Draft: January 19, 2010

ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS II. DERIVED FUNCTORS

MATTHEW EMERTON

Contents

1. Introduction 1 2. Injective and acyclic objects 2 3. The δ-functor H^{•}Ord_{P} 6 4. The case of GL_{2}(ℚ_{p}) 25 Appendix A. 41 References 49

1. Introduction

The goal of this paper, which is a sequel to [10], is to extend the functors of ordinary parts introduced in that paper to certain δ-functors. On the one hand, the δ-functors that we construct are easy to compute with in many situations; on the other hand, we expect that they coincide with the derived functors of the functors of ordinary parts. Since the functors of ordinary parts are right adjoint to the functors given by parabolic induction, we expect that the δ-functors introduced here will be useful in studying homological questions involving parabolically induced representations. As some evidence for this, we close the paper by applying them to make some Ext calculations in the category of admissible smooth representations of GL_{2}(ℚ_{p}) over a finite field of characteristic p. These calculations play a role in the mod p and p-adic Langlands programme. (See [6] and [9].)

To describe our results more precisely, let p be a prime, and A a finite Artinian local ℤ_{p}-algebra of residue characteristic p. Let G be a connected reductive padic group, and P a parabolic subgroup of G, admitting a Levi factorization P = MN. We define a δ-functor H^{•}Ord_{P} on the category of admissible smooth Grepresentations over A, taking values in the category of admissible smooth Mrepresentations over A, such that H^{0}Ord_{P} ⟶^{∼} Ord_{P} (the functor of ordinary parts defined in [10, Def. 3.1.9]). The functors H^{•}Ord_{P} are defined in terms of continuous N_{0}-cohomology, where N_{0} is a compact open subgroup of N. This continuous cohomology is quite accessible to computation, and hence so are the functors H^{•}Ord_{P}. For example, one finds that the functors H^{i}Ord_{P} vanish for i > dim N, and that H^{dim N}Ord_{P} coincides (up to a twist) with the Jacquet functor of N-coinvariants. We conjecture that H^{•}Ord_{P} is in fact a universal δ-functor, and thus coincides with the derived functors of Ord_{P}. In an appendix to this paper [11], joint with V. Paškūnas, we verify this conjecture in the case when G = GL_{2}(F), for any finite extension F of ℚ_{p}.

The author was supported in part by NSF grants DMS-0401545 and DMS-0701315.

In the case when G = GL_{2}(ℚ_{p}), P is a Borel subgroup of G, and A = k is a field of characteristic p, we explicitly evaluate the functors H^{•}Ord_{P} on all absolutely irreducible admissible smooth G-representations over k. As already mentioned, we are then able to apply the results of this calculation to compute certain Ext spaces in the category of admissible smooth G-representations over k. Some of the Ext computations that we make have also been made by Breuil and Paškūnas [4] and by Colmez [6], in both cases using different methods.

1.1. Arrangement of the paper. Section 2 of the paper is devoted to recalling some results about injective objects in certain categories of representations, as well as describing the relationship between certain Ext functors and continuous cohomology. In Section 3 we define the δ-functors H^{•}Ord_{P} , and develop some of their basic properties. In Section 4 we study the case when G = GL_{2}(ℚ_{p}) and P is a Borel subgroup in detail. Some of the more technical aspects of the calculations, which are not directly related to the general ideas of the paper, have been placed in an appendix. A second, separate, appendix, written jointly with V. Paškūnas, establishes an important homological result which is applied in Section 4.

1.2. Notation and terminology. Throughout the paper, we fix a prime p. We let Art(ℤ_{p}) denote the category of Artinian local ℤ_{p}-algebras having finite residue field. All our rings of coefficients will be objects of Art(ℤ_{p}).

We freely employ the terminology and notation that was introduced in the paper [10]. In particular, if V is a representation of a p-adic analytic group G over an object A of Art(ℤ_{p}), then we let V_{sm} (resp. V_{l.adm}) denote the G-subrepresentation of V consisting of the smooth (resp. locally admissible) vectors in V (see [10, Defs. 2.2.1, 2.2.15]).

1.3. Acknowledgments. I would like to thank Pierre Colmez, whose talk at the workshop on p-adic representations held in Montréal in September 2005 inspired me to develop the ideas of this paper, and to undertake the calculations of Section 4. I would also like to thank Adrian Iovita and Henri Darmon, as well as the Centre de Recherches Mathématiques at the University of Montréal, for organizing and hosting this wonderful workshop, during which many of the results presented here were first worked out. I thank the referee for their careful reading of the paper, which led to several corrections and clarifications of the text, and also Florian Herzig, who provided several useful corrections to and comments on an earlier version of the paper. Finally, I thank Vytas Paškūnas for his interest in this work, and for allowing [11] to be included as an appendix to this paper.

2. Injective and acyclic objects

Throughout this section, we let G denote a p-adic analytic group, and A an object of Art(ℤ_{p}).

2.1. Injective objects. In this subsection we present some basic results regarding the existence and properties of injective objects in various categories of Grepresentations over A. These properties are largely standard, and presumably well-known. We recall them here, with proofs, for the sake of completeness.

2.1.1. Proposition. Each of the full subcategories Mod_{G}^{sm}(A) and Mod_{G}^{l.adm}(A) of Mod_{G}(A) has enough injectives.

Proof. Certainly the category Mod_{G}(A) has enough injectives (since it coincides with the category of left modules over the ring A[G]). The functor V ↦ V_{sm} is right adjoint to the natural embedding Mod_{G}^{sm}(A) ↪ Mod_{G}(A). Thus it takes injective objects in Mod_{G}(A) to injective objects in Mod_{G}^{sm}(A). In particular, if V is an object of Mod_{G}^{sm}(A), and V ↪ I an A[G]-linear embedding of V into an injective object of Mod_{G}(A), then the induced embedding V ↪ I_{sm} is an embedding of V into an injective object of Mod_{G}^{sm}(A). The proof in the case of Mod_{G}^{l.adm}(A) proceeds similarly, using the functor V_{l.adm}. □

2.1.2. Proposition. If I is an injective object of Mod_{G}^{sm}(A), and H is an open subgroup of G, then I is also injective when regarded as an object of Mod_{H}^{sm}(A).

Proof. The forgetful functor Mod_{G}^{sm}(A) → Mod_{H}^{sm}(A) is right adjoint to the exact functor V ↦ A[G] ⊗_{A[H]} V. (This functor is naturally isomorphic to compact induction, V ↦ c−Ind_{H}^{G} V, and so takes smooth H-representations to smooth Grepresentations.) Thus it takes injectives to injectives. □

2.1.3. Proposition. If G is compact, then an inductive limit of injective objects of Mod_{G}^{sm}(A) is again an injective object of Mod_{G}^{sm}(A).

Proof. Let {I_{i}} be a directed system of injective object of Mod_{G}^{sm}(A), let V ↪ W be an embedding of objects of Mod_{G}^{sm}(A), and suppose given a G-equivariant map

(2.1.4)  V → lim_{\overrightarrow{i}} I_{i}.

We must show that it extends to a map W → lim_{\overrightarrow{i}} I_{i}. By the usual argument with

Zorn’s lemma, it suffices to show that if w ∈ W, and if we write W_{0} = A[G]w, then (2.1.4) extends to the G-subrepresentation V + W_{0} of W. It is evidently equivalent to show that the restriction to V_{0} := V ⋂ W_{0} of (2.1.4) extends to W_{0}. Since G is compact and W is smooth, we see that W_{0}, and thus V_{0}, is finitely generated over A, and in particular over the completed group ring A[[G]]. Since A[[G]] is Noetherian [13], V_{0} is in fact finitely presented over A[[G]]. Thus the restriction to V_{0} of (2.1.4) lifts to a map V_{0} → I_{i} for some (sufficiently large) value of i. (See the following lemma.) We may then extend this map to W_{0}, since I_{i} is injective. Composing such an extension with the natural map I_{i} → lim_{\overrightarrow{i}} I_{i}, we obtain

a G-equivariant map W_{0} → lim_{\overrightarrow{i}} I_{i} which extends the restriction to V_{0} of (2.1.4), as

required. □

The following lemma is a standard piece of algebra, whose proof we recall for the benefit of the reader.

2.1.5. Lemma. Let R be an associate ring with unit, and let M be a finitely presented left R-module. If {N_{i}} is any directed system of left R-modules, then the natural map

(2.1.6)  lim_{\overrightarrow{i}} Hom_{R}(M, N_{i}) → Hom_{R}(M, lim_{\overrightarrow{i}} N_{i})

is an isomorphism.

Proof. Fix a finite presentation R^{s} → R^{r} → M → 0 of M. Applying Hom_{R}(–, N_{i}) to this exact sequence, and taking into account the natural isomorphism

(2.1.7)  Hom_{R}(R, N) ⟶^{∼} N,

which holds for any R-module N, we obtain an exact sequence

0 → Hom_{R}(M, N_{i}) → N_{i}^{r} → N_{i}^{s}.

Passing to the inductive limit over all N_{i} yields an exact sequence

0 → lim_{\overrightarrow{i}} Hom_{R}(M, N_{i}) → lim_{\overrightarrow{i}} N_{i}^{r} → lim_{\overrightarrow{i}} N_{i}^{s}.

Applying Hom_{R}(–, lim_{\overrightarrow{i}} N_{i}) to the presentation of M, and again taking into account

the natural isomorphism (2.1.7), we obtain an exact sequence

0 → Hom_{R}(M, lim_{\overrightarrow{i}} N_{i}) → (lim_{\overrightarrow{i}} N_{i})^{r} → (lim_{\overrightarrow{i}} N_{i})^{s}.

The preceding two exact sequences fit into the diagram

\xymatrix{0 \ar[r] & lim_{\overrightarrow{i}} Hom_{R}(M, N_{i}) \ar[r] \ar[d]^{(2.1.6)} & lim_{\overrightarrow{i}} N_{i}^{r} \ar[r] \ar[d] & lim_{\overrightarrow{i}} N_{i}^{s} \ar[d] \\ 0 \ar[r] & Hom_{R}(M, lim_{\overrightarrow{i}} N_{i}) \ar[r] & (lim_{\overrightarrow{i}} N_{i})^{r} \ar[r] & (lim_{\overrightarrow{i}} N_{i})^{s},}

in which the second and third arrows are clearly isomorphisms. Thus so is the first, as claimed. □

2.1.8. Remark. If the transition maps in the directed system {N_{i}} are injective, then the map (2.1.6) will be an isomorphism provided that M is finitely generated. One can easily strengthen Lemma 2.1.5 to show that (2.1.6) is an isomorphism for every directed system {N_{i}} if and only if M is a finitely presented R-module.

2.1.9. Proposition. If G is compact, then the category Mod_{G}^{adm}(A) has enough injectives, and injective objects in this category are also injective in the category Mod_{G}^{sm}(A).

Proof. If V is an admissible smooth G-representation, then its Pontrjagin dual V^{∗} is a finitely generated A[[G]]-module. Fixing a surjection A[[G]]^{r} → V^{∗}, and dualizing, we obtain an embedding V ↪ C^{sm}(G, A^{∗})^{r}, where A^{∗} is the Pontrjagin dual of A, thought of as an A-module, and C^{sm}(G, A^{∗}) denotes the space of smooth A^{∗}-valued functions on G (regarded as a smooth admissible G-representation over A via the right regular G-action). Since A[[G]]^{r} is free of finite rank as an A[[G]]- module, one easily verifies that C^{sm}(G, A^{∗}) is injective in the category Mod_{G}^{sm}(A), and so also in the category Mod_{G}^{adm}(A). Thus any object of Mod_{G}^{adm}(A) embeds into an injective object of Mod_{G}^{adm}(A) that is also injective in Mod_{G}^{sm}(A). In particular, Mod_{G}^{adm}(A) has enough injectives. The following lemma then implies that every injective object of Mod_{G}^{adm}(A) is injective in Mod_{G}^{sm}(A). □

2.1.10. Lemma. Let C and C′ be abelian categories, and let F : C → C′ be an additive functor. Suppose that every object X of C admits a monomorphism X ↪ J for some J such that F (J) is an injective object of C′. Then if I is any injective object of C, the object F (I) is injective in C′.

Proof. Let I be an injective object of C, and choose a monomorphism I ↪ J, where F (J) is injective in C′. The injectivity of I in C implies that this monomorphism splits, and thus that I is a direct summand of J. Consequently, F (I) is a direct summand of F (J), and hence is also injective in C′. □

2.1.11. Proposition. If H is a compact subgroup of G, then any injective object of Mod_{G}^{sm}(A) is also injective as an object of Mod_{H}^{sm}(A).

Proof. Since H is compact, we may find a compact open subgroup G′ of G containing H. Appealing to Proposition 2.1.2, we see that we may replace G by G′, and thus assume that G is also compact. We do so for the remainder of the proof.

Since G is profinite, we may find a continuous section to the quotient map G → G/H, and thus find an isomorphism of right homogeneous H-spaces G ⟶^{∼} (G/H)×H (where the H-action on the target is defined via its action on the second factor). Thus we obtain an H-equivariant isomorphism

C^{sm}(G, A^{∗}) ⟶^{∼} C^{sm}(G/H, A) ⊗_{A} C^{sm}(H, A^{∗}),

where A^{∗} denotes the Pontrjagin dual of A, though of as an A-module. As we observed in the proof of Proposition 2.1.9, C^{sm}(H, A^{∗}) is injective in Mod_{H}^{sm}(A), and thus so is C^{sm}(G/H, A)⊗_{A}C^{sm}(H, A^{∗}), by Proposition 2.1.3. Thus C^{sm}(G, A^{∗}) is injective in Mod_{H}^{sm}(A). The proof of the preceding proposition shows that C^{sm}(G, A^{∗}) in injective in Mod_{G}^{sm}(A). It follows from Proposition 2.1.3 that a direct sum of any number of copies of C^{sm}(G, A^{∗}) is again injective in both Mod_{G}^{sm}(A) and Mod_{H}^{sm}(A).

