Draft: July 15, 2007 ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS OF p-adic REDUCTIVE GROUPS I. DEFINITION AND FIRST PROPERTIES

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1 Draft: July 15, 2007 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS OF p-adic REDUCTIVE ROUPS I. DEFINITION AND FIRST PROPERTIES ATTHEW EERTON Contents 1. Introduction 1 2. Representations of p-adic analytic groups 2 3. Ordinary parts Adjunction formulas 24 References Introduction This paper is the first of two in which we define and study the functors of ordinary parts on categories of admissible smooth representations of p-adic reductive groups over fields of characteristic p, as well as their derived functors. These functors of ordinary parts, which are characterized as being right adjoint to parabolic induction, play an important role in the study of smooth representation theory in characteristic p. They also have important global applications: when applied in the context of the p-adically completed cohomology spaces introduced in [5], they provide a representation-theoretic approach to Hida s theory of nearly) ordinary parts of cohomology [9, 10, 11]. The immediate applications that we have in mind are for the group L 2 : we will apply the results of our two papers both to the construction of a mod p local Langlands correspondence for the group L 2 Q p ), and to the investigation of local-global compatibility for p-adic Langlands over the group L 2 over Q. To describe our results more specifically, let be the Q p -valued points of) a connected reductive p-adic group, P a parabolic subgroup of, P an opposite parabolic to P, and = P P the corresponding Levi factor of P and P. If k is a finite field of characteristic p, we let od adm k) resp. od adm k)) denote the category of admissible smooth -representations resp. -representations) over k) od adm k). The functor of ordinary parts associated to P is then a functor Ord P : od adm k) od adm k), which is right adjoint to Ind P. In this paper, we define Ord P and study its basic properties. In the sequel [8] we investigate the derived functors of Ord P, and present the applications to the mod p k. Parabolic induction yields a functor Ind P : odadm Langlands correspondence for L 2 Q p ). The applications to the study of localglobal compatibility will form part of the arguments of [7]. We hope to discuss the applications in the context of p-adically completed cohomology in a future paper. The author was supported in part by NSF grants DS and DS

2 2 ATTHEW EERTON Let us only mention here that Theorem below is an abstract formulation of Hida s general principal that the ordinary part of cohomology should be finite over weight space Arrangement of the paper. In order to allow for maximum flexibility in applications, we actually work throughout the paper with representations defined not just over the field k, but over general Artinian local rings, or even complete local rings, having residue field k. This necessitates a development of the foundations of the theory of admissible representations over such coefficient rings. Such a development is the subject of Section 2. We also take the opportunity in that section to present some related representation theoretic notions that do not seem to be in the literature, and which will be useful in this paper, its sequel, and future applications. The functors Ord P are defined, and some of their basic properties established, in Section 3. Their characterization as adjoint functors is proved in Section Notation and terminology. Throughout the paper, we fix a prime p, as well as a finite extension E of Q p, with ring of integers O. We let F denote the residue field of O, and ϖ a choice of uniformizer of O. Let CompO) denote the category of complete Noetherian local O-algebras having finite residue fields, and let ArtO) denote the full subcategory of CompO) consisting of those objects that are Artinian or equivalently, of finite length as O-modules). If A is an object of ArtO), and V is an A-module, then we write V := Hom O V, E/O) to denote the Pontrjagin dual of V, equipped with its natural profinite topology. If V is a -representation over A, for some group, then the contragredient action makes V a -representation over A with each element of acting via a continuous automorphism). If C is any O-linear category, then for any integer i 0 resp. i = ), we let C[ϖ i ] denote the full subcategory of C consisting of objects that are annihilated by ϖ i resp. by some power of ϖ). We let C fl denote the full subcategory of C consisting of objects that are flat over O, and let C E denote the E-linearization of C. 2. Representations of p-adic analytic groups Let be a p-adic analytic group. Throughout this section, we will let A denote an object of CompO), with maximal ideal m. We denote by od A) the abelian category of representations of over A with morphisms being A-linear - equivariant maps); equivalently, od A) is the abelian category of A[]-modules, where A[] denotes the group ring of over A. In this section we introduce, and study some basic properties of, various categories of -representations, as well as of certain related categories of what we will call augmented -representations Augmented -representations. If H is a compact open subgroup of, then as usual we let A[[H]] denote the completed group ring of H over A, i.e. 1) A[[H]] := lim H A[H/H ], where H runs over all open subgroups of H. We equip A[[H]] with the projective limit topology obtained by endowing of the rings A[H/H ] appearing on the right hand side of 1) with its m-adic topology. The rings A[H/H ] are then profinite

3 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS 3 since each H/H is a finite group), and hence this makes A[[H]] a profinite, and so in particular compact, topological ring Theorem. The completed group ring A[[H]] is Noetherian. Proof. In the case when A = O, this is proved by Lazard [12]: after replacing H by an open subgroup, if necessary, Lazard constructs an exhaustive decreasing filtration F on O[[H]], which induces the ϖ-adic filtration on O, such that associated graded ring is isomorphic to r ϖo) Fp Ug), where r ϖo denotes the graded ring associated to the ϖ-adic filtration on O which is a commutative F-algebra isomorphic to F[t]), and Ug) denotes the universal enveloping algebra of the Lie algebra g of H over F p. Since this associated graded ring is Noetherian, so is O[[H]]. For general A, we may write A[[H]] := A ˆ O O[[H]], and so equip A[[H]] with an exhaustive decreasing filtration F obtained as the tensor product of Lazard s filtration on O[[H]] and the m-adic filtration on A. The associated graded ring is isomorphic to r ma) Fp Ug), where r ma denotes the graded ring associated to the m-adic filtration on A. Since r ma is Noetherian, the same is true of this tensor product, and thus A[[H]] is Noetherian, as claimed Proposition. 1) Any finitely generated A[[H]]-module admits a unique profinite topology with respect to which the A[[H]]-action on it becomes continuous. 2) Any A[[H]]-linear morphism of finitely generated A[[H]]-modules is continuous with respect to the profinite topologies on its source and target given by part 1). Proof. Since A[[H]] is Noetherian, we may find a presentation A[[H]] s A[[H]] r 0 of, for some r, s 0. Since A[[H]] is profinite, it follows that the first arrow has closed image, and hence that is the quotient of the profinite module A[[H]] r by a closed A[[H]]-submodule. If we equip with the induced quotient topology, then it becomes a profinite module. The resulting profinite topology on is clearly independent of the chosen presentation, and satisfies the requirements of the proposition Definition. If is a finitely generated A[[H]]-module, we refer to the topology given by the preceding lemma as the canonical topology on Definition. By an augmented representation of over A we mean an A[]- module equipped with an A[[H]]-module structure for some equivalently, any) compact open subgroup H of, such that the two induced A[H]-actions the first induced by the inclusion A[H] A[[H]] and the second by the inclusion A[H] A[]) coincide Definition. By a profinite augmented -representation over A we mean an augmented -representation over A that is also equipped with a profinite topology, so that the A[[H]] action on is jointly continuous 1 for some equivalently any) compact open subgroup H over A. 1 It is equivalent to ask that the profinite topology on admits a neighbourhood basis at the origin consisting of A[[H]]-submodules.

