Jacquet modules of locally analytic representations of <span class="mathjax-formula">$p$</span>-adic reductive groups I. Construction and first properties

JACQUET MODULES OF LOCALLY ANALYTIC REPRESENTATIONS OF p -ADIC REDUCTIVE GROUPS

I. CONSTRUCTION AND FIRST PROPERTIES

B

M

EMERTON

ABSTRACT. – Let G be a reductive group defined over ap-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 JP 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 usual Jacquet module functor of [Casselman W., Introduction to the theory of admissible representations of p-adic reductive groups, unpublished notes distributed by P. Sally, draft dated May 7, 1993. Available electronically at http://www.math.ubc.ca/people/faculty/cass/research.html. [5]] on the subcategory of admissible smooth G-representations. We establish several important properties of this functor.

©2006 Published by Elsevier Masson SAS

RÉSUMÉ. – SoitG un groupe réductif défini sur un corp local p-adique L, soitP un sous-groupe parabolique deGavec quotient de LeviM. On poseG=G(L),P=P(L), etM=M(L). On construit un foncteurJP de la catégorie desG-représentations localement analytiques essentiellement admissibles dans la catégorie desM-représentations localement analytiques essentiellement admissibles, que l’on appelle le foncteur de module de Jacquet attaché à P, et qui coïncide avec le foncteur de module de Jacquet usuel de [Casselman W., Introduction to the theory of admissible representations of p-adic reductive groups, unpublished notes distributed by P. Sally, draft dated May 7, 1993. Available electronically at http://www.math.ubc.ca/people/faculty/cass/research.html. [5]] sur la sous-catégorie desG-représentations admissibles lisses. On établit plusieurs propriétés importantes de ce foncteur.

©2006 Published by Elsevier Masson SAS

Contents

1 Lie-theoretic preliminaries . . . 780

2 Functional-analytic preliminaries . . . 785

3 Construction of the Jacquet module functor . . . 794

4 Some properties of the Jacquet module functor . . . 812

Acknowledgements . . . 838

References . . . 838

1The author would like to acknowledge the support of the National Science Foundation (award number DMS-0070711).

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

The goal of this paper is to introduce in the theory of locally analytic representations ofp-adic reductive groups an analogue of the Jacquet module functors that are defined in [5] for smooth representations ofp-adic reductive groups.

We refer the reader to the papers [16–18,20] of Schneider and Teitelbaum for the foundations of locally analytic representation theory. Some additional contributions to the theory have been supplied by [8,12,14]. (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. LetLbe a finite extension ofQpfor some primep, letGbe a connected reductive linear algebraic group overL, letP denote a parabolic subgroup ofG, let Ndenote the unipotent radical ofP, and writeM:=P/N(so thatMis the Levi quotient ofP). Choose an opposite parabolicPtoP; the intersectionP∩Pthen provides a lifting ofMto a Levi factor ofP(and indeed making a choice ofPis equivalent to making a choice of such a lifting ofM). WriteG:=G(L),M :=M(L), N:=N(L),P:=P(L), andP:=P(L). Fix an extensionK ofL, complete with respect to a discrete valuation extending the discrete valuation onL. All representations will be onK-vector spaces (whether or not this is explicitly stated).

We briefly recall the theory of Jacquet module functors for smooth representations [5], before turning to a discussion of the theory for locally analytic representations that is the subject of this paper.

Jacquet module functors for smooth representations. The Jacquet module of a smooth G-representationV with respect to P is defined to be the space VN of N-coinvariants ofV. The formation of Jacquet modules yields a functor from smoothG-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 ifV appears as a subrepresentation of a representation parabolically induced from a representation ofM [5, thm. 3.2.4].

Property (iii) follows from the obvious Frobenius reciprocity formula Hom_G

V,

Ind^G_PW

sm

_∼

−→Hom_M(V_N, W), (0.1)

whereV is an arbitrary smoothG-representation,Wis an arbitrary smoothM-representationW, and(Ind^G_PW)smdenotes the smooth induction ofW fromP toG. 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 representationV is at first defined as a space of coinvariants, and hence as a quotient ofV, it admits an alternative description as a subspace ofV. (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-representationU and admissible smoothG-representationV there is a canonical isomorphism

Hom_G

Ind^G_PU

sm, V _∼

−→Hom_M

U(δ), V_N , (0.2)

whereU(δ)denotes the twist ofU by the modulus character ofP(which is trivial onN, and so may be thought of as a character ofM). 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 groupH, denote byRep_la.c(H)the category of locally analyticH-representations 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 ofN-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) toRep_la.c(M)which satisfies an obvious analogue of (0.1), with smooth parabolic induction being replaced by locally analytic parabolic induction, and theHom- 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)andRep_la.c(M)(such as strong admissibility, admissibility, or essential admissibility, as defined in [17,20,8] 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 admissibleM-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 anyG-representationV, the construction ofJP(V)will depend only on the structure of V as aP-representation, and so we initially define the functor JP in the more general context of locally analyticP-representations.

In Section 3.1 we define a certain full subcategory Rep^z_la.c(M) of Rep_la.c(M). For U an object of Rep^z_la.c(M), let C_c^sm(N, U) denote the space of compactly supported lo- cally constant U-valued functions on N. The formation of this space yields a functor Cc^sm(N,–) : Rep^z_la.c(M)→Rep_la.c(P). (See Section 3.5.)