As noted in the proof of the preceding proposition, any admissible smooth representation admits an embedding into C^{sm}(G, A^{∗})^{r} for some r ≥ 0. Since any object of Mod_{G}^{sm}(A) is an inductive limit of admissible smooth representations (G is compact), any object of Mod_{G}^{sm}(A) admits an embedding into direct sum of (a possibly infinite number of) copies of C^{sm}(G, A^{∗}). Thus any object of Mod_{G}^{sm}(A) admits an embedding into an injective object of Mod_{G}^{sm}(A) that is also injective in Mod_{H}^{sm}(A). It follows that any injective object of Mod_{G}^{sm}(A) is also injective as an object of Mod_{H}^{sm}(A), by Lemma 2.1.10. □

2.2. Continuous cohomology as derived functors. Since, by Proposition 2.1.1, the category Mod_{G}^{sm}(A) has enough injectives, we may define the derived functors of the functor of G-invariants on Mod_{G}^{sm}(A). Of course, these derived functors coincide with the Ext functors Ext_{G}^{•}(\underline{1}, –) (computed in Mod_{G}^{sm}(A), of course), where \underline{1} denotes A equipped with the trivial G-action.

If V is an object of Mod_{G}^{sm}(A), then we let H^{i}(G, V) denote the cohomology of G computed using continuous cochains taking values in V, where V is equipped with its discrete topology. The functors H^{i}(G, –) form a δ-functor on Mod_{G}^{sm}(A). (Since any surjection of smooth G-representations certainly admits a set-theoretic splitting, which is automatically continuous for the discrete topologies on its source and target, one sees that a short exact sequence in Mod_{G}^{sm}(A) gives rise to a corresponding short exact sequence of continuous cochain complexes, and hence to a long exact sequence of continuous cohomology modules.) Of course for any object V of Mod_{G}^{sm}(A), we have the equalities H^{0}(G, V) = Hom_{G}(\underline{1}, V) = V^{G} (the space of G-invariants), and so the universality of derived functors as δ-functors gives rise to a natural transformation of δ-functors

(2.2.1)  Ext_{G}^{•}(\underline{1}, –) → H^{•}(G, –).

2.2.2. Proposition. The natural transformation

(2.2.3)  Ext_{G}^{1}(\underline{1}, –) → H^{1}(G, –)

is in fact a natural equivalence.

Proof. For any object V of Mod_{G}^{sm}(A), the A-module Ext_{G}^{1}(\underline{1}, V) classifies extensions of \underline{1} by V in the category Mod_{G}^{sm}(A), while it is well-known, and easily verified, that for any object V of Mod_{G}^{sm}(A), the A-module H^{1}(G, V) classifies extensions of \underline{1} by V in the category of Hausdorff topological G-modules over A. The map (2.2.3) is then the evident one, which maps an extension in Mod_{G}^{sm}(A) to the corresponding extension in the category of topological modules (each member of the exact sequence being endowed with its discrete topology). This map is easily checked to be injective. (Alternatively, this is a general fact from the theory of δ-functors.) Furthermore, any Hausdorff topological extension is easily verified to be smooth, and to necessarily be equipped with its discrete topology. Thus (2.2.3) is also surjective. This proves the proposition. □

2.2.4. Corollary. The natural transformation Ext_{G}^{2}(\underline{1}, V) → H^{2}(G, V) is injective for any object V of Mod_{G}^{sm}(A).

Proof. It is a general fact from the theory of δ-functors (and is easily verified via a dimension-shifting argument) that if the map (2.2.1) is an isomorphism in degrees i ≤ n, then it is injective in degree i = n + 1. The lemma follows from this together with the preceding proposition. □

Nothing is known (to the author at least) regarding the nature of the map (2.2.1) in degrees > 2 in general. The difficulty is that if V is an object of Mod_{G}^{sm}(A), then the natural way to efface the completed cohomology of V is via the map

(2.2.5)  V → C^{sm}(G, V),

where the target is the space of continuous (or equivalently, locally constant) V - valued functions on G, equipped with its compact-open topology, and the map sends an element v of V to the corresponding orbit function g ↦ gv; see e.g. [12, 17]. If G is not compact, then the topological G-module C^{sm}(G, V) is neither discrete nor smooth as a G-representation; thus it does not provide an effacement of the completed cohomology of V in the category Mod_{G}^{sm}(A). On the other hand, if G is compact, then C^{sm}(G, V) is a discrete, smooth G-representation, and so we obtain the following well-known result.

2.2.6. Proposition. If G is compact, then the natural transformation (2.2.1) is a natural equivalence of δ-functors.

Proof. The proof is precisely the one sketched above: namely, the reader may easily check that if V is an object of Mod_{G}^{sm}(A), regarded as a discrete topological G-representation, then C^{sm}(G, V) is discrete and smooth when equipped with its compact-open topology. (The smoothness follows from the fact that since G is compact and V is discrete, a map from G to V is continuous if and only if it is locally constant if and only if it is uniformly locally constant.) The map (2.2.5) then effaces the higher continuous cohomology of V [12, §2]. Thus H^{i}(G, –) is an effaceable δ-functor, and hence universal; the proposition follows. □

As a point of terminology, we say that an object V of Mod_{G}^{sm}(A) is G-acyclic if H^{i}(G, V) vanishes for i > 0.

3. The δ-functor H^{•}Ord_{P}

Throughout this section, we let G denote (the group of ℚ_{p}-valued points of) a connected reductive linear algebraic group over ℚ_{p}, and let P denote a parabolic subgroup of G. We also fix a Levi factorization P = MN, and let \overline{P} = M\overline{N} denote an opposite parabolic to P, chosen so that P ⋂ \overline{P} = M.

Our goal in this section is to define a certain δ-functor H^{•}Ord_{P} from the category of (locally) admissible G-representations to the category of (locally) admissible Mrepresentations, for which H^{0}Ord_{P} is naturally isomorphic to the functor Ord_{P}. We conjecture that H^{•}Ord_{P} is the universal δ-functor extending Ord_{P} (and thus coincides with the derived functor of Ord_{P} on the category Mod_{G}^{l.adm}(A)), although we cannot prove this. Nevertheless, H^{•}Ord_{P} is quite computable in some situations (as we will see in the following section), and provides a useful tool for investigating homological questions related to the functor Ord_{P}.

3.1. Some properties of Z_{M}. We let Z_{M} denote the centre of the Levi factor M of P. The group Z_{M}, together with certain of its submonoids, plays an important role in the definition of the functor Ord_{P}. In this subsection we establish some structural results about Z_{M} and some of its submonoids which we will need to apply later.

Let Z_{0} denote the maximal compact subgroup of Z_{M}, and let Z_{G} ⊂ Z_{M} denote the centre of G. Let Z_{M}^{0} denote (the group of ℚ_{p}-points of) the maximal connected subgroup of (the algebraic group underlying) Z_{M}, so that Z_{M}^{0} is (the group of ℚ_{p}-points of) a torus. Let S denote (the group of ℚ_{p}-points of) the maximal split torus in Z_{M}^{0}, and write S_{0} := S ⋂ Z_{0} to denote the maximal compact subgroup of S, and S_{G} := S ⋂ Z_{G} to denote the maximal split torus in Z_{G}. If Δ denotes the set of positive simple restricted roots appearing in the action of S on the Lie algebra of N, then there is a natural embedding

(3.1.1)  S/S_{0}S_{G} ↪ ℤ^{Δ},

with finite cokernel, given by z ↦ (ord_{p}(α(z)))_{α∈Δ}. (Here ord_{p} denotes the p-adic ordinal on ℚ_{p}^{×}.) Since Z_{M} contains Z_{M}^{0} with finite index, while Z_{M}^{0} (like any torus) is isogenous to the product of a split torus and a compact torus, we see that the natural embedding S/S_{0}S_{G} ↪ Z_{M}/Z_{0}Z_{G} has finite cokernel. Thus (noting that Z_{M}/Z_{0}Z_{G} is a finitely generated torsion-free abelian group) the embedding (3.1.1) extends to an embedding

(3.1.2)  Z_{M}/Z_{0}Z_{G} ↪ (\frac{1}{d}ℤ)^{Δ},

for an appropriately chosen d ≥ 1. We again denote this embedding

z ↦ (ord_{p}(α(z)))_{α∈Δ}.

3.1.3. Lemma. Let N_{0} be any compact open subgroup of N.

(1) If z ∈ Z_{M} has the property that zN_{0}z^{−1} ⊂ N_{0}, then ord_{p}(α(z)) ≥ 0 for all α ∈ Δ.

(2) An element z ∈ Z_{M} has the property that the conjugates z^{i}N_{0}z^{−i} form a basis of neighbourhoods of the identity of N if and only if ord_{p}(α(z)) > 0 for all α ∈ Δ.

(3) There exists C ≥ 0 such that if z ∈ Z_{M} and ord_{p}(α(z)) ≥ C for all α ∈ Δ, then zN_{0}z^{−1} ⊂ N_{0}.

(4) The intersection ⋂_{z s.t. all ordp(α(z))≤0} zN_{0}z^{−1} (where, as indicated, the intersection

is taken over all z ∈ Z_{M} such that ord_{p}(α(z)) ≤ 0 for all α ∈ Δ) is an open subgroup of N_{0}.

Proof. Decompose 𝔫 as a sum of restricted root spaces 𝔫 = ⊕_{α∈Δ}𝔫_{α}. Choose a bounded ℤ_{p}-lattice 𝔫_{1} ⊂ 𝔫, write (𝔫_{1})_{α} := 𝔫_{1} ∩ 𝔫_{α} for each α ∈ Δ, and set 𝔫_{2} := ⊕_{α∈Δ}(𝔫_{0})_{α}; then 𝔫_{2} is again a bounded ℤ_{p}-lattice in 𝔫. For any z ∈ Z_{M}, it is clear that the collection {Ad_{z^{j}}(𝔫_{2})}_{j≥0} forms a uniformly bounded collection of subsets of 𝔫 (resp. a neighbourhood basis of 0 in 𝔫) if and only if z ∈ Z_{M} satisfies ord_{p}(α(z)) ≥ 0 (resp. > 0) for all α ∈ Δ, while ⋂_{j≥0} Ad_{z^{j}}(𝔫_{2}) is open if and only if ord_{p}(α(z)) ≤ 0 for all α ∈ Δ. If Ω is any compact open subset of 𝔫, then we may find integers i ≤ j such that p^{i}Ω ⊂ 𝔫_{2} ⊂ p^{j}Ω. Thus we similarly conclude that the collection {Ad_{z^{j}}(Ω)}_{j≥0} forms a uniformly bounded collection of subsets of 𝔫 (resp. a neighbourhood basis of 0 in 𝔫) if and only if z ∈ Z_{M} satisfies ord_{p}(α(z)) ≥ 0 (resp. > 0) for all α ∈ Δ, and that ⋂_{j≥0} Ad_{z^{j}}(Ω) is open if and only if ord_{p}(α(z)) ≤ 0 for all α ∈ Δ.

Recall that exponentiation provides a canonical identification of 𝔫 with N. This identification is Z_{M}-equivariant, where Z_{M} acts on 𝔫 via the adjoint map, and on N via conjugation. From the discussion of the preceding paragraph, we conclude that for any compact open subset Ψ of N, the collection {z^{j}Ψz^{−j}}_{j≥0} forms a uniformly bounded collection of subsets of N (resp. a neighbourhood basis of the identity in N) if and only if z ∈ Z_{M} satisfies ord_{p}(α(z)) ≥ 0 (resp. > 0) for all α ∈ Δ, and that ⋂_{j≥0} z^{j}Ψz^{−j} is open if and only if ord_{p}(α(z)) ≤ 0 for all α ∈ Δ. The lemma follows upon taking Ψ to be N_{0}. □

This lemma has an evident analogue for subgroups of \overline{N}, which we now state, and which is proved in an identical fashion.

3.1.4. Lemma. Let \overline{N}_{0} be any compact open subgroup of \overline{N}.

(1) If z ∈ Z_{M} has the property that z\overline{N}_{0}z^{−1} ⊂ \overline{N}_{0}, then ord_{p}(α(z)) ≤ 0 for all α ∈ Δ.

(2) An element z ∈ Z_{M} has the property that the conjugates z^{i}\overline{N}_{0}z^{−i} form a basis of neighbourhoods of the identity of \overline{N} if and only if ord_{p}(α(z)) < 0 for all α ∈ Δ.

(3) There exists C ≤ 0 such that if z ∈ Z_{M} and ord_{p}(α(z)) ≤ C for all α ∈ Δ, then z\overline{N}_{0}z^{−1} ⊂ \overline{N}_{0}.

(4) The intersection ⋂_{z s.t. all ordp(α(z))≥0} z\overline{N}_{0}z^{−1} (where, as indicated, the intersection

is taken over all z ∈ Z_{M}^{+} such that ord_{p}(α(z)) ≥ 0 for all α ∈ Δ) is an open subgroup of \overline{N}_{0}.