4 4 ATTHEW EERTON The equivalence of the conditions some and any in the preceding definitions follows from the fact that if H 1 and H 2 are two compact open subgroups of, then H := H 1 H2 has finite index in each of H 1 and H 2. We denote by od aug A) the abelian category of augmented -representations over A, with morphisms being maps that are simultaneously -equivariant and A[[H]]-linear for some equivalently, any) compact open subgroup H of, and by pro aug od A) the abelian category of profinite augmented -representations, with morphisms being continuous A-linear -equivariant maps note that since A[H] is dense in A[[H]], any such map is automatically A[[H]]-linear for any compact open subgroup H of ). Forgetting the topology induces a forgetful functor od pro aug A) od aug A). fg aug We let od A) denote the full subcategory of od aug A) consisting of augmented -modules that are finitely generated over A[[H]] for some equivalently any) compact open subgroup H of. Proposition shows that by equipping fg aug each object of od A) with its canonical topology, we may lift the inclusion fg aug od A) od aug aug pro aug A) to an inclusion odfg A) od A). fg aug Proposition. The category od A) forms a Serre subcategory of the abelian category od aug A). In particular, it itself is an abelian category. Proof. Closure under the formation of quotients and extensions is evident, and closure under the formation of subobjects follows from Theorem Write A E := E O A, and A[[H]] E := E O A[[H]] Theorem. The ring A[[H]] E is Noetherian. Proof. This follows directly from Theorem We introduce the following analogue of Definition 2.1.4, in the context of - representations over A E Definition. An augmented representation of over A E consists of an A E []- module equipped with an A[[H]] E -module structure for some equivalently, any) compact open subgroup H of, such that the two induced A E [H]-actions the first induced by the inclusion A E [H] A[[H]] E and the second by the inclusion A E [H] A E []) coincide. fg aug We let od A E ) denote the category whose objects are augmented - modules over A E that are finitely generated over A[[H]] E for some equivalently any) compact open subgroup H of, and whose morphisms are maps that are simultaneously A E )[] and A[[H]] E -linear Lemma. The functor E O induces an equivalence of categories fg aug od A) E od fg aug A E ). Proof. Obvious Smooth -representations. In this subsection we give the basic definitions, and state the basic results, related to smooth, and admissible smooth, representations of over A. In the case when A is Artinian, our definitions will agree with the standard ones. Otherwise, they may be slightly unorthodox, but will be the most useful ones for our later purposes.

5 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS Definition. Let V be a representation of over A. We say that a vector v V is smooth if: 1) If v is fixed by some open subgroup of. 2) v is fixed by some power m i of the maximal ideal of A. We let V sm denote the subset of V consisting of smooth vectors Remark. The following equivalent way of defining smoothness can be useful: a vector v V is smooth if and only if v is annihilated by an open ideal in A[[H]], for some equivalently, any) open subgroup H of Remark. If A is Artinian, then m i = 0 for sufficiently large i, and so automatically m i v = 0 for any element v of any A-module V. Thus condition 2) can be omitted from the definition of smoothness for such A, and our definition coincides with the usual one Lemma. The subset V sm is an A[]-submodule of V. Proof. If v i V sm i = 1, 2) are fixed by open subgroups H i, and annihilated by m i1 and m i2, then any A-linear combination of v 1 and v 2 is fixed by H 1 H2, and annihilated by m max i1,i2, and hence also lies in V sm. Thus V sm is an A-submodule of V. Also, if v V sm is fixed by the open subgroup H, and annihilated by m i, then gv is fixed by ghg 1 for any g, and again annihilated by m i. Thus V sm is closed under the action of Definition. We say that a -representation V of over A is smooth if V = V sm ; that is, if every vector of V is smooth. We let od sm A) denote the full subcategory of od A) consisting of smooth -representations. Clearly od sm A) is in fact a Serre subcategory of od A) i.e. it is closed under passing to subobjects, quotients, and extensions). In particular, it is again an abelian category. The association V V sm is a left-exact functor from od A) to od sm A), which is right adjoint to the inclusion of od sm A) into od A). The following lemma provides some useful ways for thinking about smooth representations of Lemma. Let V be a -representation over A. The following conditions on V are equivalent. 1) V is smooth. 2) The A-action on V and the -action on V are both jointly continuous, when V is given its discrete topology. 3) The A-action and the -action on V are both jointly continuous when V is given its natural profinite topology. 4) For any compact open subgroup H of, the H-action on V extends to a continuous action of completed group ring A[[H]] on V. In both conditions 1) and 2), the ring A is understood to be equipped with its m-adic topology.) Proof. The straightforward and well-known) proofs are left to the reader. Lemma implies that passing to Pontrjagin duals induces an anti-equivalence 2) od sm A) anti od pro aug A).

