.3) to V N0 induces
an M + -equivariant surjective map V N0 → VN (the M + -action being defined on
the source via the operators πN0 ,m , and on the target via restricting the natural
M -action).
∼(ii) The map of (i) restricts to an M -equivariant isomorphism JP (V ) −→ VN .
Proof. For any v ∈ V , the elements v and πN0 v have the same image in VN . This
proves (i). The discussion preceding the lemma shows that (V N0 )null is the kernelof the map of (i). Thus we obtain the isomorphism of (ii). Since it is an M + equivariant map between M -modules, it is automatically M -equivariant. 
Proposition 4.3.5. For any admissible smooth representation V of G, the iso∼morphism JP (V ) −→ VN of proposition 4.3.4 (ii) is independent of the choices ofLevi factor M and compact open subgroup P0 of P that are used to compute JP (V ).
Proof. If we compute JP (V ) with respect to a different choice of M (respectivelyof P0 ), then the natural isomorphism of proposition 3.4.10 (respectively proposition 3.4.11) involves conjugation by an element of N (respectively the applicationof the projection operator πN0 ). This operation reduces to the identity of VN , andso the present proposition follows. 
We now generalize Proposition 4.3.4 so as to give a description of the Jacquet
module of certain locally algebraic G-representations.
Proposition 4.3.6. If W is a finite dimensional algebraic representation of G,
with B = EndG (W ), and if X is an admissible smooth representation of G over B,∼
then there is a natural isomorphism JP (X ⊗B W ) −→ XN ⊗B W N .
Proof. Since X is a smooth representation of G, and hence of N , the inclusion
X n ⊂ X is in fact an equality. Since W is a locally algebraic representation of G,
and hence of N, the inclusions W N ⊂ W N0 ⊂ W n are both equalities. Combining
these two observations, we obtain the equality X N0 ⊗B W N = (X ⊗B W )N0 .
If m ∈ M + , then the operator πN0 ,m on (X ⊗B W )N0 can be described in terms ofX
X
this tensor product decomposition as πN0 ,m : u⊗w 7→ πN(u)⊗mw (where πN
0 ,m
0 ,m
denotes the corresponding operator on X N0 ). Thus as an M + -representation, thespace (X ⊗B W )N0 is isomorphic to the tensor product of the M + -representation
on X N0 and the space W N , equipped with the M + -action obtained by restricting
the natural M -action on this space.