3.1.5. Lemma. If N_{0} ( resp. \overline{N}_{0}) is a compact open subgroup of N (resp. \overline{N}), and if we write Z_{M}^{+} := {z ∈ Z_{M} | zN_{0}z ⊂ N_{0}}, then we may choose z ∈ Z_{M}^{+} so that:

(1) Z_{M}^{+} together with z^{−1} generate Z_{M} as a monoid.

(2) z^{−1}\overline{N}_{0}z ⊂ \overline{N}_{0}.

(3) The descending sequence of subgroups z^{−i}\overline{N}_{0}z^{i} forms a neighbourhood basis of the identity in \overline{N}.

Proof. If z ∈ Z_{M} is chosen so that ord_{p}(α(z)) is sufficiently large for all α ∈ Δ, then it follows from Lemmas 3.1.3 (3) and 3.1.4 (2), (3) that zN_{0}z^{−1} ⊂ N_{0}, that z^{−1}\overline{N}_{0}z ⊂ \overline{N}_{0}, and that the descending sequence of subgroups z^{−i}\overline{N}_{0}z^{i} forms a neighbourhood basis of the identity in \overline{N}. If z′ is any element of Z_{M}, then ord_{p}(α(z^{i}z′)) can be made arbitrarily large for all α ∈ Δ by taking i large enough, and so, again by Lemma 3.1.3 (3), we see that z^{i}z′ ∈ Z_{M}^{+} if i is large enough. Thus z′ ∈ z^{−i}Z_{M}^{+} if i is large enough, and so z^{−1} and Z_{M}^{+} together generate Z_{M} as a monoid. □

3.2. A review of the functor Ord_{P}. Recall from [10, §3.1] that in order to define Ord_{P}, we must first choose an open subgroup P_{0} of P. We fix such a choice, and following [10, §3.1], we write M_{0} := M ⋂ P_{0}, N_{0} := N ⋂ P_{0}, and M^{+} := {m ∈ M | mN_{0}m^{−1} ⊂ N_{0}}. As in the preceding subsection, we let Z_{M} denote the centre of M, and we write Z_{M}^{+} := Z_{M} ⋂ M^{+}.

If V is any object of Mod_{P}^{sm}(A), then there is a natural action (the “Hecke action”) of M^{+} on V^{N_{0}} [10, §3.1]. We then recall from [10, Def. 3.1.9] that

Ord_{P} (V) := Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], V^{N_{0}})_{Z_{M} −fin}.

The remainder of this subsection is devoted to showing that, in some situations, we may compute Ord_{P} (V) using a single (well-chosen) element z ∈ Z_{M}^{+}, rather than the entire monoid Z_{M}^{+}. Fix an element such that Z_{M}^{+} together with z^{−1} generate Z_{M} as a monoid, as we may, by Lemma 3.1.5. Let C denote the cyclic subgroup of Z_{M} generated by z, and write C^{+} = Z_{M}^{+} ⋂ C (so C^{+} is the cyclic monoid generated by z). We regard V^{N_{0}} as an A[C^{+}]-module via the inclusion C^{+} ⊂ Z_{M}^{+}.

We adapt the terminology and notation of [10, Def. 2.3.1] in an evident way to the subgroup C of Z_{M}; namely, we say that an element in a C-representation V is locally C-finite if the A[C]-module that it generates is finitely generated over A, and we write V_{C−fin} to denote the submodule of V consisting of locally C-finite vectors.

3.2.1. Lemma. Let X denote the full subcategory of the category of A[Z_{M}^{+}]-modules consisting of modules which may be written as the union of a collection of finitely generated Z_{M}^{+}-invariant A-modules.

(1) For any object X of X, the composite

Hom_{A[C^{+}]}(A[C], X)_{C−fin} → X → A[C] ⊗_{A[C^{+}]} X,

where the first arrow is given by evaluation at 1 ∈ A[C], and the second arrow is the natural one, is an isomorphism, as is the composite

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], X)_{Z_{M} −fin} → X → A[Z_{M}] ⊗_{A[Z_{M}^{+} ]} X,

defined in the same manner.

(2) For any object X of X , the restriction map

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], X)_{Z_{M} −fin} → Hom_{A[C^{+}]}(A[C], X)_{C−fin}

is an isomorphism.

(3) The functor X ↦ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], X)_{Z_{M} −fin} is exact on the category X.

Proof. We begin by proving the first claim of (1). If X is an object of X, then by assumption we may write it as the union of finitely generated A-modules that are Z_{M}^{+}-invariant, and so in particular C^{+}-invariant. The functor Hom_{A[C^{+}]}(A[C], –) commutes with the formation of inductive limits, by (an evident analogue for C^{+} and C of) [10, Lem. 3.2.2], as does the functor A[C] ⊗_{A[C^{+}]} –. Thus, in proving (1), we may assume that X is finitely generated over A.

Let B denote the image of A[C^{+}] in End_{A}(X); then B is a finite A-algebra, and the A[C^{+}]-structure on X allows us to regard it as a B-module. We may write B as a product of local factors B = ∏_{𝔪} B_{𝔪}, where 𝔪 runs over the finite set of maximal ideals of B. We say that a maximal ideal 𝔪 is ordinary if the image of z in B does not lie in 𝔪, and write

B_{ord} := ∏_{𝔪 ordinary} B_{𝔪},  B_{nil} := ∏_{𝔪 not ordinary} B_{𝔪},

so that B = B_{ord} × B_{nil}. We also write X_{ord} := B_{ord} ⊗_{B} X and X_{nil} := B_{nil} ⊗_{B} X, so that we have the C^{+}-invariant decomposition X = X_{ord} × X_{nil}. The element z acts invertibly on X_{ord}, while some sufficiently high power of z annihilates X_{nil}. Given this, the reader may easily verify that there are isomorphisms

Hom_{A[C^{+}]}(A[C], X)_{C−fin} = Hom_{A[C^{+}]}(A[C], X) ⟶^{∼} Hom_{A[C^{+}]}(A[C], X_{ord})

⟶^{∼} X_{ord} ⟶^{∼} A[C] ⊗_{A[C^{+}]} X_{ord} ⟶^{∼} A[C] ⊗_{A[C^{+}]} X.

(Note that the equality is obvious, since X is finitely generated over A.) Thus the first claim of (1) is proved.

We turn to proving (2), and thus we again suppose that X is an arbitrary object of X. Since Z_{M}^{+} and C generate Z_{M} as a monoid, the natural map A[Z_{M}^{+}] ⊗_{A[C^{+}]} A[C] → A[Z_{M}] is an isomorphism, which in turn implies that the restriction map

(3.2.2)  Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], X) → Hom_{A[C^{+}]}(A[C], X)

is an isomorphism, and also that the natural map

(3.2.3)  A[C] ⊗_{A[C^{+}]} X → A[Z_{M}] ⊗_{A[Z_{M}^{+} ]} X

is an isomorphism. By [10, Lem 3.1.6], together with an evident variant with C^{+} and C in place of Z_{M}^{+} and Z_{M}, we see that an element ϕ in the source (resp. the target) of (3.2.2) is locally Z_{M}-finite (resp. locally C-finite) if and only if its image is finitely generated as an A-module. Since Z_{M} = Z_{M}^{+}C, and since X is a union of finitely generated Z_{M}^{+}-invariant A-modules, we conclude that the isomorphism (3.2.2) restricts to an isomorphism

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], X)_{Z_{M} −fin} ⟶^{∼} Hom_{A[C^{+}]}(A[C], X)_{C−fin},

proving (2).

We may now deduce the second claim of (1). Indeed, it follows from the first claim of (1), together with the isomorphism of (2) and the isomorphism (3.2.3). Claim (3) then follows as well, since the functor A[Z_{M}] ⊗_{A[Z_{M}^{+} ]} – is exact (as A[Z_{M}] is a localization of A[Z_{M}^{+}]). □

3.3. The definition of H^{•}Ord_{P}. Fix P_{0} as in the preceding subsection, and define N_{0}, M_{0}, M^{+}, etc., as in that subsection. Let V be an object of Mod_{P}^{sm}(A), and let V ↪ I^{•} be an injective resolution of V in the category Mod_{P}^{sm}(A). Such a resolution exists, by Proposition 2.1.1, and Proposition 2.1.11 show that such a resolution is also an N_{0}-acyclic resolution of V (since N_{0} is a compact subgroup of P). Thus (I^{•})^{N_{0}} computes the N_{0}-cohomology H^{•}(N_{0}, V). The complex (I^{•})^{N_{0}} is naturally a complex of A[M^{+}]-modules (via the Hecke M^{+}-action), and thus each cohomology module H^{•}(N_{0}, V) is also naturally an A[M^{+}]-module.

3.3.1. Definition. For any object V of Mod_{P}^{sm}(A), we define H^{•}Ord_{P} (V) := Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], H^{•}(N_{0}, V))_{Z_{M} −fin}.

Note that we may compute the M^{+}-action on H^{•}(N_{0}, V), and hence compute H^{•}Ord_{P} (V), using any resolution of V in the category Mod_{P}^{sm}(A) by objects that are acyclic as N_{0}-representations.

Our first goal is to show that the computation of H^{•}Ord_{P} is independent of the choice of P_{0}. To this end, suppose that P′_{0} is another compact open subgroup of P, giving rise to N′_{0}, (M^{+})′, and (Z_{M}^{+})′ in analogy to N_{0}, M^{+}, and Z_{M}^{+}. Write Y′ := M^{+} ⋂(M^{+})′ and Y := Z_{M}^{+} ⋂(Z_{M}^{+})′; then Y generates Z_{M} as a group [8, Prop. 3.3.2 (i)], and Y′ and Z_{M} generate M [8, Prop. 3.3.2 (ii)].

Choose z ∈ Z_{M} such that zN′_{0}z^{−1} ⊂ N_{0}. If V is an object of Mod_{P}^{sm}(A), and I^{•} is an injective resolution of V, then [10, Lem. 3.1.11] gives rise to a Y′-equivariant map

h_{N_{0},N′_{0},z} : (I^{•})^{N′_{0}} → (I^{•})^{N_{0}}.

Since Y generates Z_{M} as a group, the inclusions

Hom_{A[(Z_{M}^{+} )′]}(A[Z_{M}], H^{•}(N′_{0}, V)) ⊂ Hom_{A[Y]}(A[Z_{M}], H^{•}(N′_{0}, V))

and

Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], H^{•}(N_{0}, V)) ⊂ Hom_{A[Y]}(A[Z_{M}], H^{•}(N_{0}, V))

are equalities. The Y′-equivariant map h_{N_{0},N_{0},z} induces an M = Y′Z_{M}-equivariant map

Hom_{A[Y]}(A[Z_{M}], H^{•}(N′_{0}, V)) → Hom_{A[Y]}(A[Z_{M}], H^{•}(N_{0}, V)),

and hence an M-equivariant map

Hom_{A[(Z_{M}^{+} )′]}(A[Z_{M}], H^{•}(N′_{0}, V)) → Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], H^{•}(N_{0}, V)),

which we denote by H^{•}ı_{N_{0},N′_{0},z}. Now define

H^{•}ı_{N_{0},N′_{0}} := z^{−1}H^{•}ı_{N_{0},N′_{0},z} : Hom_{A[(Z_{M}^{+} )′]}(A[Z_{M}], H^{•}(N′_{0}, V))

→ Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], H^{•}(N_{0}, V)).

3.3.2. Proposition. (1) The map H^{•}ı_{N_{0},N′_{0}} is well-defined independently of the

choice of z.

(2) The map H^{•}ı_{N_{0},N′_{0}} is an M-equivariant isomorphism.

(3) If P′′_{0} is another choice of compact open subgroup of P, giving rise to N′′_{0}, etc., then H^{•}ı_{N_{0},N′_{0}}H^{•}ı_{N′_{0},N′′_{0}} = H^{•}ı_{N_{0},N′′_{0}}.

Proof. This is proved in an identical manner to [10, Prop. 3.1.12]. □

The preceding result shows that H^{•}Ord_{P} is defined up to canonical isomorphism independently of the choice of P_{0}. It is also independent, up to a canonical isomorphism, of the choice of Levi factor M of P. Indeed, if M′ is another choice of Levi factor of P, let H^{•}Ord′_{P} denote the analogue of H^{•}Ord_{P}, computed using M′ in place of M. There is a uniquely determined element n ∈ N such that nMn^{−1} = M′.

Let V be an object of Mod_{P}^{sm}(A), with injective resolution V ↪ I^{•}. We will compute H^{•}Ord′_{P} using P′_{0} := nP_{0}n^{−1}. Write N′_{0} := N ⋂ P′_{0} = nN_{0}n^{−1}. We find that multiplication by n induces an isomorphism (I^{•})^{N_{0}} ⟶^{∼} (I^{•})^{N′_{0}}, and hence an isomorphism H^{•}(N_{0}, V) ⟶^{∼} H^{•}(N′_{0}, V). One immediately verifies that this isomorphism in turn induces an isomorphism H^{•}Ord_{P} (V) ⟶^{∼} H^{•}Ord′_{P} (V). (See the discussion at the end of [10, §3.1].)