The following result follows from Theorem 3.5.6 below.

THEOREM 0.3. –The functorC_c^sm(N,–)admits a right-adjoint.

We define the Jacquet module functorJ_P: Rep_la.c(P)→Rep^z_la.c(M)to be the right adjoint of Cc^sm(N,–), twisted by the modulus characterδ ofP. Thus forU inRep^z_la.c(M)andV in Rep_la.c(P), there is a natural isomorphism

LP

Cc^sm(N, U), V _∼

−→ L^M

U(δ), J_P(V) . (0.4)

If we compose JP with the forgetful functor Rep_la.c(G)→Rep_la.c(P) then we obtain a functorRep_la.c(G)→Rep^z_la.c(M)(which we again denote byJ_P). Note that for any objectU ofRep^z_la.c(M), the spaceCc^sm(N, U)embeds as a closedP-invariant subspace of the parabolic inductionInd^G_PU. (It can be identified with the subspace of locally constant functions inInd^G_PU whose support lies in the open cell of P\G.) 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 functorJP takes essentially admissible locally analyticG-representa- tions to essentially admissible locally analyticM-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 functorJP is left exact and additive. On the other hand, simple examples show that it is not right exact in general. (This shows, incidentally, thatJP does not coincide with the functor of HausdorffN-coinvariants 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 thatGis quasi-split, and thatPis a Borel subgroup ofG(so that the Levi factorMis a maximal torus ofG). IfV 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 ifJ_P(V) is non-zero, then V is isomorphic to a subquotient of the locally analytic parabolic induction fromP toGof a locally analytic character ofM.

Global motivations. Our original motivation for defining the functorJ_P was not local, but global. In the paper [9], it is applied to the problem ofp-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 atpin the construction of [7], then they will get a hint of the relation between the problem ofp-adic interpolation and the construction of this paper, in which Hecke operators also play a key role.)

Some properties of the functor JP. We establish a number of additional properties of the functor JP in this paper. Before describing them, we introduce some more notation. Let Rep_es(G)(respectivelyRep_es(M)) denote the category of essentially admissible locally analytic G-representations (respectivelyM-representations). Letgdenote the Lie algebra ofG, let Z_G denote the centre ofG, and letz(g)denote the centre of the universal enveloping algebra ofg.

Similarly, letmdenote the Lie algebra ofM, letZM denote the centre ofM, and letz(m)denote the centre of the universal enveloping algebra ofm. LetZM denote the rigid analytic space that parameterizes the locallyL-analytic characters ofZM, and for anyχ∈ZM(K), and objectV of Rep_es(G), letJ_P^χ(V)denote theχ-eigenspace for theZ_M-action onJ_P(V)(which is a closed admissible subrepresentation ofJ_P(V)[8, cor. 6.4.14]).

We now summarize the main properties ofJP that are proved in this paper.

0.7. – The definition of the functor C_sm^c (N,–) : Rep^z_la.c(M)→Rep_la.c(P) depends on the choice ofP(or equivalently, on the lifting ofMto a Levi factor ofP). However, we show that the functorJP is independent, up to natural isomorphism, of the choice ofP.

0.8. – Note that there are natural actions ofZ_G×z(g)onRep_la.c(G), and ofZ_M ×z(m)on Rep^z_la.c(M). Also, sinceM is the Levi quotient ofP, there are natural injectionsZ_G→Z_M andz(g)→z(m)(the latter is the “unnormalized Harish–Chandra homomorphism”, recalled in Section 1.3 below; it depends on the choice ofP). These maps induce the upper horizontal arrow in the diagram

Z_G×z(g) Z_M×z(m)

Aut(Rep^z_la.c(M))×End(Rep^z_la.c(M))

Aut(Rep_la.c(G))×End(Rep_la.c(G)) Aut(JP)×End(JP)

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 ofJP. We prove that this diagram commutes.

0.9. – IfV is an object ofRep_la.c(P), so thatJP(V)is an object ofRep^z_la.c(M), then regard JP(V)as aP-module, by havingPact through its quotientM. For any choice of a compact open subgroupP0ofPand of a lifting ofMto a Levi factor ofP, we construct anM0N0-equivariant map

J_P(V)→V, (0.10)

functorial inV. (HereM₀=M∩P₀,N₀=N∩P₀.) Extending the terminology of [5, p. 40], we refer to this map as the “canonical lifting” determined by the given choice ofP₀ and of a lifting ofM.

0.11. – Suppose thatGis quasi-split overL, and takePto be a Borel subgroup ofG. The Levi quotient Mis then a torus, and soM =Z_M. Passing to duals induces an anti-equivalence of categories between the categoryRep_es(M), and the category of coherent rigid analytic sheaves onM. Thus, for any objectV inRep_es(G), its Jacquet moduleJ_P(V)gives rise to a coherent rigid analytic sheaf onM. The support of this sheaf is a Zariski closed subset ofM, which we denote byExp(J_P(V)). Passing to the derivative of a character induces a natural mapM→m,ˇ and hence (by restriction) a map

Exp JP(V)

→m.ˇ (0.12)

We prove that ifV is an admissible locally analytic representation ofG, then the map (0.12) has discrete fibres.

