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 be a 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 J_{P} from the category of essentially admissible locally analytic G-representations to the category of essentially admissible 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 7

2. Functional-analytic preliminaries 12

3. Construction of the Jacquet module functor 21

4. Some properties of the Jacquet module functor 40 References 67

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 that are 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 locally analytic representation introduced in [20].

We now introduce some notation that will be in force throughout this introduction. Let L be a finite extension of ℚ_{p} 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 the Levi quotient of ℙ). Choose an opposite parabolic \overline{ℙ} to ℙ; the intersection ℙ ⋂ \overline{ℙ} then provides a lifting of 𝕄 to a Levi factor of ℙ (and indeed making a choice of \overline{ℙ} is equivalent to making a choice of such a lifting of 𝕄). Write G := 𝔾(L), M := 𝕄(L), N := ℕ(L), P := ℙ(L), and \overline{P} := \overline{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).

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

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 a smooth G-representation V with respect to P is defined to be the space V_{N} of N-coinvariants of V. The formation of Jacquet modules yields a functor from smooth 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)  Hom_{G}(V, (Ind_{P}^{G} W)_{sm}) ⟶^{∼} Hom_{M}(V_{N}, W),

where V is an arbitrary smooth G-representation, W is an arbitrary smooth Mrepresentation W, and (Ind_{P}^{G} 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 as a quotient of V, it admits an alternative description as a subspace of V. (This is the theory 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)  Hom_{G}((IndG ; \overline{P} U)_{sm}, V) ⟶^{∼} Hom_{M}(U(δ), V_{N}),

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 the naive point of view of Frobenius reciprocity).

Jacquet module functors for locally analytic representations. For any locally L-analytic group H, denote by Rep_{la.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 Rep_{la.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 functor from Rep_{la.c}(G) to Rep_{la.c}(M) which satisfies an obvious analogue of (0.1), with smooth parabolic induction being replaced by locally analytic parabolic induction, and the Hom-spaces being replaced by spaces of continuous homomorphisms. (See [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 of Rep_{la.c}(G) and Rep_{la.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 J_{P} on Rep_{la.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 J_{P} (V) will depend only on the structure of V as a P -representation, and so we initially define the functor J_{P} in the more general context of locally analytic P -representations.

In subsection 3.1 we define a certain full subcategory Rep_{la.c}^{z}(M) of Rep_{la.c}(M). For U an object of Rep_{la.c}^{z}(M), let C_{c}^{sm}(N, U) denote the space of compactly supported locally constant U-valued functions on N. The formation of this space yields a functor C_{c}^{sm}(N, – ) : Rep_{la.c}^{z}(M) → Rep_{la.c}(P). (See subsection 3.5.)

The following result follows from theorem 3.5.6 below.

Theorem 0.3. The functor C_{c}^{sm}(N, – ) admits a right-adjoint.

We define the Jacquet module functor J_{P} : Rep_{la.c}(P) → Rep_{la.c}^{z}(M) to be the right adjoint of C_{c}^{sm}(N, – ), twisted by the modulus character δ of P. Thus for U in Rep_{la.c}^{z}(M) and V in Rep_{la.c}(P), there is a natural isomorphism

(0.4)  L_{P} (C_{c}^{sm}(N, U), V) ⟶^{∼} L_{M}(U(δ), J_{P} (V)).

If we compose J_{P} with the forgetful functor Rep_{la.c}(G) → Rep_{la.c}(P) then we obtain a functor Rep_{la.c}(G) → Rep_{la.c}^{z}(M) (which we again denote by J_{P}). Note that for any object U of Rep_{la.c}^{z}(M), the space C_{c}^{sm}(N, U) embeds as a closed P - invariant subspace of the parabolic induction IndG ; \overline{P} U. (It can be identified with the subspace of locally constant functions in IndG ; \overline{P} U whose support lies in the open cell of G/\overline{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 J_{P} 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 of smooth representations extend to the locally analytic setting. It follows directly from its construction that the functor J_{P} is left exact and additive. On the other hand, simple examples show that it is not right exact in general. (This shows, incidentally, that J_{P} does not coincide with the functor of Hausdorff Ncoinvariants in general.) As for the relation with parabolic induction, one might hope 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 the sequel [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 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 J_{P} (V) is non-zero, then V is isomorphic to a subquotient of the locally analytic parabolic induction from P to G of a locally analytic character of M.

Global motivations. Our original motivation for defining the functor J_{P} was not local, but global. In the paper [9], it is applied to the problem of p-adic interpolation of 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, in which Hecke operators also play a key role.)

Some properties of the functor J_{P}. We establish a number of additional properties of the functor J_{P} in this paper. Before describing them, we introduce some more notation. Let Rep_{es}(G) (respectively Rep_{es}(M)) denote the category of essentially admissible locally analytic G-representations (respectively M-representations). Let 𝔤 denote the Lie algebra of 𝔾, let Z_{G} denote the centre of G, and let 𝔷(𝔤) denote the centre of the universal enveloping algebra of 𝔤. Similarly, let 𝔪 denote the Lie algebra of 𝕄, let Z_{M} denote the centre of M, and let 𝔷(𝔪) denote the centre of the universal enveloping algebra of 𝔪. Let Ẑ_{M} denote the rigid analytic space that parameterizes the locally L-analytic characters of Z_{M}, and for any χ ∈ Ẑ_{M}(K), and object V of Rep_{es}(G), let J_{P}^{χ}(V) denote the χ-eigenspace for the Z_{M}-action on J_{P} (V) (which is a closed admissible subrepresentation of J_{P} (V) [8, cor. 6.4.14]).

We now summarize the main properties of J_{P} that are proved in this paper.

(0.7) The definition of the functor C_{sm}^{c}(N, – ) : Rep_{la.c}^{z}(M) → Rep_{la.c}(P) depends on the choice of \overline{ℙ} (or equivalently, on the lifting of 𝕄 to a Levi factor of ℙ). However, we show that the functor J_{P} is independent, up to natural isomorphism, of the choice of \overline{ℙ}.

(0.8) Note that there are natural actions of Z_{G} × 𝔷(𝔤) on Rep_{la.c}(G), and of Z_{M} × 𝔷(𝔪) on Rep_{la.c}^{z}(M). Also, since M is the Levi quotient of P, there are natural injections Z_{G} → Z_{M} and 𝔷(𝔤) → 𝔷(𝔪) (the latter is the “unnormalized Harish-Chandra homomorphism”, recalled in subsection 1.3 below; it depends on the choice of P). These maps induce the upper horizontal arrow in the diagram

\xymatrix{Z_{G} × 𝔷(𝔤) \ar[dd] \ar[r] & Z_{M} × 𝔷(𝔪) \ar[d] \\  & Aut(Rep_{la.c}^{z}(M)) × End(Rep_{la.c}^{z}(M)) \ar[d] \\ Aut(Rep_{la.c}(G)) × End(Rep_{la.c}(G)) \ar[r] & Aut(J_{P}) × End(J_{P}),}

in which the left-hand vertical arrow and upper right-hand vertical arrow arise from the natural actions referred to above, and in which the lower horizontal arrow and the lower right-hand vertical arrow are induced by the functorial nature of J_{P}. We prove that this diagram commutes.

(0.9) If V is an object of Rep_{la.c}(P), so that J_{P} (V) is an object of Rep_{la.c}^{z}(M), then regard J_{P} (V) as a P -module, by having P act through its quotient M. For any choice of a compact open subgroup P_{0} of P and of a lifting of M to a Levi factor of P, we construct an M_{0}N_{0}-equivariant map

(0.10)  J_{P} (V) → V,

functorial in V. (Here M_{0} = M ⋂ P_{0}, N_{0} = N ⋂ P_{0}.) Extending the terminology of [5, p. 40], we refer to this map as the “canonical lifting” determined by the given choice of P_{0} and of a lifting of M.

(0.11) Suppose that 𝔾 is quasi-split over L, and take ℙ to be a Borel subgroup of 𝔾. The Levi quotient 𝕄 is then a torus, and so M = Z_{M}. Passing to duals induces an anti-equivalence of categories between the category Rep_{es}(M), and the category of coherent rigid analytic sheaves on M̂. Thus, for any object V in Rep_{es}(G), its Jacquet module J_{P} (V) gives rise to a coherent rigid analytic sheaf on M̂. The support of this sheaf is a Zariski closed subset of M̂, which we denote by Exp(J_{P} (V)). Passing to the derivative of a character induces a natural map M̂ → 𝔪̌, and hence (by restriction) a map

(0.12)  Exp(J_{P} (V)) → 𝔪̌.

We prove that if V is an admissible locally analytic representation of G, then the map (0.12) has discrete fibres.

(0.13) Let W be a finite dimensional algebraic representation of G, write B = End_{G}(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 J_{P} (V) is naturally isomorphic as an M-representation to X_{N} ⊗_{B}W^{N}. In particular, J_{P} (V) is an admissible locally W^{N}-algebraic representation of M. (Taking W to be the trivial representation, we find that J_{P} restricted to admissible smooth representations coincides with the Jacquet module functor of [5].)

(0.14) Assuming that L = ℚ_{p} and that G is split over K, we give a criterion for the functor J_{P} to commute with the passage to locally algebraic vectors. More precisely, 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 V_{W −lalg} → V induces a closed embedding

J_{P} (V_{W −lalg}) → J_{P} (V)_{W^{N} −lalg}.

(Here the subscripts indicate subspaces of locally algebraic vectors, transforming locally according to W and W^{N} respectively.) If χ ∈ Ẑ_{M}(K), then we may restrict to χ-eigenspaces, and so obtain a closed embedding

(0.15)  J_{P}^{χ}(V_{W −lalg}) → J_{P}^{χ}(V)_{W^{N} −lalg}.

Let ψ denote the character through which Z_{M} acts on W^{N}. (There is such a ψ, since 𝔾, and so Z_{M}, 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 and target of (0.15) vanish.

Suppose that χ and ψ do coincide locally. If V admits a G-invariant norm, and if χ is of “non-critical slope” with respect to the irreducible representation W^{N} (in the 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 topologically of finite length, one might hope that J_{P} (V) is again admissible, of finite length. Since J_{P} is a left exact functor, it would suffice to prove this when V is topologically irreducible. 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 with which we must deal is that of analyzing compact operators on Banach modules over a 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 of development 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, and are 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. After making some preliminary constructions in subsection 4.1, in subsection 4.2 we give the 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 ℚ_{p} (for some fixed prime p), an algebraic closure \overline{L} of L, as well as an extension K of L, complete with respect to a discrete valuation extending that on L. We let ord_{L} denote the discrete valuation on L (normalized so as to take the value 1 on a uniformizer of L), ord\overline{L} denote its extension to \overline{L}, and ord_{K} its extension to K. We let O_{L} and O_{K} 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 a continuous 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 an abbreviation for “locally convex topological K-vector space”.

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 the forthcoming paper [1].

1. Lie-theoretic preliminaries

(1.1) Let 𝔾 be a split connected reductive linear algebraic group defined over the 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, and 𝔫 denote the Lie algebras of 𝔾, 𝕋, 𝔹, and ℕ respectively. We let X^{•}(𝕋)^{+} denote the 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 representation of 𝔾 over K.

(i) The subspace W^{ℕ} of W is one dimensional, and 𝕋 acts on this space through an element χ ∈ X^{•}(𝕋)^{+}.

(ii) Associating χ to W induces a bijection between the isomorphism classes of irreducible 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 on a finite dimensional K-vector space W.

(i) The subspace W^{𝔫} of W is one dimensional, and 𝔱 acts on this space through an element χ in 𝔱̌^{+}.

(ii) Associating χ to W induces a bijection between the isomorphism classes of irreducible finite dimensional representations of 𝔤 over K and the elements of 𝔱̌^{+}.

Proof. This summarizes the standard highest weight theory for the reductive Lie algebra 𝔤. □

Just as for 𝔾-representations, 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.

The following results provide the link between the irreducible representations of 𝔾 and those of 𝔤.

Theorem 1.1.3. If W is a finite dimensional algebraic representation of 𝔾 defined over 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 an irreducible 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 lies in 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. (We remark that since ℕ is connected, the inclusion W^{ℕ} ⊂ W^{𝔫} is an isomorphism for any representation W of 𝔾, and so the highest weight spaces of W, thought of alternatively as a 𝔾-representation or as a 𝔤-representation, coincide.) □

Related to the highest weight theory of irreducible representations of 𝔤 is the theory 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 of 𝔟 that is trivial on 𝔫, let K(χ) denote the field K regarded as a 𝔟-module via the character 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 a certain “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 the character 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 ad_{b}(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, \overline{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} ∈ \overline{𝔫} is a non-zero element of the (one dimensional) root space in \overline{𝔫} corresponding to the negative simple root −α, and if m_{α} = ⟨χ, α̌⟩ (where α̌ is the coroot corresponding to α), then X_{−α}^{m_{α}+1}v(χ) is annihilated by 𝔫, is fixed by the adjoint action of ℕ, and generates a U(𝔤)-submodule of Ver(χ) isomorphic to Ver(χ − (m_{α} + 1)α).

(ii) If for each positive simple root α as in (i) we let ϕ_{α} : Ver(χ − (m_{α} + 1)α) → Ver(χ) denote the homomorphism of U(𝔤)-modules obtained by mapping v(χ−(m_{α}+ 1)α) to X_{−α}^{m_{α}+1}v(χ), then the cokernel of the map

⨁_{α} Ver(χ − (m_{α} + 1)α) ⟶^{⊕_{α}ϕ_{α}} Ver(χ)

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 Verma modules: our Ver(χ) is his V (χ+ρ) (where ρ denotes one-half the sum of the positive roots). We also remark that if s_{α} denotes the simple reflection corresponding to the positive simple root α, then χ+ρ−(m_{α} +1)α = χ+ρ−⟨χ+ρ, α̌⟩α = s_{α}(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, and let 𝕄 be a Levi factor of ℙ. Choose a Borel 𝔹 of 𝔾 that is contained in ℙ, and a maximal torus 𝕋 of 𝔹 that is contained in 𝕄 (so that 𝕋 is also a maximal torus of 𝕄). We let ℕ′ denote the unipotent radical of 𝔹, let ℤ_{𝔾} denote the centre of 𝔾, and let ℤ_{𝕄} denote the centre of 𝕄. We also let 𝔤, 𝔭, 𝔫, 𝔪, 𝔟, 𝔫′, and 𝔱 denote the Lie algebras of 𝔾, ℙ, ℕ, 𝕄, 𝔹, ℕ′, and 𝕋 respectively. The intersection 𝕄 ⋂ 𝔹 is a Borel subgroup of 𝕄, with unipotent radical 𝕄 ⋂ ℕ′. Also ℕ′ = (𝕄 ⋂ ℕ′)ℕ, and so

(1.2.1)  𝔫′ = (𝔪 ⋂ 𝔫′) ⨁ 𝔫.

Let Δ(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 𝔫′). We let Δ(𝔾, 𝕋)_{s} denote the subset of Δ(𝔾, 𝕋) consisting of simple positive roots. The set Δ(𝔾, 𝕋)_{s} is a basis for a finite index sublattice of X^{•}(𝕋/ℤ_{𝔾}).

Similarly, we let Δ(𝕄, 𝕋) denote the set of positive roots of 𝕋 with respect to 𝕄 (that is, the set of characters appearing in the adjoint action of 𝕋 on 𝔪 ⋂ 𝔫′). If we let Δ(𝕄, 𝕋)_{s} denote the subset of Δ(𝕄, 𝕋) consisting of simple positive roots, then Δ(𝕄, 𝕋)_{s} is the basis of a finite index sublattice of X^{•}(𝕋/ℤ_{𝕄}).

Lemma 1.2.2. There is an equality Δ(𝕄, 𝕋) ⋂ Δ(𝔾, 𝕋)_{s} = Δ(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 𝔫′, we see that any root whose decomposition in terms of simple roots involves an element of Δ(𝔾, 𝕋)_{s} \ Δ(𝕄, 𝕋) appears in 𝔫, and so not in 𝔪 ⋂ 𝔫′. Consequently any element of Δ(𝕄, 𝕋)_{s} is simple even when regarded as an element of Δ(𝔾, 𝕋). This proves the lemma. □

We let Δ(𝔾, ℤ_{𝕄}) ⊂ X^{•}(ℤ_{𝕄}) denote the set of positive restricted roots of ℤ_{𝕄} (that is, the set of characters appearing in the adjoint action of ℤ_{𝕄} on 𝔫). We let Δ(𝔾, ℤ_{𝕄})_{s} denote the subset consisting of simple positive restricted roots. The decomposition (1.2.1) shows that the elements of Δ(𝔾, ℤ_{𝕄}) are the restrictions to ℤ_{𝕄} of the elements of Δ(𝔾, 𝕋) \ Δ(𝕄, 𝕋).

Lemma 1.2.3. Restriction of characters from 𝕋 to ℤ_{𝕄} induces a bijection

Δ(𝔾, 𝕋)_{s} \ Δ(𝕄, 𝕋)_{s} ⟶^{∼} Δ(𝔾, ℤ_{𝕄})_{s}.

Proof. Lemma 1.2.2 shows that Δ(𝔾, 𝕋)_{s} = Δ(𝔾, 𝕋)_{s} \ Δ(𝕄, 𝕋)_{s} ⋃ Δ(𝕄, 𝕋)_{s}. As already remarked, Δ(𝔾, 𝕋)_{s} is a basis for a finite index sublattice of X^{•}(𝕋/ℤ_{𝔾}), while Δ(𝕄, 𝕋)_{s} is a basis for a finite index sublattice of X^{•}(𝕋/ℤ_{𝕄}). It follows that the restriction of Δ(𝔾, 𝕋)_{s} \ Δ(𝕄, 𝕋)_{s} to ℤ_{𝕄} is a basis for a finite index sublattice of X^{•}(ℤ_{𝕄}/ℤ_{𝔾}). In particular, restriction to ℤ_{𝕄} yields an embedding

(1.2.4)  Δ(𝔾, 𝕋)_{s} \ Δ(𝕄, 𝕋)_{s} ↪ Δ(𝔾, ℤ_{𝕄}).

Any element of Δ(𝔾, ℤ_{𝕄})_{s} is the restriction to ℤ_{𝕄} of some element α ∈ Δ(𝔾, 𝕋)\ Δ(𝕄, 𝕋). We may write α = ∑_{β∈Δ(𝔾,𝕋)_{s}} n_{β}β with n_{β} ≥ 0. Since any β ∈ Δ(𝕄, 𝕋)_{s} has trivial restriction to ℤ_{𝕄}, we may omit these terms from the sum, and not alter the restriction to ℤ_{𝕄}. Thus we assume that α = ∑_{β∈Δ(𝔾,𝕋)_{s}\Δ(𝕄,𝕋)_{s}} n_{β}β. As already observed, the simple roots β appearing in this sum have non-zero restrictions to ℤ_{𝕄}. Thus, if the restriction of α to ℤ_{𝕄} is simple, then we see that α itself must be simple. Hence any element of Δ(𝔾, ℤ_{𝕄})_{s} lies in the image of (1.2.4).

Now consider an arbitrary α ∈ Δ(𝔾, 𝕋)_{s} \ Δ(𝕄, 𝕋)_{s}. The restriction α_{|ℤ_{𝕄}} is a linear combination of elements of Δ(𝔾, ℤ_{𝕄})_{s} with non-negative coefficients. By the result of the preceding paragraph, we may write α_{|ℤ_{𝕄}} = ∑_{β∈Δ(𝔾,𝕋)_{s}\Δ(𝕄,𝕋)_{s}} n_{β}β_{|ℤ_{𝕄}}, for some n_{β} ≥ 0. Since (1.2.4) induces an injection of the span of its domain into X^{•}(ℤ_{𝕄}), we deduce that α = ∑_{β∈Δ(𝔾,𝕋)_{s}\Δ(𝕄,𝕋)_{s}} n_{β}β. Since α is simple, we conclude that n_{β} = 0 if β ≠ α, while n_{α} = 1. Thus α_{|ℤ_{𝕄}} is also simple, and so the image of (1.2.4) is contained in Δ(𝔾, ℤ_{𝕄})_{s}. □

For any root α of 𝕋 in 𝔾 (that is, any element of Δ(𝔾, 𝕋) ⋃ −Δ(𝔾, 𝕋)), we let s_{α} denote the corresponding simple reflection in the Weyl group W (𝔾, 𝕋) of 𝕋 with respect 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 the centralizer of the subtorus ℤ_{𝕄} 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 s_{α}(χ) and s_{w(α)}(w(χ)) have the same restriction to ℤ_{𝕄}.

Proof. Indeed, w(s_{α}(χ)) = s_{w(α)}(w(χ)), and w acts trivially on ℤ_{𝕄}. □

Let W be an irreducible algebraic 𝔾-representation. Highest weight theory shows that W^{ℕ} is an irreducible algebraic 𝕄-representation, and hence ℤ_{𝕄} acts on W^{ℕ} through a character ψ ∈ X^{•}(ℤ_{𝕄}). This extends to the highest weight character ψ̃ ∈ X^{•}(𝕋) of W. If α is any element of Δ(𝔾, ℤ_{𝕄})_{s}, then lemma 1.2.3 shows that α lifts uniquely to α̃ ∈ Δ(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 Δ(𝔾, 𝕋).

Lemma 1.2.6. The restriction of sα̃(ψ̃ + ρ(𝔾, 𝕋)) + ρ(𝔾, 𝕋) to ℤ_{𝕄} is independent of the choice of the Borel subgroup contained in ℙ.

Proof. If 𝔹′ is any Borel subgroup contained in ℙ, then we may choose the maximal torus 𝕋 to be contained in 𝔹 ⋂ 𝔹′, and then may write 𝔹′ = w𝔹w^{−1} for some element w ∈ 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 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 the unnormalized 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 the final claim, choose an opposite parabolic \overline{ℙ} to ℙ, with unipotent radical \overline{ℕ}; the intersection ℙ ⋂ \overline{ℙ} provides a lift of 𝕄 to a Levi factor of ℙ. If \overline{𝔫} denotes the Lie algebra of \overline{ℕ}, then there is a direct sum decomposition U(𝔤) ≅ \overline{𝔫} U(\overline{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 \overline{𝔫} U(\overline{𝔭}) is trivial, we see that the elements of 𝔷(𝔤) (which are necessarily invariant under the adjoint 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′ : 𝔷(𝔤) → U(𝔪) to be the composite of the inclusion 𝔷(𝔤) ⊂ U(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′ 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′ lies in 𝔷(𝔪). By construction the kernel of γ′ 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 that splits 𝔾), and since 𝔷(𝔤) commutes with the 𝔾-action on W, we see that the kernel of γ′ annihilates W. An element of U(𝔤) that annihilates every finite-dimensional representation of 𝔾 necessarily vanishes, and thus γ′ 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 \overline{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 an action of Gal(\overline{L}/L), compatible with respect to the canonical pairing between the two of them. The natural embedding Y_{•} → X_{•} identifies Y_{•} with the sublattice of Galois invariants in X_{•}. The natural surjection X^{•} → Y^{•} identifies Y^{•} with the quotient 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 ↦ ord\overline{L}(χ(t)) is Galois invariant, and so defines a map T → ℚ ⊗_{ℤ} Y_{•}. Denote the image by Y′_{•}. Restricted to S, this yields a surjection S → Y_{•}, with kernel S^{0}. Altogether, we obtain a diagram of short exact sequences

(1.4.1)  \xymatrix{0 \ar[r] & S^{0} \ar[r] \ar[d] & S \ar[r] \ar[d] & Y_{•} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & T^{0} \ar[r] & T \ar[r] & Y′_{•} \ar[r] & 0.}

Both Y_{•} and Y′_{•} 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′_{•} 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 (as discussed in [8, §6.4]). If χ is an element of T̂(K), then the function t ↦ ord_{K}(χ(t)) is trivial on T^{0}, and so factors to yield a morphism Y′_{•} → 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′_{•} → 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 of χ 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, then c_{0}(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 c_{0}(X, K) by identifying a point x ∈ X with the function 1_{x} 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(c_{0}(X, K), V) → F(X, V), given by restricting maps to the subset X of c_{0}(X, K), is injective, and its image contains the set of maps X → V for which the image of X is contained in a bounded complete subset of V.

Proof. The discussion of [15, pp. 59–60] shows that X topologically spans c_{0}(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 with bounded 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 c_{0}(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 c_{0}(X, K) × V → W.

Proof. Suppose that (i) holds. By assumption we are given a map X → L_{s}(V, W) with equicontinuous image. We infer from [15, lem. 6.8, cor. 6.11, prop. 7.13] that the closure of the image is equicontinuous, bounded and complete. By Proposition 2.1.1, this map uniquely determines a continuous K-linear map c_{0}(X, K) → L_{s}(V, W), and thus a separately continuous K-bilinear map

(2.1.3)  c_{0}(X, K) × V → W.