Suppose now that 0 → V_{1} → V_{2} → V_{3} → 0 is a short exact sequence of objects in Mod_{P}^{sm}(A). Applying the δ-functor H^{•}(N_{0}, –) we obtain a long exact sequence

(3.3.3) 0 → H^{0}(N_{0}, V_{1}) → H^{0}(N_{0}, V_{2}) → H^{0}(N_{0}, V_{3}) → ⋯

→ H^{i}(N_{0}, V_{1}) → H^{i}(N_{0}, V_{2}) → H^{i}(N_{0}, V_{3})

→ H^{i+1}(N_{0}, V_{1}) → H^{i+1}(N_{0}, V_{2}) → H^{i+1}(N_{0}, V_{3}) → · · · .

Applying the functor Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], –)_{Z_{M} −fin} to this long exact sequence, we obtain the complex

(3.3.4) 0 → Ord_{P} (V_{1}) → Ord_{P} (V_{2}) → Ord_{P} (V_{3}) → ⋯

→ H^{i}Ord_{P} (V_{1}) → H^{i}Ord_{P} (V_{2}) → H^{i}Ord_{P} (V_{3})

→ H^{i+1}Ord_{P} (V_{1}) → H^{i+1}Ord_{P} (V_{2}) → H^{i+1}Ord_{P} (V_{3}) → · · · .

Since Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], –)_{Z_{M} −fin} is not an exact functor, we are not able to conclude that (3.3.4) is exact in general.

3.4. The main theorem. The goal of this subsection is to prove Theorem 3.4.7 below, which establishes the key properties of H^{•}(N_{0}, V) and H^{•}Ord_{P} (V) in the case when V is an admissible (or more generally, locally admissible) G-representation. In particular, it implies as a corollary that (3.3.4) is exact when the V_{i} are locally admissible G-representations. (See Corollary 3.4.8 below.)

Before proving Theorem 3.4.7, we make some preliminary constructions and prove some necessary lemmas. We fix a compact open subgroup H_{0} of G, which admits an Iwahori decomposition with respect to P and \overline{P}, i.e., writing M_{0} := H_{0} ⋂ M, N_{0} := H_{0} ⋂ N, and \overline{N}_{0} := H_{0} ⋂ \overline{N}, such that the product map \overline{N}_{0} × M_{0} × N_{0} → H_{0} is a bijection. We will take the open subgroup P_{0} with respect to which we compute H^{•}Ord_{P} to be P_{0} := H_{0} ⋂ P = M_{0}N_{0}.

Next, we apply Lemma 3.1.5 to choose an element z ∈ Z_{M}^{+} such that Z_{M} is generated as a monoid by Z_{M}^{+} together with z^{−1}. We furthermore choose z (as Lemma 3.1.5 allows us to) such that z^{−1}\overline{N}_{0}z^{1} ⊂ \overline{N}_{0}, and such that the groups z^{−i}\overline{N}_{0}z^{i} ⊂ \overline{N}_{0}, which then form a descending sequence of compact open subgroups of \overline{N}, form a neighbourhood basis of the identity in \overline{N}.

3.4.1. Lemma. There exists a descending sequence of compact open subgroups \{H_{i}\}_{i≥1} of H_{0}, each admitting an Iwahori decomposition \overline{N}_{i} × M_{i} × N_{0} ⟶^{∼} H_{i}, so that \overline{N}_{i} = z^{−i}\overline{N}_{0}z^{i}, and so that the descending sequence \{M_{i}\}_{i≥1} forms a basis of neighbourhoods of the identity in M.

Proof. Write \overline{N}_{i} := z^{−i}\overline{N}_{0}z^{i}. One sees, using the fact that z ∈ Z_{M}^{+} (so that z^{−i}N_{0}z^{i} ⊃ N_{0}, while z^{−i}M_{0}z^{i} = M_{0}), that z^{−i}H_{0}z^{i} ⋂ H_{0} = \overline{N}_{i}M_{0}N_{0} is a compact open subgroup of G which again admits an Iwahori decomposition. Let H′_{i} denote the closed subgroup of z^{−i}H_{0}z^{i} ⋂ H_{0} generated by \overline{N}_{i} and N_{0}. If h ∈ H′_{i}, then, applying the Iwahori decomposition of z^{−i}H_{0}z^{i} ⋂ H_{0}, we may write h = \overline{n}mn with \overline{n} ∈ \overline{N}_{i}, m ∈ M_{0}, and n ∈ N_{0}. Thus m = \overline{n}^{−1}hn^{−1} ∈ H′_{i}, and so H′_{i} admits the Iwahori decomposition H′_{i} = \overline{N}_{i}M′_{i}N_{0}, where M′_{i} := H′_{i} ⋂ M is a closed subgroup of M_{0}.

Fix a descending sequence of open subgroups {M_{j}}_{j≥1} of M_{0}, chosen to form a neighbourhood basis of the identity in M. For a fixed value of i, let j(i) denote the largest value of j such that M′_{i} ⊂ M_{j}. Since the groups \overline{N}_{i} form a neighbourhood basis of the identity in \overline{N}, we see that the subgroups M′_{i} become arbitrarily small as i → ∞, and so j(i) → ∞ as i → ∞.

If we write H_{i} := \overline{N}_{i}M_{j(i)}N_{0}, then one sees (using the fact that M_{j(i)} ⊂ M_{0}, so that M_{j(i)} normalizes both \overline{N}_{i} and N_{0}, together with the fact that M′_{i} ⊂ M_{j(i)}) that H_{i} is closed under multiplication. Since it is also an open subset of the profinite subgroup H_{0}, it is necessarily an open subgroup of H_{0}, which by construction admits an Iwahori decomposition of the form required by the lemma (once we relabel M_{j(i)} as M_{i}). □

Since \{\overline{N}_{i}\} (resp. M_{i}) forms a neighbourhood basis of the identity in \overline{N} (resp. M), we find that

(3.4.2)  ⋂_{i≥0} H_{i} = N_{0}.

3.4.3. Lemma. For any smooth G-representation V, there is a natural isomorphism lim_{\overrightarrow{i}} H^{•}(H_{i}, V) ⟶^{∼} H^{•}(N_{0}, V). Furthermore, if U_{i} denotes the image of H^{•}(H_{i}, V)

in H^{•}(N_{0}, V), then there exists j ≥ 0 such A[Z_{M}^{+}]U_{i} ⊂ U_{i+j} for all i ≥ 0. (Here we are regarding H^{•}(N_{0}, V) as an A[Z_{M}^{+}]-module via the Hecke Z_{M}^{+}-action.)

Proof. Let V ↪ I^{•} be an injective resolution of V in the category Mod_{G}^{sm}(A). It follows from Propositions 2.1.2 that I^{•} is an injective resolution of V in Mod_{H_{i}}^{sm}(A), for each i ≥ 0. Proposition 2.1.11 shows that we may use I^{•} to compute H^{•}(N_{0}, V). Taking into account (3.4.2), we see that there is a natural isomorphism

lim_{\overrightarrow{i}} H^{•}((I^{•})^{H_{i}}) ⟶^{∼} H^{•}((I^{•})^{N_{0}}),

and consequently, a natural isomorphism

lim_{\overrightarrow{i}} H^{•}(H_{i}, V) ⟶^{∼} H^{•}(N_{0}, V).

This proves the first claim of the lemma.

It follows from Lemmas 3.1.3 (1) and 3.1.4 (4) that ⋂_{z′∈Z_{M}^{+}} z′ \overline{N}_{0}(z′)^{−1} is an open subgroup of \overline{N}_{0}, and thus contains \overline{N}_{j} for some sufficiently large value of j. We then have that \overline{N}_{i+j} ⊂ z′ \overline{N}_{i}(z′)^{−1} for every z′ ∈ Z_{M}^{+} and every i ≥ 0, and so from [10, Lem 3.3.2] (applied with z_{0} taken to be an arbitrary element z′ of Z_{M}^{+}, I_{0} taken to be z^{−j}H_{i}z^{j} ⋂ H_{i} = \overline{N}_{i+j}M_{i}N_{0}, and I_{1} taken to be H_{i}), we conclude that A[Z_{M}^{+}](I^{•})^{H_{i}} ⊂ (I^{•})^{(H_{i} ⋂ z^{−j}H_{i}z^{j})} ⊂ (I^{•})^{H_{i+j}} for all i ≥ 0. This in turn implies that A[Z_{M}^{+}]U_{i} ⊂ U_{i+j}, completing the proof of the lemma. □

3.4.4. Lemma. If V in admissible smooth G-representation, then the cohomology modules H^{•}(H_{i}, V) are finitely generated A-modules, for any i ≥ 0.

Proof. Let V ↪ J^{•} be an injective resolution of V in the category Mod_{H_{0}}^{adm}(A). It follows from Propositions 2.1.2 and 2.1.9 that J^{•} is an injective resolution of V in Mod_{H_{i}}^{sm}(A), for each i ≥ 1, and so there are natural isomorphisms

H^{•}(H_{i}, V) ⟶^{∼} H^{•}((J^{•})^{H_{i}}).

Since J^{•} is an admissible H_{i}-representation, the complex (J^{•})^{H_{i}} is a complex of finitely generated A-modules. Its cohomology modules are thus also finitely generated A-modules. The lemma follows. □

As in Subsection 3.2, let C^{+} ⊂ Z_{M}^{+} (resp. C ⊂ Z_{M}) denote the submonoid (resp. subgroup) of Z_{M}^{+} (resp. Z_{M}) generated by z. If V is any smooth G-representation, then the Hecke Z_{M}^{+}-action on V^{N_{0}} restricts to an action of C^{+} on V^{N_{0}}. If i ≥ 0, then it follows from the fact that z^{−1}\overline{N}_{i}z ⊂ \overline{N}_{i}, together with [10, Lem. 3.3.2] (applied with z = z_{0} and I_{0} = I_{1} = H_{i}) that V^{H_{i}} is invariant under the Hecke operator h_{N_{0},z}, for each i ≥ 0. Thus V^{H_{i}} is a C^{+}-submodule of V^{N_{0}}.

Now suppose that W is a smooth H_{0}-representation, and that we are given a pair of H_{0}-equivariant maps η : V → W and ψ : W → V. (This situation will occur in the proof of Theorem 3.4.7 below.) The space of invariants W^{N_{0}} is a smooth M_{0}-representation, since M_{0} normalizes N_{0}. We may define an action of C^{+} on W^{N_{0}} by declaring z to act via the endomorphism ηh_{N_{0},z}ψ of W^{N_{0}}. Since η and ψ are H_{0}-equivariant, we find that W^{H_{i}} is also C^{+}-invariant for each i ≥ 0.

Since the C^{+}-action on W^{N_{0}} commutes with the M_{0}-action, the space

Hom_{A[C^{+}]}(A[C], W^{N_{0}})_{C−fin}

is equipped in a natural manner with an M_{0}-action, induced by the M_{0}-action on W^{N_{0}}. In this situation we have the following lemma.

3.4.5. Lemma. If W is an admissible smooth H_{0}-representation, then the M_{0}- representation Hom_{A[C^{+}]}(A[C], W^{N_{0}})_{C−fin} is smooth and admissible.

Proof. As was already noted in the proof of Lemma 3.2.1, there is an evident analogue of [10, Lem. 3.1.6], with C and C^{+} in place of Z_{M} and Z_{M}^{+}. Namely, if ϕ ∈ Hom_{A[C^{+}]}(A[C], W^{N_{0}}), then ϕ is locally C-finite if and only if im(ϕ) is a finitely generated A-submodule of W^{N_{0}}. In particular, since the M_{0}-action on W^{N_{0}} is smooth, the M_{0}-action on Hom_{A[C^{+}]}(A[C], W^{N_{0}})_{C−fin} is also smooth.

To prove admissibility, we must show that Hom_{A[C^{+}]}(A[C], W^{N_{0}})_{C−fin}^{M_{i}}, is finitely generated over A for each i ≥ 0. Since the M_{i}-action on Hom_{A[C^{+}]}(A[C], W^{N_{0}})_{C−fin} is defined by the M_{i}-action on W^{N_{0}}, we see that the natural embedding

Hom_{A[C^{+}]}(A[C], W^{M_{i}N_{0}})_{C−fin} ↪ Hom_{A[C^{+}]}(A[C], W^{N_{0}})_{C−fin}^{M_{i}}

is an isomorphism. Thus we must show that Hom_{A[C^{+}]}(A[C], W^{M_{i}N_{0}})_{C−fin} is finitely generated over A, for i ≥ 0.

Since W^{H_{i}} is a finitely generated A-module, for each i ≥ 1, it follows from [10, Lem 3.1.5] (or rather, its analogue with C and C^{+} in place of Z_{M} and Z_{M}^{+}) that Hom_{A[C^{+}]}(A[C], W^{H_{i}})_{C−fin} is finitely generated over A, for each i ≥ 0. Thus, it suffices to show that the natural embedding

(3.4.6)  Hom_{A[C^{+}]}(A[C], W^{H_{i}})_{C−fin} ↪ Hom_{A[C^{+}]}(A[C], W^{M_{i}N_{0}})_{C−fin}

is an isomorphism.

Let ϕ : A[C] → W^{M_{i}N_{0}} be an element in the target of (3.4.6). Since ϕ is locally C-finite, by assumption, its image im(ϕ) is finitely generated over A (as we noted above). Thus there is some value of j ≥ i such that im(ϕ) is invariant under \overline{N}_{j} := z^{−j}\overline{N}_{0}z^{j}, and thus under the subgroup \overline{N}_{j}M_{i}N_{0} of H_{0}. It follows from [10, Lem 3.3.2] (taking z_{0} := z, I_{0} := \overline{N}_{j−1}M_{i}N_{0}, and I_{1} := \overline{N}_{j}M_{i}N_{0}) that

h_{N_{0},z}(V^{\overline{N}_{j}M_{i}N_{0}}) ⊂ V^{\overline{N}_{j−1}M_{i}N_{0}},

and thus that

z W^{\overline{N}_{j}M_{i}N_{0}} = ηh_{N_{0},z}ψ(W^{\overline{N}_{j}M_{i}N_{0}}) ⊂ W^{\overline{N}_{j−1}M_{i}N_{0}}.