0.13. – LetW be a finite dimensional algebraic representation ofG, writeB= EndG(W), let X be an admissible smooth representation of Gover B, and set V =X ⊗B W (so that V is an admissible locallyW-algebraic representation ofG). We prove thatJP(V)is naturally isomorphic as anM-representation toXN⊗BW^N.In particular,JP(V)is an admissible locally W^N-algebraic representation ofM. (TakingW to be the trivial representation, we find thatJP

restricted to admissible smooth representations coincides with the Jacquet module functor of [5].) 0.14. – Assuming thatL=Qpand thatGis split overK, we give a criterion for the functorJP

to commute with the passage to locally algebraic vectors. More precisely, ifV is an essentially admissible locally analytic representation of G, and if W is a finite dimensional irreducible algebraic representation of G, then the closed embedding VW−lalg →V induces a closed embedding

J_P(V_W−lalg)→J_P(V)_WN−lalg.

(Here the subscripts indicate subspaces of locally algebraic vectors, transforming locally according toW andW^N respectively.) Ifχ∈ZM(K), then we may restrict toχ-eigenspaces, and so obtain a closed embedding

J_P^χ(VW−lalg)→J_P^χ(V)_W^N_−lalg. (0.15)

Letψdenote the character through whichZM acts onW^N.(There is such aψ,sinceG,and soZM,is assumed to be split overK.) Ifχandψdo not coincide locally, and so in particular if χis not locally algebraic, then both the source and target of (0.15) vanish.

Suppose thatχandψdo coincide locally. IfV admits aG-invariant norm, and ifχis of “non- critical slope” with respect to the irreducible representationW^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-representationV, while certainly essentially admissible, need not be an admissible locally analytic M-representation (as the examples of [9] show). Nevertheless, for admissible locally analytic representationV that are topologically of finite length, one might hope thatJP(V)is again admissible, of finite length. SinceJP is a left exact functor, it would suffice to prove this whenV is topologically irreducible. For those irreducible representations that admit a continuous injection into an admissible continuous representation ofG, 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 as an adjoint functor. In the preliminary Section 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 Sections 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 Section 3.4. Finally, the adjointness formula (0.4) (and so also Theorem 0.3) is established in Section 3.5.

Section 4 establishes the deeper properties of the Jacquet module functor. After making some preliminary constructions in Section 4.1, in Section 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. Section 4.3 establishes property 0.13, while property 0.14 is proved in Section 4.4 using Verma module techniques.

Notation and conventions.Throughout the paper, we fix a finite extensionLofQp(for some fixed prime p), an algebraic closure L of L, as well as an extension K of L, complete with respect to a discrete valuation extending that onL. We letordLdenote the discrete valuation on L(normalized so as to take the value1on a uniformizer ofL),ord_Ldenote its extension toL, andordKits extension toK. We letOLandOKdenote the ring of integers in each ofLandK 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 a topological group (or more generally, a topological semigroup), then we will distinguish between a topological action ofGon a topologicalK-vector spaceV (that is, an action ofGby continuous endomorphisms ofV) and a continuous action ofGonV (that is, an action for which the action mapG×V →V is jointly continuous). Also, we will often write “convexK-vector space” as an abbreviation for “locally convex topologicalK-vector space”.

1. Lie-theoretic preliminaries

1.1. LetGbe a split connected reductive linear algebraic group defined over the fieldK. We will briefly recall the highest weight theory of representations ofG, and also of its Lie algebrag.

Since G is split, we may find a split maximal torus T in G, and a Borel subgroup B containingT. We letNdenote the unipotent radical ofB. We letg,t, b,andndenote the Lie algebras ofG,T,B, andNrespectively. We letX^•(T)⁺ denote the cone of dominant weights

inX^•(T). (It consists of all weights that pair non-negatively with the coroots that are positive with respect to the chosen BorelB.)

THEOREM 1.1.1. –LetW be a finite dimensional irreducible algebraic representation ofG overK.

(i) The subspaceW^NofW is one-dimensional, andTacts on this space through an element χ∈X^•(T)⁺.

(ii) AssociatingχtoW induces a bijection between the isomorphism classes of irreducible algebraicG-representations overKand the elements ofX^•(T)⁺.

Proof. –This summarizes the standard highest weight theory for the irreducible representa- tions of the split groupG. 2

In the situation of Theorem 1.1.1, the dominant weightχis called the highest weight ofW, and any basis element ofW^Nis called a highest weight vector ofW.

If t denotes the Lie algebra of T, then ˇt (the K-dual to t) is naturally isomorphic to K⊗ZX^•(T).We say that an element ofˇtis infinitesimally integral if it assumes integral values on all positive coroots. We letˇt⁺ denote the cone of infinitesimally integral elements ofˇtthat pair non-negatively with all positive coroots.

THEOREM 1.1.2. –Suppose that we are given an irreducible representation ofgon a finite dimensionalK-vector spaceW.

(i) The subspaceWⁿofW is one-dimensional, andtacts on this space through an element χinˇt⁺.

(ii) AssociatingχtoW induces a bijection between the isomorphism classes of irreducible finite dimensional representations ofgoverKand the elements ofˇt⁺.