6 6 ATTHEW EERTON Definition. We say that a smooth -representation V over A is admissible if V H [m j ] the m i -torsion part of the subspace of H-fixed vectors in V ) is finitely generated over A for every open subgroup H of and every i Remark. If A is Artinian, so that m i = 0 for i sufficiently large, then V H [m i ] = V H for i sufficiently large, and so the preceding definition agrees with the usual definition of an admissible smooth A-module. We let od adm A) denote the full subcategory of od sm A) consisting of admissible representations Lemma. A smooth -representation V over A is admissible if and only if V is finitely generated as an A[[H]]-module for some equivalently any) compact open subgroup H of. Proof. This is well-known. Lemma shows that the anti-equivalence 2) restricts to an anti-equivalence 3) od adm A) anti od fg aug Proposition. The category od adm A) forms a Serre subcategory of the abelian category od A). In particular, it itself is an abelian category. Proof. This follows directly from the anti-equivalence 3), together with Proposition Remark. If V is any -representation over A, and j 0, then V [m j ] is a successive extension of finitely many copies of V [m]. It thus follows from the preceding result that if V is a smooth -representation over A for which V [m] is an admissible -representation over the field A/m equivalently, if the condition of Definition is assumed just in the case j = 1), then V is in fact an admissible smooth -representation. In fact, it follows from Lemma 2.2.9, together with the topological version of Nakayama s lemma, that if H is an open pro-p subgroup of so that A[[H]] is a typically non-commutative local ring), then V is admissible if and only if V H [m] is finite dimensional over A/m for one open pro-p subgroup of Definition. Let V be a -representation over A. 1) We say that an element v V is locally admissible if v is smooth, and if the smooth -subrepresentation of V generated by v is admissible. 2) We let V l.adm denote the subset of V consisting of locally admissible elements. A) Lemma. The subset V l.adm is an A[]-submodule of V. Proof. Let v i V l.adm i = 1, 2), and let W i denote the -subrepresentation of V generated by v i. Since od adm A) is a Serre subcategory of od sm A), we see that the image of the map W 1 W2 V induced by the inclusions W i V is admissible. This image certainly contains any A-linear combination of the v i, and thus V l.adm is an A-submodule of V. It is also clearly -invariant, and thus is an A[]-submodule of V Definition. We say that a -representation V of over A is locally admissible if V = V l.adm ; that is, if every vector of V is locally admissible.

7 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS 7 We let od l adm A) denote the full subcategory of od sm A) consisting of locally admissible -representations Proposition. The category od l adm A) is closed under passing to subrepresentations, quotients, and inductive limits in od sm A). In particular, it itself is an abelian category. Proof. Since od adm A) is closed under passing to subrepresentations and quotients in od sm A), the same is evidently true for od adm A). The closure under formation of inductive limits is also clear, since the property of being locally admissible is checked elementwise Remark. Clearly any finitely generated locally admissible representation is in fact admissible smooth. Since any representation is the inductive limit of finitely generated ones, we see that a representation is locally admissible if and only if it can be written as an inductive limit of admissible representations. The association V V l.adm gives rise to a left-exact functor from od A) to A), which is right adjoint to the inclusion of od l adm A) into od A). od l adm The following diagram summarizes the relations between the various categories that we have introduced: od A) ) l.adm ) sm od sm A) anti pro aug od A) od aug A) od l adm A) canonical topology od adm A) anti od fg aug A) 2.3. Some finiteness conditions. Let Z denote the centre of Definition. Let V be an object of od A). 1) We say that V is Z-finite if, writing I to denote the annihilator of V in A[Z], the quotient A[Z]/I is a finite A-algebra. 2) We say that v V is locally Z-finite if the A[Z]-submodule of V generated by v is Z-finite equivalently, is finitely generated as an A-module). We let V Z fin denote the subset of locally Z-finite elements of V. 3) We say that V is locally Z-finite over Z if every v V is locally finite over Z, i.e. if V = V Z fin Lemma. 1) For any object V of od A), the subset V Z fin is an A[]-submodule of V. 2) If V 1 and V 2 are two locally) Z-finite representations, then V 1 V2 is again locally) Z-finite. 3) Any A[]-invariant submodule or quotient of a locally) Z-finite representation is again locally) Z-finite.

8 8 ATTHEW EERTON Proof. If V 1 and V 2 are A[]-modules annihilated by the A-finite quotients A[Z]/I 1 and A[Z]/I 2 of A[Z], respectively, then V 1 V2 is annihilated by A[Z]/I 1 I2 ), which is again A-finite since it embeds into A[Z]/I 1 ) A[Z]/I 2 )). This proves 2) in the case of Z-finite representations. Now suppose that v 1 and v 2 are two elements of V Z fin, for some A[]-module V. Let I i denote the annihilator of v i in A[Z], and let W i be the -subrepresentation of V generated by v i. Since Z is the centre of, we see that the I i annihilates all of W i. Thus each W i is Z-finite, and thus so is W 1 W2. The image of the natural map W 1 W2 V contains all linear combinations of the v i, and is -invariant. Thus V Z fin is an A[]-submodule of V. Now let V 1 and V 2 be locally Z-finite A[]-modules. Then V 1 V2 ) Z fin contains V 1 and V 2, and so, since it is an A[]-submodule of V 1 V2, equals all of V 1 V2. This proves 2) in the case of locally Z-finite representations. Part 3) is evident Lemma. If V is an object of od A) which is finitely generated over A[], then V is Z-finite if and only if V is locally Z-finite. Proof. As we already observed in the proof of the preceding lemma, if an A[]- module is generated by a single locally Z-finite vector, then it is Z-finite. If V is generated by finitely many such vectors, then it is a quotient of a direct sum of finitely many Z-finite representations, and so is again Z-finite by parts 1) and 3) of the preceding lemma) Lemma. If V is an object of od A), then any locally admissible vector in V is locally Z-finite. Proof. Suppose that v V is locally admissible, let H be a compact open subgroup that fixes v, and let m i annihilate v, for some i 0. If W V denotes the A[]-submodule generated by v, then W is admissible smooth, by assumption, and is annihilated by m i, and so W H is finitely generated over A. Since W H is Z-invariant, we conclude that v is locally Z-finite, as claimed Lemma. If V is an object of od adm A) which is finitely generated over A[], then V is Z-finite. Proof. This follows directly from the preceding lemmas Lemma. Let V be an object of od sm A), and consider the following conditions on V. 1) V is of finite length as an object of od sm A), or equivalently as an object of od A)), and is admissible. 2) V is finitely generated as an A[]-module, and is admissible. 3) V is of finite length as an object of od sm A)), and is Z-finite. Then 1) implies 2) and 3). Proof. The implication 1) 2) is immediate. The implication 2) 3) then follows from this together with the preceding lemma. We make the following conjecture: Conjecture. If is a reductive p-adic group, then the three conditions of Lemma are mutually equivalent.