Iterating this observation k := j − i times, we find that

z^{k} W^{\overline{N}_{j}M_{i}N_{0}} = (ηh_{N_{0},z}ψ)^{k}(W^{\overline{N}_{j}M_{i}N_{0}}) ⊂ W^{\overline{N}_{i}M_{i}N_{0}} = W^{H_{i}}.

Since im(ϕ) = z^{k} im(ϕ) (since z^{k} ∈ A[C]^{×}), we find that im(ϕ) ⊂ W^{H_{i}}. This proves that (3.4.6) is an isomorphism, as required. □

We are now ready to state and prove the following key result.

3.4.7. Theorem. (1) If V is a locally admissible smooth G-representation, then each of the M^{+}-representations H^{•}(N_{0}, V) is a union of Z_{M}^{+}-invariant finitely generated A-submodules.

(2) If V is a (locally) admissible smooth G-representation, then each of the smooth M-representations H^{•}Ord_{P} (V) is (locally) admissible.

Proof. Suppose first that V is an admissible smooth G-representation. Lemma 3.4.3 shows that H^{•}(N_{0}, V) = ⋃_{i≥0} U_{i}, where U_{i} denotes the image of the natural map H^{•}(H_{i}, V) → H^{•}(N_{0}, V). It also shows that A[Z_{M}^{+}]U_{i} ⊂ U_{i+j}, for all i ≥ 0 and some j ≥ 0. Lemma 3.4.4 shows that U_{i+j} is a finitely generated A-module (being the image of the finitely generated A-module H^{•}(H_{i+j}, V)). Thus so is its submodule A[Z_{M}^{+}]U_{i}. Thus, we may write H^{•}(N_{0}, V) = ⋃_{i≥0} A[Z_{M}^{+}]U_{i}, exhibiting H^{•}(N_{0}, V) as a union of Z_{M}^{+}-invariant finitely generated A-submodules. This proves claim (1) for admissible V.

Continuing to assume that V is an object of Mod_{G}^{adm}(A), let V ↪ I^{•} be an injective resolution of V in the category Mod_{G}^{sm}(A), and let V ↪ J^{•} be an injective resolution of V in the category Mod_{H_{0}}^{adm}(A). It follows from Propositions 2.1.2 and 2.1.9 that I^{•} and J^{•} are both injective resolutions of V in Mod_{H_{i}}^{sm}(A), for each i ≥ 0. Thus we may find an H_{0}-equivariant chain homotopy η : I^{•} → J^{•}, as well as a homotopy inverse ψ : J^{•} → I^{•}.

Writing C to denote the cyclic subgroup of Z_{M} generated by the element z, as above, the Hecke action of M^{+} on (I^{•})^{N_{0}} restricts to an action of C^{+}, which leaves (I^{•})^{H_{i}} invariant, for each i ≥ 0. We define a corresponding action of C^{+} on (J^{•})^{N_{0}} by having z act via the composite ηh_{N_{0},z}ψ. Note that we are in the precise situation introduced in the discussion preceding Lemma 3.4.5 (taking the V and W of that discussion to be I^{•} and J^{•} respectively).

It follows from (3.4.2) that (J^{•})^{N_{0}} is the union of the C^{+}-invariant subcomplexes (J^{•})^{H_{i}}, each of which is a complex of finitely generated A-modules (since J^{•} is a complex of admissible smooth H_{0}-representations), and so Lemma 3.2.1 shows that the natural map

H^{•}(Hom_{A[C^{+}]}(A[C], (J^{•})^{N_{0}})_{C−fin}) → Hom_{A[C^{+}]}(A[C], H^{•}((J^{•})^{N_{0}}))_{C−fin}

is an isomorphism. We also have the sequence of isomorphisms

H^{•}Ord_{P} (V) := Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], H^{•}(N_{0}, V))_{Z_{M} −fin}

⟶^{∼} Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], H^{•}((I^{•})^{N_{0}}))_{Z_{M} −fin}

⟶^{∼} Hom_{A[C^{+}]}(A[C], H^{•}((I^{•})^{N_{0}}))_{C−fin}

⟶^{∼} Hom_{A[C^{+}]}(A[C], H^{•}((J^{•})^{N_{0}}))_{C−fin},

the first isomorphism following from Proposition 2.1.11 (as we noted above), the second being provided by part (2) of Lemma 3.2.1, and the third being provided by the chain homotopy η. Altogether, we obtain an M_{0}C-equivariant isomorphism

H^{•}Ord_{P} (V) ⟶^{∼} H^{•}(Hom_{A[C^{+}]}(A[C], (J^{•})^{N_{0}})_{C−fin}).

Thus (since the category of admissible smooth M_{0}-representations is abelian [10, Prop. 2.2.13]), in order to prove claim (2) for V, it suffices to show that each member of the complex Hom_{A[C^{+}]}(A[C], (J^{•})^{N_{0}})_{C−fin} is an admissible M_{0}-representation. This follows from Lemma 3.4.5, and completes the proof of (2) for admissible V.

Suppose now that V is a locally admissible smooth G-representation. We may write V as an inductive limit, V ⟶^{∼} lim_{\overrightarrow{j}} V_{j}, where each V_{j} is an admissible smooth

G-representation. We then obtain an M^{+}-equivariant isomorphism

lim_{\overrightarrow{j}} H^{•}(N_{0}, V_{j}) ⟶^{∼} H^{•}(N_{0}, V),

and hence (taking into account [10, Lem. 3.2.2]), an isomorphism

lim_{\overrightarrow{j}} Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], H^{•}(N_{0}, V_{j}))_{Z_{M} −fin}

⟶^{∼} Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], H^{•}(N_{0}, V))_{Z_{M} −fin}.

Claims (1) and (2) in the locally admissible case thus follow from the corresponding claims in the admissible case. □

3.4.8. Corollary. H^{•}Ord_{P} forms a δ-functor from the category Mod_{G}^{adm}(A) (resp. Mod_{G}^{l.adm}(A)) to the category Mod_{M}^{adm}(A) (resp. Mod_{M}^{l.adm}(A)).

Proof. We have already shown in part (2) of Theorem 3.4.7 that H^{•}Ord_{P}, when restricted to Mod_{G}^{adm}(A) (resp. Mod_{G}^{l.adm}(A)), takes values in Mod_{M}^{adm}(A) (resp. Mod_{M}^{l.adm}(A)). Part (1) of Theorem 3.4.7, together with Lemma 3.2.1, shows that the complex (3.3.4) is exact, when 0 → V_{1} → V_{2} → V_{3} is a short exact sequence in Mod_{G}^{l.adm}(A). The corollary follows. □

3.5. Some properties of H^{•}(N_{0}, –). As its title indicates, in this subsection we prove some properties of the cohomology functors H^{•}(N_{0}, –), which we will need in our subsequent study of the functor H^{•}Ord_{P}.

It will be convenient (in order to facilitate the arguments to come) to assume for the duration of this subsection that N is an arbitrary connected unipotent linear algebraic group over ℚ_{p}, of dimension d, say, and that N_{0} is a compact open subgroup of N.

If 𝔫 denotes the Lie algebra of N, then the exponential map induces an isomorphism 𝔫 ⟶^{∼} N. Under this isomorphism, the group operation on N is obtained from the Lie bracket on 𝔫, together with the Cambpell–Hausdorff formula. We let 𝔫_{0} denote the preimage of N_{0} under this isomorphism.

3.5.1. Definition. We say that N_{0} is a standard compact open subgroup of N, or simply standard, if 𝔫_{0} is a ℤ_{p}-Lie subalgebra of 𝔫.

3.5.2. Lemma. The identity of N admits a neighbourhood basis of standard compact open subgroups.

Proof. Let 𝔫_{0} be any bounded ℤ_{p}-lattice in 𝔫 (i.e. 𝔫_{0} is a finitely generated ℤ_{p}- submodule of 𝔫 which spans 𝔫 over ℚ_{p}). Since 𝔫_{0} is finitely generated over ℤ_{p}, and spans 𝔫 over ℚ_{p}, one sees that

[𝔫_{0}, 𝔫_{0}] ⊂ p^{−m}𝔫_{0}

for m ≥ 0. Hence, for any integer n, there is an inclusion

[p^{n}𝔫_{0}, p^{n}𝔫_{0}] ⊂ p^{n−m} · p^{n}𝔫_{0},

and thus the ℤ_{p}-submodules p^{n}𝔫_{0}, for n ≥ m, form a basis of neighbourhoods of 0 in 𝔫 consisting of ℤ_{p}-Lie subalgebras of 𝔫. Since every Lie bracket in p^{n}𝔫_{0} is in fact divisible by p^{n−m}, we see that if n is sufficiently large (depending on the dimension d of N) then p^{n}𝔫_{0} will exponentiate to a standard compact open subgroup of N (i.e. applying the Cambpell–Hausdorff formula to multiply elements of exp(p^{n}𝔫_{0}) will not introduce any denominators). This proves the lemma. □

Since N is unipotent, it admits a surjective homomorphism N → ℚ_{p}. The kernel N′ of this homomorphism is again unipotent. The image N′′_{0} of N_{0} under this homomorphism is a compact open subgroup of ℚ_{p}, and hence is isomorphic to ℤ_{p}. Thus if we write N′_{0} := N_{0} ⋂ N, then we have a short exact sequence

(3.5.3)  0 → N′_{0} → N_{0} → N′′_{0} → 0,

where N′′_{0} is one-dimensional and N′_{0} is again a compact open subgroup of a unipotent group. Since any compact open subgroup of ℚ_{p} is standard, it is clear that if N_{0} is standard, then so is N′_{0}.

3.5.4. Lemma. If V is any smooth N_{0}-representation over A, then H^{•}(N_{0}, V) vanishes in degrees greater than d.

Proof. If d = 1, and if we let γ denote a topologically generator of N_{0}, then it is well-known that H^{•}(N_{0}, V) can be computed by the complex

(3.5.5)  V ⟶^{γ−1} V.

Thus the lemma holds in this case. The general case then follows by applying the Hochschild–Serre spectral sequence to the short exact sequence (3.5.3), and using induction on d. □

If L is any finite rank free ℤ_{p}-module, then we write det L to denote the top non-vanishing exterior power of L (so det L = ⋀^{d} L if L has rank d), and L^{∨} to denote the ℤ_{p}-dual of L, i.e. L^{∨} := Hom_{ℤ_{p}}(L, ℤ_{p}).

3.5.6. Proposition. If N_{0} is standard, corresponding to the Lie subalgebra 𝔫_{0} of 𝔫, then for V any smooth N_{0}-representation over A, there is a natural isomorphism

H^{d}(N_{0}, V) ⟶^{∼} V_{N_{0}} ⊗_{ℤ_{p}} det 𝔫_{0}^{∨}.

Proof. Suppose first that N_{0} is one-dimensional, topologically generated by an element γ. In this case the Lie bracket on 𝔫_{0} is trivial, and the exponential map 𝔫_{0} ⟶^{∼} N_{0} is in fact an isomorphism of groups (with the source being given the group structure underlying its structure as a ℤ_{p}-module). The choice of γ induces isomorphisms 𝔫_{0} ⟶^{∼} N_{0} ⟶^{∼} ℤ_{p}, and hence a dual isomorphism ℤ_{p} ⟶^{∼} 𝔫_{0}^{∨}, while the complex (3.5.5) gives rise to an isomorphism

H^{1}(N_{0}, V) ⟶^{∼} V/(γ − 1)V ⟶^{∼} V_{N_{0}}.

Combining this isomorphism with the preceding one, we obtain an isomorphism

H^{1}(N_{0}, V) ⟶^{∼} V_{N_{0}} ⊗_{ℤ_{p}} 𝔫_{0}^{∨},

which the reader may verify is independent of the choice of γ.

We now turn to the general case. Applying the Hochschild–Serre spectral sequence to (3.5.3), and taking into account Lemma 3.5.4, we obtain an isomorphism

H^{d}(N_{0}, V) ⟶^{∼} H^{1}(N′′_{0}, H^{d−1}(N′_{0}, V)).

If we let 𝔫′_{0} (resp. 𝔫′′_{0}) denote the preimage of N′_{0} (resp. N′′_{0}) under the exponential map, then by induction we have a natural isomorphism

(3.5.7)  H^{d−1}(N′_{0}, V) ⟶^{∼} V_{N′_{0}} ⊗_{ℤ_{p}} det(𝔫′_{0})^{∨},

and hence an isomorphism

H^{d}(N_{0}, V) ⟶^{∼} (V_{N′_{0}} ⊗_{ℤ_{p}} det(𝔫′_{0})^{∨})_{N′′_{0}} ⊗_{ℤ_{p}} (𝔫′′_{0})^{∨}.