Since the closed O_{K}-lattice spanned by X is the unit ball of c_{0}(X, K) (with respect to the sup norm on this space), the unit ball of c_{0}(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 c_{0}(X, K) × V → W, then we see that the entire unit ball of c_{0}(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 are two 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 and only 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 {ϕ^{n} | n ≥ 0} of endomorphisms of V is equicontinuous.

For any element x in K^{×}, denote by K ⟨⟨xt⟩⟩ 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 ⟨⟨t⟩⟩.) Similarly denote by K ⟨⟨xt, xt^{−1}⟩⟩ 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 ⟨⟨xt, xt^{−1}⟩⟩ = 0.) For any K-Banach algebra A, denote by A ⟨⟨xt⟩⟩ the completed tensor product A ˆ⊗_{K} K ⟨⟨xt⟩⟩ (and similarly for A ⟨⟨xt, x^{−1}t⟩⟩).

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 ⟨⟨t⟩⟩-module with respect to which t acts via ϕ, and this module structure is unique if it exists.

Proof. Since the map 1_{n} ↦ t^{n} induces a topological isomorphism c_{0}(ℕ, K) ⟶^{∼} K ⟨⟨t⟩⟩, proposition 2.1.2 shows that ϕ is power-bounded if and only if there is a jointly continuous map

(2.1.7)  K ⟨⟨t⟩⟩ × V → V

such that (t^{n}, v) ↦ ϕ^{n}(v) for every v ∈ V, and that ϕ uniquely determines (2.1.7) (assuming that it exists). Since K [t] is dense in K ⟨⟨t⟩⟩, we see immediately that (2.1.7) (if it exists) gives V the structure of a K ⟨⟨t⟩⟩-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 id_{U} ⊗ ϕ is a power-bounded endomorphism of U ⊗_{K,π} V.

(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 ⟨⟨t⟩⟩- module structure of proposition 2.1.6, then α and β are K ⟨⟨t⟩⟩-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

\xymatrix{U \ar[r]^{ψ} \ar[d] & U \ar[d] \\ V \ar[r]^{ϕ} & V.}

If ψ is power-bounded, then so is ϕ, and the surjection U → V is furthermore a homomorphism of K ⟨⟨t⟩⟩-modules.

Proof. Part (i) follows immediately from part (i) of lemma 2.1.4.

As for part (ii), write X := {ϕ^{n} | n ≥ 0} and Y := {ψ^{n} | n ≥ 0}. From the definition of ϕ and ψ we see that Y = α ∘ X ∘ β ⋃{id_{V} }, and that X = β ∘ Y ∘ α ⋃{id_{U}}, and so lemma 2.1.4 (ii) and (iii) show that X is equicontinuous if and only if Y is so. Assuming that they are equicontinuous, and noting that α∘ϕ = ψ∘α and that β ∘ψ = ϕ∘β, we see that α and β are K [t]-linear, and hence K ⟨⟨t⟩⟩-linear, by continuity.

Part (iii) follows from lemma 2.1.4 (iii) and (iv) and the definition of the K ⟨⟨t⟩⟩- module 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 be continuous R-linear maps such that the composites ϕ = β ∘ α and ψ = α ∘ β are each equal to an element t ∈ R. (In the situation of proposition 2.1.8 (ii), if ϕ and ψ are power-bounded, then these hypotheses are satisfied for any R admitting a continuous injection into K ⟨⟨t⟩⟩ whose image contains K [t].) If S is a topological R-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 id_{S} ⊗ ϕ of S ⊗_{R,π} U and id_{S} ⊗ ψ of S ⊗_{R,π} V are both topological isomorphisms, since t is a unit in S. The proposition thus follows from the 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 L_{b}(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.

(Note that since we are assuming that K is discretely valued, any K-Banach space is isomorphic to c_{0}(X, K) for some set X [15, prop. 10.1]. Thus the orthonormalizable A-Banach modules are precisely those of the form A ˆ⊗_{K} V, for some K-Banach 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-Banach space. If we are given A-linear continuous maps A ˆ⊗_{K} V → M and N → M with the image of the first map being contained in the image of the second then we may find an A-linear map A ˆ⊗_{K} V → N that gives a commutative diagram

\xymatrix{A ˆ⊗_{K} V \ar[d] \ar[dr] \\ N \ar[r] & M.}

Proof. Let P denote the image of the map N → M, equipped with the topology that it inherits as a subspace of M, and let P_{1} 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, and the closed graph theorem shows that the induced map V → P_{1} 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 → P_{1}. Composing such a splitting with the continuous map V → P_{1} induces a map of K-Banach spaces V → N, which in turn induces a map of A-Banach modules A ˆ⊗_{K} V → N. This is the required map. □

Lemma 2.2.2. If V is a K-Banach space and M is an A-Banach module, then a continuous A-linear map A ˆ⊗_{K} V → M has its image contained in a finitely generated A-submodule of M if and only if it may be factored as a composite of continuous A-linear maps A ˆ⊗_{K} V → A^{r} → M for some natural number r.

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 morphism ϕ : A ˆ⊗_{K} V → A ˆ⊗_{K} W is called A-compact if we may write it as the limit in L_{b}(A ˆ⊗_{K} V, A ˆ⊗_{K} W) of a sequence {ϕ_{n}} of A-linear maps A ˆ⊗_{K} V → A ˆ⊗_{K} W, each of whose image is contained in a finitely generated A-submodule of A ˆ⊗_{K} W.

(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

\xymatrix{A ˆ⊗_{K} V \ar[r] \ar[d] & A ˆ⊗_{K} W \ar[d] \\ M \ar[r]^{ϕ} & N,}

in which the left hand vertical arrow is surjective, and the upper horizontal arrow is A-compact in the sense of (i).

The compatibility of parts (i) and (ii) of the preceding definition, in the case of orthonormalizable A-Banach modules, is a consequence of the following lemma.

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 A ˆ⊗_{K} V → M and A ˆ⊗_{K} W → N, the latter being surjective, then we may lift ϕ to a morphism A ˆ⊗_{K} V → A ˆ⊗_{K} W that is A-compact in the sense of definition 2.2.3 (i).

Proof. Follow definition 2.2.3 (ii) and form a commutative diagram

\xymatrix{A ˆ⊗_{K} V_{1} \ar[r] \ar[d] & A ˆ⊗_{K} W_{1} \ar[d] \\ M \ar[r]^{ϕ} & N}

in which V_{1} and W_{1} 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). Use lemma 2.2.1 to lift the maps A ˆ⊗_{K} V → M and A ˆ⊗_{K} W_{1} → N to A-linear maps A ˆ⊗_{K} V → A ˆ⊗_{K} V_{1} and A ˆ⊗_{K} W_{1} → A ˆ⊗_{K} W respectively. Composing these lifts with the upper horizontal map of the commutative diagram yields a map A ˆ⊗_{K} V → A ˆ⊗_{K} W which lifts ϕ and satisfies the conditions of definition 2.2.3 (i). □

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 just for 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 not orthonormalizable, 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 M_{1} → M, M → N and N → N_{1} are continuous Alinear maps of A-Banach modules, the second being A-compact, then the composite M_{1} → M → N → N_{1} is A-compact.

(ii) If A → B is a continuous homomorphism of K-Banach algebras, and if M → N is an A-compact map of A-Banach modules, then the induced map B ˆ⊗_{A} M → B ˆ⊗_{A} N is B-compact.

(iii) If we are given a commutative diagram of continuous A-linear maps between A-Banach modules

\xymatrix{M \ar[d] \ar[r] & N \ar[d] \\ P \ar[r] & Q,}

in which the upper horizontal arrow is A-compact and the left-hand vertical arrow is surjective, then the bottom horizontal arrow is also A-compact.

Proof. For part (i), note that the maps M_{1} → M and N_{1} → N induce the top horizontal arrows of the commutative diagram

\xymatrix{A ˆ⊗_{K} M_{1} \ar[r] \ar[d] & A ˆ⊗_{K} M \ar[d] & A ˆ⊗_{K} N \ar[r] \ar[d] & A ˆ⊗_{K} N_{1} \ar[d] \\ M_{1} \ar[r] & M \ar[r] & N \ar[r] & N_{1},}

in which the vertical arrows are the natural maps induced by the A-module structures on each of M, M_{1}, N, and N_{1} respectively. Lemma 2.2.4 shows that we may lift the A-compact map M → N to a map ϕ : A ˆ⊗_{K} M → A ˆ⊗_{K} N satisfying definition 2.2.3 (i). If we fill in the above diagram with ϕ, then the composite of the upper horizontal arrows yields a map A ˆ⊗_{K} M_{1} → A ˆ⊗_{K} N_{1} that again satisfies definition 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 there is a natural isomorphism B ˆ⊗_{A}(A ˆ⊗_{K} V) ⟶^{∼} B ˆ⊗_{K} V, and take into account 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 ⟨⟨xt⟩⟩-module (with t acting via ϕ on M),

then K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K ⟨⟨xt⟩⟩} M is finitely generated as an A ⟨⟨xt, xt^{−1}⟩⟩-module.

Proof. Choose a continuous A-linear surjection A ˆ⊗_{K} V → M for some K-Banach space V (e.g. take V = M). Lemma 2.2.4 shows that we may lift ϕ to an A-linear endomorphism ψ of A ˆ⊗_{K} V that satisfies definition 2.2.3 (i).

Proposition 2.1.10 shows that xψ is power-bounded if ord_{K}(x) is large enough. Proposition 2.1.8 (iii) implies that xϕ is then also power-bounded, and that the surjection A ˆ⊗_{K} V → M is K ⟨⟨xt⟩⟩-linear. Since it is also A-linear, it is an A ⟨⟨xt⟩⟩- linear surjection. It thus induces an A ⟨⟨xt, xt^{−1}⟩⟩-linear surjection

K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K ⟨⟨xt⟩⟩} A ˆ⊗_{K} V → K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K ⟨⟨xt⟩⟩} M.

Hence, if the source of this map is finitely generated as an A ⟨⟨xt, xt^{−1}⟩⟩-module, then the same will be true of its target. In summary, we see that (i) holds, and that in proving (ii), we may replace M with A ˆ⊗_{K} V and ϕ with ψ.

We thus suppose that M is an orthonormalizable A-Banach module, equipped with an A-linear endomorphism ϕ that may be written as the limit in L_{b}(M, M) of a sequence of A-linear maps ϕ_{n}, 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, then K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K ⟨⟨xt⟩⟩} M is a finitely generated A ⟨⟨xt, xt^{−1}⟩⟩-module.

Choose n sufficiently large so that the element x^{−1}(ϕ − ϕ_{n}) of L(M, M) is topo-

logically nilpotent. Applying lemma 2.2.2, we factor ϕ_{n} as M ⟶^{ϕ′_{n}} A^{r} ⟶^{ϕ′′_{n}} M for some r ≥ 0. Define a continuous A-linear map ξ : M → A ⟨⟨xt, xt^{−1}⟩⟩^{r} via the formula ξ(m) = ∑_{i=0}^{∞} x^{i}t^{−i−1}ϕ′_{n}(x^{−i}(ϕ − ϕ_{n})^{i}(m)). (This expression is well-defined, since x^{−1}(ϕ − ϕ_{n}) is topologically nilpotent.) The map ξ induces a continuous A ⟨⟨xt, xt^{−1}⟩⟩-linear map ξ′ : K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K} M → A ⟨⟨xt, xt^{−1}⟩⟩^{r}, while ϕ′′_{n} induces a continuous A ⟨⟨xt, xt^{−1}⟩⟩-linear map ξ′′ : A ⟨⟨xt, xt^{−1}⟩⟩^{r} → K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K ⟨⟨xt⟩⟩} M. One computes that ξ′′ ∘ ξ′ coincides with the quotient map K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K} M → K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K ⟨⟨xt⟩⟩} M. Thus ξ′′ is a surjection, and so K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K ⟨⟨xt⟩⟩} M is finitely generated over A ⟨⟨xt, xt^{−1}⟩⟩. □

To conclude this subsection, let us note the following strengthened version of the preceding proposition, in the special case where A = K.

Proposition 2.2.7. If ϕ is a compact endomorphism of a K-Banach space V, then for 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 ⟨⟨xt⟩⟩-module (with t acting via ϕ on V),

then K ⟨⟨xt, xt^{−1}⟩⟩ ˆ⊗_{K ⟨⟨xt⟩⟩} V is a finite dimensional K-vector space.

Proof. Part (i) is proposition 2.2.6 (i) in the case A = K, while (ii) is a restatement of the usual theory of compact operators on K-Banach spaces [21]. □

(2.3) This subsection extends the theory of compact operators to topological modules 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 ⟶^{∼} lim_{\overrightarrow{n}} A_{n}, where \{A_{n}\}_{n≥0} is a directed sequence of K-Banach algebras, with transition

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 locally convex, 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 tensor product B ˆ⊗_{A} M is a compact type topological B-module.

Proof. By definition, B ˆ⊗_{A} M is a Hausdorff quotient of the completed tensor product B ˆ⊗_{K} M. The latter space is of compact type [8, prop. 1.1.32], and thus so is the former. □

Fix a compact type topological K-algebra A. The following definition, while a little 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 → N_{1} → N, where N_{1} is a compact type topological A-module satisfying the following condition: there is a continuous K-linear map of convex K-vector spaces V → N_{1}, with V a convex K-vector space of compact type, such that the natural map A ˆ⊗_{K} V → N_{1} is surjective, and such that the composite V → N_{1} → N is a compact map of convex K-vector spaces.

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 topological A-modules, and if M_{1} → M and N → N_{1} are any two continuous A-linear maps of compact type topological A-modules, then the composite M_{1} → M → N → N_{1} 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, then the induced map B ˆ⊗_{A} M → B ˆ⊗_{A} N is B-compact.

(iii) If we are given a commutative diagram of continuous A-linear maps between compact type topological A-modules

\xymatrix{M \ar[d] \ar[r] & N \ar[d] \\ P \ar[r] & Q,}

in which the upper horizontal arrow is A-compact and the left-hand vertical arrow is 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 → N_{1} → N, as in definition 2.2.8. Thus we assume that this factorization is A-linear, that N_{1} is a topological A-module of compact type, and that there is a map V → N_{1} with V of compact type for which the induced map A ˆ⊗_{K} V → N_{1} is surjective, and for which the composite V → N is compact. Let Q_{1} denote the compact type A-module obtained as the quotient of N_{1} by the kernel of the composite N_{1} → N → Q. The kernel of the surjection M → P is contained in the kernel of the composite M → N_{1} → Q_{1}. Since P is a strict quotient of 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 → Q_{1} → Q. The surjection A ˆ⊗_{K} V → N_{1} induces a surjection A ˆ⊗_{K} V → Q_{1}, and the composite V → N_{1} → Q_{1} 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 {r_{n}}_{n≥0} of real numbers converging to 1 from above. Letting R_{n}(x) denote the Tate algebra of rigid analytic functions defined over K that converge on the disk |t| ≤ r_{n}|x|^{−1}, define K ⟨⟨xt⟩⟩^{†} := lim_{\overrightarrow{n}} R_{n}(x). Each R_{n}(x) is

naturally a Banach space, and if we equip K ⟨⟨xt⟩⟩^{†} with its locally convex inductive limit topology, it becomes a compact type topological K-algebra. Letting S_{n}(x) denote the Tate algebra of rigid analytic functions over A on the annulus r_{n}^{−1}|x| ≤ |t| ≤ r_{n}|x|^{−1}, we may similarly define the compact type algebra K ⟨⟨xt, xt^{−1}⟩⟩^{†} = lim_{\overrightarrow{n}} S_{n}(x). Both K ⟨⟨xt⟩⟩^{†} and K ⟨⟨xt, xt^{−1}⟩⟩^{†} are independent (up to canonical

isomorphism) of the choice of sequence {r_{n}}.

If A = lim_{\overrightarrow{n}} A_{n} is a compact type topological K-algebra (written as the locally con-

vex inductive limit, with compact and injective transition maps, of the K-Banach algebras A_{n}), then we write A ⟨⟨xt⟩⟩^{†} := A ˆ⊗_{K} K ⟨⟨xt⟩⟩^{†} ⟶^{∼} lim_{\overrightarrow{n}} A_{n} ˆ⊗_{K} R_{n}(x) and

A ⟨⟨xt, xt^{−1}⟩⟩^{†} := A ˆ⊗_{K} K ⟨⟨xt, xt^{−1}⟩⟩^{†} ⟶^{∼} lim_{\overrightarrow{n}} A_{n} ˆ⊗_{K} S_{n} (the isomorphisms fol-

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 sense of definition 2.1.5.

(ii) If we regard M as a topological K ⟨⟨xt⟩⟩^{†}-module, with t acting via ϕ on M, then K ⟨⟨xt, xt^{−1}⟩⟩^{†} ˆ⊗_{K ⟨⟨xt⟩⟩^{†}} M is finitely generated as an A ⟨⟨xt, xt^{−1}⟩⟩^{†}-module.

Proof. By assumption, we may factor the map ϕ in the form M → M_{1} → M, where M_{1} is a compact type A-module for which there is a surjection π : A ˆ⊗_{K} V → M_{1} for some compact type K-vector space V, with the additional property that the composite V → M_{1} → 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 ⟨⟨xt⟩⟩^{†} and K ⟨⟨xt, xt^{−1}⟩⟩^{†}), we see that it suffices to prove the proposition with M_{1} in place of M, and with the composite M_{1} → M → M_{1} (which we denote by ϕ_{1}) in place of ϕ.

Write A ⟶^{∼} lim_{\overrightarrow{n}} A_{n} as in definition 2.3.1, and write V ⟶^{∼} lim_{\overrightarrow{n}} V_{n} as the lo-

cally convex inductive limit of a sequence of Banach spaces with compact injective transition maps. By [8, prop. 1.1.32], the natural map lim_{\overrightarrow{n}} A_{n} ˆ⊗_{K} V_{n} → A ˆ⊗_{K} V

is a topological isomorphism. For each natural number n, let W_{n} denote the K-Banach space obtained as the quotient of A_{n} ˆ⊗_{K} V_{n} by the kernel of the composite A_{n} ˆ⊗_{K} V_{n} → A ˆ⊗_{K} V → M_{1}. For each value of n there is a continuous injection ι_{n} : W_{n} → M_{1}, and the induced map lim_{\overrightarrow{n}} W_{n} → M_{1} is a topological isomorphism.

Since the composite V ⟶ M_{1} ⟶^{ϕ_{1}} M_{1} is compact, it factors through a continuous map α_{m} : V → W_{m}, for some m [3, prop. 1, p. I.20]. At the expense of replacing m by m + 1 if necessary, we may assume that α_{m} is a compact map (since the map W_{m} → W_{m+1} is compact). If U denotes the image of W_{m} in M_{1}, then (since it is A-linear, and so in particular A_{m}-linear) the map ϕ_{1} restricts to an A_{m}-linear endomorphism of U. The closed graph theorem then implies that this endomorphism of U lifts to a continuous A_{m}-linear endomorphism ψ_{m} of W_{m} [8, prop. 1.1.2], which by proposition 2.2.5 is A_{m}-compact (in the sense of definition 2.2.3), since α_{m} is compact.

Proposition 2.1.10 implies that xψ_{m} is power-bounded if x has sufficiently large valuation. Thus the sequence of maps {x^{n}ψ_{m}^{n}}_{n≥0} is equicontinuous, and thus so is the sequence of maps {ι_{m} ∘ x^{n}ψ_{m}^{n} ∘ α_{m}}_{n≥0} from V to M_{1}, by lemma 2.1.4 (iii). Extending these sequence of maps by A-linearity to a sequence of maps A ˆ⊗_{K} V → M_{1}, we obtain precisely the sequence of composites {x^{n}ϕ_{1}^{n+1} ∘ π}_{n≥0}, which is thus 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 sequence of map {x^{n}ϕ_{1}^{n+1}}_{n≥0} is equicontinuous, and hence that xϕ_{1} is power bounded.

If m′ ≥ m, then ϕ_{1} equally well induces a compact map ψ_{m′} : W_{m′} → W_{m′}.

Since ϕ_{1} is A-linear (and so in particular A_{m′}-linear), since A_{m′} ˆ⊗_{K} V_{m′} subjects onto W_{m′}, and since ϕ_{1}(V_{m′}) ⊂ W_{m}, we find that ψ_{m′} admits a factorization into continuous A_{m′}-linear maps W_{m′} ⟶^{h_{m}} A_{m′} ˆ⊗_{A_{m}} W_{m} ⟶^{j_{m}} W_{m′}, where j_{m} is induced by the natural map W_{m} → W_{m′}, and h_{m} is defined so that ψ_{m′} = j_{m} ∘ h_{m}. (We have also appealed again to [8, prop. 1.1.2].) The composite h_{m} ∘ j_{m} is then equal to id_{A_{m′}} ˆ⊗ ψ_{m}. It thus follows from proposition 2.1.8 (i) and (iii) that if xψ_{m} is power-bounded, then the same is true of xψ_{m′}.

For each natural number n, let r_{n}, R_{n}(x) and S_{n}(x) be as defined in the discussion preceding the statement of the proposition. Proposition 2.2.6 implies that S_{m}(x) ˆ⊗_{R_{m}(x)} W_{m} (respectively S_{m′}(x) ˆ⊗_{R_{m′} (x)} W_{m′}) is a finitely generated A_{m} ˆ⊗_{K} S_{m}(x) module (respectively A_{m′} ˆ⊗_{K} S_{m′}(x) module). Proposition 2.1.9 implies that the continuous A_{m′}-linear map

(A_{m′} ˆ⊗_{K} S_{m′}(x)) ˆ⊗_{A_{m} ˆ⊗_{K} S_{m}(x)} S_{m}(x) ˆ⊗_{R_{m}(x)} W_{m} → S_{m′}(x) ˆ⊗_{R_{m′} (x)} W_{m′}

induced by j_{m} is an isomorphism (of finitely generated A_{m′} ˆ⊗_{K} S_{m′}(x)-modules). Passing to the locally convex inductive limit in m now shows that the completed tensor product K ⟨⟨xt, xt^{−1}⟩⟩^{†} ˆ⊗_{K ⟨⟨xt⟩⟩^{†}} M_{1} is a finitely generated module over A ⟨⟨xt, xt^{−1}⟩⟩, as required. □

It is convenient to note that in the situation of part (ii) of the preceding proposition, the natural map

K ⟨⟨xt, xt^{−1}⟩⟩^{†} ˆ⊗_{K [t]} M → K ⟨⟨xt, xt^{−1}⟩⟩^{†} ˆ⊗_{K ⟨⟨xt⟩⟩^{†}} M

is an isomorphism. Indeed, this follows immediately from the fact that K [t] is dense in K ⟨⟨xt⟩⟩^{†}.

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 ⟨⟨xt⟩⟩^{†}-module (with t acting via ϕ on M),

then K ⟨⟨xt, xt^{−1}⟩⟩^{†} ˆ⊗_{K ⟨⟨xt⟩⟩^{†}} V is a finite dimensional K-vector space.

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 Rep_{top.c}(G) denote the category 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 Rep_{top.c}(G), then the direct product U × V, with the diagonal G-action, is again an object of Rep_{top.c}(G).

(ii) If V is an object of Rep_{top.c}(G), and if U is a closed G-invariant subspace of V, then U is again an object of the category Rep_{top.c}(G), as is the quotient V/U.

(iii) If {V_{n}}_{n≥0} is an inductive sequence of objects in Rep_{top.c}(G), having injective transition maps, then the locally convex inductive limit lim_{\overrightarrow{n}} V_{n} again lies in

Rep_{top.c}(G).

(iv) If U and V are two objects Rep_{top.c}(G), then U ˆ⊗_{K} V (with the diagonal action of G) is again an object of Rep_{top.c}(G).

Proof. This is immediate, once one recalls the following facts: a finite direct product of 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; the locally convex inductive limit of any inductive sequence of convex K-vector spaces of compact type having injective transition maps is again of compact type; the completed tensor product of two convex K-vector spaces of compact type is again of compact type [8, prop. 1.1.32 (i)]. □

The preceding lemma implies that Rep_{top.c}(G) is an additive category, admitting kernels, cokernels, images and coimages. More precisely, if ϕ : U → V is a morphism in Rep_{top.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 quotient of U; the image of ϕ is the closure of ϕ(U), regarded as a subspace of V; and the cokernel 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 Rep_{la.c}(G) denote the full subcategory of Rep_{top.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 Rep_{la.c}(G), then the direct product U × V, with the diagonal G-action, is again an object of Rep_{la.c}(G).