9 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS Theorem. Conjecture holds in the following two cases: 1) is a torus. 2) = L 2 Q p ). Proof. Suppose first that is a torus, so that = Z. Let V be an object of od sm A). Taking into account Lemma 2.3.5, we see that each of conditions 2) and 3) of Lemma implies that V is both finitely generated over A[Z] and Z-finite. These conditions, taken together, imply in their turn that V is finitely generated as an A-module. In particular, it is a finite length object of od adm A). Thus conditions 2) and 3) of Lemma each imply condition 1). Suppose now that = L 2 Q p ). The results of [1] and [2] imply that any irreducible smooth -representation that is Z-finite is admissible, and thus condition 3) of Lemma implies condition 1). It remains to show that the same is true of condition 2) of Lemma To this end, let V be an object of od adm A) that is finitely generated over A[]. We must show that it is of finite length. Write H = L 2 Z p )Z. Let S V be a finite set that generates V over A[], let W be the A[H]-submodule of V generated by S. Since V is smooth by assumption), and Z-finite by Lemma 2.3.5), we see that W is finitely generated as an A-module, annihilated by some power of m. Since W contains the generating set S, the H-equivariant embedding W V induces a -equivariant surjection c Ind H W V. Thus it suffices to show that any admissible quotient V of c Ind H W is of finite length as an A[]-module. Taking into account Proposition , we see in fact that it suffices to prove the analogous result with W replaced by one of its finitely many) Jordan-Hölder factors, which will be a finite dimensional irreducible representation of H over k := A/m. Extending scalars to a finite extension of k, if necessary, we may furthermore assume that W is an absolutely irreducible representation of H over k, and this we do from now on. Since V is a quotient of c Ind H W, the evaluation map 4) Hom k[] c Ind H W, V ) k c Ind H W V is surjective. Lemma below shows that Hom k[] c Ind H W, V ) is a finite dimensional k-vector space. It is also naturally a module over the endomorphism algebra H := End k[] c Ind H W ). In [1] it is proved that H = k[t ] where T denotes a certain specified endomorphism whose precise definition need not concern us here). Thus Hom k[] c Ind H W, V ) is a finite dimensional k[t ]-module, and hence is a direct sum of k[t ]-modules of the form k[t ]/ft )k[t ], for various nonzero polynomials ft ) k[t ]. Taking into account the surjection 4), we see that it suffices to show that any -representation of the form c Ind H W )/ft )c Ind H W ) is of finite length. Further extending scalars, if necessary, we may assume that ft ) factors completely in k[t ] as a product of linear factors, and hence are reduced to showing that for any scalar λ k, the quotient c Ind H W )/T λ)c Ind H W ) is of finite length. This, however, follows from the results of [1] and [2]. Thus the theorem is proved in the case of L 2 Q p ) Lemma. Let U and V be smooth representations of over A. Suppose that U is finitely generated over A[], and that V is admissible. Then Hom A[] U, V ) is a finitely generated A-module.

10 10 ATTHEW EERTON Proof. Let W U be a finitely generated A-submodule that generates U over A[], let H be a compact open subgroup of that fixes W, and let i 0 be such that m i annihilates W. Since U is a smooth representation of over A, such H and i exist.) Restricting homomorphisms to U induces an embedding 5) Hom A[] U, V ) Hom A W, V H [m i ]). Since V is an admissible smooth -representation, its submodule V H is finitely generated over A, and so the target of 5) is finitely generated over A. The same is thus true of the source, and the lemma follows ϖ-adically admissible -representations. In this subsection we introduce a class of representations related to the considerations of [14] Definition. We say that a -representation V over A is ϖ-adically continuous if: 1) V is ϖ-adically separated and complete. 2) The O-torsion subspace V [ϖ ] of V is of bounded exponent i.e. is annihilated by a sufficiently large power of ϖ). 3) The -action V V is continuous, when V is given its ϖ-adic topology. 4) The A-action A V V is continuous, when A is given its m-adic topology and V is given its ϖ-adic topology Remark. Conditions 3) and 4) of the definition can be expressed more succinctly as follows: for any i 0, the -action on V/ϖ i V is smooth, in the sense of We let od ϖ cont A) denote the full subcategory of od A) consisting of ϖ- adically continuous representations of over A. Any object V of od ϖ cont A) sits in a canonical exact sequence 0 V [ϖ ] V V fl 0, where V fl is the maximal O-torsion free quotient of V. By assumption V [ϖ ] = V [ϖ i ] for i sufficiently large, and so for any j i, tensoring the above short exact sequence by O/ϖ j over O induces a short exact sequence 0 V [ϖ ] = V [ϖ i ] V/ϖ j V V fl /ϖ j V fl 0, while multiplication by ϖ i induces an isomorphism V fl ϖ i V, and hence embeddings V fl /ϖ j i V/ϖ j V. Thus we see that both V [ϖ ] and V fl are again objects of od ϖ cont A), and the study of of the category od ϖ cont A) can largely be separated into the study of O-torsion objects, and O-torsion free objects Proposition. Let 0 V 1 V 2 V 3 0 be a short exact sequence in od A). 1) If V 1 and V 3 are objects of od A), then so is V 2. 2) Suppose furthermore that V 2 is an object of od ϖ cont A). Then the following are equivalent: a) V 1 is closed in the ϖ-adic topology of V, and V 3 [ϖ ] has bounded exponent. b) V 3 lies in od ϖ cont A).