Since det(𝔫′_{0})^{∨} is a rank one free ℤ_{p}-module, the N′′_{0}-action on it is necessarily trivial, and hence

(V_{N′_{0}} ⊗_{ℤ_{p}} det(𝔫′_{0})^{∨})_{N′′_{0}} ⟶^{∼} (V_{N′_{0}})_{N′′_{0}} ⊗_{ℤ_{p}} det(𝔫′_{0})^{∨} ⟶^{∼} V_{N_{0}} ⊗_{ℤ_{p}} det(𝔫′_{0})^{∨}.

Thus we may rewrite (3.5.7) as an isomorphism

(3.5.8)  H^{d}(N_{0}, V) ⟶^{∼} V_{N_{0}} ⊗_{ℤ_{p}} det(𝔫′_{0})^{∨} ⊗_{ℤ_{p}} (𝔫′′_{0})^{∨}.

The short exact sequence (3.5.3) gives rise to a short exact sequence

0 → 𝔫′_{0} → 𝔫_{0} → 𝔫′′_{0} → 0,

and thus there is an isomorphism

det(𝔫′_{0})^{∨} ⊗_{ℤ_{p}} (𝔫′′_{0})^{∨} ⟶^{∼} det 𝔫_{0}^{∨}.

Combining this isomorphism with (3.5.8) gives the isomorphism of the lemma. We leave to the reader the task of verifying that this isomorphism is natural (and, in particular, is independent of the choice of dévissage (3.5.3)). □

Let N_{1} ⊂ N_{0} be an inclusion of standard compact open subgroups of N, corresponding under exponential map to the inclusion 𝔫_{1} ⊂ 𝔫_{0} of ℤ_{p}-Lie subalgebras of 𝔫. This inclusion induces an embedding of rank one free ℤ_{p}-modules

det 𝔫_{0}^{∨} ↪ det 𝔫_{1}^{∨},

whose cokernel has order equal to the index [𝔫_{0} : 𝔫_{1}]. Thus this embedding induces an isomorphism

det 𝔫_{0}^{∨} ⟶^{∼} [𝔫_{0} : 𝔫_{1}] det 𝔫_{1}^{∨}.

Multiplication by [𝔫_{0} : 𝔫_{1}], composed with the inverse of this isomorphism, thus induces an isomorphism

(3.5.9)  det 𝔫_{1}^{∨} ⟶^{∼} det 𝔫_{0}^{∨}.

3.5.10. Lemma. If V is a smooth N_{0}-module, then we have the following commutative diagram

\xymatrix{H^{d}(N_{1}, V) \ar[d]^{cores} \ar[r]^{∼} & V_{N_{1}} ⊗_{ℤ_{p}} det 𝔫_{1}^{∨} \ar[d] \\ H^{d}(N_{0}, V) \ar[r]^{∼} & V_{N_{0}} ⊗_{ℤ_{p}} det 𝔫_{0}^{∨},}

in which the horizontal arrows are the isomorphisms of Proposition 3.5.6, the lefthand vertical arrow is given by corestriction, and the right-hand vertical arrow is the tensor product of the natural surjection V_{N_{1}} → V_{N_{0}} and the isomorphism (3.5.9).

Proof. To begin with, suppose that N_{0}, and hence N_{1}, has dimension one. Write I := [N_{0} : N_{1}]. Let γ denote a topological generator of N_{0}, so that γ^{I} is a topological generator of N_{1}. The explicit description of the isomorphism of Proposition 3.5.6 given in the proof of that proposition shows that we may rewrite the diagram in the statement of the lemma in the form

(3.5.11)  \xymatrix{H^{1}(N_{1}, V) \ar[d]^{cores} \ar[r] & V/(γ − 1)V \ar[d] \\ H^{1}(N_{0}, V) \ar[r] & V/(γ^{I} − 1)V,}

in which the left-hand vertical arrow is given by corestriction, the right-hand vertical arrow is given by the natural surjection, and the horizontal isomorphisms are given by using the complexes

V ⟶^{γ−1} V

and

V ⟶^{γ^{I}−1} V

to compute N_{0} cohomology and N_{1} cohomology respectively. Now 1, γ, …, γ^{I−1} serve as coset representatives for N_{0} in N_{1}, and so cores : V^{N_{1}} → V^{N_{0}} is given by multiplication by ∑_{i=0}^{I−1} γ^{i}. The diagram

[figure: V; γ−1; 1+...+γ^{I−1}; V; V; γ^{I}−1; V]

evidently commutes, and so we see that cores : H^{1}(N_{1}, V) → H^{1}(N_{0}, V) corresponds to the natural surjection V/(γ − 1)V → V/(γ^{I} − 1)V. Thus (3.5.11) does indeed commute.

We now consider the general case. Intersecting the members of the short exact sequence (3.5.3) induces a short exact sequence

(3.5.12)  0 → N′_{1} → N_{1} → N′′_{1} → 0.

If (3.5.3) is the image under the exponential map of

(3.5.13)  0 → 𝔫′_{0} → 𝔫_{0} → 𝔫′_{0} → 0,

then (3.5.12) is the image of

0 → 𝔫′_{1} → 𝔫_{1} → 𝔫′_{1} → 0,

obtained by intersecting (3.5.13) with 𝔫_{1}.

The Hochschild–Serre spectral sequence induces isomorphisms

H^{d}(N_{0}, V) ⟶^{∼} H^{1}(N′′_{0}, H^{d−1}(N′_{0}, V))

and

H^{d}(N_{1}, V) ⟶^{∼} H^{1}(N′′_{1}, H^{d−1}(N′_{1}, V)),

in terms of which corestriction may be written as the composite

(3.5.14) H^{1}(N′′_{1}, H^{d−1}(N′_{1}, V)) → H^{1}(N′′_{1}, H^{d−1}(N′_{0}, V))

→ H^{1}(N′′_{0}, H^{d−1}(N′_{0}, V)),

in which the first arrow is induced by applying the functor H^{1}(N′′_{1}, –) to the corestriction map H^{d−1}(N′_{1}, V) → H^{d−1}(N′_{0}, V), and the second arrow is given by corestriction from N′′_{1} to N′′_{0}.

As noted in the proof of Proposition 3.5.6, there are isomorphisms

(V_{N′_{1}} ⊗_{ℤ_{p}} (det 𝔫′_{1})^{∨})_{N′′_{1}} ⊗ (𝔫′′_{1})^{∨} ⟶^{∼} V_{N_{1}} ⊗_{ℤ_{p}} det 𝔫_{1}^{∨}

and

(V_{N′_{0}} ⊗_{ℤ_{p}} (det 𝔫′_{0})^{∨})_{N′′_{0}} ⊗ (𝔫′′_{0})^{∨} ⟶^{∼} V_{N_{0}} ⊗_{ℤ_{p}} det 𝔫_{0}^{∨}.

One easily sees that with respect to these isomorphisms, the right-hand vertical column of the diagram in the statement of the lemma coincides with the composite

(3.5.15) (V_{N′_{1}} ⊗_{ℤ_{p}} (det 𝔫′_{1})^{∨})_{N′′_{1}} ⊗_{ℤ_{p}} (𝔫′′_{1})^{∨} → (V_{N′_{0}} ⊗_{ℤ_{p}} (det 𝔫′_{0})^{∨})_{N′′_{1}} ⊗_{ℤ_{p}} (𝔫′′_{1})^{∨}

→ (V_{N′_{0}} ⊗_{ℤ_{p}} (det 𝔫′_{0})^{∨})_{N′′_{0}} ⊗_{ℤ_{p}} (𝔫′′_{0})^{∨},

where the first arrow is induced from the map

V_{N′_{1}} ⊗_{ℤ_{p}} (det 𝔫′_{1})^{∨} → V_{N′_{0}} ⊗_{ℤ_{p}} (det 𝔫′_{0})^{∨},

obtained by tensoring the natural surjection with the analogue of the isomorphism (3.5.9) for the inclusion 𝔫′_{1} ⊂ 𝔫′_{0}, by passing to N′′_{1}-invariants and tensoring with (𝔫′′_{1})^{∨}, while the second arrow is the tensor product of the natural surjection

(V_{N′_{0}} ⊗_{ℤ_{p}} (det 𝔫′_{0})^{∨})_{N′′_{1}} → (V_{N′_{0}} ⊗_{ℤ_{p}} (det 𝔫′_{0})^{∨})_{N′′_{0}}

with the analogue of the isomorphism (3.5.9) for the inclusion 𝔫′′_{1} ⊂ 𝔫′′_{0}. Consider now the diagram

\xymatrix{H^{1}(N′′_{1}, H^{d−1}(N′_{1}, V)) \ar[d] \ar[r]^{∼} & (V_{N′_{1}} ⊗_{ℤ_{p}} (det 𝔫′_{1})^{∨})_{N′′_{1}} ⊗ (𝔫′′_{1})^{∨} \ar[d] \\ H^{1}(N′′_{1}, H^{d−1}(N′_{0}, V)) \ar[d] \ar[r]^{∼} & (V_{N′_{0}} ⊗_{ℤ_{p}} (det 𝔫′_{0})^{∨})_{N′′_{1}} ⊗ (𝔫′′_{1})^{∨} \ar[d] \\ H^{1}(N′′_{0}, H^{d−1}(N′_{0}, V)) \ar[r]^{∼} & (V_{N′_{0}} ⊗_{ℤ_{p}} (det 𝔫′_{0})^{∨})_{N′′_{0}} ⊗ (𝔫′′_{0})^{∨},}

in which the horizontal isomorphisms are provided by Proposition 3.5.6 (applied variously to N′′_{1}, N′_{1}, N′_{0}, and N′′_{0}), the left-hand column is equal to (3.5.14), and the right-hand column is equal to (3.5.15).

By induction on d, each of the squares commutes, and thus so does the outer square. On the other hand, we have already identified the two columns of this outer square with two vertical arrows of the diagram in the statement of the lemma. The explicit description of the isomorphism of Proposition 3.5.6 given in the proof of that proposition further allows us to identify the horizontal arrows of this diagram with the horizontal arrows in the diagram in the statement of the lemma. Thus the diagram in the statement of the lemma commutes, as claimed. □

3.6. An application to Jacquet modules. We employ our standard notation (G, P, M, N, etc.). Let d denote the dimension of N.

3.6.1. Proposition. The functors H^{•}Ord_{P} on Mod_{P}^{sm}(A) vanish in degrees greater than d.

Proof. This follows immediately from Lemma 3.5.4 together with the definition of H^{•}Ord. □

In the remainder of this subsection we demonstrate a relation between the functor H^{d}Ord_{P} and the so-called Jacquet module functor associated to P. Recall that if V is a smooth G-representation, then the Jacquet module of V with respect to P is defined to be the space of coinvariants V_{N}. It is naturally an M-representation (since M normalizes N), and is characterized by the adjunction formula

Hom_{G}(V, Ind_{P}^{G} U) ⟶^{∼} Hom_{M}(V_{N}, U)

for any smooth representation U of M.

The action by conjugation of M on N induces an action (the so-called adjoint action) of M on 𝔫, the Lie algebra of N. Passing to top exterior powers, we obtain an action of M on det 𝔫. Since det 𝔫 is one-dimensional, this action is given by a character α : M → ℚ_{p}^{×}. If we let | | : ℚ_{p}^{×} → p^{ℤ} denote the p-adic absolute value (normalized so that |p| = p^{−1}), then we write α^{u} := α|α| : M → ℤ_{p}^{×} for the unit part of α. Of course, we can and do also regard α^{u} as an A-valued character of M, by composing it with the natural map ℤ_{p}^{×} → A^{×}.

Choose N_{0} to be a standard compact open subgroup of N, so that N_{0} = exp 𝔫_{0}, as in the setting of Proposition 3.5.6. If we set M_{0} := {m ∈ M | mN_{0}m^{−1} = N_{0}}, then M_{0} is a compact open subgroup of M, and so P_{0} := M_{0}N_{0} is a compact open subgroup of P with respect to which we may compute the functor H^{•}Ord_{P} .

Under the exponential map 𝔫 ⟶^{∼} N, the action of M on N by conjugation corresponds to the adjoint action of M on 𝔫. In particular, if m ∈ M^{+} (so that by definition mN_{0}m^{−1} ⊂ N_{0}) then Ad_{m} restricts to an injective endomorphism of 𝔫_{0}, and hence det Ad_{m} induces an injective endomorphism

det Ad_{m} : det 𝔫_{0} ↪ det 𝔫_{0}.

Of course, this action is given simply by multiplication by α(m). Since its source and target are free of rank one over ℤ_{p}, its cokernel is cyclic of order |α(m)|^{−1}. Thus composing det Ad_{m} with multiplication by |α(m)|, we obtain an isomorphism

det 𝔫_{0} ⟶^{∼} det 𝔫_{0},

which is given by multiplication by α^{u}(m).

3.6.2. Proposition. There is a natural isomorphism of functors

V_{N} ⟶^{∼} H^{d}Ord_{P} (V) ⊗ α^{u}

on the category Mod_{G}^{l.adm}(A).

Proof. We compute H^{d}Ord_{P} with respect to the subgroup P_{0} constructed above. Proposition 3.5.6 gives a natural isomorphism H^{d}(N_{0}, V) ⟶^{∼} V_{N_{0}} ⊗_{ℤ_{p}} det 𝔫_{0}^{∨}. Via this isomorphism we may transfer the Hecke M^{+}-action on the left-hand side of this isomorphism to the right-hand side, and we will begin by describing this action explicitly.