64

MATTHEW EMERTON∼

Proposition 3.2.9 thus yields an M -equivariant isomorphism ((X ⊗B W )N0 )fs −→(X N0 )fs ⊗B W N . The proposition now follows from the definition of the functor
JP , together with proposition 4.3.4 (ii). 
Recall that a locally algebraic representation V of G is said to be an admissible
locally algebraic G-representation if it becomes an admissible locally analytic Grepresentation when endowed with its finest convex topology [8, def. 6.3.9]. We
will always regard an admissible locally algebraic representation as being an objectof Repad (G), by endowing it with its finest convex topology. By [8, prop. 6.3.11],∼any
L admissible locally algebraic G-representation admits an isomorphism V −→n Xn ⊗Bn Wn , where Wn runs over a sequence of isomorphism class representatives
for the irreducible algebraic representations of G, Bn = EndG (Wn ), and Xn is anadmissible smooth representation of G over Bn . Proposition L
4.3.6 together with∼lemma 3.4.7 (i) and (iv) thus yields an isomorphism JP (V ) −→ n (Xn )N ⊗Bn WnN .
(4.4) In this subsection we will establish property (0.14). As above we fix acompact open subgroup P0 of P , as well as a Levi factor M of P , and we define N0 ,+M0 , M + and ZM
as they were defined preceding corollary 3.3.3. If V is a locallyanalytic representation of G, we define the Hecke operators πN0 ,m on V N0 as indefinition 3.4.2. We also assume throughout this section that L = Qp . (This is not
such a serious loss of generality, since restriction of scalars allows us to regard a
reductive group over any finite extension of Qp as a reductive group over Qp .)Let S denote the maximal subtorus of ZM that splits over Qp , and as in subsection 1.4, let Y • (respectively Y• ) denote the character lattice (respectively thecocharacter lattice) of S. If χ is a character of ZM , then the discussion of that
subsection allows us to define the element slope(χ) of Q ⊗Z Y • .
Let ∆(G, S) denote the set of positive restricted roots of S (i.e. the characters of
S appearing in the adjoint action of S on the Lie algebra of N), and let ∆(G, S)sdenote the subset of positive simple restricted roots. Let R• denote the sublatticeof Y • spanned by ∆(G, S)s , and let (Q ⊗Z R• )≥0 denote the Q≥0 -invariant cone in
Q ⊗Z R• generated by the elements of ∆(G, S)s .
Lemma 4.4.1. If χ is an element of ẐM , then χ satisfies the inequality |χ(a)| ≤ 1+
for every a ∈ ZM
if and only if slope(χ) ∈ (Q ⊗Z R• )≥0 .
Proof. Taking T = ZM in the diagram (1.4.1) yields a surjection ord : ZM → Y•0 ,where Y•0 is a sublattice of Q ⊗Z Y• containing Y• with finite index. Consider thesubmonoid(Y•0 )+ = {y ∈ Y•0 | hy, ri ≥ 0 for all r ∈ (Q ⊗Z R• )≥0 }+of Y•0 . The image of ZM
under the map ord is contained in (Y•0 )+ , and is cofinal in this submonoid (when the latter is directed by the relation of divisibility; compare the proof of lemma 3.3.1). This cofinality, together with the equa+tion ord(χ(a)) = hord(a), slope(χ)i, shows that ord(χ(a)) ≥ 0 for all a ∈ ZM
0 +if and only if hy, slope(χ)i ≥ 0 for all y ∈ (Y• ) , which holds if and only ifslope(χ) ∈ (Q ⊗Z R• )≥0 , proving the lemma. As in the preceding subsections, we let δ denote the modulus character of P ,regarded as a smooth character of M . If we write ρ := ρ(G, S) to denote one-halfthe sum of the elements of ∆(G, S), each counted with the multiplicity with respect
to which it appears in the adjoint action of S on N, then slope(δ|S ) = −ρ. (This iswhere the assumption that L = Qp is used.)JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