(ii) If V is an object of Rep_{la.c}(G), and if U is a closed G-invariant subspace of V, then U is again an object of the category Rep_{la.c}(G), as is the quotient V/U.

(iii) If {V_{n}}_{n≥0} is an inductive sequence of objects in Rep_{la.c}(G), having injective transition maps, then the locally convex inductive limit lim_{\overrightarrow{n}} V_{n} again lies in

Rep_{top.c}(G).

(iv) If U and V are two objects Rep_{la.c}(G), then U ˆ⊗_{K} V (with the diagonal action of G) is again an object of Rep_{la.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 Rep_{la.c}(G) is an additive category that admits kernels, cokernels, images and coimages. Furthermore, these coincide with the corresponding constructions in the larger category Rep_{top.c}(G).

Continuing to suppose that G is locally L-analytic, and also supposing that the centre Z_{G} of G is topologically finitely generated, we let Rep_{es}(G) denote the full subcategory of Rep_{la.c}(G) consisting of essentially admissible locally analytic representations of G (as defined in [8, def. 6.4.9]). We let Rep_{ad}(G) denote the full subcategory of Rep_{es}(G) consisting of admissible locally analytic representations of G (as defined in [20]; see also [8, def. 6.1.1]).

Theorem 3.1.3. Each of Rep_{es}(G) and Rep_{ad}(G) is an abelian category. Furthermore, the construction of direct products, kernels, cokernels, images and coimages coincide with their construction in Rep_{top.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 Rep_{es}(G) is necessarily strict, with closed image.

There is one last full subcategory of Rep_{la.c}(G) that we will consider, to be denoted by Rep_{la.c}^{z}(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 the union of an increasing sequence of Z_{G}-invariant BH-subspaces.

As in [8, §6.4], let Ẑ_{G} denote the rigid analytic space of locally L-analytic characters on Z_{G}, and let C^{an}(Ẑ_{G}, K) denote the nuclear Fréchet algebra of K-valued rigid analytic functions on Ẑ_{G}. Evaluation at points of Z_{G} induces a natural map Z_{G} → C^{an}(Ẑ_{G}, K), with image lying in the group of invertible elements of C^{an}(Ẑ_{G}, K). It follows from [8, prop. 6.4.7] that Rep_{la.c}^{z}(G) consists of those objects V in Rep_{la.c}(G) for which the Z_{G}-action on V extends to a C^{an}(Ẑ_{G}, K)-module structure on V, with the map C^{an}(Ẑ_{G}, 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 Rep_{la.c}^{z}(G), then the direct product U × V, with the diagonal G-action, is again an object of Rep_{la.c}^{z}(G).

(ii) If V is an object of Rep_{la.c}^{z}(G), and if U is a closed G-invariant subspace of V, then U is again an object of the category Rep_{la.c}^{z}(G), as is the quotient V/U.

(iii) If {V_{n}}_{n≥0} is an inductive sequence of objects in Rep_{top.c}(G), having injective transition maps, then the locally convex inductive limit lim_{\overrightarrow{n}} V_{n} again lies in

Rep_{la.c}^{z}(G).

(iv) If U and V are two objects Rep_{la.c}^{z}(G), then U ˆ⊗_{K} V (with the diagonal action of G) is again an object of Rep_{la.c}^{z}(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 Rep_{es}(G) is an object of Rep_{la.c}^{z}(G).

(3.2) If Z is an abelian group and Z^{+} is a submonoid of Z, then the group algebra K[Z] contains the monoid algebra K[Z^{+}] as a subalgebra, and the functor Hom_{Z^{+}}(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 Hom_{Z^{+}}(K[Z], M) → M given by evaluation at the identity element of K[Z] realizes the Z-module Hom_{Z^{+}}(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^{+} generates Z as a group, and contains a compact open subgroup of Z. Recall the categories Rep_{la.c}^{z}(Z) and Rep_{top.c}(Z^{+}), defined in subsection 3.1. There is an obvious forgetful functor Rep_{la.c}^{z}(Z) → Rep_{top.c}(Z^{+}). We will construct a right adjoint to this functor.

Definition 3.2.1. If V is any object of Rep_{top.c}(Z^{+}), then we write

V_{fs} := L_{b,Z^{+}}(C^{an}(Ẑ, K), V).

(The right hand side denotes the subspace of L_{b}(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 V_{fs} as the “finite slope part” of V.

If V is an object of Rep_{top.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 V_{fs} is a closed subspace of the Hausdorff space L_{b}(C^{an}(Ẑ, K), V). The natural action of Z on C^{an}(Ẑ, K) induces an action of Z on L_{b}(C^{an}(Ẑ, K), V), and since Z is abelian, V_{fs} is obviously Z-invariant. Thus V_{fs} is equipped with a natural Z-action.

The formation of V_{fs} 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)  V_{fs} → V.

The following lemma provides a description of the strong dual of V_{fs}.

Lemma 3.2.3. If V is an object of Rep_{top.c}(Z^{+}), then there is a natural isomor-

phism (V_{fs})′_{b} ⟶^{∼} C^{an}(Ẑ, K) ˆ⊗_{K[Z^{+}]} V′_{b}.

Proof. It follows from [15, cor. 18.8] and [8, prop. 1.1.32] that L_{b}(C^{an}(Ẑ, K), V) is a compact type space whose strong dual is isomorphic to C^{an}(Ẑ, K) ˆ⊗_{K} V′_{b}. Since V_{fs} is defined to the the closed subspace of L_{b}(C^{an}(Ẑ, K), V) consisting of Z^{+}-equivariant maps, it follows that its strong dual (V_{fs})′_{b} is naturally isomorphic to the Hausdorff quotient C^{an}(Ẑ, K) ˆ⊗_{K[Z^{+}]} V′_{b} of C^{an}(Ẑ, K) ˆ⊗_{K} V′_{b}. □

Proposition 3.2.4. (i) If V is an object of Rep_{top.c}(Z^{+}), then V_{fs} is an object of Rep_{la.c}^{z}(Z). Thus V ↦ V_{fs} yields a functor Rep_{top.c}(Z^{+}) → Rep_{la.c}^{z}(Z).

(ii) For any objects W and V of Rep_{la.c}^{z}(Z) and Rep_{top.c}(Z^{+}) respectively, the map (3.2.2) induces a natural topological isomorphism

(3.2.5)  L_{b,Z}(W, V_{fs}) ⟶^{∼} L_{b,Z^{+}}(W, V).

Proof. To prove (i), note that the isomorphism of lemma 3.2.3 shows that (V_{fs})′_{b} is a topological C^{an}(Ẑ, K)-module. It follows from [8, 6.4.7] that V_{fs} lies in Rep_{la.c}^{z}(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 structure of 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 maps

L_{b}(W, V) ⟶ L_{b}(C^{an}(Ẑ, K) ⊗_{K,ı} W, V) → L_{b}(W, L_{b}(C^{an}(Ẑ, K), V)),

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 the source to a Z^{+}-equivariant map in the target. Since W and V_{fs} are both Z-modules, the inclusion L_{b,Z}(W, V_{fs}) ⊂ L_{b,Z^{+}}(W, V_{fs}) is an equality, and so we obtain a map L_{b,Z^{+}}(W, V) → L_{b,Z}(W, V_{fs}). It is easily checked to provide an inverse to (3.2.5). □

The preceding result implies in particular that V ↦ V_{fs} is a right adjoint to the forgetful functor Rep_{la.c}^{z}(Z) → Rep_{top.c}(Z^{+}).

Proposition 3.2.6. (i) The functor V ↦ V_{fs} is additive.

(ii) If U → V is an injective map in the category Rep_{top.c}(Z^{+}), then the morphism U_{fs} → V_{fs} induced by functoriality is again injective.

(iii) If U → V is a closed embedding in the category Rep_{top.c}(Z^{+}), then the morphism U_{fs} → V_{fs} induced by functoriality is again a closed embedding. Furthermore, in this situation, the morphism V_{fs}/U_{fs} → (V/U)_{fs} induced by functoriality is a continuous injection.

(iv) If {V_{n}}_{n≥0} is an inductive sequence in the category Rep_{top.c}(Z^{+}) having injective transition maps, and if we form the locally convex inductive limit V = lim_{\overrightarrow{n}} V_{n}

(an object of Rep_{top.c}(Z^{+}), by lemma 3.1.1), then the natural map lim_{\overrightarrow{n}} (V_{n})_{fs} → V_{fs}

is a topological isomorphism.

Proof. Parts (i) and (ii) are immediate from the construction of U_{fs} and V_{fs}.

Let U → V be a closed embedding as in (iii). Applying L_{b,Z^{+}}(C^{an}(Ẑ, K), – ) to the short exact sequence 0 → U → V → (V/U) → 0 yields the sequence of continuous Z-equivariant maps

0 → L_{b,Z^{+}}(C^{an}(Ẑ, K), U) → L_{b,Z^{+}}(C^{an}(Ẑ, K), V) → L_{b,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_{\overrightarrow{n}} V_{n} as in (iv), then the natural map

lim_{\overrightarrow{n}} L_{b,Z^{+}}(C^{an}(Ẑ, K), V_{n}) → L_{b,Z^{+}}(C^{an}(Ẑ, K), V) is a continuous bijection. Since

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 Rep_{top.c}(Z^{+}) be equipped with a locally analytic H-representation that commutes with the Z^{+}-action on V.

(i) The H-action on V_{fs} induced by functoriality is also locally analytic.

(ii) If 𝔥 denotes the Lie algebra of H, then the action of U(𝔥) on V_{fs}, induced by functoriality from the U(𝔥)-action on the locally analytic H-representation V, coincides with the U(𝔥)-action on V_{fs}, obtained by regarding it as a locally analytic H-representation via (i).

Proof. Lemma 3.2.3 yields an isomorphism C^{an}(Ẑ, K) ˆ⊗_{K[Z^{+}]} V′_{b} ⟶^{∼} (V_{fs})′_{b}. It follows from [17, cor. 3.3] that the H-action induced on the source of this isomorphism by the H-action on V extends to a topological D^{la}(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 D^{la}(H, K)-module structure. Thus (again by [17, cor. 3.3]) the H-action on V_{fs} 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 Rep_{la.c}^{z}(Z), regarded via the forgetful functor as an object of Rep_{top.c}(Z^{+}), the natural map (3.2.2) is a topological isomorphism.

Proof. This follows from the universal property satisfied by V_{fs}, together with the fact that for any objects V and W of Rep_{la.c}^{z}(Z), the natural map L_{Z^{+}}(W, V) → L_{Z}(W, V) is an isomorphism. □

Proposition 3.2.9. If V and W are objects of Rep_{top.c}(Z^{+}) and Rep_{la.c}^{z}(Z) respectively, then there is a natural isomorphism V_{fs} ˆ⊗_{K} W ⟶^{∼} (V ˆ⊗_{K} W)_{fs}.

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 U_{1} and U_{2}, there is an isomorphism

(3.2.10)  L(U_{1} ˆ⊗_{K} W′_{b}, U_{2}) ⟶^{∼} L(U_{1}, U_{2} ˆ⊗_{K} W).

Now V ˆ⊗_{K} W is an object of Rep_{top.c}(Z^{+}), while if U is an object of Rep_{la.c}^{z}(Z) then so is U ˆ⊗_{K} W (as follows from part (iv) of lemmas 3.1.2 and 3.1.4 respectively). Taking into account the universal property of V_{fs} and (V ⊗_{K} W)_{fs}, and applying (3.2.10), we obtain natural isomorphisms

L_{Z}(U, V_{fs} ˆ⊗_{K} W) ⟶^{∼} L_{Z}(U ˆ⊗_{K} W′_{b}, V_{fs}) = L_{Z^{+}}(U ˆ⊗_{K} W′_{b}, V)

⟶^{∼} L_{Z^{+}}(U, V ˆ⊗_{K} W) ⟶^{∼} L_{Z}(U, (V ˆ⊗_{K} W)_{fs}).

Thus we obtain the required natural isomorphism V_{fs} ˆ⊗_{K} W ⟶^{∼} (V ˆ⊗_{K} W)_{fs}. □

Proposition 3.2.11. If V is an object of Rep_{top.c}(Z^{+}) and if X is any collection of continuous Z^{+}-equivariant endomorphisms of V, then the closed embedding V^{X} → V induces an isomorphism (V^{X})_{fs} ⟶^{∼} (V_{fs})^{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 L_{b,Z^{+}}(C^{an}(Ẑ, K), V^{X}) ⟶^{∼} L_{b,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 Rep_{top.c}(Z^{+}), then we define

V^{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 through the 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 Rep_{la.c}^{z}(Z). Note also that the preceding construction and notation applies in particular to objects of Rep_{la.c}^{z}(Z), which may be regarded as objects of Rep_{top.c}(Z^{+}), by restricting the Z-action to a Z^{+}-action.

Proposition 3.2.12. If χ ∈ Ẑ(K) and if V is an object of Rep_{top.c}(Z^{+}), then the map (3.2.2) induces an isomorphism (V_{fs})^{χ} ⟶^{∼} V^{χ}.

Proof. The universal property of V_{fs}, applied to the map V^{χ} → V, yields a map V^{χ} → V_{fs}, and hence a map V^{χ} → (V_{fs})^{χ}. On the other hand, the map (3.2.2) certainly induces a map (V_{fs})^{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 relationship of divisibility. (That is, for a pair of elements z_{1}, z_{2} ∈ Z^{+}, we have z_{1} < z_{2} if there exists z ∈ Z^{+} such that z_{2} = zz_{1}.) If z ∈ Z^{+} and V is an object of Rep_{top.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 annihilated by the element z ∈ Z^{+}. If z_{1}, z_{2} ∈ Z^{+} such that z_{1} < z_{2}, then V^{z_{1}} ⊂ V^{z_{2}}.

Definition 3.2.13. If V is an object of Rep_{top.c}(Z^{+}), then we define V_{null} := lim_{\overrightarrow{z∈Z^{+}}} V^{z}, the locally convex inductive limit being indexed by the monoid Z^{+}, directed,

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 V_{null}, and a continuous Z^{+}-equivariant injection

(3.2.14)  V_{null} → V.

Proposition 3.2.15. (i) If V is an object of Rep_{top.c}(Z^{+}), then V_{null} is also an object of Rep_{top.c}(Z^{+}).

(ii) Suppose that {V_{n}}_{n≥0} is a Z^{+}-equivariant inductive sequence objects of Rep_{top.c}(Z^{+}), having injective transition maps. If V denotes the locally convex inductive limit V = lim_{\overrightarrow{n}} V_{n} (also an object of Rep_{top.c}(Z^{+}), by lemma 3.1.1), then

the natural map lim_{\overrightarrow{n}} (V_{n})_{null} → V_{null} is a topological isomorphism.

Proof. By assumption, Z^{+} contains a compact open subgroup Z_{0} of Z, and the inductive limit in definition 3.2.13 may be regarded as being indexed by Z^{+}/Z_{0}, rather than by Z^{+}. The quotient Z^{+}/Z_{0} is a subsemigroup of the finitely generated group Z/Z_{0}, 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 Rep_{top.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 Rep_{top.c}(Z^{+}). Thus part (iii) of the same lemma, when combined with the observation of the preceding paragraph, shows that V_{null} is again an object of Rep_{top.c}(Z^{+}). This proves (i).

To prove (ii), suppose that V = lim_{\overrightarrow{n}} V_{n} is the locally convex inductive limit of

an inductive sequence of objects of Rep_{top.c}(Z^{+}) having injective transition maps. Clearly the natural map lim_{\overrightarrow{n}} (V_{n})_{null} → V_{null} is a continuous bijection. Since source

and target are both of compact type, it is necessarily a topological isomorphism. □

Definition 3.2.16. If V is an object of Rep_{top.c}(Z^{+}), we say that V is null if the natural map V_{null} → V is a continuous bijection (or equivalently, since source and target are spaces of compact type, a topological isomorphism).

Lemma 3.2.17. If V is a null object of Rep_{top.c}(Z^{+}), then V_{fs} = 0.

Proof. Proposition 3.2.6 (iv), combined with the observation in the first paragraph of the proof of proposition 3.2.15, shows that for any object V of Rep_{top.c}(Z^{+}), the natural map lim_{\overrightarrow{z∈Z^{+}}} (V^{z})_{fs} → (V_{null})_{fs} is an isomorphism. Thus it suffices to

show that (V^{z})_{fs} = 0 for any object V of Rep_{top.c}(Z^{+}). Proposition 3.2.11 gives an isomorphism (V^{z})_{fs} ⟶^{∼} (V_{fs})^{z}. The target of this isomorphism is clearly trivial, since the z-action on V_{fs} arises from an action of the group Z. □

Lemma 3.2.18. If V is a finite dimensional object of Rep_{top.c}(Z^{+}), then the natural map V_{null} ⨁ V_{fs} → 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 Rep_{top.c}(Z^{+}), then the natural map V_{fs} = L_{b,Z^{+}}(C^{an}(Ẑ, K), V) → L_{b,Y} (C^{an}(Ẑ, K), V) is a topological isomorphism.

Proof. The map in question is an inclusion of closed subspaces of L_{b}(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^{+}, then we may find y, y′ ∈ Y such that zy = y′. If f ∈ C^{an}(Ẑ, K), then (taking into account that ϕ is Y -equivariant) we compute

ϕ(zf) = ϕ(y′y^{−1}f) = y′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 Rep_{top.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 ϕ and 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 action of z.

Then the map L_{b}(C^{an}(Ẑ, K), U) → L_{b}(C^{an}(Ẑ, K), V) induced by ϕ restricts to a Z-equivariant topological isomorphism U_{fs} ⟶^{∼} V_{fs}.

Proof. The map ϕ induces a map ϕ̃ : L_{b,Y} (C^{an}(Ẑ, K), U) → L_{b,Y} (C^{an}(Ẑ, K), V) which is evidently Z-equivariant, and whose source and target are equal to U_{fs} and V_{fs} respectively, by lemma 3.2.19. To show that ϕ̃ is an isomorphism, it suffices to show 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 ⊂ L_{b,Y} (C^{an}(Ẑ, K), U^{z}) = (U^{z})_{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 V_{fs}, then define f̃: C^{an}(Ẑ, K) → U via f̃(α) = ψ ∘ f(z^{−1}α). Since ψ is Y -equivariant, one sees that f̃ is Y -equivariant, and hence lies in U_{fs}. Furthermore, the composite ϕ ∘ f̃ is equal to f. Thus ϕ̃ is surjective. □

Now let A be a K-Fréchet algebra, written as a projective limit A ⟶^{∼} lim_{\overleftarrow{n}} A_{n},

where each A_{n} is a compact type topological K-algebra, each of the transition maps A_{n+1} → A_{n} is compact, and the map A → A_{n} has dense image for each n. Suppose also that the projective system A_{n} 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 {Ẑ_{n}}_{n≥0} of admissible open affinoid subdomains of Ẑ, with the property that each inclusion Z_{n} ⊂ Z_{n+1} is relatively compact. There is then an isomorphism C^{an}(Ẑ, K) ⟶^{∼} lim_{\overleftarrow{n}} C^{an}(Ẑ_{n}, K)^{†} (where

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

(3.2.22)  C^{an}(Ẑ, K) ˆ⊗_{K} A → lim_{\overleftarrow{n}} (C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K} A_{n}).

(Here we are using the fact that the projective sequences {C^{an}(Ẑ, K)^{†}}_{n≥0} and {A_{n}}_{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 Rep_{top.c}(Z^{+}). Then the Amodule structure on V_{fs} induced by functoriality makes V_{fs} an A-module, and the multiplication map A × V_{fs} → A is again separately continuous.

Proof. By [8, prop. 1.1.35], the action of A on V induces a continuous map A → L_{b,Z^{+}}(V, V), which when composed with (3.2.2) yields a continuous map A → L_{b,Z^{+}}(V_{fs}, V). Composing this map with the inverse of the isomorphism (3.2.5) gives a continuous map A → L_{b,Z}(V_{fs}, V_{fs}). The A-module structure induced on V_{fs} by this map is precisely the A-module structure on V induced by the functoriality of the construction of finite slope parts, and so the multiplication map A×V_{fs} → V_{fs} is separately continuous, as claimed. □

In the context of the preceding lemma, the A-module structure on V_{fs} induces a topological A-module structure on its strong dual (V_{fs})′_{b} [8, prop. 1.2.14]. Since V_{fs} is an object of Rep_{la.c}^{z}(Z), the same reference shows that (V_{fs})′_{b} is also a topological C^{an}(Ẑ, K)-module. Thus (V_{fs})′_{b} is in fact a topological C^{an}(Ẑ, K) ˆ⊗_{K} A-module.

Proposition 3.2.24. In the above situation, suppose given the following data:

(i) For each n ≥ 0 a compact type topological A_{n}-module U_{n}, equipped with an A_{n}-linear action of Z^{+}, as well as an A_{n+1}[Z^{+}]-linear transition map U_{n+1} → U_{n}, such that the induced A_{n}[Z^{+}]-linear map A_{n} ˆ⊗_{A_{n+1}} U_{n+1} → U_{n} is A_{n}-compact (in the sense of definition 2.3.3);

(ii) An element z ∈ Z^{+}, such that for each n ≥ 0, the map U_{n} → U_{n} induced by z factors through the transition map A_{n} ˆ⊗_{A_{n+1}} U_{n+1} → U_{n}, so as to give a commutative diagram

\xymatrix{A_{n} ˆ⊗_{A_{n+1}} U_{n+1} \ar[r] \ar[d]_{id ˆ⊗ z} & U_{n} \ar[d]^{z} \ar[dl] \\ A_{n} ˆ⊗_{A_{n+1}} U_{n+1} \ar[r] & U_{n}.}

(iii) An A[Z^{+}]-equivariant isomorphism V′_{b} ⟶^{∼} lim_{\overleftarrow{n}} U_{n}.

Then we may conclude that (V_{fs})′_{b} is a coadmissible C^{an}(Ẑ, K) ˆ⊗_{K} A-module (in the sense of [8, def. 1.2.8]).

Proof. The isomorphisms V′_{b} ⟶^{∼} lim_{\overleftarrow{n}} U_{n} and C^{an}(Ẑ, K) ⟶^{∼} lim_{\overleftarrow{n}} C^{an}(Ẑ_{n}, K)^{†}, to-

gether with [8, prop. 1.1.29] and lemma 3.2.3, yield isomorphisms

(V_{fs})′_{b} ⟶^{∼} C^{an}(Ẑ, K) ˆ⊗_{K[Z^{+}]} V′_{b} ⟶^{∼} lim_{\overleftarrow{n}} C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K[Z^{+}]} U_{n}.

Thus to prove the proposition, it suffices to prove that:

(a) For each n ≥ 0, the natural map

(3.2.25) C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K} A_{n} ˆ⊗_{C^{an}(Ẑ_{n+1},K)^{†} ˆ⊗_{K} A_{n+1}} C^{an}(Ẑ_{n+1}, K)^{†} ˆ⊗_{K[Z^{+}]} U_{n+1}

→ C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K[Z^{+}]} U_{n}

is an isomorphism;

(b) C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K[Z^{+}]} U_{n} is a finitely generated C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K} A_{n}-module, for any n.

To prove (a), note that the map (3.2.25) may be written more simply as

(3.2.26)  C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(A_{n} ˆ⊗_{A_{n+1}} U_{n+1}) → C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K[Z^{+}]} U_{n}.

Let z be as in hypothesis (ii). Since the image of z in C^{an}(Ẑ_{n}, 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 Ẑ_{n} is an affinoid domain, the rigid analytic functions on Ẑ_{n} induced by each of z and z^{−1} are bounded, and so for x ∈ K^{×} with ord_{K}(x) sufficiently large, the map K[z, z^{−1}] → C^{an}(Ẑ_{n}, K)^{†} extends to a map K ⟨⟨xz, xz^{−1}⟩⟩^{†} → C^{an}(Ẑ_{n}, K)^{†}. Hypotheses (i) and (ii) together imply that z acts as an A_{n}-compact operator on U_{n}, and thus proposition 2.3.5 (i) implies that (perhaps after increasing the lower bound on ord_{K}(x)) the operator xz is a power-bounded endomorphism of U_{n}, and hence that U_{n} is naturally a K ⟨⟨xz⟩⟩^{†}-module. For such values of x we obtain a surjection

C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K ⟨⟨xz,xz^{−1}⟩⟩^{†}} K ⟨⟨xz, xz^{−1}⟩⟩^{†} ˆ⊗_{K ⟨⟨xz⟩⟩^{†}} U_{n}

⟶^{∼} C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K ⟨⟨xz⟩⟩^{†}} U_{n}

⟶^{∼} C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K[z]} U_{n} → C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K[Z^{+}]} U_{n}.