If m ∈ M^{+} then the Hecke action of m on H^{d}(N_{0}, V) is the equal to the composite H^{d}(N_{0}, V) ⟶^{∼} H^{d}(mN_{0}m^{−1}, V) → H^{d}(N_{0}, V), in which the first isomorphism is induced by conjugation by m, and the second by corestriction. We may make this composite into the left-hand column of the following commutative diagram:

\xymatrix{H^{d}(N_{0}, V) \ar[d]^{∼} \ar[r]^{∼} & V_{N_{0}} ⊗_{ℤ_{p}} det 𝔫_{0}^{∨} \ar[d]^{∼} \\ H^{d}(mN_{0}m^{−1}, V) \ar[r]^{∼} \ar[d]^{cores} & V_{mN_{0}m^{−1}} ⊗_{ℤ_{p}} (det Ad_{m})(det 𝔫_{0})^{∨} \ar[d] \\ H^{d}(N_{0}, V) \ar[r]^{∼} & V_{N_{0}} ⊗_{ℤ_{p}} det 𝔫_{0}^{∨},}

in which the horizontal isomorphisms are those of Proposition 3.5.6, the upper right-hand vertical arrow is given by the tensor product of the isomorphism

(3.6.3)  V_{N_{0}} ⟶^{∼} V_{mN_{0}m^{−1}}

induced by multiplication by m and the inverse of the isomorphism

det Ad_{m}^{∨} : (det Ad_{m})(det 𝔫_{0})^{∨} ⟶^{∼} det 𝔫_{0}^{∨},

and the lower right-hand vertical arrow is given by the tensor product of the natural surjection

(3.6.4)  V_{mN_{0}m^{−1}} → V_{N_{0}},

and the isomorphism (3.5.9) (taking 𝔫_{1} to be Ad_{m}(𝔫_{0})). The commutativity of the upper square is immediate, while the commutativity of the lower square follows from Lemma 3.5.10.

We may equip V_{N_{0}} with an M^{+}-representation by defining the action of m on V_{N_{0}} to be the composite of (3.6.4) and (3.6.3). We may equip det 𝔫_{0}^{∨} with an M^{+}- representation by defining the action of m on det 𝔫_{0}^{∨} to be the composite of the isomorphism (3.5.9), with 𝔫_{1} taken to be Ad_{m}(𝔫_{0}), and the isomorphism

(det Ad_{m}^{∨})^{−1} : det 𝔫_{0}^{∨} ⟶^{∼} ((det Ad_{m})(det 𝔫_{0}))^{∨}.

The discussion preceding the statement of the proposition shows that this action of M^{+} on det 𝔫_{0}^{∨} is given by the character (α^{u})^{−1}.

By construction, the composite of the two vertical arrows of the above diagram coincides with the tensor product action of m on V_{N_{0}} ⊗_{ℤ_{p}} det 𝔫_{0}^{∨}, and thus this commutative diagram exhibits a natural M^{+}-equivariant isomorphism

H^{d}(N_{0}, V) ⟶^{∼} V_{N_{0}} ⊗_{ℤ_{p}} det 𝔫_{0}^{∨} ⟶^{∼} V_{N_{0}} ⊗_{ℤ_{p}} (α^{u})^{−1},

which we may rewrite as an isomorphism

(3.6.5)  V_{N_{0}} ⟶^{∼} α^{u} ⊗ H^{d}(N_{0}, V).

Now fix an element z ∈ Z_{M}^{+} such that z^{−1} and Z_{M}^{+} generate Z_{M} as a monoid, as we may be Lemma 3.1.5. We may then form the increasing chain of compact open subgroups

N_{0} ⊂ z^{−1}N_{0}z ⊂ ⋯ ⊂ z^{−i}N_{0}z^{i} ⊂ ⋯;

Lemma 3.1.3 (2) shows that N = ⋃_{i=0}^{∞} z^{−i}N_{0}z^{i}. Thus there is a natural isomor-

phism lim_{\overrightarrow{i}} V_{z^{−i}N_{0}z^{i}} ⟶^{∼} V_{N}, with the transition map V_{z^{−i}N_{0}z^{i}} → V_{z^{−(i+1)}N_{0}z^{i+1}}

being the natural projection. If we let C^{+} (resp. C) denote the submonoid (resp. subgroup) of Z_{M} generated by z, then we may rewrite the preceding isomorphism more succinctly in the form

(3.6.6)  A[C] ⊗_{A[C^{+}]} V_{N_{0}} ⟶^{∼} V_{N}.

Part (1) of Theorem 3.4.7, implies that H^{d}(N_{0}, V) is a union of finitely generated, Z_{M}^{+}-invariant A-modules. Thus Lemma 3.2.1 shows that

H^{d}Ord_{P} (V) := Hom_{A[Z_{M}^{+} ]}(A[Z_{M}], H^{d}(N_{0}, V))

⟶^{∼} Hom_{A[C_{M}^{+} ]}(A[C_{M}], H^{d}(N_{0}, V)) ⟶^{∼} A[C_{M}] ⊗_{A[C_{M}^{+} ]} H^{d}(N_{0}, V).

Combining this with the isomorphism (3.6.5), we find that

A[C] ⊗_{A[C^{+}]} V_{N_{0}} ⟶^{∼} α^{u} ⊗ H^{d}Ord_{P} (V).

Combining this in turn with the isomorphism (3.6.6) proves the proposition. □

3.6.7. Corollary. If V is a (locally) admissible (finitely generated) smooth Grepresentation over A, then V_{N} is a (locally) admissible (finitely generated) smooth M-representation over A.

Proof. The (local) admissibility of V_{N} follows from part (2) of Theorem 3.4.7, which shows that H^{d}Ord_{P} (V) is (locally) admissible if V is. If V is finitely generated over A[G], then writing G = ΩP for some compact open subset Ω of G, we see that V is in fact finitely generated over A[P], and thus that V_{N} is finitely generated over A[M]. □

3.6.8. Remark. In the representation theory of p-adic reductive groups on complex vector spaces, locally admissible smooth representations were introduced by Harish-Chandra under the name of quasi-admissible smooth representations (see [7]), and it is shown in [7] that in this context, the Jacquet functor takes quasi-admissible smooth G-representations to quasi-admissible smooth M-representations. In [5] the corresponding result is proved for admissible smooth representations.

3.7. The relation to derived functors. We write R^{•}Ord_{P} to denote the derived functors of the functor Ord_{P} on the category Mod_{G}^{l.adm}(A). Since the derived functors may be characterized as the universal δ-functor whose 0th term is equivalent to Ord_{P}, there is a natural transformation (of δ-functors)

(3.7.1)  R^{•}Ord_{P} → H^{•}Ord_{P}

extending the equivalence R^{0}Ord_{P} ⟶^{∼} H^{0}Ord_{P} ⟶^{∼} Ord_{P} . We make the following conjecture regarding this transformation.

3.7.2. Conjecture. The natural transformation (3.7.1) is a natural equivalence.

The derived functors are also characterized by their effaceability. Thus Conjecture 3.7.2 is equivalent to conjecturing that the functors H^{≥1}Ord_{P} are effaceable.

In an appendix, joint with Paškūnas [11], we prove Conjecture 3.7.2 in the case when G = GL_{2}(F), for some finite extension F of ℚ_{p}, and P is a Borel subgroup of G.

3.7.3. Remark. The natural transformation R^{1}Ord_{P} → H^{1}Ord_{P} is an injection. (This is a general property of the natural transformation from a universal δ-functor to any other δ-functor which is an equivalence on 0th terms.)

In [10, Thm. 4.4.6], we proved that Ord_{P} : Mod_{G}^{adm}(A) → Mod_{M}^{adm}(A) is right adjoint to the functor IndG ; \overline{P} : Mod_{M}^{adm}(A) → Mod_{G}^{adm}(A). (Note that IndG ; \overline{P} does indeed induce a functor between these categories, by [10, Lem 4.1.7].) Since both functors commute with the formation of inductive limits [10, Lems. 3.2.2, 4.1.4], the same statement is true with Mod_{G}^{adm}(A) and Mod_{M}^{adm}(A) replaced by Mod_{G}^{l.adm}(A) and Mod_{M}^{l.adm}(A). Since IndG ; \overline{P} is furthermore exact [10, Prop. 4.1.5], we find that Ord_{P} takes injective objects in Mod_{G}^{l.adm}(A) to injective objects in Mod_{M}^{l.adm}(A). Thus the adjointness between Ord_{P} and IndG ; \overline{P} extends to a derived adjointness, i.e., for any two objects U of Mod_{M}^{l.adm}(A) and V of Mod_{G}^{l.adm}(A), there is a natural quasi-isomorphism

R Hom_{M}(U, ROrd_{P} (V))⟶^{∼} R Hom_{G}(IndG ; \overline{P} U, V).

More concretely, there is a spectral sequence

(3.7.4)  E_{2}^{i,j} := Ext_{M}^{i}(U, R^{j}Ord_{P} (V))⇒ Ext_{G}^{i+j}(IndG ; \overline{P} U, V).

(Here the R Hom and Ext functors are computed in the categories of locally admissible representations.)

The spectral sequence (3.7.4) gives rise in particular to an exact sequence

(3.7.5) 0 → Ext_{M}^{1}(U, Ord_{P} (V)) → Ext_{G}^{1}(IndG ; \overline{P} U, V) → Hom_{M}(U, R^{1}Ord_{P} (V))

→ Ext_{M}^{2}(U, Ord_{P} (V)) → Ext_{G}^{2}(IndG ; \overline{P} U, V).

As we noted in Remark 3.7.3, there is a natural injection

R^{1}Ord_{P} (V) ↪ H^{1}Ord_{P} (V),

and so from (3.7.5) we obtain an exact sequence

(3.7.6) 0 → Ext_{M}^{1}(U, Ord_{P} (V)) → Ext_{G}^{1}(IndG ; \overline{P} U, V) → Hom_{M}(U, H^{1}Ord_{P} (V)),

which one can combine with a computation of H^{1}Ord_{P} (V) in various cases to obtain an upper bound on the size of Ext_{G}^{1}(IndG ; \overline{P} U, V).

3.7.7. Remark. The maps in (3.7.6) have a concrete interpretation. Given an extension of locally admissible smooth M-representations

0 → Ord_{P} (V) → E → U → 0,

applying the exact functor IndG ; \overline{P} yields an extension of smooth G-representations

0 → IndG ; \overline{P} Ord_{P} (V) → IndG ; \overline{P} E → IndG ; \overline{P} U → 0,

which we may push forward along the canonical map IndG ; \overline{P} Ord_{P} (V) → V to obtain an extension

0 → V → E′ → IndG ; \overline{P} U → 0.

Thus we have constructed a map

Ext_{M}^{1}(U, Ord_{P} (V))→ Ext_{G}^{1}(IndG ; \overline{P} U, V),

which, up to a sign, coincides with the first non-trivial arrow in (3.7.6). On the other hand, given an extension

0 → V → E′ → IndG ; \overline{P} U → 0

of locally admissible smooth G-representations, we may apply the δ-functor H^{•}Ord_{P} to obtain an exact sequence

0 ⟶ Ord_{P} (V) ⟶ Ord_{P} (E′) ⟶ Ord_{P} (IndG ; \overline{P} U) ⟶^{δ} H^{1}Ord_{P} (V) → · · · .

In particular, the boundary map δ yields a map

Ord_{P} (IndG ; \overline{P} U) → H^{1}Ord_{P} (V).

Composing this with the canonical isomorphism U ⟶^{∼} Ord_{P} (IndG ; \overline{P} U) of [10, Cor. 4.3.5] we obtain an M-equivariant map U → H^{1}Ord_{P} (V). Thus we have constructed a map

Ext_{G}^{1}(IndG ; \overline{P} U, V) → Hom_{M}(U, H^{1}Ord_{P} (V)),

which, up to a sign, coincides with the second non-trivial arrow in (3.7.6).

3.7.8. Remark. An unfortunate complication in studying the homological algebra of admissible smooth G-representations is that there are (at least) three relevant categories in which one might compute the Ext spaces of a pair of admissible smooth representations: namely Mod_{G}^{adm}(A), Mod_{G}^{l.adm}(A), and Mod_{G}^{sm}(A) (in the first, via Yoneda Exts, and in the second and third either via Yoneda Exts, or equivalently, via derived functors). While working in the first of these categories might seem to be the most natural choice, working in the second has the advantage of putting at ones disposal the formalism of derived functors (since, by Proposition 2.1.1, it has enough injectives), and certain useful results, such as Frobenius reciprocity, are most naturally available when one works in the category of all smooth representations. It follows from [10, Prop. 2.2.13] that one obtains the same value of Ext^{1}, no matter which of the three categories one computes in. One might hope that this is true for all Ext^{i}, but one does not know this at present. Thus we will be careful below to specify in which categories any values of Ext^{i} (i > 1) are computed.

4. The case of GL_{2}(ℚ_{p})

In this section we set G := GL_{2}(ℚ_{p}), and we work in the category Mod_{G}^{adm}(k), where k is a finite field of characteristic p. We take P (resp. \overline{P}) to be the Borel subgroup of G consisting of upper triangular (resp. lower triangular) matrices, so that M is the maximal torus consisting of diagonal matrices. We will compute H^{•}Ord_{P} (V) for all absolutely irreducible objects V of this category, and then, as an application, we will compute the spaces Ext_{G}^{1}(V_{1}, V_{2}) for various pairs of absolutely irreducible objects V_{1}, V_{2}. As we remarked in the introduction, similar computations have been performed by Breuil and Paškūnas [4] (who in fact consider the case of GL_{2}(F) for any finite extension F of ℚ_{p}) and Colmez [6], using different methods. Furthermore, in [15] Paškūnas completes the computation of the spaces Ext_{G}^{1}(V_{1}, V_{2}) for absolutely irreducible V_{1} and V_{2} (at least for p > 2), by treating the cases when V_{1} is supersingular and V_{2} is supersingular or a twist of the Steinberg (cases which are not treated here, nor in [4] or [6].)