Proof. –This summarizes the standard highest weight theory for the reductive Lie alge- brag. 2

Just as forG-representations, the dominant weightχis called the highest weight ofW, and any basis element ofWⁿis called a highest weight vector ofW.

The following results provide the link between the irreducible representations ofGand those ofg.

THEOREM 1.1.3. –IfW is a finite dimensional algebraic representation ofGdefined overK then the induced representation ofgis also irreducible.

Proof. –This follows from the fact thatGis connected. 2

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 ofGif and only if the highest weight ofW lies inX^•(T). In this case, there is a uniquely determined finite dimensional algebraic representation ofGonW lifting the giveng-action.

Proof. –This follows from Theorem 1.1.3, together with the fact that irreducible represen- tations of both G and g are determined by their highest weights. (We remark that since N is connected, the inclusion W^N⊂Wⁿ is an isomorphism for any representation W of G, and so the highest weight spaces ofW, thought of alternatively as aG-representation or as a g-representation, coincide.) 2

Related to the highest weight theory of irreducible representations ofgis the theory of Verma modules ofg,which we now recall.

DEFINITION 1.1.5. – Letχ∈ˇtbe a character oft. Regardingχ as a character ofbthat is trivial onn, letK(χ)denote the fieldKregarded as ab-module via the characterχ. We define the Verma module of highest weightχto be the leftU(g)-moduleVer(χ) =U(g)⊗U(b)K(χ).

We refer to the element1⊗1ofVer(χ)as the highest weight vector ofVer(χ), and denote it byv(χ). The lie algebrabacts onv(χ)through the characterχ.

In addition to the action ofg, the Verma moduleVer(χ)is equipped with a certain “adjoint”

action of the algebraic groupB, which we now explain. The adjoint action of the groupGon ginduces a natural adjoint action ofGonU(g). NowU(b)is invariant under the restriction of this adjoint action toB, as is the characterχ. ThusVer(χ)is naturally equipped with an adjoint action ofB.

We can differentiate the adjointB-action onVer(χ)to obtain an adjointb-action. Of course balso acts onVer(χ)via restricting the leftg-module structure onVer(χ). The adjoint action and the left-module action ofbare related in the following way: ifb∈bandY ∈Ver(χ)then adb(Y) +χ(b)Y =bY.

We will require the following structure theorem for Verma modules whose highest weight lies inˇt⁺. (In the statement of theorem,ndenotes the opposite nilpotent tondetermined byt.)

PROPOSITION 1.1.6. –Letχlie inˇt⁺.

(i) If α is a positive simple root of T over K, if X_−α∈n is a non-zero element of the (one-dimensional)root space inncorresponding to the negative simple root−α, and if m_α=χ,αˇ(whereαˇis the coroot corresponding toα), thenX₋^m_α^α+1v(χ)is annihilated by n, is fixed by the adjoint action of N, and generates aU(g)-submodule of Ver(χ) isomorphic toVer(χ−(mα+ 1)α).

(ii) If for each positive simple rootαas in (i)we letφα: Ver(χ−(mα+ 1)α)→Ver(χ) denote the homomorphism ofU(g)-modules obtained by mappingv(χ−(mα+ 1)α)to X_−α^mα+1v(χ), then the cokernel of the map

α

Ver

χ−(m_α+ 1)α

αφα

−→ Ver(χ)

is a finite dimensional irreducible representation ofgof highest weightχ.

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,αˇα=s_α(χ+ρ). 2

1.2. We let G be a split connected reductive linear algebraic group over K, let P be a parabolic subgroup ofG, let Ndenote the unipotent radical ofP, and letM be a Levi factor ofP. Choose a BorelBofGthat is contained inP, and a maximal torusTofBthat is contained inM(so thatTis also a maximal torus ofM). We letN denote the unipotent radical ofB, let Z_Gdenote the centre ofG, and letZ_Mdenote the centre ofM. We also letg,p,n,m,b,n,and tdenote the Lie algebras ofG,P,N,M,B,N,andTrespectively. The intersectionM∩Bis a Borel subgroup ofM, with unipotent radicalM∩N. AlsoN= (M∩N)N,and so

n= (m∩n)⊕n.

(1.2.1)

Let Δ(G,T)⊂X^•(T)denote the set of positive roots ofT with respect to G(that is, the set of characters appearing in the adjoint action ofTonn). We letΔ(G,T)sdenote the subset of Δ(G,T) consisting of simple positive roots. The setΔ(G,T)s is a basis for a finite index sublattice ofX^•(T/Z_G).

Similarly, we let Δ(M,T)denote the set of positive roots of T with respect toM (that is, the set of characters appearing in the adjoint action ofTonm∩n). If we letΔ(M,T)sdenote the subset ofΔ(M,T)consisting of simple positive roots, thenΔ(M,T)sis the basis of a finite index sublattice ofX^•(T/ZM).

LEMMA 1.2.2. –There is an equalityΔ(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 nis an ideal ofp,and so in particular ofn, we see that any root whose decomposition in terms of simple roots involves an element ofΔ(G,T)_s\Δ(M,T)appears inn,and so not inm∩n. Consequently any element ofΔ(M,T)_sis simple even when regarded as an element ofΔ(G,T).