(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 of this surjection is a finitely generated C^{an}(Ẑ_{n}, K)^{†} ˆ⊗_{K} A_{n}-module (again after increasing the lower bound on ord_{K}(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 Rep_{top.c}(Z^{+}). By assumption, Z^{+} contains an open subgroup Z_{0} of Z. We let Rep_{la.c}^{z}(Z^{+}) denote the full subcategory of Rep_{top.c}(Z^{+}) consisting of objects which may be written as a union of an increasing sequence of Z^{+}-invariant BH-subspaces, and on which Z_{0} acts through a locally analytic representation (for one, or equivalently any, choice of such open subgroup Z_{0}).

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, and such that Y Z^{+} = Z. If V is an object of Rep_{top.c}(Z^{+}), then by restricting the Z^{+}-action to Y^{+}, we may also regard V as an object of Rep_{top.c}(Y^{+}). We write V_{Z−fs} to denote the finite slope part of V as a Z^{+}-representation, and write V_{Y −fs} to denote the finite slope part of V as a Y^{+}-representation. There is a natural Y -equivariant map

(3.2.27)  V_{Z−fs} → V_{Y −fs}

(constructed either by a consideration of the universal property of V_{Y −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 V_{Y −fs}, and so V_{Y −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 Rep_{la.c}^{z}(Z^{+}), then the map (3.2.27) is a Z-equivariant topological isomorphism.

Proof. Let W be an object of Rep_{la.c}^{z}(Z). The universal property of V_{Y −fs} yields an isomorphism L_{Y} (W, V_{Y −fs}) ⟶^{∼} L_{Y^{+}}(W, V), and hence (recalling that Z = Y Z^{+}) an isomorphism L_{Z}(W, V_{Y −fs}) ⟶^{∼} L_{Z^{+}}(W, V). The proposition will follow provided that we prove that V_{Y −fs} lies in Rep_{la.c}^{z}(Z).

Proposition 3.2.7 (i) shows that Z_{0} acts locally analytically on V_{Y −fs}. By assumption (and [8, prop. 1.1.2]), we may write V ≅ lim_{\overrightarrow{n}} V_{n}, where each V_{n} is a K-Banach

space equipped with a topological Z^{+}-action. Also, we may write Ŷ = ⋃_{n} Ŷ_{n}, where {Ŷ_{n}} is an increasing sequence of affinoid subdomains of Ŷ, and hence write C^{an}(Ŷ , K) ⟶^{∼} lim_{\overleftarrow{n}} C^{an}(Ŷ_{n}, K); each C^{an}(Ŷ_{n}, K) is a K-Banach space. From

[3, prop. 1, p. I.20] we conclude that the natural map lim_{\overrightarrow{n}} L_{b}(C^{an}(Ŷ_{n}, K), V_{n}) →

L_{b}(C^{an}(Ẑ, K), V) is a Y -equivariant continuous bijection, and so we obtain a continuous bijection lim_{\overrightarrow{n}} L_{b,Y} (C^{an}(Ŷ_{n}, K), V_{n}) → V_{Y −fs}. Since each of C^{an}(Ŷ_{n}, K) and

V_{n} is a Banach space, the same is true of the spaces L_{b,Y} (C^{an}(Ŷ_{n}, K), V_{n}). The action of Z^{+} on V_{n} and of Y on C^{an}(Ŷ_{n}, K) induces a topological action of Z = Y Z^{+} on L_{b,Y} (C^{an}(Ŷ_{n}, K), V_{n}). Thus V_{Y −fs} is indeed an object of Rep_{la.c}^{z}(Z). □

The following lemma is helpful for verifying the hypothesis of the preceding proposition, in the context of proposition 3.2.24.

Lemma 3.2.29. If V ∈ Rep_{top.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 an increasing sequence of Z^{+}-invariant BH-subspaces.

Proof. Appealing to [8, prop. 1.1.2], we may write V ⟶^{∼} lim_{\overrightarrow{n}} W_{n} as the limit of an

inductive sequence of Fréchet spaces, each equipped with a topological Z^{+}-action, such that the transition maps W_{n} → W_{n+1} and the maps W_{n} → 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 {W_{n}} so that each transition map factors as W_{n} → X_{n} → W_{n+1}, where X_{n} is a Banach space. (See [8, props. 1.1.2 and 1.1.10].) Let Z_{0} be a compact open subgroup of Z contained in Z^{+}. The quotient Z^{+}/Z_{0} is a submonoid of the finitely generated abelian group Z/Z_{0}, and so is itself a finitely generated monoid. Fix a subset S of Z^{+} that is the union of Z_{0} and a finite set whose image in Z^{+}/Z_{0} is a generating set. Since Z_{0} is compact, the set S acts equicontinuously on each of the spaces W_{n}.

Let A_{n+1} be an open neighbourhood of the origin of W_{n+1}, and choose an open neighbourhood B_{n+1} of the origin of W_{n+1} such that SB_{n+1} ⊂ A_{n+1}. If we let A_{n} (respectively B_{n}) denote the preimage in W_{n} of A_{n+1} (respectively B_{n+1}), then since the map W_{n} → W_{n+1} factors through the Banach space X_{n}, and is Z^{+}-equivariant, the open subsets A_{n} and B_{n} of W_{n} are commensurable with one another (i.e. there exists x ∈ K^{×} such that xA_{n} ⊂ B_{n} ⊂ x^{−1}A_{n}), and SB_{n} ⊂ A_{n}. Thus if Y_{n} denote the Banach space obtained by completing W_{n} with respect to the gauge of A_{n} (or equivalently, of B_{n}), then the S-action on W_{n} extends to an equicontinuous S-action on Y_{n}, and hence to a topological Z^{+}-action. By construction, the map W_{n} → X_{n} factors through the Z^{+}-equivariant map W_{n} → Y_{n}. Thus we obtain an isomorphism V ⟶^{∼} lim_{\overrightarrow{n}} Y_{n}, where each Y_{n} is a Banach space

equipped with a topological Z^{+}-action. This proves the lemma. □

In the context of proposition 3.2.24, dualizing condition (iii) yields an isomorphism V ⟶^{∼} lim_{\overrightarrow{n}} (U_{n})′_{b}. Since each U_{n} is of compact type, each of the strong duals

(U_{n})′_{b} 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 a connected 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 of ℙ, so that ℙ = 𝕄ℕ. Write G := 𝔾(L), P := ℙ(L), M := 𝕄(L), and N := ℕ(L). Let ℤ_{𝔾} (respectively ℤ_{𝕄}) denote the centre of 𝔾 (respectively 𝕄), and write Z_{G} := ℤ_{𝔾}(L) (respectively Z_{M} := ℤ_{𝕄}(L)), so that Z_{G} (respectively Z_{M}) is the centre of G (respectively M). Taking T to be Z_{M} in the discussion of subsection 1.4, let Z_{M}^{0} denote the maximal compact subgroup of Z_{M}, and choose a sublattice Λ of Z_{M} which splits the bottom row of (1.4.1).

Lemma 3.3.1. If N_{0} and N′_{0} are two compact open subgroups of N, and if we write Λ′ := {λ ∈ Λ | λN′_{0}λ^{−1} ⊂ N_{0}}, then Λ′ 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 ord_{L}(α(λ)) > C for every positive restricted root α of Z_{M}, then λ ∈ Λ′. (See for example [5, prop. 1.4.3].) The subsemigroup of elements satisfying this inequality generates Λ as a group. □

Proposition 3.3.2. Let {(N_{i}, N′_{i})}_{1≤i≤n} be any finite sequence of pairs of compact open subgroups of N.

(i) If Y = {z ∈ Z_{M} | zN_{i}z^{−1} ⊂ N′_{i} for all 1 ≤ i ≤ r}, then Y generates Z_{M} as a group.

(ii) If Y′ = {m ∈ M | mN_{i}m^{−1} ⊂ N′_{i} for all 1 ≤ i ≤ r}, then Y′ and Z_{M} together generate M as a semigroup. In particular, Y′ generates M as a group.

Proof. If N_{0} denotes the compact open subgroup of N generated by all the N_{i}, and N′_{0} denotes the intersection of all the N′_{i}, then it suffices to prove the proposition with the sequence of pairs {(N_{i}, N′_{i})} replaced by the single pair (N_{0}, N′_{0}). Let N′′_{0} denote the compact open subgroup of N generated by the elements z_{0}n_{0}z_{0}^{−1}, as z_{0} and n_{0} range over all elements of Z_{M}^{0} and N_{0} respectively. Lemma 3.3.1 gives a subsemigroup Λ′ of Λ that generates Λ as a group, such that λN′′_{0}λ^{−1} ⊂ N_{0} for λ ∈ Λ′. Then Λ′Z_{M}^{0} ⊂ Y, and Λ′Z_{M}^{0} generates Z_{M} as a group. This proves (i).

If m ∈ M, then lemma 3.3.1 applied to the pair mN_{0}m^{−1} and N′_{0} of compact open subgroups of N gives an element z ∈ Z_{M} such that z(mN_{0}m^{−1})z^{−1} ⊂ N′_{0}. Thus zm ∈ Y′, and so m ∈ z^{−1}Y′ ⊂ Z_{M}Y′. This proves (ii).

Since, by definition, Y = Y′ ⋂ Z_{M} ⊂ Y′, it follows from (i) and (ii) together that Y′ generates M as a group. □

Fix a compact open subgroup P_{0} of P, and write M_{0} := M ⋂ P_{0}, N_{0} := N ⋂ P_{0}, M^{+} := {m ∈ M | mN_{0}m^{−1} ⊂ N_{0}}, and Z_{M}^{+} := M^{+} ⋂ Z_{M}. There are inclusions M_{0} ⊂ M^{+} and Z_{G} ⊂ Z_{M}^{+}.

Corollary 3.3.3. The abelian group Z_{M} is generated by its submonoid Z_{M}^{+}.

Proof. This is a special case of proposition 3.3.2. □

Definition 3.3.4. Let M^{+} ×_{Z_{M}^{+}} Z_{M} denote the quotient of the monoid M^{+} × Z_{M} by the equivalence relation generated by the relations (mz^{+}, z) ∼ (m, z^{+}z), for any m ∈ M^{+}, z^{+} ∈ Z_{M}^{+} and z ∈ Z_{M}.

The morphism of monoids M^{+} × Z_{M} → M defined by (m, z) ↦ mz clearly factors through M^{+} ×_{Z_{M}^{+}} Z_{M}, yielding a morphism

(3.3.5)  M^{+} ×_{Z_{M}^{+}} Z_{M} → M.

Proposition 3.3.6. The morphism (3.3.5) is an isomorphism.

Proof. Suppose that (m_{1}, z_{1}) and (m_{2}, z_{2}) are two elements of the product M^{+} × Z_{M}, such that m_{1}z_{1} and m_{2}z_{2} are equal as elements of M. Corollary 3.3.3 implies that we may find an element z ∈ Z_{M}^{+} such that z_{1}z_{2}^{−1}z also lies in Z_{M}^{+}, yielding the relations (m_{1}, z_{1}) ∼ (m_{1}z_{1}z_{2}^{−1}z, z^{−1}z_{2}) = (m_{2}z, z^{−1}z_{2}) ∼ (m_{2}, z_{2}). Thus (m_{1}, z_{1}) and (m_{2}, z_{2}) are related by the kernel of the canonical map M^{+} × Z_{M} → M^{+} ×_{Z_{M}^{+}} Z_{M}, and (3.3.5) is injective. Proposition 3.3.2 (ii) shows that M^{+} and Z_{M} 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 the notation 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].

Definition 3.4.1. Let V be a locally analytic representation of P. If N_{0} is any compact open subgroup of N, then we define the projection operator π_{N_{0}} : V^{𝔫} → V^{N_{0}} as follows: π_{N_{0}}v = ∫_{N_{0}} nv dn, for all v ∈ V^{𝔫}. (The measure dn is Haar measure on N, normalized so that N_{0} has measure one.)

More concretely, if v ∈ V^{𝔫}, then v is fixed by some subgroup N′_{0} of N_{0}, and so π_{N_{0}}v = [N_{0} : N′_{0}]^{−1} ∑_{n∈N_{0}/N′_{0}} nv.

We now fix a compact open subgroup P_{0} of P, and define M_{0}, N_{0}, M^{+} and Z_{M}^{+} as they were defined preceding corollary 3.3.3. That corollary shows that Z_{M}^{+} generates Z_{M} as a group. Also, Z_{M}^{+} contains the compact open subgroup Z_{M} ⋂ P_{0} of Z_{M}.

Definition 3.4.2. If m ∈ M^{+}, define π_{N_{0},m} : V^{𝔫} → V^{N_{0}} as follows: π_{N_{0},m}(v) = π_{N_{0}}(mv), for all v ∈ V^{𝔫}.

The alternative formula π_{N_{0},m}(v) = mπ_{m^{−1}N_{0}m}(v) is also useful. Note that the restriction of π_{N_{0},m} to V^{N_{0}} induces an endomorphism of V^{N_{0}}.

Lemma 3.4.3. If m ∈ M^{+}, then the endomorphism π_{N_{0},m} of V^{N_{0}} is continuous.

Proof. Since m ∈ M^{+}, we have an inclusion mN_{0}m^{−1} ⊂ N_{0}. The operator π_{N_{0},m}

is defined by the formula π_{N_{0},m}(v) = \frac{1}{[N_{0} : mN_{0}m^{−1}]} ∑_{n∈N_{0}/mN_{0}m^{−1}} nmv, for any

v ∈ V^{N_{0}}. Since N acts via continuous automorphisms of V, the lemma follows. □

Lemma 3.4.4. If m, m′ ∈ M^{+}, then π_{N_{0},m}π_{N_{0},m′} = π_{N_{0},mm′}.

Proof. Since m ∈ M^{+} there is an inclusion N_{0} ⊂ m^{−1}N_{0}m. Thus π_{m^{−1}N_{0}m}π_{N_{0}} = π_{m^{−1}N_{0}m}, and so we compute, for any v ∈ V,

π_{N_{0},m}π_{N_{0},m′}(v) = mπ_{m^{−1}N_{0}m}π_{N_{0}}(m′v)

= mπ_{m^{−1}N_{0}m}(m′v) = π_{N_{0}}(mm′v) = π_{N_{0},mm′}(v),

as claimed. □

The preceding two lemmas show that the operators π_{N_{0},m} define a topological action of the monoid M^{+} on V^{N_{0}}. In particular, we obtain a topological action of Z_{M}^{+} on V^{N_{0}}. Suppose now that V is of compact type (so that V lies in the category Rep_{la.c}(P) defined in subsection 3.1). The closed subspace V^{N_{0}} of V is then also of compact type, and is equipped with a topological Z_{M}^{+}-action. Thus it is an object of the category Rep_{top.c}(Z_{M}^{+}), and following definition 3.2.1 (with Z = Z_{M} and Z^{+} = Z_{M}^{+}), we may take its finite slope part.

Definition 3.4.5. For any object V of Rep_{la.c}(P), we define J_{P} (V) := (V^{N_{0}})_{fs}.

Proposition 3.2.4 shows that V^{N_{0}} is an object of Rep_{la.c}^{z}(Z_{M}), and so in particular is a convex K-vector space of compact type, equipped with a locally analytic representation of Z_{M}.

Proposition 3.4.6. If V is an object of Rep_{la.c}(P), then the locally analytic Z_{M}-representation on J_{P} (V) extends in a natural way to a locally analytic Mrepresentation.

Proof. Since Z_{M}^{+} lies in the centre of M^{+}, the M^{+}-action on V^{N_{0}} induces an M^{+}- action on (V^{N_{0}})_{fs}. Proposition 3.3.6 then implies that the M^{+}-action and Z_{M} action on (V^{N_{0}})_{fs}-action together determine an M-action on (V^{N_{0}})_{fs}. It remains to show that this action is locally analytic. Since M_{0} is compact open in M, it suffices to show that the M_{0}-action on (V^{N_{0}})_{fs} is locally analytic.

The group M_{0} normalizes N_{0}. This implies that M_{0} ⊂ M^{+}, and that for m ∈ M_{0}, the operator π_{N_{0},m} on V^{N_{0}} coincides with multiplication by m. Since V^{N_{0}} is a closed M_{0}-invariant subspace of V, which is locally analytic as a P -representation, and so in particular as an M_{0}-representation, we see that the M_{0}-action on V^{N_{0}} is locally analytic. The operators π_{N_{0},m} commute with the action of Z_{M}^{+} on V^{N_{0}}, and so lemma 3.2.7 (i) shows that the M_{0}-action on (V^{N_{0}})_{fs} is locally analytic. □

Thus J_{P} defines a functor Rep_{la.c}(P) → Rep_{la.c}^{z}(M), which in subsection 3.5 will be characterized as an adjoint functor. Since any object of Rep_{la.c}(G) gives rise to an object of Rep_{la.c}(P) by restricting the group action from G to P, we also obtain a functor J_{P} : Rep_{la.c}(G) → Rep_{la.c}^{z}(M), which in subsection 4.2 will be shown to restrict to a functor Rep_{es}(G) → Rep_{es}(M). The remainder of this subsection is devoted to proving some more elementary properties of J_{P}.

Lemma 3.4.7. (i) The functor J_{P} is additive.

(ii) If U → V is a continuous injection in Rep_{la.c}(P), then the morphism induced by functoriality, J_{P} (U) → J_{P} (V), is again injective.

(iii) If U → V is a closed embedding in Rep_{la.c}(P), then the morphism induced by functoriality, J_{P} (U) → J_{P} (V), is again a closed embedding. Furthermore, in this situation, the morphism J_{P} (V)/J_{P} (U) → J_{P} (V/U) induced by functoriality is a continuous injection.

(iv) If {V_{n}}_{n≥0} is an inductive sequence in the category Rep_{la.c}(P), having injective transition maps, and if V denotes the locally convex inductive limit V = lim_{\overrightarrow{n}} V_{n}

(an object of Rep_{la.c}(P), by lemma 3.1.2), then the natural map lim_{\overrightarrow{n}} J_{P} (V_{n}) →

J_{P} (V) is a topological isomorphism.

Proof. The additivity of the formation of N_{0}-invariants is clear. If U → V is injective (respectively a closed embedding), then the same is true of the induced map U^{N_{0}} → V^{N_{0}}. Also, if V = lim_{\overrightarrow{n}} V_{n}, then V^{N_{0}} = lim_{\overrightarrow{n}} V_{n}^{N_{0}} (taking into account the

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 J_{P} (V) = (V^{N_{0}})_{fs} → V^{N_{0}}, provided by (3.2.2), with the inclusion V^{N_{0}} → V, we obtain a natural map

(3.4.8)  J_{P} (V) → V.

It is immediate from the construction that this map is M_{0}N_{0}-equivariant (where the compact open subgroup M_{0}N_{0} of P is regarded as acting on J_{P} (V) through its quotient M_{0}). As will become clear in subsection 4.3, this is a generalization of the so-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 Z_{M}, and V is an object of Rep_{la.c}(P), then we write

V^{N_{0},Z_{M}^{+} =χ} = {v ∈ V^{N_{0}} | π_{N_{0},z}v = χ(z)v for all z ∈ Z_{M}^{+}}.

As in the introduction, we let J_{P}^{χ}(V) denote the closed subrepresentation of J_{P} (V) on which Z_{M} acts through the character χ.

Proposition 3.4.9. If V is an object of Rep_{la.c}(P), then (3.4.8) induces an iso-

morphism J_{P}^{χ}(V) ⟶^{∼} V^{N_{0},Z_{M}^{+} =χ}.

Proof. This follows from proposition 3.2.12. □

We now determine the extent to which the functor J_{P} depends on the choice of the Levi factor M and the open subgroup P_{0} of P.

If M′ is another lift of M to a Levi subgroup of P, then we may write M′ = nMn^{−1} for some (uniquely determined) n ∈ N. If we set N′_{0} = nN_{0}n^{−1} and (M′)^{+} = nM^{+}n^{−1}, then (M′)^{+} = {m′ ∈ M′ | m′N′_{0}(m′)^{−1} ⊂ N′_{0}}. Thus for any object V of Rep_{la.c}(P), we may define Hecke operators π_{N′_{0},m′} : V^{N′_{0}} → V^{N′_{0}} for m′ ∈ (M′)^{+}, and so regard V^{N′_{0}} as an (M′)^{+}-module.

Let us use the isomorphism given by conjugation by n to identify M and M′ (and hence M^{+} and (M′)^{+}). (This is the same identification that arises by identifying each of M and M′ with the Levi quotient of P.)

Proposition 3.4.10. For any object V of Rep_{la.c}(P), multiplication by n induces an isomorphism of M^{+}-modules V^{N_{0}} ⟶^{∼} V^{N′_{0}}, and hence an isomorphism (V^{N_{0}})_{fs} ⟶^{∼} (V^{N′_{0}})_{fs} of objects in Rep_{la.c}(M).

Proof. This is immediate. □

Proposition 3.4.10 shows that the functor J_{P} 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 P_{0}.

Proposition 3.4.11. If P′_{0} is an open subgroup of P_{0}, and if N′_{0} = N ⋂ P′_{0}, then the projection π_{N_{0}} : V^{N′_{0}} → V^{N_{0}} induces an M-equivariant topological isomorphism

(V^{N′_{0}})_{fs} ⟶^{∼} (V^{N_{0}})_{fs}.

Proof. Choose z ∈ Z_{M}^{+} such that zN_{0}z^{−1} ⊂ N′_{0}, and write ϕ := π_{N_{0}} : V^{N′_{0}} → V^{N_{0}}, ψ := π_{N′_{0},z} : V^{N_{0}} → V^{N′_{0}}. Let (M^{+})′ denote the submonoid of M that conjugates N′_{0} into itself, and write Y′ = M^{+} ⋂(M^{+})′. One checks easily that ϕ and ψ intertwine the Y′-action on each of V^{N′_{0}} and V^{N_{0}}. (Here m ∈ Y′ acts on these spaces through the operators π_{N′_{0},m} and π_{N_{0},m} respectively.) Also, ψ ∘ ϕ = π_{N′_{0},z}, while ϕ ∘ ψ = π_{N_{0},z}. Proposition 3.3.2 (i) shows that Y := Y′ ⋂ Z_{M} generates Z_{M} as a group, and thus, by proposition 3.2.20, we see that ϕ induces a Z_{M}-equivariant isomorphism (V^{N′_{0}})_{fs} ⟶^{∼} (V^{N_{0}})_{fs} that intertwines the Y′-action on its source and target. Since proposition 3.3.2 (ii) shows that Y′ generates M as a group, this isomorphism is in fact M-equivariant. □

Although the preceding results show that the functor J_{P} is determined, up to a natural isomorphism, independently of any particular choice of Levi factor and compact open subgroup of P, the “canonical lifting” (3.4.8) does in general depend on 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 J_{P}, and hence also the canonical lifting, does not require any choices.

Proposition 3.4.12. Suppose that V is an object of Rep_{la.c}^{z}(G). (For example, V could be an object of Rep_{es}(G).) Taking P to be G, the morphism J_{G}(V) → V provided by (3.4.8) is a topological isomorphism.

Proof. In this case N_{0} = N = 1, Z_{G}^{+} = Z_{G}, and lemma 3.2.8 gives the result. □

We finish this subsection by establishing property (0.8) of the introduction.

Proposition 3.4.13. The diagram of (0.8) is commutative.

Proof. Although M^{+} is not a group, it contains the compact open subgroup M_{0} = M ⋂ P_{0}. Thus we write Rep_{la.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 M_{0}. Note that Z_{M}^{+} × 𝔷(𝔪) acts as symmetries of the category Rep_{la.c}(M^{+}).

The functor J_{P} is the composite of two functors: the functor V ↦ V^{N_{0}}, mapping Rep_{la.c}(G) to Rep_{la.c}(M^{+}), and the functor W ↦ W_{fs}, mapping Rep_{la.c}(M^{+}) to Rep_{la.c}(M). Thus to verify that the diagram of (0.8) commutes, it suffices to verify the commutativity of the diagrams

(3.4.14)  \xymatrix{Z_{G} × 𝔷(𝔤) \ar[d] \ar[r] & Z_{M}^{+} × 𝔷(𝔪) \ar[d] \\ Aut(V) × End(V) \ar[r] & Aut(V^{N_{0}}) × End(V^{N_{0}}),}

(where V is any object of Rep_{la.c}(G), and the bottom horizontal arrow is induced by the functoriality of the formation of N_{0}-invariants), and

(3.4.15)  \xymatrix{Z_{M}^{+} × 𝔷(𝔪) \ar[d] \ar[r] & Z_{M} × 𝔷(𝔪) \ar[d] \\ Aut(W) × End(W) \ar[r] & Aut(W_{fs}) × End(W_{fs}),}

(where W is any object of Rep_{la.c}(M^{+}), and the bottom arrow is induced by the functoriality of the formation of finite slope parts). Here we are taking into account the fact that the map Z_{G} → Z_{M} factors through the inclusion Z_{M}^{+} → Z_{M}.