11 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS 11 Furthermore, if these equivalent conditions hold, then V 1 is also an object of od ϖ cont A), and the ϖ-adic topology on V 2 induces the ϖ-adic topology on V 1. Proof. Suppose we are in the situation of 1). Since V 2 is an extension of ϖ- adically complete modules, it is again ϖ-adically complete. If we choose i so large that V 1 [ϖ i ] = V 1 [ϖ ] and V 3 [ϖ i ] = V 3 [ϖ ], then evidently V 2 [ϖ 2i ] = V 2 [ϖ ], and so the O-torsion of V 2 is of bounded exponent. Thus V 2 lies in od ϖ cont A). Suppose now that we are in the situation of 2). The surjection V 3 V 2 induces a surjection V 2 /ϖ i V 2 V 3 /ϖ i V 3 for each i 0. Thus V 3 /ϖ i V 3 is a smooth A[]- representation for each i 0, since this is true of V 2 /ϖ i V 2 by assumption. Since V 2 is assumed to be ϖ-adically complete and separated, we see furthermore that V 1 is closed in V 2 if and only if V 3 which is isomorphic to the quotient V 2 /V 1 ) is ϖ-adically complete and separated. From this we conclude the equivalence of 2)a) and 2)b). It remains to show that, if these equivalent conditions hold, then V 1 is an object of od ϖ cont A), and the ϖ-adic topology on V 2 induces the ϖ-adic topology on V 1. Since V 2 [ϖ ] is of bounded exponent, by assumption, the same is true of V 1 [ϖ ]. We have already shown in 1) that od ϖ cont A) is closed under formation of extensions, and so it suffices to show that V 1 ) fl is an object of od ϖ cont A). Clearly V 2 /V 1 [ϖ ]) lies in od ϖ cont A) since it is an extension of V 2 ) fl by V 2 [ϖ ]/V 1 [ϖ ]), and so replacing V 1 by V 1 ) fl and V 2 by V 2 /V 1 [ϖ ], we may assume that V 1 is O-torsion free. Choose i 0 so that V 3 [ϖ i ] = V 3 [ϖ ]. For each j i, our original exact sequence yields an exact sequence Thus and so V 3 [ϖ i ] = V 3 [ϖ j ] V 1 /ϖ j V 1 V 2 /ϖ j V 2. ϖ i V 1 ϖ j V 2 ) ϖ j V 1, ϖ j V 1 V 1 ϖ j V 2 ϖ j i V 1, since V 1 is O-torsion free. Thus the ϖ-adic topology on V 2 induces the ϖ-adic topology on V 1. Since V 1 is the kernel of the A[]-linear and hence O-linear) map V 2 V 3, and since V 2 and V 3 are ϖ-adically complete and separated, we see that V 1 is complete and separated in the topology induced by the ϖ-adic topology on V 2, and so is in fact ϖ-adically complete and separated. Thus V 1 lies in od ϖ cont A) Corollary. The category od ϖ cont A) is closed under the formation of kernels, images, and extensions in the abelian category od A). Proof. The closure of od ϖ cont A) under the formation of extensions is a restatement of part 1) of the preceding proposition. Suppose now that 0 V 1 V 2 V 3 is an exact sequence of objects in od A), with V 2 and V 3 lying in od ϖ cont A). Let V 3 denote the image of V 2 in V 3. Since V 3 [ϖ ] has bounded exponent, by assumption, so does V 3[ϖ ]. From part 2) of the preceding proposition, we conclude that both V 1 and V 3 lies in od ϖ cont A). This proves the closure of od ϖ cont A) under the formation of kernels and images.

12 12 ATTHEW EERTON Definition. We define an object V of od ϖ cont A) to be admissible if the induced -representation on V/ϖV )[m] which is then a smooth -representation over A/m, by Remark 2.4.2) is in fact an admissible smooth -representation over A/m Remark. Since admissible smooth representations form a Serre subcategory of od A), we see that if V is a p-adically admissible -representation over A, then in fact V/ϖ i V is an admissible smooth representation for every i 0; cf. Remark We let od ϖ adm A) denote the full subcategory of od ϖ cont A) consisting of admissible representations Remark. When A is an object of ArtO), the categories od ϖ cont A) and od sm A) coincide, as do the categories od ϖ adm A) and od adm A) Lemma. If V is any object of od ϖ adm A), then each of V [ϖ ] and V fl is also an object of od ϖ adm A). Proof. By assumption V [ϖ ] = V [ϖ i ] for i sufficiently large, and so for any j i, tensoring the above short exact sequence by O/ϖ j over O induces a short exact sequence 0 V [ϖ ] = V [ϖ i ] V/ϖ j V V fl /ϖ j V fl 0, while multiplication by ϖ i induces an isomorphism V fl ϖ i V, and hence embeddings V fl /ϖ j i V/ϖ j V. Thus we see that both V [ϖ ] and V fl are again objects of od ϖ adm A) Proposition. For any i 0, the functor V V induces an A-linear antiequivalence of categories od ϖ cont A)[ϖ i pro aug ] od A)[ϖ i ], which restricts to an anti-equivalence of categories od ϖ adm A)[ϖ i fg aug ] od A)[ϖ i ]. Proof. This follows by passing to ϖ i -torsion in the anti-equivalences 2) and 3) Corollary. The category od ϖ adm A)[ϖ ] is a Serre subcategory of the abelian category od A). In particular, od ϖ adm A)[ϖ ] is an abelian category. Proof. This follows directly from the proposition, together with the evident fact fg aug that od A)[ϖ pro aug ] is a Serre subcategory of od A) Proposition. The functor V Hom O V, O) induces an A-linear antiequivalence of categories od ϖ adm A) fl fg aug od A) fl. Proof. As is explained in the proof of [14, Thm. 1.2], the functor V Hom O V, O), the latter space being equipped with the weak topology i.e. the topology of pointwise convergence), induces an equivalence of categories between the category of ϖ-adically separated and complete flat O-modules and the category of compact linear-topological flat O-modules. The reader may now easily check that this functor induces the required equivalence Proposition. The category od ϖ adm A) is closed under the formation of kernels, images, cokernels, and extensions in the abelian category od A). In particular, od ϖ adm A) is an abelian category.