65

Lemma 4.4.2. Let V be an object of Repla.c (P ), and suppose that V admits a+
P -invariant norm. If χ ∈ ẐM is such that V N0 ,ZM =χ is non-zero, then for every+a ∈ ZM
we have |δ(a)−1 χ(a)| ≤ 1. Equivalently, ρ + slope(χ) lies in (Q ⊗Z R• )≥0 .
Proof. Let || – || denote the P -invariant norm on V , whose existence we are as++
N0 ,ZM=χsuming, and let v be a non-zero.P
If a ∈ ZM
then χ(a)v =P vector in V
−1 −1
πN0 ,a v = [N0 : aN0 a ]
nav
=
δ(a)
n∈N0 /aN0 a−1n∈N0 /aN0 a−1 nav, henceP−1|δ(a) χ(a)| ||v|| = || n∈N0 /aN0 a−1 nav || ≤ ||v||, and so |δ(a)−1 χ(a)| ≤ 1. The
equivalent formulation in terms of slopes follows from lemma 4.4.1. For the remainder of the subsection, assume that G is split over K, and fix a finite
dimensional irreducible algebraic representation W of G over K. As was noted insubsection 1.2, highest weight theory shows that the space W N of N-invariants of W
is an irreducible algebraic representation of M, and so ZM must act on W N througha character ψ ∈ X • (ZM ), which we also regard as an element of ẐM (K). We also
fix a smooth (i.e. locally constant) character θ ∈ ẐM (K), and write χ = ψθ.
Let us for the moment choose a Borel subgroup B of G, defined over K, and let
T be a maximal torus ofTthis Borel subgroup (again, defined over K), contained in
M. The intersection M B is then a Borel subgroup of M. Let n0 denote the Liealgebra of the unipotent radical of B, and let n0 denote the Lie algebra of unipotent
radical of the opposite Borel to B (with respect to the chosen torus T).
Since G is split over K, the highest weight theory recalled in subsection 1.1
applies to W . Let ψ̃ ∈ X • (T) denote the highest weight of the representation W
with respect T
to B; this is also the highest weight of the M-representation W N withrespect to M B. Note that ψ̃|ZM = ψ. If α is a simple root of ZM acting on n, thendenote by α̃ the (unique, by lemma 1.2.3) simple root of of T acting on n0 that lifts
α. If ρ̃ := ρ(G, T) denotes one-half the sum of the positive roots of T acting on n0 ,then ρ̃|S = ρ.
Definition 4.4.3. We say that χ = ψθ is of critical slope with respect to therepresentation W N if for some positive simple root α ∈ ∆(G, ZM ), the elementsα̃ (ψ̃ + ρ̃)|S + slope(θ) + ρ of Q ⊗Z Y • in fact lies in (Q ⊗Z R• )≥0 . Otherwise, we saythat χ is of non-critical slope with respect to W N . (By lemma 1.2.6 this definition
is independent of the choice of Borel subgroup B.)
Let V be an object of Repla.c (P ) admitting a P -invariant norm. If the eigenspace+0of V N ,ZM =χ is non-zero, then lemma 4.4.2 shows that ψ|S + ρ + slope(θ) + ρ liesin (Q ⊗Z R• )≥0 . The element sα̃ (ψ̃ + ρ̃)|S + slope(θ) + ρ thus also lies in Q ⊗Z R• .
The key point of definition 4.4.3 in this case is thus whether or not this element
actually lies in the positive cone of Q ⊗Z R• .
If G is quasi-split over Qp , and if P is chosen to be a Borel subgroup of G definedover Qp , then ZM = M = T (using the above notation), and so ψ̃ = ψ. In this casethe character χ thus determines the representation W N (indeed, the latter is justequal to the character ψ), and so we may say that the character χ is of critical or
non-critical slope, without making explicit reference to the representation W N .
+If V is an object of Repla.c (G), then V N0 ,ZM =χ is a locally analytic represen+tation of M0 . Thus we may consider its subspace (V N0 ,ZM =χ )W N −lalg of locally
W N -algebraic vectors (as defined in [8, prop.-def. 4.2.2]; note that if we hadn’t chosen χ so that it coincided with ψ locally, then this space would necessarily vanish).

66

MATTHEW EMERTON+

We may also consider its subspace (V N0 ,ZM =χ )m∩b=dψ̃ , the subspace of vectors onTwhich the Lie algebra m ∩ b of B M acts through the character dψ̃ of its quotientt (the Lie algebra of T).
The proof of the following proposition is somewhat analogous to a computation
appearing in the proof of [17, prop. 6.2] (for the special case G = GL2 ).
Proposition 4.4.4. Let α̃ be a positive simple root of T acting on n0 , and let
X−α̃ ∈ n0 be an element of the root space n−α̃ corresponding to the negative root−α̃. Let α̃ˇdenote the corresponding coroot, and write m = hψ̃, α̃ˇi. If V is a locallym+1
analytic representation of G then the action of X−
on V N0 restricts to a map
α̃++
N0 ,ZM=χ m∩b=dψ̃
N0 ,ZM=α̃−m−1 χ
(V
)
→V
.
+

Proof. Let v be an element of (V N0 ,ZM =χ )m∩b=dψ̃ . The action of g on v determinesa U (g)-module morphism Ver(ψ̃) → V, given by mapping the highest weight vectorv(ψ̃) of Ver(ψ̃) to v. (Here Ver(ψ̃) denotes the Verma module of highest weight ψ̃,m+1
as in definition 1.1.5.) Proposition 1.1.6 shows that the element X−
α̃ (1 ⊗ v(ψ̃))
of Ver(ψ̃) is fixed by the adjoint action of N . Thus if n ∈ N0 we see thatm+1
m+1 −1m+1nX−v = φ(Adn X−α̃ v = nX−α̃ nα̃ (1 ⊗ v(ψ̃)))m+1
m+1= φ(X−α̃ (1 ⊗ v(ψ̃))) = X−α̃ v,+
m+1
N0. Furthermore, for any a ∈ ZM
we compute thatand so X−α̃ v lies in V