Let us begin by verifying the commutativity of (3.4.14). This diagram is the direct product of two diagrams, the first involving Z_{G} and Z_{M}^{+}, and the second involving 𝔷(𝔤) and 𝔷(𝔪). We verify the commutativity of each factor separately.

If V is an object of Rep_{la.c}(G), then the action of Z_{G} on V preserves V^{N_{0}}. Thus for any z ∈ Z_{G}, the operator π_{N_{0},z} coincides with the action of z on V^{N_{0}}. Recalling that the action of Z_{M}^{+} on V^{N_{0}} is defined via the operators π_{N_{0},z}, the commutativity of the first factor of (3.4.14) follows.

Let γ′ : 𝔷(𝔤) → 𝔷(𝔪) denote the the map of proposition 1.3.3. If X ∈ 𝔷(𝔤), then we may write X = γ′(X) + X′, with X′ ∈ U(𝔤)𝔫. If v ∈ V^{N_{0}} ⊂ V^{𝔫}, then we see that Xv = γ′(X)v. Recalling that the U(𝔪)-action, and hence the 𝔷(𝔪)-action, on V^{N_{0}} is defined by differentiating the M_{0}-action on this space, the commutativity of the second factor of (3.4.14) follows.

We now turn to establishing the commutativity of (3.4.15). Again, we regard this diagram as the product of two factors, and treat each factor separately. We must first show that the Z_{M}^{+}-action on W_{fs}, induced by functoriality from the Z_{M}^{+}-action on W, coincides with the Z_{M}^{+}-action on W_{fs} obtained from regarding Z_{M}^{+} as a submonoid of Z_{M}. This follows from the definition of W_{fs} as L_{b,Z_{M}^{+}} (C^{an}(Ẑ_{M}, K), W), or alternatively, from the universal property of W_{fs}.

Secondly, we must show that the 𝔷(𝔪)-action on W_{fs}, induced by functoriality from the 𝔷(𝔪)-action on W, coincides with the 𝔷(𝔪)-action on W_{fs}, obtained by differentiating the M_{0}-action on W_{fs}. This follows from lemma 3.2.7 (ii), and so we are done. □

The preceding result implies in particular that when V is an object of Rep_{la.c}(G), the canonical lifting (3.4.8) is both Z_{G} and 𝔷(𝔤)-equivariant (if Z_{G} and 𝔷(𝔤) act on J_{P} (V) through the maps Z_{G} → Z_{M} and 𝔷(𝔤) → 𝔷(𝔪) considered above).

(3.5) In this subsection we characterize J_{P} as an adjoint functor. We begin by describing the functor to which it is adjoint. As in the preceding section we fix a parabolic subgroup P of G, and a Levi factor M of P, so that P = MN.

For U an object of Rep_{la.c} M, let C_{c}^{sm}(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 C_{c}^{sm}(N, U) as the inductive limit

(3.5.1)  C_{c}^{sm}(N, U) ⟵^{∼} lim_{\overrightarrow{N_{0},N_{1}}} F(N_{1}/N_{0}, U),

where N_{0} is an open subgroup of the compact open subgroup N_{1} of N, and the inductive limit is taken as N_{0} shrinks to the identity while N_{1} grows without bound. The quotient N_{1}/N_{0} is finite, and so F(N_{1}/N_{0}, U) (the space of U-valued functions on N_{1}/N_{0}) is isomorphic to a finite direct sum of copies of U. In this way, C_{c}^{sm}(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 C_{c}^{sm}(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 C_{c}^{sm}(N, U) equips this space with a smooth (and so locally analytic) action of N. We extend this to an action of P = MN by defining the following action of an element mn (with m ∈ M and n ∈ N) on an element f of C_{c}^{sm}(N, U):

(mnf)(n′) = mf(m^{−1}n′mn)

(the right-hand side denoting the action of m on the element f(m^{−1}n′mn) of U). One easily checks that this does yield an action of P on C_{c}^{sm}(N, U).

If N_{0} ⊂ N_{1} are compact open subgroups of N, then a sufficiently small open subgroup of M will leave both N_{0} and N_{1} invariant under conjugation. The description (3.5.1) of C_{c}^{sm}(N, U) and the fact that M acts locally analytically on U then implies that C_{c}^{sm}(N, U) is an object of Rep_{la.c} P. Its construction is clearly functorial in U, and so defines a functor Rep_{la.c} M → Rep_{la.c} P. We will show that the functor J_{P} is right adjoint (up to a twist) to (the restriction to Rep_{la.c}^{z} M of) the functor C_{c}^{sm}(N, – ).

As usual, we denote by δ the modulus character of P, which describes the effect of 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 [N_{0} : mN_{0}m^{−1}]^{−1}, for any compact open subgroup N_{0} of N. For an object U of Rep_{la.c} M, we let U(δ) denote the object of Rep_{la.c} M obtained by twisting the M-action on U by the character δ.

Lemma 3.5.2. If U is an object of Rep_{la.c}^{z} M, then there is a natural isomorphism

J_{P} (C_{c}^{sm}(N, U)) ⟶^{∼} U(δ).

Proof. Fix a compact open subgroup P_{0} of P, with respect to which we will compute J_{P}, and let M_{0}, M^{+}, N_{0}, Z_{M}^{+} and so on have the same meanings as in subsection 3.4. Composing the map (3.4.8) (with V taken to be C_{c}^{sm}(N, U)) with the map induced by evaluating at the identity of N yields a continuous map

(3.5.3)  J_{P} (C_{c}^{sm}(N, U)) → U.

We will show that this map is a topological isomorphism of vector spaces, that intertwines the action of M up to a twist by δ.

If u is an object of U, let f_{u} denote the element of C_{c}^{sm}(N, U) that is constant on N_{0}, with value equal to u, and that vanishes elsewhere. Clearly f_{u} is invariant under N_{0}, and one easily computes that π_{N_{0},m}(f_{u}) = δ(m)f_{mu} = f_{δ(m)mu} for each m ∈ M^{+}. Thus the association of f_{u} to u induces an M^{+}-equivariant closed embedding

(3.5.4)  U(δ) → (C_{c}^{sm}(N, U))^{N_{0}},

and hence (taking into account proposition 3.2.6 (iii) and lemma 3.2.8) an Mequivariant closed embedding

(3.5.5)  U(δ) → J_{P} (C_{c}^{sm}(N, U)).

It is easily checked that (3.5.5) provides an inverse to (3.5.3), and hence we have obtained our required isomorphism. (It is easily seen that this isomorphism is independent of the choice of the compact open subgroup P_{0} 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 Rep_{la.c} P and U is an object of Rep_{la.c}^{z} M, then passing to Jacquet modules yields a natural isomorphism

(3.5.7)  L_{P} (C_{c}^{sm}(N, U), V) ⟶^{∼} L_{M}(U(δ), J_{P} (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 of lemma 3.5.2, we fix P_{0}, N_{0} and so on, and consider the corresponding canonical lifting J_{P} (V) → V^{N_{0}}. If we are given ϕ ∈ L_{M}(U(δ), J_{P} (V)), we may compose it with the canonical lifting to obtain a map ϕ̃ : U(δ) → V^{N_{0}}, which intertwines the action of M^{+} on the source with the action of the Hecke operators π_{N_{0},m} (for m ∈ M^{+}) on the target. For any z ∈ Z_{M}^{+} and f ∈ C^{sm}(N_{0}, U) define I_{z}(f) ∈ V via

I_{z}(f) = δ(z)^{−1} ∫_{N} n^{−1}zϕ̃(z^{−1}f(n))dn.

(Here z^{−1} acts on the values f(n) via the action of Z_{M} ⊂ M on U, and dn is Haar measure on N, normalized to give N_{0} measure one.) One checks (using the equivariance property of ϕ̃, and the fact that it takes values in V^{N_{0}}) that if z ∈ Z_{M}^{+} is chosen such that f is locally constant on the left cosets of zN_{0}z^{−1} in N, then I_{z}(f) = I_{zz′}(f) for any z′ ∈ Z_{M}^{+}. Thus the net I_{z}(f) (indexed by the elements of Z_{M}^{+}, directed by the relation of divisibility) is eventually constant, and in particular the limit I(f) := lim_{\overrightarrow{z∈Z{+ \atop M}}} I_{z}(f) exists. The map f ↦ I(f) is easily seen to yield an

element of L_{P} (C_{c}^{sm}(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 N_{0}M^{+} ⊂ P, and let C^{sm}(N_{0}, U) denote the subspace of C_{c}^{sm}(N, U) consisting of locally constant U-valued functions on N_{0}. One sees that P^{+} is a subsemigroup of P, that the P -action on C_{c}^{sm}(N, U) restricts to a P^{+}-action on C^{sm}(N_{0}, U), and that the embedding C^{sm}(N_{0}, U) → C_{c}^{sm}(N, U) induces an isomorphism

K[N] ⊗_{K[N_{0}]} C^{sm}(N_{0}, U) ⟶^{∼} C_{c}^{sm}(N, U),

and so also (since M^{+} generates M as a group) an isomorphism

(3.5.8)  L_{P} (C_{c}^{sm}(N, U), V) ⟶^{∼} L_{P^{+}}(C^{sm}(N_{0}, U), V).

The map (3.5.4) induces a map

(3.5.9)  L_{P^{+}}(C^{sm}(N_{0}, U), V) → L_{M^{+}}(U(δ), V^{N_{0}}).

Composing this on the left with (3.5.8), and on the right with the isomorphism

L_{M^{+}}(U(δ), V^{N_{0}}) ⟶^{∼} L_{M}(U(δ), J_{P} (V))

induced by the adjointness property of (V^{N_{0}})_{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 N_{0} obtained as the image of (3.5.4) clearly generate C^{sm}(N_{0}, U) as a P^{+}-representation. □

The adjointness isomorphism of the preceding theorem does not depend on the choice of the compact open subgroup P_{0} of P that is used to compute the functor J_{P}, since this is true of the isomorphism of lemma 3.5.2. However, it does depend on the choice of the Levi factor M of P (or equivalently, on the particular choice of \overline{P}), since the very definition of the functor C_{c}^{sm}(N, – ) depends on this choice.

If we take U = J_{P} (V) in theorem 3.5.6 then the identity map of J_{P} (V) to itself gives rise to a P -equivariant morphism

(3.5.10)  C_{c}^{sm}(N, J_{P} (V)) → V.

If we fix a compact open subgroup P_{0} of P, then the constant J_{P} (V)-valued functions on N_{0} form an M_{0}N_{0}-invariant subspace of the source of (3.5.10), which is isomorphic to J_{P} (V) itself (regarded as an M_{0}N_{0}-representation by having this product act through its quotient M_{0}). Thus (3.5.10) induces an M_{0}N_{0}-equivariant map J_{P} (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.

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 a locally 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 subgroup of G. Since H is Zariski dense in ℍ, there is at most one rigid analytic group structure 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 to an 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′ ⊂ H is an inclusion of open subgroups of G each of which is equipped with the structure of an analytic open subgroup of G, then we say that it is an inclusion of analytic open subgroups if it lifts to a rigid analytic map ℍ′ → ℍ on the associated rigid analytic groups. (Since H′ and H are Zariski dense in ℍ′ 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 subgroups of G. Since the group structure on G is locally analytic, the image of this map is cofinal in the directed set of all open subgroups of G.

We now introduce a particular class of analytic open subgroups of G (the “good analytic open subgroups”), obtained by applying the Campbell-Baker-Hausdorff formula to certain O_{L}-Lie subalgebras of 𝔤. We begin with some terminology. By an O_{L}-lattice in 𝔤 we mean an O_{L}-submodule 𝔥 of 𝔤 which is free of finite rank as an O_{L}-module, and which spans 𝔤 over L. The gauge of 𝔥 defines a norm on 𝔤, and we 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 sublattice of 𝔤, we mean an O_{L}-lattice 𝔥 in 𝔤 that is also an O_{L}-Lie subalgebra of 𝔤. Note that if 𝔥 is any O_{L}-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-Campbell-Hausdorff formula converges on 𝔥 and so defines a rigid analytic group structure on 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 subgroups of 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′ and H are two good analytic open subgroups of G, corresponding to Lie sublattices 𝔥′ and 𝔥 of 𝔤 respectively, then H′ ⊂ H if and only if 𝔥′ ⊂ 𝔥, and if these equivalent conditions hold, then the inclusion H′ ⊂ 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 open subgroup H of G. For any real number 0 < r < 1, we let ℍ_{r} denote the rigid analytic closed ball of radius r in 𝔤 with respect to the gauge of 𝔥. The open subdomain ℍ_{r} of 𝔥 is in fact a rigid analytic subgroup of ℍ, and if r equals the absolute value of an element of the algebraic closure \overline{L} of L, then ℍ_{r} is furthermore affinoid [2, thm. 6.1.5/4]. (Note that such elements r are dense in the interval (0, 1).) We let H_{r} denote the locally analytic group of L-valued points of ℍ_{r}. If we write ℍ^{∘} = ⋃_{0<r<1} ℍ_{r}, then ℍ^{∘} is a strictly σ-affinoid rigid analytic open subgroup of ℍ (i.e. it is the union of an increasing sequence of open affinoid subdomains, each relatively 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 Ad_{g}(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 that gHg^{−1} ⊂ H if and only if Ad_{g}(𝔥) ⊂ 𝔥, and that if these equivalent conditions hold, then the action of g on H by conjugation extends to a rigid analytic endomorphism of ℍ, and also of the rigid analytic groups ℍ_{r}, 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 G_{0} of G. The space C^{la}(G_{0}, K) is a compact type space equipped with a locally analytic action of G_{0}. (In fact, C^{la}(G_{0}, K) is the most fundamental example of a strongly admissible locally analytic G_{0}-representation.) Let H be a good analytic open subgroup of G, contained in G_{0}, such that G_{0} 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 ℍ_{r}, for each r ∈ (0, 1), and of ℍ^{∘}. (We say that G_{0} “normalizes” these rigid analytic groups.) Following [8, def. 3.4.1], we define

(4.1.1) C^{la}(G_{0}, K)_{ℍ^{∘}−an} = lim_{\overleftarrow{r}} C^{la}(G_{0}, K)_{ℍ_{r}−an} ⟶^{∼} lim_{\overleftarrow{r}} ⨁_{g∈G_{0}/H_{r}} C^{an}(gℍ_{r}, K).

(Here r ranges through those elements of the interval (0, 1) that are absolute values of elements of \overline{L}. If g ∈ G_{0}, then we let gℍ_{r} denote the coset gH_{r}, endowed with the structure of a rigid analytic space via the locally L-analytic isomorphism H_{r} ⟶^{∼} gH_{r} obtained via multiplication by g, together with the equality H_{r} = ℍ_{r}(L). The second isomorphism then follows from the natural isomorphism C^{la}(G_{0}, K) ⟶^{∼} ⨁_{g∈G_{0}/H_{r}} C^{la}(gH_{r}, K) together with [8, cor. 3.3.26].) Since the inclusion ℍ_{r} ⊂ ℍ_{r′} is relatively compact if r < r′, we see that C^{la}(G_{0}, K)_{ℍ^{∘}−an} is the projective limit of K-Banach spaces under compact transition maps, and so is a nuclear Fréchet space. (See [8, prop. 2.1.16], for example.) Since G_{0} normalizes each ℍ_{r}, we see that C^{la}(G_{0}, K)_{ℍ^{∘}−an} is naturally equipped with a continuous G_{0}-action.

If D(ℍ^{∘}, G_{0}) denotes the strong dual to C^{la}(G_{0}, K)_{ℍ^{∘}−an}, then D(ℍ^{∘}, G_{0}) is a locally convex topological K-vector space of compact type. In fact the definition of C^{la}(G_{0}, K)_{ℍ_{an}^{∘}} induces an isomorphism

(4.1.2) D(ℍ^{∘}, G_{0}) := (C^{la}(H, K)_{ℍ^{∘}−an})′_{b} ⟶^{∼} lim_{\overrightarrow{r}} ( ⨁_{g∈G_{0}/H_{r}} C^{an}(gℍ_{r}, K))′_{b}

⟶^{∼} lim_{\overrightarrow{r}} ⨁_{g∈G_{0}/ℍ_{r}} δ_{g} ∗ D^{an}(ℍ_{r}, K),

where D^{an}(ℍ_{r}, K) denotes the strong dual to C^{an}(ℍ_{r}, K), and δ_{g} denotes the δfunction supported at g. Each of the spaces D^{an}(ℍ_{r}, K) is naturally a K-Banach algebra [8, cor. 5.1.8]. Since G_{0} normalizes each ℍ_{r}, each of the direct sums ⨁_{g∈G_{0}/H_{r}} δ_{g} ∗ D^{an}(ℍ_{r}, K) also has a natural structure of K-Banach algebra. Finally, the space D(ℍ^{∘}, G_{0}) 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^{∘}, G_{0}), while a priori separately continuous, is in fact jointly continuous, since D(ℍ^{∘}, G_{0}) is of compact type [8, prop. 1.1.31].)

We now return to the setting of the introduction, in which G is supposed to be 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 a neighbourhood basis of the identity consisting of a descending sequence {H_{n}}_{n≥0} of good analytic open subgroups with certain additional nice properties. Among these is the existence of certain rigid analytic Iwahori decompositions for each of the groups H_{n}. We begin by explaining what we mean by this.

Let H_{n} be a good analytic open subgroup belonging to our sort-for neighbourhood basis, let ℙ be a parabolic subgroup of 𝔾, let \overline{ℙ} be a parabolic subgroup opposite to ℙ, and let 𝕄 = ℙ ⋂ \overline{ℙ} be the common Levi factor of each of ℙ and \overline{ℙ}. Let ℕ (respectively \overline{ℕ}) denote the unipotent radical of ℙ (respectively \overline{ℙ}). Write P, \overline{P}, M, N, and \overline{N} to denote the corresponding groups of L-valued points. Set M_{n} := H_{n} ⋂ M, N_{n} := H_{n} ⋂ N, and \overline{N}_{n} := H_{n} ⋂ \overline{N}, and let 𝕄_{n}, ℕ_{n}, and \overline{ℕ}_{n} denote the rigid analytic Zariski closure of M_{n}, N_{n}, and \overline{N}_{n} in ℍ_{n} (the rigid analytic group underlying the good analytic subgroup H_{n}). Let 𝔥_{n} denote the Lie sublattice of 𝔤 corresponding to H_{n}, and write 𝔪_{n} := 𝔥_{n} ⋂ 𝔪, 𝔫_{n} := 𝔥_{n} ⋂ 𝔫, and \overline{𝔫}_{n} := 𝔥_{n} ⋂ \overline{𝔫}.

Definition 4.1.3. We will say that the good analytic open subgroup H_{n} admits a rigid analytic Iwahori decomposition with respect to ℙ and \overline{ℙ} if:

(i) Under the identification H_{n} = 𝔥_{n}, the groups M_{n}, N_{n}, and \overline{N}_{n} are identified with 𝔪_{n}, 𝔫_{n}, and \overline{𝔫}_{n} respectively. Thus M_{n}, N_{n}, and \overline{N}_{n} coincide with the good analytic open subgroups of M, N, and \overline{N} obtained by exponentiating 𝔪_{n}, 𝔫_{n}, and \overline{𝔫}_{n}, respectively, and 𝕄_{n}, ℕ_{n}, and \overline{ℕ}_{n} coincide with the rigid analytic groups underlying M_{n}, N_{n}, and \overline{N}_{n}, respectively.

(ii) The rigid analytic map

(4.1.4)  \overline{ℕ}_{n} × 𝕄_{n} × ℕ_{n} → ℍ_{n}

induced by multiplication in ℍ_{n} is in fact a rigid analytic isomorphism.

Note that if (4.1.4) is an isomorphism, then the corresponding map

(4.1.5)  \overline{N}_{n} × M_{n} × N_{n} → H_{n}

obtained 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 Levi factor of ℙ_{∅}. Denote by \overline{ℙ}_{∅} the opposite parabolic to ℙ_{∅}, chosen so that 𝕄_{∅} := ℙ_{∅} ⋂ \overline{ℙ}_{∅}.

The following result refines [5, prop. 1.4.4].

Proposition 4.1.6. We may find a decreasing sequence {H_{n}}_{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 H_{n+1} ⊂ H_{n} extends to a relatively compact rigid analytic map ℍ_{n+1} ⊂ ℍ_{n}.

(ii) For each n ≥ 0, the subgroup H_{n} of H_{0} is normal.

The remaining properties refer to any pair ℙ and \overline{ℙ} of opposite parabolic subgroups of 𝔾, chosen so that ℙ contains ℙ_{∅} and \overline{ℙ} contains \overline{ℙ}_{∅}. We employ the notation introduced in the discussion preceding definition 4.1.3.

(iii) Each H_{n} admits a rigid analytic Iwahori decomposition with respect to ℙ and \overline{ℙ}.

(iv) If z ∈ Z_{M} is such that z^{−1}\overline{N}_{0}z ⊂ \overline{N}_{0}, then z^{−1}\overline{N}_{n}z ⊂ \overline{N}_{n} for each n ≥ 0.

(v) If z ∈ Z_{M} is such that zN_{0}z^{−1} ⊂ N_{0}, then zN_{n}z^{−1} ⊂ N_{n} for each n ≥ 0. (vi) We may find z ∈ Z_{M} such that z^{−1}\overline{N}_{0}z ⊂ \overline{N}_{0} and zN_{0}z^{−1} ⊂ N_{0}, and

such that, for each n ≥ 0, the embedding of part (iv) factors through the inclusion \overline{N}_{n+1} ⊂ \overline{N}_{n}.

Proof. Let 𝔞_{∅} denote the Lie algebra of 𝔸_{∅}, let 𝔪_{∅} denote Lie algebra of 𝕄_{∅}, and write 𝔤 = 𝔪_{∅} ⊕ ⨁_{α∈±Δ(𝔾,𝔸_{∅})} 𝔤_{α}, where Δ(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 𝔤_{α} the corresponding restricted root space. Fix an O_{L}-lattice 𝔥′ in 𝔤, fix a ∈ K^{×} of sufficiently large valuation, and set

𝔥 := (a𝔥′ ⋂ 𝔪_{∅}) ⊕ ⨁_{α∈±Δ(𝔾,𝔸_{∅})} (a𝔥′ ⋂ 𝔤_{α}) .

More generally, if π is a uniformizer of L, set 𝔥_{n} := π^{n}𝔥. Then, provided that the valuation of a was chosen large enough, we see that each 𝔥_{n} is a Lie sublattice of 𝔤 which exponentiates to a good analytic open subgroup H_{n} of G. Since the sequence {𝔥_{n}} forms a neighbourhood basis of zero in the Lie algebra 𝔤, we see that {H_{n}} forms a neighbourhood basis of the identity in G. As above, for each n ≥ 0, let ℍ_{n} denote the rigid analytic group underlying H_{n}. By construction, the inclusion H_{n+1} → H_{n} extends to a rigid analytic embedding ℍ_{n+1} → ℍ_{n}, for each value of n, and (again by construction) this embedding is relatively compact. Thus we have established (i). Property (ii) is clear from our construction.

By construction, the natural map

(4.1.7)  (𝔥_{n} ⋂ 𝔪_{∅}) ⊕ ⨁_{α∈±Δ(𝔾,𝔸_{∅})} (𝔥_{n} ⋂ 𝔤_{α}) ⟶^{∼} 𝔥_{n}

is an isomorphism, for each n. Suppose now that ℙ is a parabolic subgroup of 𝔾 that contains ℙ_{∅}, that \overline{ℙ} is the corresponding opposite parabolic, chosen to contain \overline{ℙ}_{∅}, 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 \overline{ℕ} denote the unipotent radical of \overline{ℙ}. We write 𝔪, 𝔫, and \overline{𝔫} for the Lie algebras of 𝕄, ℕ, and \overline{ℕ}. The structure theory of parabolic subgroups of reductive groups shows that there is a subset Φ of Δ(𝔾, 𝔸_{∅}) such that 𝔪 = 𝔪_{∅} ⊕ ⨁_{α∈±Φ} g_{α}, 𝔫 = ⨁_{α∈Δ(𝔾,𝔸_{∅})\Φ} 𝔤_{α}, and \overline{𝔫} = ⨁_{α∈Δ(𝔾,𝔸_{∅})\Φ} g_{−α}. It thus follows from the the fact that (4.1.7) is an isomorphism that the natural map

(4.1.8)  \overline{𝔫}_{n} ⨁ 𝔪_{n} ⨁ 𝔫_{n} → 𝔥_{n}

is an isomorphism, for each n ≥ 0. If we fix a value of n, then each of \overline{𝔫}_{n}, 𝔪_{n}, and 𝔫_{n} exponentiates to a good analytic open subgroup \overline{N}_{n} of \overline{N}, M_{n} of M, and N_{n} of N, respectively. Following the convention introduced above, we write \overline{ℕ}_{n}, 𝕄_{n}, and ℕ_{n} to denote the rigid analytic groups underlying the good analytic subgroups \overline{N}_{n}, M_{n}, and N_{n}. These groups are also equal to the Zariski closures in ℍ_{n} of \overline{N}_{n}, M_{n}, and N_{n}, respectively. Increasing the valuation of a if necessary, we furthermore find that \overline{N}_{n} = H_{n} ⋂ \overline{N}, that M_{n} = H_{n} ⋂ M, that N_{n} = H_{n} ⋂ N, and (taking into account the fact that (4.1.8) is an isomorphism) that the natural rigid analytic map (4.1.4) is an isomorphism. Altogether, we have established (iii).