13 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS 13 Proof. Suppose first that 0 V 1 V 2 V 3 0 is a short exact sequence of objects in od A), with V 1 and V 3 being objects of od ϖ adm A). It follows from Corollary that V 2 is an object of od ϖ cont A). For any i 0, we have the exact sequence V 1 /ϖ i V 2 V 2 /ϖ i V 2 V 3 /ϖ i V 3 0. Since V 1 /ϖ i V 1 and V 3 /ϖ i V 3 are assumed to be admissible smooth -representations over A, the same is true of V 2 /ϖ i V 3, by Proposition Thus V 2 is an object of od ϖ adm A). Suppose now that 0 V 1 V 2 V 3 is an exact sequence of objects of od A), with V 2 and V 3 being objects of od ϖ adm A). Let V 3 denote the image of V 2 in V 3. It follows from Corollary that V 1 and V 3 are objects of od ϖ cont A). Since V 3 [ϖ ] lies in od ϖ adm A), so does V 3[ϖ ], by Corollary Choose i 0 so that V 3[ϖ i ] = V 3[ϖ ]. For each j i, our given exact sequence yields an exact sequence V 3[ϖ i ] = V 3[ϖ j ] V 1 /ϖ j V 1 V 2 /ϖ j V 2 V 3/ϖ j V 3 0. Since the first and fourth terms of the exact sequence lie in od adm A), we conclude from Proposition that the other two terms do also. We conclude that V 1 and V 3 are both objects of od ϖ adm. Suppose finally that V 1 V 2 V 3 0 is an exact sequence of objects of od A), with V 1 and V 2 being objects of od ϖ adm A). We wish to show that V 3 is also an object of od ϖ adm A). Applying Hom O, O) to our exact sequence, we then obtain the exact sequence 0 Hom O V3 ) fl, O ) Hom O V2 ) fl, O ) Hom O V1 ) fl, O ) Hom O V 3 [ϖ ], E/O) Hom O V 2 [ϖ ], E/O) Propositions and imply that, for any compact open subgroup H of, each of Hom O V1 ) fl, O ), Hom O V2 ) fl, O ), and Hom O V 2 [ϖ ], E/O) is a finitely generated A[[H]]-module. Since A[[H]] is Noetherian, the same is therefore true of Hom O V3 ) fl, O ) and Hom O V 3 [ϖ ], E/O). Thus V 3 ) fl and V 3 [ϖ ] are both object of od ϖ adm A) by the same propositions), and thus V 3 is also being an extension of the first by the second) Proposition. If 0 V 1 V 2 V 3 0 is a short exact sequence in od A), and if V 2 is an object of od ϖ adm A), then the following are equivalent: 1) V 1 is closed in the ϖ-adic topology of V, and V 3 [ϖ ] has bounded exponent. 2) V 3 lies in od ϖ cont A). 3) V 3 lies in od ϖ adm A). If these equivalent conditions hold, then V 1 is also an object of od ϖ adm A), and the ϖ-adic topology on V 2 induces the ϖ-adic topology on V 1. Proof. The equivalence of 1) and 2) follows from part 2) of Proposition 2.4.3, and clearly 2) implies 3). On the other hand, if 2) holds, then V 3 /ϖ i V 3 is a quotient of V 2 /ϖ i V 2 for any i 0. The latter is an admissible A[]-representation by assumption, and thus so is the former. Consequently V 3 is an object of od ϖ adm A), and so 2) implies 3). The remaining claims of the proposition follow from Proposition ) together with Proposition We have obvious equivalences of E-linearized categories od ϖ adm A) fl E fg aug A) E and od A) fl fg aug E od A) E, and so E-linearizing the od ϖ adm

14 14 ATTHEW EERTON anti-equivalence of the preceding proposition yields an A E -linear anti-equivalence of categories 6) od ϖ adm A) fl E fg aug od A) fl E. We will reinterpret the source and target of 6) more concretely, after introducing some further definitions and establishing some simple lemmas Definition. A unitary action of A E [] on an E-Banach space V consists of an A E []-module structure on V extending its E-vector space structure) satisfying the following conditions: 1) The -action on V is unitary i.e. V contains a bounded open -invariant lattice) and continuous i.e. the action map V V is continuous). 2) The induced A-action A V V is continuous when A is equipped with its m-adic topology Lemma. If V is an E-Banach space equipped with a unitary A E []-action, then the set of A[]-invariant bounded open lattices in V is cofinal in the set of all bounded open lattices in V. Proof. Condition 2) of Definition together with the Banach-Steinhaus theorem see e.g. Prop together with Example 2) on p. 40 of [13]) implies that the action of A on V is equicontinuous. If we let V 0 be any open bounded lattice in V, then the A[]-submodule V 1 of V generated by V 0 is again an open lattice, and is bounded, since the -action is unitary and the A-action is equicontinuous. The A[]-submodules ϖ i V 1 for i 0) are then also A[]-invariant bounded open lattices in V. As these form a neighbourhood basis of the zero element of V since V is a Banach space) the lemma follows Lemma. Let V be an E-Banach space equipped with a unitary A E []- action. 1) If V 0 is any A[]-invariant bounded open lattice of V, then the the induced -action on the A-module V 0 /ϖv 0 is smooth. 2) If furthermore the V 0 /ϖv 0 is an admissible representation of A over for one choice of V 0, then it is so for any other choice. Proof. If V 0 is an A[]-invariant bounded open lattice in V, then the actions of and A on V 0 /ϖv 0 are continuous when the latter space is equipped with its discrete topology, which is to say, the A[]-action on V 0 /ϖv 0 is smooth. Suppose now that V 1 is a second such lattice. Scaling V 1 if necessary, we may assume that V 1 V 0. For some n > 0, we then also have the inclusion ϖ n V 0 ϖv 1, so that V 1 /ϖv 1 is a subquotient of V 0 /ϖ n V 0. If the A[]-action on V 0 /ϖv 0 is admissible, then, since V 0 /ϖ n V 0 is a successive extension of finitely many copies of V 0 /ϖv 0, it is again an object of od adm A), by Proposition It follows from the same proposition that its subquotient V 1 /ϖv 1 is an object of od adm A), as claimed Definition. Let V be an E-Banach space. An admissible unitary action of A E [] on V consists of a unitary A E []-action on V such that for some or equivalently any) A[]-invariant bounded bounded lattice V 0 in V, the induced -action on V 0 /ϖv 0 ) which is necessarily smooth) makes this representation an admissible smooth -representation over A. The parenthetical claims follow from Lemma )