This proves the lemma. 2

We letΔ(G,Z_M)⊂X^•(Z_M)denote the set of positive restricted roots ofZ_M(that is, the set of characters appearing in the adjoint action ofZ_Monn). We letΔ(G,Z_M)sdenote the subset consisting of simple positive restricted roots. The decomposition (1.2.1) shows that the elements ofΔ(G,Z_M)are the restrictions toZ_Mof the elements ofΔ(G,T)\Δ(M,T).

LEMMA 1.2.3. –Restriction of characters fromTtoZMinduces a bijection Δ(G,T)s\Δ(M,T)s−→∼ Δ(G,ZM)s.

Proof. –Lemma 1.2.2 shows thatΔ(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/ZG),while Δ(M,T)sis a basis for a finite index sublattice ofX^•(T/ZM).It follows that the restriction ofΔ(G,T)s\ Δ(M,T)stoZMis a basis for a finite index sublattice ofX^•(ZM/ZG).In particular, restriction toZMyields an embedding

Δ(G,T)s\Δ(M,T)s→Δ(G,Z_M).

(1.2.4)

Any element ofΔ(G,Z_M)sis the restriction toZ_Mof some elementα∈Δ(G,T)\Δ(M,T).

We may writeα=

β∈Δ(G,T)snββwithnβ0. Since anyβ∈Δ(M,T)shas trivial restriction to Z_M, we may omit these terms from the sum, and not alter the restriction to Z_M. Thus we assume thatα=

β∈Δ(G,T)s\Δ(M,T)snββ.As already observed, the simple rootsβappearing in this sum have non-zero restrictions toZ_M. Thus, if the restriction ofαtoZ_Mis simple, then we see thatαitself must be simple. Hence any element ofΔ(G,Z_M)slies in the image of (1.2.4).

Now consider an arbitrary α∈ Δ(G,T)s \ Δ(M,T)s. The restriction α_|ZM is a linear combination of elements of Δ(G,ZM)s with non-negative coefficients. By the result of the preceding paragraph, we may write α_|ZM =

β∈Δ(G,T)s\Δ(M,T)snββ_|ZM, for some nβ 0.

Since (1.2.4) induces an injection of the span of its domain into X^•(ZM), we deduce that α=

β∈Δ(G,T)s\Δ(M,T)sn_ββ. Since αis simple, we conclude that n_β = 0 if β =α, while n_α= 1.Thusα_|ZM is also simple, and so the image of (1.2.4) is contained inΔ(G,ZM)_s. 2

For any rootαofTinG(that is, any element ofΔ(G,T)∪ −Δ(G,T)), we letsαdenote the corresponding simple reflection in the Weyl groupW(G,T)of Twith respect toG. Note that the Weyl group W(M,T)of Twith respect toMnaturally embeds as a subgroup of the Weyl groupW(G,T). Indeed, it is identified with the centralizer of the subtorusZMofTinW(G,T).

LEMMA 1.2.5. –Ifw∈W(M,T) andαis a root ofTinG, then for anyχ∈X^•(T), the characterssα(χ)ands_w(α)(w(χ))have the same restriction toZ_M.

Proof. –Indeed,w(sα(χ)) =sw(α)(w(χ)), andwacts trivially onZM. 2

LetW be an irreducible algebraicG-representation. Highest weight theory shows thatW^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 characterψ˜∈X^•(T)ofW. Ifαis any element ofΔ(G,Z_M)s,then Lemma 1.2.3 shows thatαlifts uniquely toα˜∈Δ(G,T)s.Note thatψand αare both well-determined, independent of the choice ofBcontained inP, whileψ˜andα˜ in general depend on this choice.

Letρ(G,T)denote one-half of the sum of the elements ofΔ(G,T).

LEMMA 1.2.6. –The restriction ofsα˜( ˜ψ+ρ(G,T)) +ρ(G,T)toZM is independent of the choice of the Borel subgroup contained inP.

Proof. –IfBis any Borel subgroup contained inP, then we may choose the maximal torusT to be contained inB∩B, and then may writeB=wBw⁻¹for some elementw∈W(M,T). The corresponding lift ofα(respectivelyψ) is thus equal tow( ˜α)(respectivelyw( ˜ψ)). Alsoρ(G,T) is replaced byw(ρ(G,T)). The lemma is now seen to be a consequence of Lemma 1.2.5. 2

1.3. Let G denote a connected reductive linear algebraic group over K, let P denote a parabolic subgroup ofG, letNdenote the unipotent radical ofP, and writeM=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 letz(g)(respectivelyz(m)) denote the centre of the universal enveloping algebraU(g) ofg(respectively the centre of the universal enveloping algebraU(m)ofm). In this subsection we recall the construction of the unnormalized Harish–Chandra homomorphismz(g)→z(m).

We use the inclusionp⊂gto regard the universal enveloping algebraU(p)as a subalgebra of U(g).

PROPOSITION 1.3.1. –The sumU(p)+U(g)nis a subalgebra ofU(g), which containsU(g)n as an ideal, and which contains the centrez(g)ofU(g)as a subalgebra.