Note that \overline{N}_{0} is closed under conjugation by z^{−1}, for some z ∈ Z_{M}, if and only if 𝔥 ⋂ 𝔫 is closed under the action of Ad_{z^{−1}}. This in turn is true if and only if π^{n}𝔥 ⋂ 𝔫 is closed under the action of Ad_{z^{−1}}, which holds if and only if \overline{N}_{n} is closed under conjugation by z^{−1}. Thus (iv) is proved, and (v) is proved similarly. The theory of roots shows that we may find z ∈ Z_{M} such that Ad_{z}(𝔥 ⋂ 𝔫) ⊂ 𝔫 and Ad_{z}(𝔥 ⋂ \overline{𝔫}) ⊂ π\overline{𝔫}. This establishes (vi). □

In the context of the preceding proposition, we may apply the construction of (4.1.2). Namely, taking the pair (G_{0}, H) to be the pair (H_{0}, H_{n}) for each n ≥ 0 in turn, we obtain for each n ≥ 0 a compact type topological K-algebra D(ℍ_{n}^{∘}, H_{0}) as the strong dual to the nuclear Fréchet space C^{la}(H_{0}, K)_{ℍ_{n}^{∘}−an}. The natural isomorphism C^{la}(H_{0}, K) ⟶^{∼} lim_{\overrightarrow{n}} C^{la}(H_{0}, K)_{ℍ_{n}^{∘}−an} induces a natural isomorphism

(4.1.9)  D^{la}(H_{0}, K) ⟶^{∼} lim_{\overleftarrow{n}} D(ℍ_{n}^{∘}, H_{0}).

(Compare the proof of [8, prop. 5.3.1].) We may similarly apply the construction of (4.1.2) to the pair (M_{0}, M_{n}), for each n ≥ 0. Thus we obtain for each n a compact type topological K-algebra D(𝕄_{n}^{∘}, M_{0}), and a natural isomorphism

(4.1.10)  D^{la}(M_{0}, K) ⟶^{∼} lim_{\overleftarrow{n}} D(𝕄_{n}^{∘}, M_{0}).

The closed embedding M_{0} → H_{0} gives rise to a closed embedding of K-Fréchet algebras

(4.1.11)  D^{la}(M_{0}, K) → D^{la}(H_{0}, 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}^{∘}, M_{0}) → D(ℍ_{n}^{∘}, H_{0}),

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, and define certain additional good analytic open subgroups of G. Let Z^{+} denote the submonoid of Z_{M} consisting of those z such that zN_{0}z ⊂ N_{0} and z^{−1}\overline{N}_{0}z ⊂ \overline{N}_{0}. An evident variant of proposition 3.3.2 (i) shows that Z^{+} generates Z_{M} as a group.

If n ≥ 0, and if z ∈ Z^{+}, then write H(z)_{n} := (z^{−1}H_{n}z) ⋂ H_{n}. Since H_{n} admits the Iwahori decomposition (4.1.5), we find that z^{−1}H_{n}z admits the corresponding Iwahori decomposition z^{−1}\overline{N}_{n}z ×M_{n} ×z^{−1}N_{n}z ⟶^{∼} z^{−1}H_{n}z, and thus (taking into 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}\overline{N}_{n}z × M_{n} × N_{n} ⟶^{∼} H(z)_{n}.

If 𝔥_{n} is the O_{L}-Lie subalgebra of 𝔤 corresponding to the good analytic open subgroup H_{n}, then we see that 𝔥(z)_{n} := Ad_{z^{−1}}𝔥_{n} ⋂ 𝔥_{n} is an O_{L}-Lie subalgebra of 𝔥_{n}, and that the identification of 𝔥_{n} with H_{n} yields an identification of 𝔥(z)_{n} with H(z)_{n}. This latter identification gives H(z)_{n} the structure of a good analytic open subgroup of H_{n}; 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}\overline{ℕ}_{n}z × 𝕄_{n} × ℕ_{n} ⟶^{∼} ℍ(z)_{n}.

(Recall that z^{−1}\overline{ℕ}_{n}z denotes the image of \overline{ℕ}_{n} 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 topological K-algebras

D^{la}(H(z)_{0}, K) ⟶^{∼} lim_{\overleftarrow{n}} D(ℍ(z)_{n}^{∘}, H(z)_{0}).

The inclusion H(z)_{0} ⊂ H_{0} induces a continuous homomorphism of K-Fréchet algebras

D^{la}(H(z)_{0}, K) → D^{la}(H_{0}, K),

and a consequent isomorphism

D^{la}(H_{0}, K) ≅ ⨁_{h∈H(z)_{0}\H_{0}} D^{la}(H(z)_{0}, K) ∗ δ_{h}.

The same inclusion H(z)_{0} ⊂ H_{0}, 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}^{∘}, H_{0}).

We define

D(ℍ(z)_{n}^{∘}, H_{0}) := D(ℍ(z)_{n}^{∘}, H(z)_{0}) ⊗_{D^{la}(H(z)_{0},K)} D^{la}(H_{0}, K)

⟶^{∼} ⨁_{h∈H(z)_{0}\H_{0}} D(ℍ(z)_{n}^{∘}, H_{0}) ∗ δ_{h}.

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

(4.1.16)  D(ℍ(z)_{n}^{∘}, H_{0}) → D(ℍ_{n}^{∘}, H_{0}).

(The left D(ℍ(z)_{n}^{∘}, H(z)_{0})-module structure on the target is defined via (4.1.15), and the right D^{la}(H_{0}, K)-module structure via the homomorphism D^{la}(H_{0}, K) → D(ℍ_{n}^{∘}, H_{0}) 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}^{∘}, M_{0}) → 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}^{∘}, M_{0}) on D(ℍ(z)_{n}^{∘}, H(z)_{0}), and also on D(ℍ(z)_{n}^{∘}, H_{0}). 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}^{∘}, M_{0})-modules.

Finally, suppose that z ∈ Z^{+} is chosen so that z^{−1}\overline{N}_{n}z ⊂ \overline{N}_{n+1}. We then define H(z)_{n,n+1} := (z^{−1}H_{n}z) ⋂ H_{n+1}. Just as with the open subgroup H(z)_{n}, we see that H(z)_{n,n+1} admits the structure of a good analytic open subgroup associated to the O_{L}-Lie subalgebra Ad_{z^{−1}}𝔥_{n} ⋂ 𝔥_{n+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}\overline{ℕ}_{n}z × 𝕄_{n+1} × ℕ_{n+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}), D^{la}(H_{0}, K))-bimodule

D(ℍ(z)_{n,n+1}^{∘}, H_{0}) := D(ℍ(z)_{n,n+1}^{∘}, H(z)_{0}) ⊗_{D^{la}(H(z)_{0},K)} D^{la}(H_{0}, K)

⟶^{∼} ⨁_{h∈H(z)_{0}\H_{0}} D(ℍ(z)_{n,n+1}^{∘}, H_{0}) ∗ δ_{h}.

Analogously to (4.1.17), there is a closed embedding of topological K-algebras

D(𝕄_{n+1}^{∘}, M_{0}) → D(ℍ(z)_{n}^{∘}, H(z)_{0}),

and thus D(ℍ(z)_{n,n+1}^{∘}, H_{0}) may also be regarded as a (D(𝕄_{n+1}^{∘}, M_{0}), D^{la}(H_{0}, K))- bimodule. The diagram of inclusions of good analytic open subgroups

\xymatrix{H(z)_{n,n+1} \ar[r] \ar[d] & H(z)_{n} \ar[d] \\ H_{n+1} \ar[r] & H_{n}}

induces a diagram of continuous maps of topological (D(𝕄_{n+1}^{∘}, M_{0}), D^{la}(H_{0}, K))- bimodules

(4.1.19)  \xymatrix{D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} D(ℍ(z)_{n,n+1}^{∘}, H_{0}) \ar[r] \ar[d] & D(ℍ(z)_{n}^{∘}, H_{0}) \ar[d] \\ D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} D(ℍ_{n+1}^{∘}, H_{0}) \ar[r] & D(ℍ_{n}^{∘}, H_{0}) .}

(4.2) In this subsection we prove theorem 0.5, that is, that J_{P} restricts to a functor Rep_{es}(G) → Rep_{es}(M). Equivalently, we will show that if V is an object of Rep_{es}(G), then J_{P} (V)′_{b} is a coadmissible C^{an}(Ẑ_{M}, K) ˆ⊗_{K} D^{la}(M_{0}, K)-module. Our 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 𝕄 of ℙ, let \overline{ℙ} denote the opposite parabolic to ℙ with respect to 𝕄 (so that 𝕄 = ℙ ⋂ \overline{ℙ}), and let \overline{ℕ} denote the unipotent radical of \overline{ℙ}. Also let \overline{P} := \overline{ℙ}(L) and \overline{N} := \overline{ℕ}(L). We apply proposition 4.1.6 to an opposite pair of minimal parabolic subgroups of 𝔾 that are compatible with ℙ and \overline{ℙ}, so as to find a decreasing sequence {H_{n}}_{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 \overline{ℙ} taken to be the particular parabolic subgroups just introduced.) We will use the compact type algebras D(ℍ_{n}^{∘}, H_{0}) and D(𝕄_{n}^{∘}, M_{0}) introduced in subsection 4.1, as well as the (D(𝕄_{n}^{∘}, M_{0}), D^{la}(H_{0}, K))-bimodules D(ℍ(z)_{n}^{∘}, H_{0}) (for n ≥ 0, and z ∈ Z^{+}) and the (D(𝕄_{n+1}^{∘}, M_{0}), D^{la}(H_{0}, K))-bimodules D(ℍ(z)_{n,n+1}^{∘}, H_{0}) (for n ≥ 0 and z ∈ Z^{+} such that z^{−1}\overline{N}_{n}z ⊂ \overline{N}_{n+1}).

Choose an exhaustive increasing sequence {Ẑ_{G,n}}_{n≥0} of admissible open affinoid subdomains of Ẑ_{G}, such that each inclusion Ẑ_{G,n} ⊂ Ẑ_{G,n+1} is relatively compact. If C^{an}(Ẑ_{G,n}, K)^{†} denotes the space of overconvergent (in Ẑ_{G}) rigid analytic functions on Ẑ_{G,n}, then C^{an}(Ẑ_{G,n}, K)^{†} is a locally convex topological K-algebra of compact type, and there is a natural isomorphism

(4.2.1)  C^{an}(Ẑ_{G}, K) ⟶^{∼} lim_{\overleftarrow{n}} C^{an}(Ẑ_{G,n}, K)^{†}.

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

C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(M_{0}, K) ⟶^{∼} lim_{\overleftarrow{n}} C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}).

This isomorphism describes a weak Fréchet-Stein structure (in the sense of [8, def. 1.2.6]) on the nuclear Fréchet algebra C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(M_{0}, K).

Recall that Z_{M}^{+} denotes the submonoid of Z_{M} consisting of elements z for which zN_{0}z^{−1} ⊂ N_{0}, and that the operators π_{N_{0},z} (for z ∈ Z_{M}^{+}) define an action of Z_{M}^{+} on V^{N_{0}}. We let Z^{+} denote the submonoid of Z_{M}^{+} consisting of elements z such that z^{−1}\overline{N}_{0}z ⊂ \overline{N}_{0}. As mentioned in the preceding section, Z^{+} generates Z_{M} as a group, and so lemma 3.2.19 shows that we may compute J_{P} (V) as the finite slope part of V^{N_{0}}, regarded as a Z^{+}-module. We will apply proposition 3.2.24 to V^{N_{0}}, taking A to be the nuclear Fréchet algebra C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(M_{0}, K), and A_{n} to be C^{an}(Ẑ_{G,N}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}), for each n ≥ 0.

For proposition 3.2.24 to apply, we must introduce the following additional data:

(4.2.2) For each n ≥ 0, a C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})-module U_{n} as well as a C^{an}(Ẑ_{G,n+1}, K)^{†} ˆ⊗_{K} D(𝕄_{n+1}^{∘}, M_{0})-linear transition map U_{n+1} → U_{n}, and a C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(M, K)-linear topological isomorphism (V^{N_{0}})′_{b} ⟶^{∼} lim_{\overrightarrow{n}} U_{n}.

(4.2.3) A C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})-linear action of Z^{+} on each U_{n}, such that the transition maps U_{n+1} → U_{n} and the maps (V^{N_{0}})′_{b} → U_{n} are Z^{+}-equivariant.

We must then establish the following hypotheses:

(4.2.4) For each n ≥ 0, the induced map

C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{C^{an}(Ẑ_{G,n+1},K)^{†} ˆ⊗_{K} D(𝕄_{n+1}^{∘},M_{0})} U_{n+1} → U_{n}

is C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})-compact (in the sense of definition 2.3.3).

(4.2.5) There exists z ∈ Z^{+} such that the endomorphism of each U_{n} induced by z factors through the map of (4.2.4), so as to render commutative the diagram

\xymatrix{C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{C^{an}(Ẑ_{G,n+1},K)^{†} ˆ⊗_{K} D(𝕄_{n+1}^{∘},M_{0})} U_{n+1} \ar[r] \ar[d]_{id ˆ⊗ z} & U_{n} \ar[d]^{z} \ar[dl] \\ C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{C^{an}(Ẑ_{G,n+1},K)^{†} ˆ⊗_{K} D(𝕄_{n+1}^{∘},M_{0})} U_{n+1} \ar[r] & U_{n}.}

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

C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(H_{0}, K) ⟶^{∼} lim_{\overleftarrow{n}} C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(ℍ_{n}^{∘}, H_{0}).

This isomorphism describes a weak Fréchet-Stein structure (in the sense of [8, def. 1.2.6]) on the nuclear Fréchet algebra C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(H_{0}, K).

The strong dual V′_{b} is a coadmissible C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(H_{0}, K)-module (since V is an object of Rep_{es}(G)) and so [8, thm. 1.2.11 (i)] yields a natural isomorphism

V′_{b} ⟶^{∼} lim_{\overleftarrow{n}} C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{C^{an}(Ẑ_{G},K) ˆ⊗_{K} D^{la}(H_{0},K)} V′_{b},

where each of the tensor products

C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{C^{an}(Ẑ_{G},K) ˆ⊗_{K} D^{la}(H_{0},K)} V′_{b}

is a finitely generated topological C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(ℍ_{n}^{∘}, H_{0})-module. Since the strong dual to V^{N_{0}} is naturally identified with the Hausdorff N_{0}-coinvariants (V′_{b})_{N_{0}} of the strong dual to V, we obtain an isomorphism

(4.2.6) (V^{N_{0}})′_{b} ⟶^{∼}

lim_{\overleftarrow{n}} (C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{C^{an}(Ẑ_{G},K)} (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}) .

(Here (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} denotes the Hausdorff N_{0}-coinvariants of the completed tensor product D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b}.)

For each n ≥ 0, define

U_{n} := (C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{C^{an}(Ẑ_{G},K)} (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H,K)} V′_{b})_{N_{0}}) .

The completed tensor product D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b} is naturally a topological D(ℍ_{n}^{∘}, H_{0})-module, and hence is also (taking into account (4.1.12)) a topological D(𝕄_{n}^{∘}, M_{0})-module. As M_{0} normalizes N_{0}, the space (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} inherits a natural quotient D(𝕄_{n}^{∘}, M_{0})-module structure (this is “dual” to the fact that V^{N_{0}} is an M_{0}-invariant subspace of V), and hence U_{n} is naturally a compact type topological C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})-module for each value of n. Furthermore, the isomorphism (4.2.6) may be rewritten as an isomorphism

(V^{N_{0}})′_{b} ⟶^{∼} lim_{\overrightarrow{n}} U_{n}.

We next turn to explaining the action of the monoid Z^{+} on the modules U_{n}. 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}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b} → D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b}

of left D(ℍ(z)_{n}^{∘}, H(z)_{0})-modules. The preceding discussion generalizes to show that the D(𝕄_{n}^{∘}, M_{0})-action on D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b} induced by (4.1.17) descends to an action of D(𝕄_{n}^{∘}, M_{0}) on (D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}, and that (4.2.7) induces a map of D(𝕄_{n}^{∘}, M_{0})-modules

(4.2.8)  (D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} → (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}.

For each z ∈ Z^{+}, we are going to define a continuous D(𝕄_{n}^{∘}, M_{0})-linear map

(4.2.9)  (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} → (D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}},

in such a way that the diagram

(4.2.10)  \xymatrix{(V^{N_{0}})′_{b} \ar[r]^{∼} \ar[d]^{(π_{N_{0},z})′} & (V′_{b})_{N_{0}} \ar[r]^{v′↦1⊗v′} & (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} \ar[d]^{(4.2.9)} \\ (V^{N_{0}})′_{b} \ar[r]^{∼} & (V′_{b})_{N_{0}} \ar[r]^{v′↦1⊗v′} \ar[dr]^{v′↦1⊗v′} & (D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} \ar[d]^{(4.2.8)} \\  &  & (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}}

commutes. (Here, as indicated on the diagram, the left-hand vertical arrow is given by the action of z ∈ Z^{+} on (V′_{b})_{N_{0}}, obtained as the transpose of the operator π_{N_{0},z} on V^{N_{0}}.) Since the horizontal arrows have dense image (because the maps D^{la}(H_{0}, K) → D(ℍ_{n}^{∘}, H_{0}) and D^{la}(H_{0}, K) → D(ℍ(z)_{n}^{∘}, H_{0}) have dense image, as one sees by noting that their strong duals are the injections C(H_{0}, K)_{ℍ_{n}^{∘}−an} → C^{la}(H_{0}, K) and C(H_{0}, K)_{ℍ(z)_{n}^{∘}−an} → C^{la}(H_{0}, K) respectively), the map (4.2.9) is uniquely determined, if it exists.

Lemma 4.2.11. The morphism (4.2.9) can be constructed so as to make (4.2.10) commute.

Proof. Let λ : H_{0} → \overline{N}_{0} × M_{0} =: \overline{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 N_{0} (we use the symbol x rather than n so as to avoid confusion with the index n of the analytic open subgroup H_{n}), then

(4.2.12)  h ↦ z^{−1}λ(x^{−1}λ(h^{−1}))^{−1}z

describes a locally analytic map H_{0} → z^{−1}\overline{P}_{0}z ⊂ H_{0}. This map induces a continuous linear endomorphism s_{x,z} of C^{la}(H_{0}, K), defined for f ∈ C^{la}(H_{0}, K) via

(s_{x,z}f)(h) = f(z^{−1}λ(x^{−1}λ(h^{−1}))^{−1}z).

We regard H_{0} as the locally analytic group underlying the rigid analytic group

(4.2.13)  ∐_{h∈H_{0}/ℍ_{n}^{∘}(L)} hℍ_{n}^{∘}.

(Note that proposition 4.1.6 (ii) implies that we obtain the same decomposition of H_{0} 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}(H_{0}, K)_{ℍ_{n}^{∘}−an}. We simultaneously regard H_{0} as the locally analytic space underlying the rigid analytic space

(4.2.14)  ∐_{h∈H_{0}/ℍ(z)_{n}^{∘}(L)} hℍ(z)_{n}^{∘}.

The space of rigid analytic K-valued functions on this rigid analytic space is precisely C^{la}(H_{0}, K)_{ℍ(z)_{n}^{∘}−an}.

Since H_{n} admits a rigid analytic Iwahori decomposition, the restriction to H_{n} of λ induces a rigid analytic map ℍ_{n} → \overline{ℕ}_{n} × 𝕄_{n}, and hence also a rigid analytic map ℍ_{n}^{∘} → \overline{ℕ}_{n}^{∘} × 𝕄_{n}^{∘}. Thus (taking into account proposition 4.1.6 (iv) as well) the map (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 s_{x,z} induces a continuous linear map C^{la}(H_{0}, K)_{ℍ(z)_{n}^{∘}−an} → C^{la}(H_{0}, K)_{ℍ_{n}^{∘}−an}. Passing to the transpose, we obtain a continuous linear map s′_{x,z} : D(ℍ_{n}^{∘}, H_{0}) → D(ℍ(z)_{n}^{∘}, H_{0}). Since (4.2.12) is left M_{0}- equivariant, this map is in fact D(𝕄_{n}^{∘}, M_{0})-linear.

As in subsection 3.5, let δ denote the modulus character of P. Let x run over a set of coset representatives for zN_{0}z^{−1} in N_{0}, and define a map

(4.2.15)  D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{K} V′_{b} → D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{K} V′_{b}

via

(4.2.16)  μ ⊗ v′ ↦ δ(z) ∑_{x} s′_{x,k}(μ) ⊗ z^{−1}x^{−1}v′.

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

(4.2.17)  K[H_{0}] ⊗_{K} V′_{b} → K[H_{0}] ⊗_{K} V′_{b}.

In lemma 4.2.19 below we will establish the commutativity of the diagram

(4.2.18)  \xymatrix{K[H_{0}] ⊗_{K} V′_{b} \ar[r] \ar[d]^{(4.2.17)} & (V′_{b})_{N_{0}} \ar[r]^{∼} & (V^{N_{0}})′_{b} \ar[d]^{(π_{N_{0},z})′} \\ K[H_{0}] ⊗_{K} V′_{b} \ar[r] & (V′_{b})_{N_{0}} \ar[r]^{∼} & (V^{N_{0}})′_{b}}

(in which the left horizontal arrows are induced by the action of H_{0} on V′_{b}, together with the quotient map V′_{b} → (V′_{b})_{N_{0}}). In particular, the map (4.2.17) descends to a map (V′_{b})_{N_{0}} → (V′_{b})_{N_{0}}. Since K[H_{0}] ⊗_{K} V′_{b} is dense in each of D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{K} V′_{b} and D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{K} V′_{b}, we infer that (4.2.15) similarly descends to a map (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} → (D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}. If we take this map to be our sought-after morphism (4.2.9), then the commutativity of (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^{N_{0}}, v′ ∈ V′_{b}, and h ∈ H_{0}, there is an equality

(4.2.20)  δ(z) ∑_{x}⟨s′_{x,z}(δ_{h}) ⊗ z^{−1}x^{−1}v′, v⟩ = ⟨hv′, π_{N_{0},z}(v)⟩.

(Here we have denoted by δ_{h} the element h ∈ H_{0} regarded as an element of the group ring K[H_{0}], since this element is identified with the delta function supported at h under the natural embedding K[H_{0}] → D^{la}(H_{0}, K); it has nothing to do with the modulus δ.) It is an immediate consequence of the definition of s_{x,z} that s′_{x,z}(δ_{h}) = δ_{z^{−1}λ(x^{−1}λ(h^{−1}))^{−1}z}. Thus we compute

δ(z) ∑_{x}⟨s′_{x,z}(δ_{h}) ⊗ z^{−1}x^{−1}v′, v⟩ = δ(z) ∑_{x}⟨δ_{z^{−1}λ(x^{−1}λ(h^{−1}))^{−1}z} ⊗ z^{−1}x^{−1}v′, v⟩

= δ(z) ∑_{x}⟨v′, xzz^{−1}λ(x^{−1}λ(h^{−1}))zv⟩ = δ(z) ∑_{x}⟨v′, xλ(x^{−1}λ(h^{−1}))zv⟩

= δ(z) ∑_{x}⟨v′, h^{−1}hxλ(x^{−1}λ(h^{−1}))zv⟩.

Lemma 4.2.21 below shows that as x ranges over a set of right coset representatives of zN_{0}z^{−1} in N_{0}, the same is true of the elements hxλ(x^{−1}λ(h^{−1})). Thus if we take into account the definition of π_{N_{0},z}, we find that

δ(z) ∑_{x}⟨s′_{x,z}(δ_{h}) ⊗ x^{−1}z^{−1}v′, v⟩ = ⟨v′, h^{−1}π_{N_{0},z}v⟩.