15 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS Remark. It follows from Remark that in the preceding definition, it is equivalent to require that V 0 /ϖv 0 )[m] be an admissible smooth representation of over A/m Remark. Suppose that A = O. The notion of a unitary admissible action of E[] on an E-Banach space V, as given by the above definition, coincides with the notion of a unitary representation of on V which is admissible in the sense of [14]. Let od u A E ) denote the category of E-Banach spaces equipped with unitary A E []-actions, with morphisms being continuous A E -linear -equivariant maps. Let od u adm A E ) denote the full subcategory of od u A E ) consisting of E- Banach spaces equipped with unitary admissible A E []-actions Lemma. The functor V E O V induces an equivalence of categories od ϖ adm A) E od u adm A E ). Proof. In the proof of [14, Thm. 1.2] it is observed that the functor V E O V induces an equivalence of categories between the E-linearization of the category of ϖ-adically separated and complete flat O-modules with morphisms being O-linear maps) and the category of E-Banach spaces with morphisms being continuous E- linear maps). The reader can easily check that this functor induces the equivalence of the lemma Remark. The reader may easily check, in Definition 2.4.1, that condition 2) is equivalent to the apparently weaker condition that the map V V be left-continuous i.e. that the map V defined by g gv is continuous for each v V ) when V is equipped with its ϖ-adic topology, and that condition 3) is equivalent to the apparently weaker condition that the map A V V be leftcontinuous when A is equipped with its m-adic topology and V is equipped with its ϖ-adic topology. Similarly, using the Banach-Steinhaus theorem, the reader may check, in Definition , that condition 1) is equivalent to the apparently weaker condition that the -action be separately continuous, and that condition 2) is equivalent to the apparently weaker condition that the A-action be separately continuous when A is equipped with is m-adic topology). If V is an E-Banach space, we let V denote the continuous dual space to V Proposition. The functor V V induces an anti-equivalence of categories od u adm A E ) fg aug od A E ). Proof. This follows by E-linearizing the anti-equivalence of Proposition , and taking into account the equivalences of Lemmas and Corollary. The category od u adm A E ) is closed under the formation of closed subrepresentations, Hausdorff quotients, and strict extensions in the category od u A E ), and is furthermore abelian. In particular, any morphism in the category has closed image. Proof. Theorem shows that A[[H]] E is Noetherian for any compact open subgroup of, and thus that od A E ) is closed under passing to subobjects, as fg aug well as quotients and extensions, in the category of all augmented representations of over A E ; in particular, it is abelian. Proposition then implies that the

16 16 ATTHEW EERTON analogous results hold for the category od u adm A E ) regarded as a full subcategory of the exact category of all E-Banach spaces equipped with an A E []-action satisfying conditions 1) and 2) of Lemma ) Remark. In the case when A is finite over O, the preceding corollary is due to Schneider and Teitelbaum [14]. 3. Ordinary parts In this section we take to be a p-adic reductive group by which we always mean the group of Q p -points of a connected reductive algebraic group over Q p ). We suppose that P is a parabolic subgroup of, with unipotent radical N, that is a Levi factor of P so that P = N), and that P is an opposite parabolic to P, with unipotent radical N, chosen so that = P P. This condition uniquely determines P.) Our goal in this section is to define the functor of ordinary parts, which for any object A of CompO)) is a functor Ord P : od adm A) od adm A). In the following section we will show that it is right adjoint to parabolic induction Ind P : odadm A) od adm A). In fact, it is technically convenient to define Ord P in a more general context, as a functor Ord P : od sm P A) od sm A), and this we do in Subsection 3.1. In Subsection 3.2 we establish some elementary properties of Ord P. In Subsection 3.3, we prove that Ord P, when applied to admissible smooth -representations, yields admissible smooth -representations. Finally, Subsection 3.4 extends the main definitions and results of the preceding subsections to the context of ϖ-adically continuous and ϖ-adically admissible representations over A. The functor Ord P is analogous to the locally analytic Jacquet functor J P studied in [4] and [6], and many of the constructions and arguments of this section and the next are suitable modifications of the constructions and arguments found in those papers. Typically, the arguments become simpler. As a general rule, the theory of Ord P is more elementary than the theory of the Jacquet functors J P, just as the theory of p-adic modular forms is more elementary in the ordinary case than in the more general finite slope case.) 3.1. The definition of Ord P. We fix, P, P, etc., as in the preceding discussion. We let A denote an object of CompO) Definition. If V is a representation of N over A, and if N 1 N 2 are compact open subgroups of N, then let h N2,N 1 : V N1 V N2 denote the operator defined by h N2,N 1 v) := n N 2/N 1 nv. Here we are using n to denote both an element of N 2 /N 1, and a choice of coset representative for this element in N 2. Since v V N1, the value nv is well-defined independently of the choice of coset representative.) Lemma. If N 1 N 2 N 3 are compact open subgroups of N, then h N3,N 2 h N2,N 1 = h N3,N 1. Proof. This is immediate from the definition of the operators. Fix a compact open subgroup P 0 of P, and set 0 := P 0, N 0 := N P 0, and := {m mn 0 m 1 N 0 }. Let Z denote the centre of, and write Z := Z. Note that each of 0 and N 0 is a subgroup of, while and Z are submonoids of.

17 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS Definition. If V is a representation of P over A, then for any m, we define h N0,m : V N0 V N0 via the formula h N0,mv) := h N0,mN 0m 1mv) Lemma. If m 1, m 2, then h N0,m 1m 2 = h N0,m 1 h N0,m 2. Proof. We compute that h N0,m 1m 2 = h N0,m 1m 2N 0m 1 2 m 1 1 m 1 m 2 = h N0,m 1N 0m 1h m 1N 0m 1,m 1m 2N 0m 1 2 m 1 1 = h N0,m 1N 0m 1m 1h N0,m 2N 0m 1 2 m 1 m 2 m 2 = h N0,m 1 h N0,m 2. If m 0, then since in this case mn 0 m 1 = N 0 ) the operator h N0,m coincides with the given action of m on V N0. In general, the preceding lemma shows that the operators h N0,m induce an action of the monoid on V N0, which we refer to as the Hecke action of. Regarding V N0 as a module over the monoid algebra A[ ], and so in particular its central subalgebra A[Z ], via this action, we may consider the A[Z ]-module Hom A[Z ]A[Z ], V N0 ), as well as its submodule Hom A[Z ]A[Z ], V N0 ) Z fin Lemma. Let W be an A[Z ]-module. If φ Hom A[Z ]A[Z ], W ) Z fin, then the image imφ) of φ is a finitely generated A-submodule of W, which is invariant under the action of Z. Proof. Let ev 1 : Hom A[Z ]A[Z ], W ) Z fin W denote the map given by evaluation at 1 Z. For any φ Hom A[Z ]A[Z ], W ) Z fin, if U denotes the A[Z ]-submodule of Hom A[Z ]A[Z ], W ) Z fin generated by φ, then clearly ev 1 U) coincides with the image imφ) of φ. Since φ is locally Z -finite, we see that U is finitely generated over A, and hence the same is true of imφ). Since z 1 φz 2 ) = φz 1 z 2 ) for z 1 Z and z 2 Z, we see that imφ) is invariant under the action of Z Lemma. Let W be an A[ ]-module. 1) The A[Z ]-module structure on Hom A[Z ]A[Z ], W ) extends naturally to an A[]-module structure, characterized by the property that for any z Z, the map Hom A[Z ]A[Z ], W ) W induced by evaluation at z is -equivariant. 2) If the 0 -action on W is furthermore smooth, then the induced -action on Hom A[Z ]A[Z ], W ) Z fin is smooth. Proof. From [4, Prop ], we see that the restriction map induces an isomorphism 7) Hom A[] A[], W ) Hom A[Z ]A[Z ], W ). The source of this isomorphism has a natural A[]-module structure, which we may then transport to the target. This A[]-module structure clearly satisfies the condition of 1). Suppose now that W is 0 -smooth, and let φ Hom A[Z ]A[Z ], W ) Z fin. Since imφ) is a finitely generated A-submodule of W, by Lemma 3.1.5, we may find an open ideal I in A[[ 0 ]] which annihilates imφ), and hence which annihilates φ