Proof. –The first two claims follow from the fact thatnis an ideal inp. To prove the final claim, choose an opposite parabolic Pto P, with unipotent radical N; the intersection P∩P provides a lift of M to a Levi factor of P. If ndenotes the Lie algebra of N, then there is a direct sum decompositionU(g)∼=nU(p)⊕U(m)⊕U(g)n.IfAis the maximal split torus in the centre of M, then since the adjoint A-action on U(g) preserves this decomposition, and since theA-invariant subspace ofnU(p)is trivial, we see that the elements ofz(g)(which are necessarily invariant under the adjointA-action) must lie in the direct sumU(m)⊕U(g)n= U(p) +U(g)n. 2

We define the “unnormalized Harish–Chandra homomorphism” γ:z(g)→U(m)to be the composite of the inclusionz(g)⊂U(p) +U(g)nof the preceding proposition with the projection U(p) +U(g)n=U(m)⊕U(g)n→U(m).

PROPOSITION 1.3.2. –The mapγis injective with image lying inz(m)⊂U(m).

Proof. –Sincez(g)lies in the centre ofU(p) +U(g)n, its image underγ lies inz(m). By construction the kernel ofγis contained inU(g)n. IfW is any finite dimensional representation of G, thenW^Nis annihilated byU(g)n. Since W is generated as aG-representation byW^N (apply highest weight theory over an extension ofK that splitsG), and sincez(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 Gnecessarily vanishes, and thus γ is injective. 2

1.4. We recall some basic structure theory of tori over the local field L. If T is a torus defined overL, letSdenote the maximal split subtorus ofT. LetX_•(respectivelyX^•) denote the cocharacter lattice (respectively the character lattice) of T over L, and similarly let Y_• (respectively Y^•) denote the cocharacter lattice (respectively the character lattice) ofS. Each ofX^•andX_•is equipped with an action ofGal(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^• identifiesY^• with the quotient by its torsion subgroup of the group of Galois coinvariants ofX^•.

WriteS:=S(L)andT:=T(L), and letS⁰(respectivelyT⁰) denote the maximal compact subgroup of S (respectivelyT). If t∈T, then the element of Hom(X^•,Q) defined byχ→ ord_L(χ(t))is Galois invariant, and so defines a mapT→Q⊗_ZY_•.Denote the image byY_•. Restricted toS, this yields a surjectionS→Y_•, with kernelS⁰. Altogether, we obtain a diagram of short exact sequences

0 S⁰ S Y_• 0

0 T⁰ T Y_• 0

(1.4.1)

Both Y_• andY_• 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 toT.)

We letTdenote the rigid analytic variety of locallyL-analytic characters ofT (as discussed in [8, §6.4]). Ifχis an element ofT(K), then the functiont→ord_K(χ(t))is trivial onT⁰,and so factors to yield a morphismY_•→Q.

DEFINITION 1.4.2. – Ifχ∈T(K), then we define slope(χ)to be the element ofQ⊗_ZY^• corresponding to the morphismY_•→Qdescribed in the preceding paragraph.

We remark that ifχin fact happens to be an element ofX^•(that is, an algebraic character of T that can be defined overK), thenslope(χ)is equal to the image ofχinY^•. (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. IfXis a set, thenc0(X, K)denotes the K-Banach space of functionsf:X →K whose values become arbitrarily small outside sufficiently large finite subsets ofX(topologized via the sup norm). We embedXintoc0(X, K) by identifying a pointx∈X with the function1xthat is equal to1at the pointxand vanishes onX\ {x}.For anyK-vector spaceV, we letF(X, V)denote theK-vector space ofV-valued functions onX.

PROPOSITION 2.1.1. –If V is a Hausdorff convex K-vector space, then the natural map L(c0(X, K), V)→ F(X, V), given by restricting maps to the subsetXofc0(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 ofV.

Proof. –The discussion of [15, pp. 59–60] shows that X topologically spans c0(X, K), implying the claimed injectivity. It is also proved there that if V is quasi-complete then the map under consideration has image equal to the set of mapsX→V with bounded image. IfV is arbitrary, and ifX→V has image contained in a bounded complete subset ofV, then this map factors as a compositeX→U→V in which the second arrow is continuous andK-linear, with source equal to a Banach space (and so in particular quasi-complete). Thus the first arrow extends to a continuousK-linear mapc0(X, K)→U,which when composed with the second arrow yields the required extension of the given mapX→V. 2

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 fromV toW. The following are equivalent:

(i) The collection of mapsXis equicontinuous.

(ii) The mapX×V →Wthat describes the action of the maps inX extends(in a necessarily unique fashion)to a jointly continuous mapc₀(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 continuousK-linear mapc₀(X, K)→ Ls(V, W),and thus a separately continuous K-bilinear map

c0(X, K)×V →W.

(2.1.3)

Since the closedOK-lattice spanned byX is the unit ball ofc₀(X, K)(with respect to the sup norm on this space), the unit ball ofc₀(X, K)maps to an equicontinuous subset ofL(V, W). It follows directly that (2.1.3) is jointly continuous.

Conversely, if we are given a jointly continuous mapc0(X, K)×V →W,then we see that the entire unit ball ofc0(X, K)(and so in particular the setX) induces an equicontinuous set of mappingsV →W. 2

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 toV and from U to W respectively, then the collectionX ⊗Y :={x⊗y|x∈X, y∈Y} of maps T⊗K,πU→V ⊗K,πW is again equicontinuous.

(ii) IfX andY are two equicontinuous collections ofK-linear mapsU→V then so is the unionX∪Y.