This establishes (4.2.20). □

Lemma 4.2.21. If x ranges over a set of right coset representatives of zN_{0}z^{−1} in N_{0}, then for any h ∈ H_{0}, the elements hxλ(x^{−1}λ(h^{−1})) also range through a set of right coset representatives of zN_{0}z^{−1} in N_{0}.

Proof. Define ρ : H_{0} → N_{0} via the equation h = λ(h)ρ(h) for any h ∈ H_{0}. (Thus the product λ × ρ yields the isomorphism (4.1.5) for n = 0.) Note that hλ(h^{−1}) = ρ(h^{−1})^{−1} ∈ N_{0}, and thus that the lemma is true if and only if the elements

λ(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 zN_{0}z^{−1} in N_{0}. Write \overline{p} = λ(h^{−1}) ∈ \overline{P}_{0}. Since the inverses of a system of right coset representatives form a system of left coset representatives, it is equivalent to show that, as x ranges over a set of left coset representatives of zN_{0}z^{−1} in N_{0}, the elements ρ(x\overline{p}) range over such a set of left coset representatives.

Let x and x′ be two elements of N_{0} that lie in the given set of left coset representatives of zN_{0}z^{−1} in N_{0}. Then x\overline{p} and x′\overline{p} are both elements of H_{0}, and so we may write x\overline{p} = λ(x\overline{p})ρ(x\overline{p}) and x′\overline{p} = λ(x′\overline{p})ρ(x′\overline{p}). Thus

x(x′)^{−1} = λ(x\overline{p})ρ(x\overline{p})ρ(x′\overline{p})^{−1}λ(x′\overline{p})^{−1},

and so (zN_{0}z^{−1})x = (zN_{0}z^{−1})x′ if and only if x(x′)^{−1} ∈ zN_{0}z^{−1}, which holds if and only if λ(x\overline{p})ρ(x\overline{p})ρ(x′\overline{p})^{−1}λ(x′\overline{p})^{−1} ∈ zN_{0}z^{−1}, which holds if and only if ρ(x\overline{p})ρ(x′\overline{p})^{−1} lies in λ(x\overline{p})^{−1}zN_{0}z^{−1}λ(x′\overline{p}) ∩ N_{0}.

Now H_{0} ∩ zH_{0}z^{−1} = \overline{P}_{0}zN_{0}z^{−1}. Thus \overline{p}′zN_{0}z^{−1}\overline{p}′′ ∩ N_{0} ⊂ H_{0} ∩ zH_{0}z^{−1} ∩ N_{0} = zN_{0}z^{−1} for any pair of elements \overline{p}′ and \overline{p}′′ of \overline{P}_{0}. Hence (zN_{0}z^{−1})x = (zN_{0}z^{−1})x′ implies that ρ(x\overline{p})ρ(x′\overline{p})^{−1} ∈ zN_{0}z^{−1}. Conversely, if ρ(x\overline{p})ρ(x′\overline{p})^{−1} ∈ zN_{0}z^{−1}, then x(x′)^{−1} ∈ λ(x\overline{p})zN_{0}z^{−1}λ(x′\overline{p}) ∩ N_{0} ⊂ zN_{0}z^{−1}. This proves the lemma. □

Let ϕ_{z} denote the composite of (4.2.8) with (4.2.9). (Thus ϕ_{z} is the composite of the two right-hand vertical arrows in the diagram (4.2.10).) As observed, ϕ_{z} is uniquely determined by the fact that it makes the diagram (4.2.10) commute. Thus, since the operators (π_{N_{0},z})′ define a topological action of Z^{+} on (V′_{b})_{N_{0}}, we see that the operators ϕ_{z} define a topological action of Z^{+} on (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}. Taking the completed tensor product with C^{an}(Ẑ_{G,n}, K)^{†} over C^{an}(Ẑ_{G}, K), we obtain continuous C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})-linear endomorphisms id ˆ⊗ ϕ_{z} of U_{n} = C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{C^{an}(Ẑ_{G},K)}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}, defining the required topological action of Z^{+} on U_{n}.

This completes our specification of the data (4.2.2) and (4.2.3). Since V is a locally analytic G-representation, the closed subspace V^{N_{0}} is certainly a locally analytic M_{0}-representation, and so also a locally analytic representation of Z_{0} := M_{0} ⋂ Z^{+}. Together with the remarks following the proof of lemma 3.2.29, this shows that V^{N_{0}} lies in the category Rep_{la.c}^{z}(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 M_{0} normalizes N_{0}, we see that the space of Hausdorff N_{0}-coinvariants D(ℍ_{n}^{∘}, H_{0})_{N_{0}} is naturally a compact type D(𝕄_{n}^{∘}, M_{0})-module, for each n ≥ 0.

Proposition 4.2.22. For any n ≥ 0, the continuous D(𝕄_{n}^{∘}, M_{0})-linear map

(4.2.23)  D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} D(ℍ_{n+1}^{∘}, H_{0})_{N_{0}} → D(ℍ_{n}^{∘}, H_{0})_{N_{0}}

induced by the natural map D(ℍ_{n+1}^{∘}, H_{0})_{N_{0}} → D(ℍ_{n}^{∘}, H_{0})_{N_{0}} is a D(𝕄_{n}^{∘}, M_{0})- compact map (in the sense of definition 2.3.3).

Proof. For each n ≥ 0 the Iwahori decomposition of H_{0}, and the rigid analytic Iwahori decomposition of ℍ_{n}^{∘}, give rise to a topological isomorphism

D(ℍ_{n}^{∘}, H_{0}) ⟶^{∼} D\overline{ℕ}_{n}^{∘}, \overline{N}_{0}) ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{K} D(ℕ_{n}^{∘}, N_{0})

(where each of the factors in the target is defined via (4.1.2)), and hence an isomorphism of D(𝕄_{n}^{∘}, M_{0})-modules

(4.2.24)  D(ℍ_{n}^{∘}, H_{0})_{N_{0}} ⟶^{∼} D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0}) ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}),

provided that the D(𝕄_{n}^{∘}, M_{0})-module structure on the target of this map is defined by having M_{0} act on the first factor via the action of M_{0} on N_{0} given by conjugation, and on the second factor by regarding D(𝕄_{n}^{∘}, M_{0}) 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 product D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0}) ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}), equipped with the D(𝕄_{n}^{∘}, M_{0})-module structure that appears in (4.2.24), and the same completed tensor product, equipped with the evident D(𝕄_{n}^{∘}, M_{0})-structure defined by the D(𝕄_{n}^{∘}, M_{0})-structure on the second factor. Composing this isomorphism with (4.2.24) yields an isomorphism of D(𝕄_{n}^{∘}, M_{0})-modules

(4.2.25)  D(ℍ_{n}^{∘}, H_{0})_{N_{0}} ⟶^{∼} D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0}) ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}),

in which the D(𝕄_{n}^{∘}, M_{0})-module structure on the target is defined simply by the D(𝕄_{n}^{∘}, M_{0})-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

\xymatrix{D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} D(ℍ_{n+1}^{∘}, H_{0})_{N_{0}} \ar[d]^{(4.2.23)} \ar[r]^{∼^{id ˆ⊗ (4.2.25)}} & D(\overline{ℕ}_{n+1}^{∘}, \overline{N}_{0}) ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) \ar[d] \\ D(ℍ_{n}^{∘}, H_{0})_{N_{0}} \ar[r]^{∼^{(4.2.25)}} & D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0}) ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}).}

To see that (4.2.23) is D(𝕄_{n}^{∘}, M_{0})-compact, it now suffices (by lemma 2.3.4 (ii) and the remark following definition 2.3.3) to note that the map \overline{ℕ}_{n+1} → \overline{ℕ}_{n} is relatively compact (by proposition 4.1.6 (i)), and thus that the map D(\overline{ℕ}_{n+1}^{∘}, \overline{N}_{0}) → D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0}) is a compact map of convex K-vector spaces. □

Corollary 4.2.26. For each n ≥ 0, the map

C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{C^{an}(Ẑ_{G,n+1},K)^{†} ˆ⊗_{K} D(𝕄_{n+1}^{∘},M_{0})} U_{n+1} → U_{n}

of (4.2.4) is a C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})-compact map (in the sense of definition 2.3.3).

Proof. By assumption V′_{b} is a coadmissible C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(H_{0}, K)-module, and so we may find a surjection

(4.2.27) C^{an}(Ẑ_{G,n+1}, K)^{†} ˆ⊗_{K} D(ℍ_{n+1}^{∘}, H_{0})^{r}

→ C^{an}(Ẑ_{G,n+1}, K)^{†} ˆ⊗_{K} D(ℍ_{n+1}^{∘}, H_{0}) ˆ⊗_{C^{an}(Ẑ_{G},K) ˆ⊗_{K} D^{la}(H_{0},K)} V′_{b},

for some natural number r, and hence also a surjection

C^{an}(Ẑ_{G,n+1}, K)^{†} ˆ⊗_{K} D(ℍ_{n+1}^{∘}, H_{0})_{N_{0}}^{r} → U_{n+1}.

Taking the completed tensor product of (4.2.27) with C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(ℍ_{n}^{∘}, H_{0}) over C^{an}(Ẑ_{G,n+1}, K)^{†} ˆ⊗_{K} D(ℍ_{n+1}^{∘}, H_{0}) yields a surjection

C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(ℍ_{n}^{∘}, H_{0})^{r}

→ C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{C^{an}(Ẑ_{G},K) ˆ⊗_{K} D^{la}(H_{0},K)} V′_{b},

and hence also a surjection

C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(ℍ_{n}^{∘}, H_{0})_{N_{0}}^{r} → U_{n}.

The morphism of (4.2.4) may thus be covered by a morphism

C^{an}(Ẑ_{G,n}, K) ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} D(ℍ_{n+1}^{∘}, H_{0})_{N_{0}}^{r}

→ C^{an}(Ẑ_{G,n}, K) ˆ⊗_{K} D(ℍ_{n}^{∘}, H_{0})_{N_{0}}^{r}

of C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(ℍ(z)_{n}^{∘}, H_{0})-modules, and so the corollary follows from (ii) 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}\overline{N}_{n}z ⊂ \overline{N}_{n+1}, then the endomorphism of U_{n} induced by z factors through the map of (4.2.4), in such a way that the diagram of (4.2.5) commutes.

Proof. The action of z on U_{n} is defined as id ˆ⊗ ϕ_{z}, where ϕ_{z} is the endomorphism of (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} defined as the composite of the D(𝕄_{n}^{∘}, M_{0})-linear maps (4.2.8) and (4.2.9). We will prove the proposition by showing that id ˆ⊗ (4.2.8) factors through the map of (4.2.4). To this end we consider the diagram

(4.2.29)

\xymatrix{D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} U(z)_{n,n+1} \ar[r] \ar[d] & U(z)_{n} \ar[d]_{_{id ˆ⊗ (4.2.8)}} \\ C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{C^{an}(Ẑ_{G,n+1},K)^{†} ˆ⊗_{K} D(𝕄_{n+1}^{∘},M_{0})} U_{n+1} \ar[r]^{(4.2.4)} & U_{n}}

induced by the diagram (4.1.19), where to ease typesetting we have written

U(z)_{n,n+1} := C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{C^{an}(Ẑ_{G},K)} (D(ℍ(z)_{n,n+1}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}

and similarly

U(z)_{n} := C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{C^{an}(Ẑ_{G},K)} (D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} .

We will show below that the natural map

(4.2.30) D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{C^{an}(Ẑ_{G},K)} (D(ℍ(z)_{n,n+1}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} S)_{N_{0}}

⟶ C^{an}(Ẑ_{G,n}, K)^{†} ˆ⊗_{C^{an}(Ẑ_{G},K)} (D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} S)_{N_{0}}

is an isomorphism for any coadmissible C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(H_{0}, K)-module S. In particular, the upper horizontal arrow of (4.2.29) is an isomorphism, and inverting this isomorphism then yields the required factorization of id ˆ⊗ (4.2.8).

We turn to proving that (4.2.30) is an isomorphism. To ease notation, we denote by M(z)_{n,n+1} the topological (D(𝕄_{n+1}^{∘}, M_{0}), C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(H_{0}, K))- bimodule C^{an}(Ẑ_{G,n}, K) ˆ⊗_{K} D(ℍ(z)_{n,n+1}^{∘}, H_{0}), and denote by M(z)_{n} the topological (D(𝕄_{n}^{∘}, M_{0}), C^{an}(Ẑ_{G}, K) ˆ⊗_{K}D^{la}(H_{0}, K))-bimodule C^{an}(Ẑ_{G,n}, K) ˆ⊗_{K}D(ℍ(z)_{n}^{∘}, H_{0}). Since C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(H_{0}, K) is a Fréchet-Stein algebra, we may find a continuous K-algebra homomorphism C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(H_{0}, K) → B for some Noetherian Banach algebra B such that the action of C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(H_{0}, K) on each of the topological modules M(z)_{n,n+1} and M(z)_{n} factors through a corresponding continuous action of B, and such that S_{1} := B ˆ⊗_{C^{an}(Ẑ_{G},K) ˆ⊗_{K} D^{la}(H_{0},K)} S is a finitely presented B-Banach module. It thus suffices to show that the natural map

(4.2.31)  D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} (M(z)_{n,n_{1}} ˆ⊗_{B} S_{1})_{N_{0}} → (M(z)_{n} ˆ⊗_{B} S_{1})_{N_{0}}

is an isomorphism for any finitely presented B-Banach module S_{1}.

The construction of (4.2.31) is evidently functorial in S_{1}. If we apply this construction to the members of a finite presentation B^{r} → B^{s} → S_{1} → 0 of S_{1}, we obtain the diagram

\xymatrix{D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} (M(z)_{n,n+1}^{r})_{N_{0}} \ar[r] \ar[d] & (M(z)_{n}^{r})_{N_{0}} \ar[d] \\ D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} (M(z)_{n,n+1}^{s})_{N_{0}} \ar[r] \ar[d] & _{(M(z)_{n}^{s})_{N_{0}}} \ar[d] \\ D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} (M(z)_{n,n+1} ˆ⊗_{B} S_{1})_{N_{0}} \ar[r] \ar[d] & _{(M(z)_{n} ˆ⊗_{B} S_{1})_{N_{0}}} \ar[d] \\ 0 & 0}

(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 have formed completed tensor products, and then passed to Hausdorff N_{0}-coinvariants), but they are exact in the category of Hausdorff locally convex K-spaces (i.e. the third member of each column is isomorphic to the quotient of the second member of that column by the closure of the image of the first member). Thus to show that (4.2.31) is an isomorphism, it suffices to show that the natural map

D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} (M(z)_{n,n+1})_{N_{0}} → (M(z)_{n})_{N_{0}}

is an isomorphism. For this, it suffices in turn to note that the natural map

D(𝕄_{n}^{∘}, M_{0}) ˆ⊗_{D(𝕄_{n+1}^{∘},M_{0})} D(ℍ(z)_{n,n+1}^{∘}, H_{0})_{N_{0}} → D(ℍ(z)_{n}^{∘}, H_{0})_{N_{0}}

is 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 Rep_{es}(G), then the Jacquet module J_{P} (V) is an object of Rep_{es}(M). Thus J_{P} induces an additive, left exact functor Rep_{es}(G) → Rep_{es}(M).

Proof. Proposition 4.1.6 (vi) yields z ∈ Z^{+} such that z^{−1}\overline{N}_{n}z ⊂ \overline{N}_{n+1} for each n ≥ 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 J_{P} (V)′_{b} is a coadmissible module over the nuclear Fréchet algebra C^{an}(Ẑ_{M}, K) ˆ⊗_{K} C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(M_{0}, K). Since the two actions of C^{an}(Ẑ_{G}, K) on J_{P} (V)′_{b} (one through the embedding C^{an}(Ẑ_{G}, K) → C^{an}(Ẑ_{M}, K) and the other through the first factor of C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(M_{0}, K)) coincide, the action of the algebra C^{an}(Ẑ_{M}, K) ˆ⊗_{K} C^{an}(Ẑ_{G}, K) ˆ⊗_{K} D^{la}(M_{0}, K) on J_{P} (V)′_{b} factors through its quotient C^{an}(Ẑ_{M}, K) ˆ⊗_{K} D^{la}(M_{0}, K) (in which the two copies of C^{an}(Ẑ_{G}, K) described above are identified). Thus J_{P} (V)′_{b} is a coadmissible C^{an}(Ẑ_{M}, K) ˆ⊗_{K} D^{la}(M_{0}, K)-module, and so J_{P} restricts to a functor Rep_{es}(G) → Rep_{es}(M), as required. That this functor is additive and left exact follows from lemma 3.4.7. □

We now suppose that 𝔾 is quasi-split, and that ℙ is a Borel subgroup of 𝔾, and turn to proving (0.11). The Levi factor 𝕄 is then a torus, and so M = Z_{M}.

Proposition 4.2.33. If V is an object of Rep_{ad}(G), then the map Exp(J_{P} (V)) → 𝔪̌ has discrete fibres.

Proof. Write M̂ = ⋃ M̂_{n} as the a union of affinoid subdomains. We have to show that the intersection of Exp(J_{P} (V)) with any of the affinoid subdomains M̂_{n} has a finite fibre over any point of 𝔪̌. For this, it suffices to show that for any character χ ∈ 𝔪̌, the tensor product (C^{an}(M̂_{n}, K)^{†} ˆ⊗_{C^{an}(M̂,K)} J_{P} (V)′_{b})_{𝔪=χ} is finite dimensional for 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.

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

(4.2.34)  D(𝕄_{n}^{∘}, M_{0}) → C^{an}(M̂_{n}, K)^{†}.

Since V is admissible as a locally analytic G-representation (rather than just essentially admissible) we may strengthen the isomorphism (4.2.6) as follows:

(V^{N_{0}})′_{b} ⟶^{∼} lim_{\overleftarrow{n}} (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}.

Take A_{n} to be D(𝕄_{n}^{∘}, M_{0}) and U_{n} to be (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}, for n ≥ 0; variants of corollary 4.2.26 and proposition 4.2.28 then show that the hypotheses of proposition 3.2.24 are satisfied. The proof of that proposition thus yields an isomorphism

J_{P} (V)′_{b} = ((V^{N_{0}})_{fs})′_{b} ⟶^{∼} C^{an}(M̂, K) ˆ⊗_{K[Z^{+}]}(V^{N_{0}})′_{b} ⟶^{∼}

C^{an}(M̂, K) ˆ⊗_{K[Z^{+}]} lim_{\overleftarrow{n}} (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} ⟶^{∼}

lim_{\overleftarrow{n}} C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}},

where the space C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} is finitely generated as a C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})-module, for each n ≥ 0. The continuous homomorphism (4.2.34) induces a continuous surjection

C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) → C^{an}(M̂_{n}, K)^{†},

and by construction the C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})-action on

C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}

factors through this surjection. Thus we see that for each n ≥ 0, the space

C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}

is finitely generated as a C^{an}(M̂_{n}, K)^{†}-module. (In short, we have just repeated the argument of theorem 4.2.32, proving that J_{P} (V) lies in Rep_{es}(M), in the particular situation of the current proposition.)

We deduce from [8, thm. 1.2.11 (i)] that there is a topological isomorphism

C^{an}(M̂_{n}, K)^{†} ˆ⊗_{C^{an}(M̂,K)} J_{P} (V)′_{b} ⟶^{∼} C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}},

and hence a topological isomorphism

(C^{an}(M̂_{n}, K)^{†} ˆ⊗_{C^{an}(M̂,K)} J_{P} (V)′_{b})_{𝔪=χ}

⟶^{∼} (C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}})_{𝔪=χ},

for each n ≥ 0.

If n is chosen large enough, then χ integrates to a rigid analytic character χ̃ of 𝕄_{n}^{∘}, and the natural map (C^{an}(M̂_{n}, K)^{†})_{𝔪=χ} → (C^{an}(M̂_{n}, K)^{†})_{M_{n}^{∘}=χ̃} is an isomorphism (as follows from the fact that the natural map D^{la}(M_{0}, K) → C^{an}(M̂, K) factors through the map (4.2.34)). Thus there is an isomorphism

(C^{an}(M̂_{n}, K)^{†} ˆ⊗_{C^{an}(M̂,K)} J_{P} (V)′_{b})_{𝔪=χ} ⟶^{∼} (C^{an}(M̂_{n}, K)^{†} ˆ⊗_{C^{an}(M̂,K)} J_{P} (V)′_{b})_{M_{n}^{∘}=χ̃},

and hence an isomorphism

(4.2.35) (C^{an}(M̂_{n}, K)^{†} ˆ⊗_{C^{an}(M̂,K)} J_{P} (V)′_{b})_{𝔪=χ} ⟶^{∼}

(C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}})_{M_{n}^{∘}=χ̃} ⟶^{∼}

C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(D(𝕄_{n}^{∘}, M)_{M_{n}^{∘}=χ̃} ˆ⊗_{D(𝕄_{n}^{∘},M)}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}).

If z ∈ Z^{+} is such that ϕ_{z} induces a D(𝕄_{n}^{∘}, M_{0})-compact endomorphism of (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}, then by lemma 2.3.4 (ii), we see that ϕ_{z} induces a D(𝕄_{n}^{∘}, M)_{M_{n}^{∘}=χ̃}-compact endomorphism of

D(𝕄_{n}^{∘}, M)_{M_{n}^{∘}=χ̃} ˆ⊗_{D(𝕄_{n}^{∘},M)}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}.

Since M_{n}^{∘} has finite index in M_{0}, we see that D(𝕄_{n}^{∘}, M)_{M_{n}^{∘}=χ̃} is in fact a finite dimensional K-algebra, and thus that ϕ_{z} induces a compact endomorphism of D(𝕄_{n}^{∘}, M)_{M_{n}^{∘}=χ̃} ˆ⊗_{D(𝕄_{n}^{∘},M)}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}}. Proposition 2.3.6 implies that

C^{an}(M̂_{n}, K)^{†} ˆ⊗_{K[Z^{+}]}(D(𝕄_{n}^{∘}, M)_{M_{n}^{∘}=χ̃} ˆ⊗_{D(𝕄_{n}^{∘},M)}(D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}})

is finite dimensional over K. Taking into account (4.2.35), this completes the proof of 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 Rep_{ad}(G). If there is a compact open subgroup H of G such that V ⟶^{∼} C^{la}(H, K)^{r} as an H representation, for some natural number r (so that in particular V is a strongly admissible locally analytic representation of G), then Exp(J_{P} (V)) is equidimensional of dimension d, where d is the dimension of the maximal torus 𝕄 of 𝔾.

Proof. The key point is that our assumption on V will allow us to reduce to a situation in which we may apply the relative theory of compact operators developed in [6], for compact operators on orthonormalizable Banach modules. A careful explanation of how we make this reduction is somewhat involved, and accounts for the length of the following argument.

If H′ is an open subgroup of H, then there is an H′-equivariant isomorphism C^{la}(H, K)^{r} ⟶^{∼} C^{la}(H′, K)^{r[H:H′]}, and so in the statement of the proposition, we may replace H by any such open subgroup H′. In particular, after relabelling the members of the sequence {H_{n}} if necessary, we may assume that the open subgroup H appearing in the statement of the proposition is equal to H_{0}.

We choose z ∈ Z^{+} such that z^{−1}\overline{ℕ}_{n}z ⊂ \overline{ℕ}_{n+1} for each n ≥ 0, and such that z^{−1} and Z^{+} together generate the group Z_{M}. (A consideration of the proof of proposition 4.1.6 (vi) shows that we may do this.) Let Y^{+} (respectively Y) denote the submonoid (respectively subgroup) of Z_{M} generated by z. The discussion preceding the statement of proposition 4.2.22 shows that V^{N_{0}} lies in Rep_{la.c}^{z}(Z^{+}), and so by proposition 3.2.28 there is a natural isomorphism

J_{P} (V) ⟶^{∼} L_{b,Y^{+}}(C^{an}(Ŷ , K), V^{N_{0}}).

Dualizing, we obtain an isomorphism

(4.2.37)  J_{P} (V)′_{b} ⟶^{∼} K {{z, z^{−1}}} ˆ⊗_{K[z]}(V^{N_{0}})′_{b},

where K {{z, z^{−1}}} denotes the ring of entire functions on Ŷ (which is isomorphic to 𝔾_{m}, thought of as a rigid analytic space).

As in the proof of proposition 4.2.33, we set U_{n} := (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} , for each n ≥ 0. Also, set U(z)_{n} = (D(ℍ(z)_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0},K)} V′_{b})_{N_{0}} for each value of n. There is an isomorphism

(4.2.38)  (V^{N_{0}})′_{b} ⟶^{∼} lim_{\overleftarrow{n}} U_{n}.