18 18 ATTHEW EERTON itself. See Remark ) Thus the -action on Hom A[Z ]A[Z ], W ) Z fin is smooth, proving 2). We now give the main definition of this section Definition. If V is a smooth representation of P over A, then we write Ord P V ) := Hom A[Z ]A[Z ], V N0 ) Z fin, and refer to Ord P V ) as the P -ordinary part of V or just the ordinary part of V, if P is understood). Lemma shows that Hom A[Z ]A[Z ], V N0 ), and hence Ord P V ), is naturally an -representation over A. Since the Hecke 0 -action on V N0 coincides with the given 0 -action, it is smooth, and so the same lemma shows that the -action on Ord P V ) is smooth. Thus the formation of Ord P V ) yields a functor from od sm P A) to od sm A) Definition. We define the canonical lifting 8) Ord P V ) V N0 to be the composite of the inclusion Ord P V ) Hom A[Z ]A[Z ], V N0 ) with the map Hom A[Z ]A[Z ], V N0 ) V N0 given by evaluation at the element 1 A[Z ]. Our choice of terminology is motivated by that of [3, 4]. Although the definition of the functor Ord P depends on the choice of compact open subgroup P 0 of P, we now show that it is independent of this choice, up to natural isomorphism. To this end, suppose that P 0 is an open subgroup of P 0. Write 0 := P 0, N 0 := N P 0, ) := {m mn 0n 1 N 0 }, Z ) := ) Z. We define the Hecke action of ) on V N 0 in analogy to the Hecke action of on V N Proposition. Composition with the operator h N0,N 0 : V N 0 V N 0 induces an isomorphism Hom A[Z ) ] A[Z ], V N 0 ) HomA[Z ]A[Z ], V N0 ). Proof. We write Y := ), and Y := Z Z ) = Y Z. The notation is chosen to be compatible with that of [4, Prop ].) Since Y generates Z as a group [4, Prop i)], the inclusions Hom A[Z ) ] A[Z ], V N 0 ) Hom A[Y ] A[Z ], V N 0 ) and HomA[Z ]A[Z ], V N0 ) Hom A[Y ] A[Z ], V N0 ) are in fact equalities. Thus composition with h N0,N 0 does induce a map φ : Hom A[Z ) ] A[Z ], V N 0 ) HomA[Z ]A[Z ], V N0 ). If m Y, then we compute: 9) h N0,mh N0,N 0 = h N 0,mN 0m 1mh N 0,N 0 = h N 0,mN 0m 1h mn 0m 1,mN 0 m 1m = h N0,mN 0 m 1m = h N 0,N 0 h N 0,mN 0 m 1m = h N 0,N 0 h N 0,m. Thus h N0,N 0 : V N0 V N 0 intertwines the Hecke action of Y on each of its source and target. Since Y and Z together generate [4, Prop ii)], we see that the map φ is -equivariant.

19 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS 19 If z Y is such that zn 0 z 1 N 0, then we define a map 10) V N0 V N 0 via the formula v h N 0,zN 0z 1zv). A computation similar to that of 9) taking into account the fact that z is central in ) shows that 10) also intertwines the Hecke action of Y on its source and target. Thus 10) induces an -equivariant map ψ : Hom A[Z ]A[Z ], V N0 ) Hom A[Z ) ] A[Z ], V N 0 ). One furthermore computes that h N0,N 0 h N 0,zN0z 1z = h N 0,z, while h N 0,zN 0z 1zh N 0,N 0 = h N 0,z. Thus the composite φψ resp. the composite φψ) coincides with the automorphism of Hom A[Z ]A[Z ], V N0 ) resp. Hom A[Z ) ] A[Z ], V N 0 )) induced by the action of z, and in particular both φψ and ψφ are isomorphisms. This implies that φ is an isomorphism, as required. The preceding result shows that Ord P is well-defined, up to canonical isomorphism, independently of the choice of the open subgroup P 0 of P. It is similarly independent of the choice of the Levi factor of P. Indeed, if is another Levi factor of P, then we may canonically identify and via the canonical isomorphisms P/N. Furthermore, there is a uniquely determined element n N such that nn 1 =, and the isomorphism induced by conjugation by n yields the same identification of and. Via this canonical identification, we may and do regard any representation equally well as an -representation. iven a choice of compact open subgroup P 0 P, write P 0 := np 0 n 1, 0 := n 0 n 1 = nn 1 P 0, N 0 := nn 0 n 1 = N P 0, ) := n n 1 = {m m N 0m ) 1 N 0}, and Z := nz n 1 = ) Z. We define the Hecke action of ) on V N 0 in analogy to the Hecke action of on V N0, and denote by Ord P : od A) od A) the functor V Hom A[Z ] A[Z ], V N 0 )Z fin. The analogue of Lemma for P 0 and shows that Hom A[Z ] A[Z ], V N 0 ), and so also Hom A[Z ] A[Z ], V N 0 )Z fin, is naturally an -representation, and hence an -representation.) Thus Ord P is the analogue of the functor Ord P, but computed using the Levi factor of P rather than. Note that we have already seen that this functor is well-defined independently of the particular choice of compact open subgroup P 0 used to compute it.) V N 0, which intertwines ultiplication by n provides an isomorphism V N0 the operator h m on V N0 with the operator h nmn 1 on V N 0, for each m. Consequently, multiplication by n induces an -equivariant isomorphism Hom A[Z ]A[Z ], V N0 ) Hom A[Z ] A[Z ], V N 0 ), and hence an -equivariant isomorphism Ord P V ) Ord P V ). This shows that the functor Ord P is well-defined, up to canonical isomorphism, independently of the choice of Levi factor of P.