(iii) IfX is an equicontinuous collection ofK-linear mapsU →V, and ifα:T →U and β:V →W are continuousK-linear maps, then the setβ◦X◦α:={β◦x◦α|x∈X}

is an equicontinuous collection of mapsT→W.

(iv) IfX is a collection ofK-linear mapsU→V, andα:T→U is a continuousK-linear strict surjection, thenX is an equicontinuous collection of maps if and only ifX◦α:=

{x◦α|x∈X}is an equicontinuous collection of mapsT→V.

Proof. –All four parts of the lemma are immediate. 2

DEFINITION 2.1.5. – Letφbe a continuousK-linear endomorphism of a convex K-vector space V. We say thatφ is power-bounded if the set{φⁿ|n0} of endomorphisms of V is equicontinuous.

For any elementxinK^×, denote byKxtthe Tate algebra overKof power series intthat converge on the closed disk|t||x|⁻¹. (Ifx= 1, then we omit it from the notation, and write

simplyKt.) Similarly denote byKxt, xt⁻¹the Tate algebra overKof power series intand t⁻¹that converge on the closed annulus|x||t||x|⁻¹. (If|x|>1then setKxt, xt⁻¹= 0.) For anyK-Banach algebraA, denote byAxtthe completed tensor productA⊗KKxt(and similarly forAxt, x⁻¹t).

PROPOSITION 2.1.6. –A continuous K-linear endomorphismφof a quasi-complete Haus- dorff convex K-vector space V is power-bounded if and only ifV admits the structure of a topologicalKt-module with respect to whichtacts viaφ, and this module structure is unique if it exists.

Proof. –Since the map 1n→tⁿ induces a topological isomorphism c0(N, K)−→^∼ Kt, Proposition 2.1.2 shows that φ is power-bounded if and only if there is a jointly continuous map

Kt ×V →V (2.1.7)

such that(tⁿ, v)→φⁿ(v)for everyv∈V, and thatφuniquely determines (2.1.7) (assuming that it exists). SinceK[t]is dense inKt, we see immediately that (2.1.7) (if it exists) givesV the structure of aKt-module. 2

PROPOSITION 2.1.8. –

(i) If φ is a power-bounded endomorphism of a convex K-vector space V, and if U is a second convex K-vector space, then id_U ⊗φ is a power-bounded endomorphism of U⊗K,πV.

(ii) Let α:U →V and β:V →U be a pair of maps between quasi-complete Hausdorff convexK-vector spaces. Let φ:=β◦αand ψ:=α◦β. Then φis a power-bounded endomorphism ofU if and only if ψ is a power-bounded endomorphism ofV. If these equivalent conditions hold, and ifU andV are equipped with theKt-module structure of Proposition2.1.6, thenαandβareKt-linear maps.

(iii) LetU →V be a strict surjection of Hausdorff quasi-complete convexK-vector spaces, and letψandφbe endomorphisms ofU andV respectively that fit into a commutative diagram

U ^ψ U

V ^φ V

If ψ is power-bounded, then so is φ, and the surjection U →V is furthermore a homomorphism ofKt-modules.

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

As for part (ii), write X :={φⁿ|n0} and Y :={ψⁿ|n0}. 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 thatX is equicontinuous if and only ifY is so. Assuming that they are equicontinuous, and noting thatα◦φ=ψ◦αand thatβ◦ψ=φ◦β,we see thatαand β areK[t]-linear, and henceKt-linear, by continuity.

Part (iii) follows from Lemma 2.1.4(iii) and (iv) and the definition of the Kt-module structure onU andV in terms ofψandφrespectively. 2

PROPOSITION 2.1.9. –Let R be a locally convex topological K-algebra, letU and V be locally convex topologicalR-modules, and letα:U→V andβ:V →U be continuousR-linear maps such that the compositesφ=β◦αandψ=α◦βare each equal to an elementt∈R.(In

the situation of Proposition2.1.8(ii), ifφandψare power-bounded, then these hypotheses are satisfied for anyRadmitting a continuous injection intoKtwhose image containsK[t].)IfS is a topologicalR-algebra in which the elementtbecomes invertible, then the naturalS-linear mapS⊗R,πU→S⊗R,πV induced byαis a topological isomorphism.

Proof. –The endomorphisms idS ⊗φ of S⊗R,πU and idS ⊗ψ of S ⊗R,π V are both topological isomorphisms, sincetis a unit inS. The proposition thus follows from the formulas β◦α=φandα◦β=ψ. 2

PROPOSITION 2.1.10. –Ifφis a continuous endomorphism of aK-Banach spaceV, thenxφ is power-bounded ifx∈Khas sufficiently large valuation.

Proof. –Fix a norm defining the topology ofV; this determines a corresponding operator norm onLb(V, V).For anyx∈Kfor whichxφhas operator norm at most one, all powers ofxφagain have operator norm at most one, and soxφis power bounded. 2

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 toc0(X, K)for some set X [15, prop. 10.1]. Thus the orthonormalizableA-Banach modules are precisely those of the form A⊗KV,for someK-Banach spaceV.) We fix theK-Banach algebraAfor the duration of this subsection.