By assumption, V ⟶^{∼} C^{la}(H_{0}, K)^{r}. The untwisting isomorphism (4.2.25) thus yields isomorphisms of D(𝕄_{n}^{∘}, M_{0})-modules

(4.2.39)  U_{n} ⟶^{∼} D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0})^{r} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})

and

(4.2.40)  U(z)_{n} ⟶^{∼} D(z^{−1}\overline{ℕ}_{n}^{∘}z, \overline{N}_{0})^{r} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}),

where D(𝕄_{n}^{∘}, M_{0}) acts on the targets of these isomorphisms via its action on the right hand factor in the completed tensor product. We may fit these isomorphisms into the following commutative diagram:

\xymatrix{U_{n} \ar[dd]_{ϕ_{z}} \ar[r]^{∼^{(4.2.39)}} \ar[d]_{(4.2.9)} & D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0})^{r} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) \ar[d]^{α_{n}} \\ U(z)_{n} \ar[r]^{∼^{(4.2.40)}} \ar[d]_{(4.2.8)} & D(z^{−1}\overline{ℕ}_{n}^{∘}z, \overline{N}_{0})^{r} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) \ar[d]^{β_{n}} \\ U_{n} \ar[r]^{∼^{(4.2.39)}} & D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0})^{r} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}),}

where the vertical arrow labelled β_{n} is obtained by taking the completed tensor product of the natural map

(4.2.41)  D(z^{−1}ℕ_{n}^{∘}z, \overline{N}_{0})^{r} → D(ℕ_{n}^{∘}, \overline{N}_{0})^{r}

(induced by the embedding z^{−1}ℕ_{n}^{∘}z → ℕ_{n}^{∘}) with D(𝕄_{n}^{∘}, M_{0}) 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

D(z^{−1}ℕ_{n}^{∘}z, \overline{N}_{0})^{r} ⟶^{ı_{n,1}} W_{n} ⟶^{ı_{n,2}} D(ℕ_{n}^{∘}, \overline{N}_{0})^{r},

where W_{n} is a Banach space. Consequently, the map β_{n} factors as

D(z^{−1}ℕ_{n}^{∘}z, \overline{N}_{0})^{r} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}) ⟶^{β_{n,1}} W_{n} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0})

⟶^{β_{n,2}} D(ℕ_{n}^{∘}, \overline{N}_{0})^{r} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}),

where β_{i,n} := ı_{i,n} ˆ⊗ id. The composite β_{n} ∘ α_{n} defines a D(𝕄_{n}^{∘}, M_{0})-linear action of z (and thus Y^{+}) on D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0})^{r} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}), compatible with the isomorphism (4.2.39) and the action of z (and thus Y^{+}) on U_{n} defined by ϕ_{z}. The composite β_{n,1} ∘ α ∘ β_{n,2} defines a D(𝕄_{n}^{∘}, M_{0})-linear action of z (and thus Y^{+}) on W_{n} ˆ⊗_{K} D(𝕄_{n}^{∘}, M_{0}). By construction, the maps β_{n,1} ∘ α_{n} and β_{n,2} are Y^{+}- equivariant and D(𝕄_{n}^{∘}, M_{0})-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) J_{P} (V)′_{b} ⟶^{∼} lim_{\overleftarrow{n}} K \{\{z, z^{−1}\}\} ˆ⊗_{K[z]} U_{n}

⟶^{∼} lim_{\overleftarrow{n}} K \{\{z, z^{−1}\}\} ˆ⊗_{K[z]} D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0})^{r} ˆ⊗_{K} D(M_{n}^{∘}, M_{0})

⟶^{∼} lim_{\overleftarrow{n}} K \{\{z, z^{−1}\}\} ˆ⊗_{K[z]} W_{n} ˆ⊗_{K} D(M_{n}^{∘}, M_{0})

(the third isomorphism being induced by the maps β_{n,1} ∘ α_{n}).

The isomorphism D^{la}(M_{0}, K) ⟶^{∼} C^{an}(M̂_{0}, K) of [8, prop. 6.4.6] shows that D^{la}(M_{0}, K) is equal to the Fréchet algebra of rigid analytic functions on the rigid analytic space M̂_{0}. Writing M̂_{0} as the union of an increasing sequence of admissible open affinoid subdomains, we may write D^{la}(M_{0}, K) ⟶^{∼} lim_{\overleftarrow{n}} A_{n}, where

{A_{n}} is a projective sequence of Tate algebras over K. We also have the isomorphism D^{la}(M_{0}, K) ⟶^{∼} lim_{\overleftarrow{n}} D(𝕄_{n}^{∘}, M_{0}). The two projective sequences \{A_{n}\} and

{D(𝕄_{n}^{∘}, M_{0})} are cofinal [8, prop. 1.2.7]. Passing to subsequences, and relabelling if necessary, we may assume that the map D^{la}(M_{0}, K) → A_{n} factors through the map D^{la}(M_{0}, K) → D(𝕄_{n}^{∘}, M_{0}), for each n. Tensoring the nth term of the projective sequence with A_{n} over D(𝕄_{n}^{∘}, M_{0}), the isomorphism (4.2.42) yields an isomorphism

(4.2.43)  J_{P} (V)′_{b} ⟶^{∼} lim_{\overleftarrow{n}} K \{\{z, z^{−1}\}\} ˆ⊗_{K[z]} W_{n} ˆ⊗_{K} A_{n}.

Since ı_{n,2} is compact (being a map from a Banach space to the compact type space D(𝕄_{n}^{∘}, M_{0})), we find that z acts on W_{n} ˆ⊗_{K} A_{n} through an A_{n}-compact map.

In order to show that the support of J_{P} (V)′_{b} is equidimensional of dimension d, it follows from (4.2.43) that it suffices to show that this is true for each of the A_{n}-modules

K {{z, z^{−1}}} ˆ⊗_{K[z]} W_{n} ˆ⊗_{K} A_{n}.

Since W_{n} ˆ⊗_{K} A_{n} is an orthonormalizable A_{n}-Banach module on which z acts via an A_{n}-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 notation of the preceding subsections.

Let V be an admissible smooth representation of G defined over K. If we equip V with its finest convex topology, then it becomes an object of Rep_{ad}(G) [8, prop. 6.3.2]. Our first goal is to show that J_{P} (V) is isomorphic as an Mrepresentation to the N-coinvariants V_{N} of V. In order to compute J_{P} (V), we fix a lift of the Levi quotient M to a subgroup of P, as well as a compact open subgroup P_{0} of P. Let \overline{P} denote the opposite parabolic to P (chosen so that P ⋂ \overline{P} = M) and let \overline{N} denote the unipotent radical of \overline{P}.

Lemma 4.3.1. The Z_{M}^{+}-representation V^{N_{0}} is the locally convex inductive limit of an increasing sequence of finite dimensional Z_{M}^{+}-invariant subspaces.

Proof. Let v ∈ V^{N_{0}}, and choose an open subgroup H_{1} of G that leaves v invariant. Shrinking H_{1} if necessary, we may assume that H_{1} admits an Iwahori decomposition H_{1} = \overline{N}_{1}M_{1}N_{1} with N_{1} ⊂ N_{0}. Set Y := \{y ∈ Z_{M}^{+} | yN_{1}y^{−1} ⊂ N_{1}, y^{−1}\overline{N}_{1}y ⊂ \overline{N}_{1}\}. Since V is an admissible smooth G-representation, the space V^{H_{1}} is finite dimensional. Furthermore, the Hecke operators π_{N_{1},y} for y ∈ Y leave V^{H_{1}} invariant. (If we let π_{H_{1}} denote averaging over H_{1}, then a consideration of the Iwahori decomposition of H_{1} shows that π_{H_{1}}(yv_{1}) = π_{N_{1}}(yv_{1}) =: π_{N_{1},y}(v_{1}) for any y ∈ Y and v_{1} ∈ V^{H_{1}}.) Hence the span of the set {π_{N_{1},y}(v) | y ∈ Y } is finite dimensional. A variant of proposition 3.3.2 (i) shows that Y generates Z_{M} as a group, and so we may find a finite subset {z_{1}, …, z_{n}} of Z_{M}^{+} so that Z_{M}^{+} = ⋃_{i=1}^{n} z_{i}Y. Since π_{N_{0},y} = π_{N_{0}}π_{N_{1},y} for y ∈ Y, we conclude that the span of the set {π_{N_{0},z}(v) | z ∈ Z_{M}^{+}} is also finite dimensional, and hence that V^{N_{0}} is the union of finite dimensional Z_{M}^{+}-invariant subspaces. Since V, and hence V^{N_{0}}, is equipped with its finest convex topology, we see that V^{N_{0}} is even isomorphic to the locally convex inductive limit of these invariant subspaces. □

Proposition 4.3.2. (i) The natural map (V^{N_{0}})_{null} ⨁ J_{P} (V) → V^{N_{0}} is an isomorphism.

(ii) Each of (V^{N_{0}})_{null} and J_{P} (V) is an M^{+}-invariant subspace of V^{N_{0}}.

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, J_{P} (V) = (V^{N_{0}})_{fs}). Part (ii) follows by functoriality of the formation of (V^{N_{0}})_{null} and J_{P} (V). □

We now recall some ideas from [5, §3]. Since V is a smooth representation of G, it is in particular smooth as a representation of N, and consequently, for any compact open subgroup N′_{0} of N the map π_{N′_{0}} 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 π_{N′_{0}}v = 0 for some compact open subgroup N′_{0} of N; it is easily checked that V (N) is equal to the kernel of the natural map

(4.3.3)  V → V_{N}

[5, prop. 3.2.1]. (This description of the kernel shows that the functor V ↦ V_{N}, which is right exact by construction, is in fact an exact functor on smooth Nrepresentations [5, prop. 3.2.3].)

By proposition 3.3.2, any compact open subgroup of N is contained in z^{−1}N_{0}z for some z ∈ Z_{M}^{+}. The equation π_{N_{0},z} = zπ_{z^{−1}N_{0}z} thus shows that V (N) may also be described as the subspace of V consisting of those vectors that are annihilated by π_{N_{0},z} for some element z ∈ Z_{M}^{+}. In particular, V (N) ⋂ V^{N_{0}} = (V^{N_{0}})_{null}.

Proposition 4.3.4. (i) The restriction of the morphism (4.3.3) to V^{N_{0}} induces an M^{+}-equivariant surjective map V^{N_{0}} → V_{N} (the M^{+}-action being defined on the source via the operators π_{N_{0},m}, and on the target via restricting the natural M-action).

(ii) The map of (i) restricts to an M-equivariant isomorphism J_{P} (V) ⟶^{∼} V_{N}.

Proof. For any v ∈ V, the elements v and π_{N_{0}}v have the same image in V_{N}. This proves (i). The discussion preceding the lemma shows that (V^{N_{0}})_{null} is the kernel of 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 isomorphism J_{P} (V) ⟶^{∼} V_{N} of proposition 4.3.4 (ii) is independent of the choices of Levi factor M and compact open subgroup P_{0} of P that are used to compute J_{P} (V).

Proof. If we compute J_{P} (V) with respect to a different choice of M (respectively of P_{0}), then the natural isomorphism of proposition 3.4.10 (respectively proposition 3.4.11) involves conjugation by an element of N (respectively the application of the projection operator π_{N_{0}}). This operation reduces to the identity of V_{N}, and so 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 = End_{G}(W), and if X is an admissible smooth representation of G over B, then there is a natural isomorphism J_{P} (X ⊗_{B} W) ⟶^{∼} X_{N} ⊗_{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^{N_{0}} ⊂ W^{𝔫} are both equalities. Combining these two observations, we obtain the equality X^{N_{0}} ⊗_{B} W^{N} = (X ⊗_{B} W)^{N_{0}}.

If m ∈ M^{+}, then the operator π_{N_{0},m} on (X⊗_{B} W)^{N_{0}} can be described in terms of this tensor product decomposition as π_{N_{0},m} : u⊗w ↦ π_{N_{0},m}^{X}(u)⊗mw (where π_{N_{0},m}^{X} denotes the corresponding operator on X^{N_{0}}). Thus as an M^{+}-representation, the space (X ⊗_{B} W)^{N_{0}} is isomorphic to the tensor product of the M^{+}-representation on X^{N_{0}} and the space W^{N}, equipped with the M^{+}-action obtained by restricting the natural M-action on this space.

Proposition 3.2.9 thus yields an M-equivariant isomorphism ((X ⊗_{B} W)^{N_{0}})_{fs} ⟶^{∼} (X^{N_{0}})_{fs} ⊗_{B} W^{N}. The proposition now follows from the definition of the functor J_{P}, 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 object of Rep_{ad}(G), by endowing it with its finest convex topology. By [8, prop. 6.3.11], any admissible locally algebraic G-representation admits an isomorphism V ⟶^{∼} ⨁_{n} X_{n}⊗_{B_{n}}W_{n}, where W_{n} runs over a sequence of isomorphism class representatives for the irreducible algebraic representations of 𝔾, B_{n} = End_{G}(W_{n}), and X_{n} is an admissible smooth representation of G over B_{n}. Proposition 4.3.6 together with lemma 3.4.7 (i) and (iv) thus yields an isomorphism J_{P} (V) ⟶^{∼} ⨁_{n}(X_{n})_{N} ⊗_{B_{n}} W_{n}^{N}.

(4.4) In this subsection we will establish property (0.14). As above we fix a compact open subgroup P_{0} of P, as well as a Levi factor M of P, and we define N_{0}, M_{0}, M^{+} and Z_{M}^{+} as they were defined preceding corollary 3.3.3. If V is a locally analytic representation of G, we define the Hecke operators π_{N_{0},m} on V^{N_{0}} as in definition 3.4.2. We also assume throughout this section that L = ℚ_{p}. (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 ℚ_{p} as a reductive group over ℚ_{p}.)

Let 𝕊 denote the maximal subtorus of ℤ_{𝕄} that splits over ℚ_{p}, and as in subsection 1.4, let Y^{•} (respectively Y_{•}) denote the character lattice (respectively the cocharacter lattice) of 𝕊. If χ is a character of Z_{M}, then the discussion of that subsection allows us to define the element slope(χ) of ℚ ⊗_{ℤ} Y^{•}.

Let Δ(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 Δ(𝔾, 𝕊)_{s} denote the subset of positive simple restricted roots. Let R^{•} denote the sublattice of Y^{•} spanned by Δ(𝔾, 𝕊)_{s}, and let (ℚ ⊗_{ℤ} R^{•})^{≥0} denote the ℚ^{≥0}-invariant cone in ℚ ⊗_{ℤ} R^{•} generated by the elements of Δ(𝔾, 𝕊)_{s}.

Lemma 4.4.1. If χ is an element of Ẑ_{M}, then χ satisfies the inequality |χ(a)| ≤ 1 for every a ∈ Z_{M}^{+} if and only if slope(χ) ∈ (ℚ ⊗_{ℤ} R^{•})^{≥0}.

Proof. Taking T = Z_{M} in the diagram (1.4.1) yields a surjection ord : Z_{M} → Y′_{•}, where Y′_{•} is a sublattice of ℚ ⊗_{ℤ} Y_{•} containing Y_{•} with finite index. Consider the submonoid

(Y′_{•})^{+} = {y ∈ Y′_{•} | ⟨y, r⟩ ≥ 0 for all r ∈ (ℚ ⊗_{ℤ} R^{•})^{≥0}}

of Y′_{•}. The image of Z_{M}^{+} under the map ord is contained in (Y′_{•})^{+}, 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 equation ord(χ(a)) = ⟨ord(a), slope(χ)⟩, shows that ord(χ(a)) ≥ 0 for all a ∈ Z_{M}^{+} if and only if ⟨y, slope(χ)⟩ ≥ 0 for all y ∈ (Y′_{•})^{+}, which holds if and only if slope(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-half the sum of the elements of Δ(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 is where the assumption that L = ℚ_{p} is used.)

Lemma 4.4.2. Let V be an object of Rep_{la.c}(P), and suppose that V admits a P -invariant norm. If χ ∈ Ẑ_{M} is such that V^{N_{0},Z_{M}^{+} =χ} is non-zero, then for every a ∈ Z_{M}^{+} 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 assuming, and let v be a non-zero vector in V^{N_{0},Z_{M}^{+} =χ}. If a ∈ Z_{M}^{+} then χ(a)v = π_{N_{0},a}v = [N_{0} : aN_{0}a^{−1}]^{−1} ∑_{n∈N_{0}/aN_{0}a^{−1}} nav = δ(a) ∑_{n∈N_{0}/aN_{0}a^{−1}} nav, hence |δ(a)^{−1}χ(a)| ||v|| = || ∑_{n∈N_{0}/aN_{0}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 in subsection 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 ℤ_{𝕄} must act on W^{ℕ} through a character ψ ∈ X^{•}(ℤ_{𝕄}), which we also regard as an element of Ẑ_{M}(K). We also fix a smooth (i.e. locally constant) character θ ∈ Ẑ_{M}(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 of this Borel subgroup (again, defined over K), contained in 𝕄. The intersection 𝕄 ⋂ 𝔹 is then a Borel subgroup of 𝕄. Let 𝔫′ denote the Lie algebra of the unipotent radical of 𝔹, and let \overline{𝔫}′ 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 to 𝔹; this is also the highest weight of the 𝕄-representation W^{ℕ} with respect to 𝕄 ⋂ 𝔹. Note that ψ̃_{|ℤ_{𝕄}} = ψ. If α is a simple root of ℤ_{𝕄} acting on 𝔫, then denote by α̃ the (unique, by lemma 1.2.3) simple root of of 𝕋 acting on 𝔫′ that lifts α. If ρ̃:= ρ(𝔾, 𝕋) denotes one-half the sum of the positive roots of 𝕋 acting on 𝔫′, then ρ̃_{|𝕊} = ρ.

Definition 4.4.3. We say that χ = ψθ is of critical slope with respect to the representation W^{ℕ} if for some positive simple root α ∈ Δ(𝔾, ℤ_{𝕄}), the element sα̃(ψ̃+ρ̃)_{|𝕊} +slope(θ)+ρ of ℚ⊗_{ℤ} Y^{•} in fact lies in (ℚ⊗_{ℤ} R^{•})^{≥0}. Otherwise, we say that 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 Rep_{la.c}(P) admitting a P -invariant norm. If the eigenspace of V^{N^{0},Z_{M}^{+} =χ} is non-zero, then lemma 4.4.2 shows that ψ_{|𝕊} + ρ + slope(θ) + ρ lies in (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 ℚ_{p}, and if ℙ is chosen to be a Borel subgroup of 𝔾 defined over ℚ_{p}, then ℤ_{𝕄} = 𝕄 = 𝕋 (using the above notation), and so ψ̃ = ψ. In this case the character χ thus determines the representation W^{ℕ} (indeed, the latter is just equal 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 Rep_{la.c}(G), then V^{N_{0},Z_{M}^{+} =χ} is a locally analytic representation of M_{0}. Thus we may consider its subspace (V^{N_{0},Z_{M}^{+} =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).

We may also consider its subspace (V^{N_{0},Z_{M}^{+} =χ})^{𝔪∩𝔟=dψ̃}, the subspace of vectors on which the Lie algebra 𝔪 ∩ 𝔟 of 𝔹 ⋂ 𝕄 acts through the character dψ̃ of its quotient 𝔱 (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 = GL_{2}).

Proposition 4.4.4. Let α̃ be a positive simple root of 𝕋 acting on 𝔫′, and let X_{−α̃} ∈ \overline{𝔫}′ be an element of the root space \overline{𝔫}_{−α̃} corresponding to the negative root −α̃. Let α̃ˇ denote the corresponding coroot, and write m = ⟨ψ̃, α̃ˇ ⟩. If V is a locally analytic representation of G then the action of X_{−α̃}^{m+1} on V^{N_{0}} restricts to a map (V^{N_{0},Z_{M}^{+} =χ})^{𝔪∩𝔟=dψ̃} → V^{N_{0},Z_{M}^{+} =α̃^{−m−1}χ}.

Proof. Let v be an element of (V^{N_{0},Z_{M}^{+} =χ})^{𝔪∩𝔟=dψ̃}. The action of 𝔤 on v determines a U(𝔤)-module morphism Ver(ψ̃) → V, given by mapping the highest weight vector v(ψ̃) of Ver(ψ̃) to v. (Here Ver(ψ̃) denotes the Verma module of highest weight ψ̃, as in definition 1.1.5.) Proposition 1.1.6 shows that the element X_{−α̃}^{m+1}(1 ⊗ v(ψ̃)) of Ver(ψ̃) is fixed by the adjoint action of N. Thus if n ∈ N_{0} we see that

nX_{−α̃}^{m+1}v = nX_{−α̃}^{m+1}n^{−1}v = ϕ(Ad_{n}X_{−α̃}^{m+1}(1 ⊗ v(ψ̃)))

= ϕ(X_{−α̃}^{m+1}(1 ⊗ v(ψ̃))) = X_{−α̃}^{m+1}v,

and so X_{−α̃}^{m+1}v lies in V^{N_{0}}. Furthermore, for any a ∈ Z_{M}^{+} we compute that

π_{N_{0},a}X_{−α̃}^{m+1}v = [N_{0} : aN_{0}a^{−1}]^{−1} ∑_{n∈N_{0}/aN_{0}a^{−1}} naX_{−α̃}^{m+1}v
= [N_{0} : aN_{0}a^{−1}]^{−1} ∑ Ad_{n}Ad_{a}X_{−α̃}^{m+1}nav
= [N_{0} : aN_{0}a^{−1}]^{−1} ∑ α̃(a)^{−m−1}X_{−α̃}^{m+1}nav
= α̃(a)^{−m−1}X_{−α̃}^{m+1}[N_{0} : aN_{0}a^{−1}]^{−1} ∑ nav
= α̃(a)^{−m−1}X_{−α̃}^{m+1}π_{N_{0},a}v
= α̃(a)^{−m−1}χ(a)X_{−α̃}^{m+1}v.

(The third equality follows from the fact that X_{−α̃} is an eigenvector for the action of Ad_{a}, with eigenvalue α̃^{−1}, together with the fact, already used above, that X_{−α̃}^{m+1}v is fixed by the adjoint action of N, while the final equality follows from the assumption that v lies in V^{N_{0},Z_{M}^{+} =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 Rep_{la.c}(G), and suppose that V admits a G-invariant norm. If χ is of non-critical slope, then the space (V^{N_{0},Z_{M}^{+} =χ})_{W^{ℕ}−lalg} consists of locally W -algebraic vectors of V.

Proof. Let v be an element of (V^{N_{0},Z_{M}^{+} =χ})_{W^{ℕ}−lalg}. If U denotes the 𝔪-submodule of V^{N_{0}} generated by v, then by assumption U is isomorphic to a finite direct sum of 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^{N_{0},Z_{M}^{+} =χ})^{𝔪∩𝔟=ψ̃}. Replacing v by such a vector, if necessary, we may assume that v is a non-locally algebraic element of (V^{N_{0},Z_{M}^{+} =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 from proposition 1.1.6 (ii) that we may find a positive simple root α̃ of 𝕋 acting on 𝔫′, and an element X_{−α̃} ∈ \overline{𝔫}_{−α̃}, such that X_{−α̃}^{m+1}v ≠ 0. (Here the notation is that of proposition 4.4.4. In particular, m = ⟨ψ̃, α̃ˇ ⟩, and so m + 1 = ⟨ψ̃ + ρ̃, α̃ˇ ⟩.) Since v is W^{ℕ}-locally algebraic, the simple root α̃ must appear in 𝔫, and thus restricts to an element α ∈ Δ(𝔾, ℤ_{𝕄}). By proposition 4.4.4 we thus obtain a nonzero element of V^{N_{0},Z_{M}^{+} =α̃^{−m−1}χ}. Applying lemma 4.4.2 to α̃^{−m−1}χ, we find that ρ − (m + 1)α̃_{|𝕊} + slope(χ) lies in (ℚ ⊗_{ℤ} R^{•})^{+}.

We compute that

ρ − (m + 1)α̃_{|𝕊} + slope(χ) = ρ − ⟨ψ̃ + ρ̃, α̃ˇ ⟩α̃_{|𝕊} + ψ_{|𝕊} + slope(θ)

= ψ_{|𝕊} + ρ − ⟨ψ̃ + ρ̃, α̃ˇ ⟩α̃_{|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

critical slope, contradicting the hypothesis of the theorem. Hence V_{W^{ℕ}−lalg}^{N_{0},Z_{M}^{+} =χ} must consist entirely of locally algebraic elements. □

Property (0.14) follows from preceding theorem, together with proposition 3.4.9.

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