X

m+1
−1 −1
πN0 ,a X−]
α̃ v = [N0 : aN0 an∈N0 /aN0−1 −1

X= [N0 : aN0 a−1 ]−1X= [N0 : aN0 a

]

m+1
naX−
α̃ v
a−1

m+1
Adn Ada X−α̃ navm+1
α̃(a)−m−1 X−α̃ navX
m+1−1 −1= α̃(a)−m−1 X−]navα̃ [N0 : aN0 am+1= α̃(a)−m−1 X−
α̃ πN0 ,a v
m+1
= α̃(a)−m−1 χ(a)X−α̃ v.

(The third equality follows from the fact that X−α̃ is an eigenvector for the actionof Ada , with eigenvalue α̃−1 , together with the fact, already used above, thatm+1X−α̃ v is fixed by the adjoint action of N , while the final equality follows from the+assumption that v lies in V N0 ,ZM =χ .) This proves the proposition. We can now prove the main result of this subsection.
Theorem 4.4.5. Let V be an object of Repla.c (G), and suppose that V admits a+
G-invariant norm. If χ is of non-critical slope, then the space (V N0 ,ZM =χ )W N −lalgconsists of locally W -algebraic vectors of V .
+Proof. Let v be an element of (V N0 ,ZM =χ )W N −lalg . If U denotes the m-submoduleof V N0 generated by v, then by assumption U is isomorphic to a finite direct sumof copies of W N . Thus if v is not locally algebraic, then we may find a highest
weight vector in the m-representation U which is also not locally algebraic. Such+a vector lies in (V N0 ,ZM =χ )m∩b=ψ̃ . Replacing v by such a vector, if necessary, weJACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS

67+

may assume that v is a non-locally algebraic element of (V N0 ,ZM =χ )m∩b=ψ̃ , which
is nevertheless W N -locally algebraic under the action of M .
As in the proof of proposition 4.4.4, the vector v determines a U (g)-module
morphism Ver(ψ̃) → V, given by mapping the highest weight vector v(ψ̃) of Ver(ψ̃)
to v. Since v is not locally algebraic, corollary 1.1.4 shows that this map does
not factor through the finite dimensional g-representation W . It thus follows fromproposition 1.1.6 (ii) that we may find a positive simple root α̃ of T acting on n0 ,m+1
and an element X−α̃ ∈ n−α̃ , such that X−α̃ v 6= 0. (Here the notation is thatof proposition 4.4.4. In particular, m = hψ̃, α̃ˇi, and so m + 1 = hψ̃ + ρ̃, α̃ˇi.)
Since v is W N -locally algebraic, the simple root α̃ must appear in n, and thus
restricts to an element α ∈ ∆(G, ZM ). By proposition 4.4.4 we thus obtain a non+−m−1
χzero element of V N0 ,ZM =α̃
. Applying lemma 4.4.2 to α̃−m−1 χ, we find thatρ − (m + 1)α̃|S + slope(χ) lies in (Q ⊗Z R• )+ .
We compute thatρ − (m + 1)α̃|S + slope(χ) = ρ − hψ̃ + ρ̃, α̃ˇiα̃|S + ψ|S + slope(θ)= ψ|S + ρ − hψ̃ + ρ̃, α̃ˇiα̃|S + slope(θ) + ρ = sα̃ (ψ̃ + ρ̃)|S + slope(θ) + ρ(the last equality following from the definition of sα̃ ). Thus we see that χ is of
N ,Z + =χ

M
critical slope, contradicting the hypothesis of the theorem. Hence VW 0N −lalgconsist entirely of locally algebraic elements. 

mustProperty (0.14) follows from preceding theorem, together with proposition 3.4.9.
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Northwestern University
Department of Mathematics
2033 Sheridan Rd.
Evanston, IL 60208-2730, USA
E-mail address: emerton@math.northwestern.edu