LEMMA 2.2.1. –LetMandN beA-Banach modules, and letV be aK-Banach space. If we are givenA-linear continuous mapsA⊗KV →M andN→M with the image of the first map being contained in the image of the second then we may find anA-linear mapA⊗KV →N that gives a commutative diagram

A⊗KV

N M

Proof. –Let P denote the image of the mapN →M, equipped with the topology that it inherits as a subspace of M, and let P1 denote the same abstract vector space, but equipped with the (Banach space) topology that it inherits as a quotient ofN. By assumption the given mapV →M factors through the inclusionP ⊂M, and the closed graph theorem shows that the induced mapV →P1 is again continuous [8, prop. 1.1.2]. By [15, prop. 10.5(a)] (and its proof) we may find a continuousK-linear splitting of the surjectionN→P1.Composing such a splitting with the continuous mapV →P1induces a map ofK-Banach spacesV →N,which in turn induces a map ofA-Banach modulesA⊗KV →N.This is the required map. 2

LEMMA 2.2.2. –IfV is aK-Banach space andMis anA-Banach module, then a continuous A-linear mapA⊗KV →M has its image contained in a finitely generatedA-submodule ofMif and only if it may be factored as a composite of continuousA-linear mapsA⊗KV →A^r→M for some natural numberr.

Proof. –The if direction is immediate. Its converse follows from Lemma 2.2.1. 2 DEFINITION 2.2.3. –

(i) IfV andW areK-Banach spaces, then anA-linear morphismφ:A⊗KV →A⊗KW is calledA-compact if we may write it as the limit inLb(A⊗KV, A⊗KW)of a sequence {φn} of A-linear maps A⊗KV →A⊗KW, each of whose image is contained in a finitely generatedA-submodule ofA⊗KW.

(ii) IfM andNareA-Banach modules, then a continuousA-linear mapφ:M→Nis called A-compact if there exist K-Banach spaces V and W and a commutative diagram of continuousA-linear maps

A⊗KV A⊗KW

M ^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 orthonormal- izableA-Banach modules, is a consequence of the following lemma.

LEMMA 2.2.4. –Letφ:M →N beA-compact in the sense of Definition2.2.3(ii). IfV and W areK-Banach spaces, and if we are given continuousA-linear morphismsA⊗KV →M and A⊗KW →N, the latter being surjective, then we may liftφto a morphismA⊗KV → A⊗KW that isA-compact in the sense of Definition2.2.3(i).

Proof. –Follow Definition 2.2.3(ii) and form a commutative diagram A⊗KV1 A⊗KW1

M ^φ N

in whichV₁ andW₁ areK-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⊗KV →M andA⊗KW₁→N toA-linear mapsA⊗KV →A⊗KV₁ and A⊗KW₁→A⊗KW respectively. Composing these lifts with the upper horizontal map of the commutative diagram yields a mapA⊗KV →A⊗KWwhich liftsφand satisfies the conditions of Definition 2.2.3(i). 2

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 orthonormalizableA-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 anA-compact morphism between orthonormalizable A-Banach modules agrees with the notion of a completely continuousA-linear map introduced in [6]. However, ifM andN 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 continuousA-linear maps betweenA-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) IfM1→M,M→NandN→N1are continuousA-linear maps ofA-Banach modules, the second beingA-compact, then the compositeM1→M→N→N1isA-compact.

(ii) IfA→B is a continuous homomorphism ofK-Banach algebras, and ifM →N is an A-compact map ofA-Banach modules, then the induced mapB⊗AM →B⊗AN is B-compact.

(iii) If we are given a commutative diagram of continuousA-linear maps betweenA-Banach modules

M N

P Q

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

Proof. –For part (i), note that the mapsM1→M andN1→N induce the top horizontal arrows of the commutative diagram

A⊗KM1 A⊗KM A⊗KN A⊗KN1

M₁ M N N₁

in which the vertical arrows are the natural maps induced by theA-module structures on each of M, M1, N, and N1 respectively. Lemma 2.2.4 shows that we may lift the A-compact map M →N to a map φ:A⊗KM →A⊗KN 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⊗KM₁→A⊗KN₁ 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 isA-compact, as claimed.

Part (ii) follows from the definition, once we note that for a K-Banach space V there is a natural isomorphismB⊗A(A⊗KV)−→^∼ B⊗KV, and take into account Lemma 2.2.2, while part (iii) follows directly from the definition. 2

We can now prove the main result of this subsection.

PROPOSITION 2.2.6. –Letφbe anA-compact endomorphism of anA-Banach moduleM. If x∈K^×has sufficiently large valuation then:

(i) The endomorphismxφofM is power-bounded, in the sense of Definition2.1.5.

(ii) If we regard M as a topological Kxt-module (with t acting via φ on M), then Kxt, xt⁻¹⊗KxtM is finitely generated as anAxt, xt⁻¹-module.

Proof. –Choose a continuousA-linear surjectionA⊗KV →M for someK-Banach spaceV (e.g. takeV =M). Lemma 2.2.4 shows that we may liftφto anA-linear endomorphismψof A⊗KV that satisfies Definition 2.2.3(i).

Proposition 2.1.10 shows that xψ is power-bounded if ordK(x) is large enough. Proposi- tion 2.1.8(iii) implies thatxφis then also power-bounded, and that the surjectionA⊗KV →M