4.1. Absolutely irreducible representations. We begin with some generalities regarding irreducible and absolutely irreducible admissible smooth representations of a p-adic analytic group H over the finite field k.

4.1.1. Lemma. If H is a p-adic analytic group and V is a non-zero irreducible object of Mod_{G}^{adm}(k), then End_{k[H]}(V) is a finite field extension of k.

Proof. If V is irreducible, then in particular it is finitely generated, and hence (being smooth) is generated by V^{H_{0}} for some compact open subgroup H_{0} of V. Thus restriction induces an embedding End_{k[H]}(V) ↪ End_{k}(V^{H_{0}}). Since V is admissible, the space V^{H_{0}} is finite dimensional over k, and hence the target of the above embedding is finite dimensional. Thus End_{k[H]}(V) is finite dimensional k. It is also a division ring (since V is irreducible), and thus is a finite dimensional division algebra over k. Since k is finite, so is End_{k[H]}(V), and so it is in fact commutative, and hence is a field. □

4.1.2. Lemma. If H is a p-adic analytic group, and V is an irreducible admissible smooth representation of H over k, then for any field extension l of k, the Hrepresentation l ⊗_{k} V is a direct sum of finitely many irreducible H-representations over l, and the natural map l ⊗_{k} End_{k[H]}(V) → End_{l[H]}(l ⊗_{k} V) is an isomorphism.

Proof. Since V is irreducible, it is in particular finitely generated over k[H], and so the functor Hom_{k[H]}(V, –) commutes with the formation of inductive limits with respect to an inductive system having injective transition maps. Thus we see that

Hom_{l[H]}(l ⊗_{k} V, l ⊗_{k} V) ⟶^{∼} Hom_{k[H]}(V, l ⊗_{k} V) ⟶^{∼} l ⊗_{k} Hom_{k[H]}(V, V),

proving the second claim of the lemma.

Now suppose that W is an l[H]-submodule of l ⊗_{k} V. The inclusion of W into l ⊗_{k} V induces an l ⊗_{k} End_{k[H]}(V)-linear embedding

(4.1.3)  Hom_{k[H]}(V, W) ↪ Hom_{k[H]}(V, l ⊗_{k} V) ⟶^{∼} l ⊗_{k} End_{k[H]}(V).

Tensoring on the right with V over End_{k[H]}(V), we obtain an l[H]-linear map

(4.1.4) Hom_{k[H]}(V, W) ⊗_{End_{k[H]}(V)} V → l ⊗_{k} End_{k[H]}(V) ⊗_{End_{k[H]}(V)} V ⟶^{∼} l ⊗_{k} V,

which is again an embedding, since by the previous lemma End_{k[H]}(V) is a field. The image of this map is precisely the image of the natural evaluation map

(4.1.5)  Hom_{k[H]}(V, W) ⊗_{End_{k[H]}(V)} V → W.

Since l ⊗_{k} V is semi-simple as a k[H]-representation, and is in fact just a direct sum of copies of the irreducible representation V, we see that as a k[H]-representation, W is also isomorphic to a direct sum of copies of V. Hence (4.1.5) is surjective, and thus the image of (4.1.4) is equal to W. Thus we may factor (4.1.4) as

(4.1.6)  Hom_{k[H]}(V, W) ⊗_{End_{k[H]}(V)} V ⟶^{∼} W ↪ l ⊗_{k} V.

Since k is finite, and so separable, the l-algebra l ⊗_{k} End_{k[H]}(V) is separable, and so (4.1.3) must split as a map of l ⊗_{k} End_{k[H]}(V)-modules. Thus the embedding (4.1.4) must be split as a map of l[H]-modules. A consideration of (4.1.6) then shows that W must be a direct summand of l ⊗_{k} V as an l[H]-module.

Clearly, if W_{1} ⊊ W_{2} is a strict inclusion of l[H]-submodules of l ⊗_{k} V, then we have a corresponding inclusion Hom_{k[H]}(V, W_{1}) ⊂ Hom_{k[H]}(V, W_{2}) of ideals in l ⊗_{k} End_{k[H]}(V), which, by a consideration of (4.1.6), is seen to also be strict. Since l ⊗_{k} End_{k[H]}(V) is a finite dimensional l-algebra, any strictly increasing chain of ideals in this algebra is of some uniformly bounded length. Another consideration of (4.1.6) then shows that any strictly increasing chain of l[H]-submodules of l⊗_{k} V must be of finite length. Combining this with the conclusion of the preceding paragraph, we infer that l ⊗_{k} V is indeed the direct sum of finitely many irreducible l[H]-modules, completing the proof of the lemma. □

4.1.7. Lemma. Let H be a p-adic analytic group. If V is a non-zero irreducible object of Mod_{G}^{adm}(k), then the following are equivalent:

(1) End_{k[H]}(V) = k.

(2) l ⊗_{k} V is irreducible (as a representation of H over l) for any finite field extension l of k.

(3) l⊗_{k} V is irreducible (as a representation of H over l) for any field extension l of k.

Proof. Suppose that (1) holds, and that l is an extension of k. Lemma 4.1.2 implies that l⊗_{k} V is a semi-simple representation of H over l, and that End_{l[H]}(l⊗_{k} V) = l. It follows that in fact l ⊗_{k} V is irreducible, and thus that (3) holds.

On the other hand, if (1) does not hold, then Lemma 4.1.1 shows that End_{k[H]}(V) is a proper finite extension l of k. Lemma 4.1.2 then shows that End_{l[H]}(l ⊗_{k} V) = l⊗_{k}l, which is not a field. Thus (2) does not hold either. Since clearly (3) implies (2), the lemma follows. □

4.1.8. Definition. If H is a p-adic analytic group, then we say that an object V of Mod_{H}^{adm}(k) is absolutely irreducible if it is irreducible, and furthermore satisfies the equivalent conditions of Lemma 4.1.7.

The following lemma shows that the study of irreducible admissible smooth representations may always be reduced to the study of absolutely irreducible ones by making a finite extension of scalars.

4.1.9. Lemma. If V is an irreducible object of Mod_{H}^{adm}(k), then there is a finite extension l of k such l ⊗_{k} V is a finite direct sum of absolutely irreducible objects of Mod_{H}^{adm}(l).

Proof. If we write l = End_{k[H]}(V), then Lemma 4.1.1 shows that l is a finite extension of k, and Lemma 4.1.2 shows that l⊗_{k}V is a semi-simple H-representation

and that End_{l[H]}(l ⊗_{k} V) ⟶^{∼} l ⊗_{k} l. Now l ⊗_{k} l ⟶^{∼} ∏_{i} l_{i}, where each l_{i} is an

isomorphic copy of l. We conclude that l ⊗_{k} V = ∏_{i} V_{i}, where each V_{i} is irreducible and satisfies End_{l[H]}(V_{i}) = l. Thus each V_{i} is in fact absolutely irreducible, and the lemma is proved. □

We now recall a result from [1], which provides a (partial) classification of the absolutely irreducible objects of Mod_{G}^{adm}(k). (Recall that G := GL_{2}(ℚ_{p}).)

4.1.10. Proposition. If V is a non-zero absolutely irreducible object of Mod_{G}^{adm}(k), then V has precisely one of the following forms:

(1) V ⟶^{∼} IndG ; \overline{P} χ_{1} ⊗ χ_{2}, where χ_{1} ≠ χ_{2} are distinct characters ℚ_{p}^{×} → k^{×} that are uniquely determined by V.

(2) V ⟶^{∼} (χ ∘ det) ⊗ St, where St denotes the Steinberg representation and χ : ℚ_{p}^{×} → k^{×} is a uniquely determined character.

(3) V ⟶^{∼} χ ∘ det for some uniquely determined character χ : ℚ_{p}^{×} → k^{×}.

(4) V is supersingular, which is to say, V is neither a quotient nor a subobject of any representation of the form IndG ; \overline{P} U, with U an object of Mod_{M}^{l.adm}(k).

Proof. The paper [1] is set in a slightly different context to the one that we are working in; they classify smooth irreducible representations over the algebraically closed field \overline{k} which furthermore admit a central character.

If V is an absolutely irreducible object of Mod_{G}^{adm}(k), then Lemma 4.1.7 (1) implies that V admits a central character, while part (3) of the same lemma implies that \overline{k} ⊗_{k} V remains irreducible. The results of [1] then imply that precisely one of the following holds:

(1′) \overline{k} ⊗_{k} V ⟶^{∼} IndG ; \overline{P} χ_{1} ⊗ χ_{2}, where χ_{1} ≠ χ_{2} are distinct characters ℚ_{p}^{×} → \overline{k}^{×}

that are uniquely determined by V. (2′) \overline{k} ⊗_{k} V ⟶^{∼} (χ ∘ det) ⊗ St, where St denotes the Steinberg representation

and χ : ℚ_{p}^{×} → \overline{k}^{×} is a uniquely determined character.

(3′) \overline{k} ⊗_{k} V ⟶^{∼} χ ∘ det for some uniquely determined character χ : ℚ_{p}^{×} → \overline{k}^{×}. (4′) \overline{k} ⊗_{k} V is supersingular, which is to say, V is neither a quotient nor a

subobject of any representation of the form IndG ; \overline{P} χ_{1} ⊗ χ_{2} for any pair of

characters χ_{1}, χ_{2} : ℚ_{p}^{×} → \overline{k}^{×}.

Since in each of cases (1′), (2′), and (3′) the characters are uniquely determined, a simple descent argument then shows that in fact these characters take values in k^{×}, and that V satisfies the corresponding condition (1), (2), or (3) of the Proposition.

Suppose then that \overline{k} ⊗_{k} V satisfies condition (4′), and consider a map

IndG ; \overline{P} U → V,

where U lies in Mod_{M}^{l.adm}(k); we will show that this map necessarily vanishes. Since U is the inductive limit of finitely generated admissible smooth M-representations, and since parabolic induction commutes with the formation of inductive limits, it is no loss of generality to assume that U is fact finitely generated and admissible, and hence finite dimensional (since M is a torus). Extending scalars to \overline{k}, we obtain a map

(4.1.11)  IndG ; \overline{P}(\overline{k} ⊗_{k} U) → \overline{k} ⊗_{k} V,

and it suffices to show that (4.1.11) vanishes. Since \overline{k} ⊗_{k} U is finite dimensional representation of M over \overline{k}, it is a successive extension of characters, and thus IndG ; \overline{P}(\overline{k} ⊗_{k} U) is a successive extension of inductions of characters. The vanishing of (4.1.11) thus follows from condition (4′).

A similar argument shows that any map V → IndG ; \overline{P} U, with U an object of Mod_{M}^{l.adm}(k), must vanish. Thus if \overline{k} ⊗_{k} V satisfies condition (4′), then V satisfies condition (4) of the lemma. □

4.1.12. Remark. The absolutely irreducible supersingular representations are completely classified in [3], but we will not need this classification for our computations.

4.2. Computations of H^{•}Ord_{P} (V). Fix a ring of coefficients A in Art(ℤ_{p}), with residue field k. Suppose that U is a locally admissible smooth representation of M over A. We may also regard U as a P -representation via the natural surjection P → P/N ⟶^{∼} M. We equip C_{c}^{sm}(N, A) with the P -action described in [10, §4.1], and equip C_{c}^{sm}(N, A) ⊗_{A} U with the tensor product P -action (making it a smooth P -representation).

4.2.1. Remark. There is a natural isomorphism of P -representations

(4.2.2)  C_{c}^{sm}(N) ⊗_{A} U ⟶^{∼} c−Ind_{M}^{P} U.

To construct this isomorphism, note first that restricting functions from P to N induces an isomorphism

(4.2.3)  c−Ind_{M}^{P} U ⟶^{∼} C_{c}^{sm}(N, U).

Secondly, there is a natural isomorphism

(4.2.4)  C_{c}^{sm}(N, A) ⊗_{A} U ⟶^{∼} C_{c}^{sm}(N, U).

Composing the inverse of (4.2.3) with (4.2.4) yields the desired isomorphism (4.2.2).

We let w denote the involution of M induced by the non-trivial element of the Weyl group of M in G; concretely, w ( \begin{matrix}a & 0 \\ 0 & d \end{matrix}) := ( \begin{matrix}d & 0 \\ 0 & a \end{matrix}). If U is an M-representation, we let U^{w} denote the twist of the representation U by w, i.e. the U^{w} has the same underlying A-module as U, and writing ·^{U} to denote the action on U, and ·^{U^{w}} to denote the action on U^{w}, we have m ·^{U^{w}} u := w(m) ·^{U} u for all m ∈ M and u ∈ U.

If χ_{1}, χ_{2} : ℚ_{p}^{×} → A^{×} are two smooth characters, we write χ_{1} ⊗ χ_{2} to denote the character of M defined by (χ_{1} ⊗ χ_{2}) ( \begin{matrix}a & 0 \\ 0 & d \end{matrix}) := χ_{1}(a)χ_{2}(d). If α^{u} : M → A^{×} denotes the character defined in Subsection 3.6, then α^{u} = ε ⊗ ε^{−1}, where ε denotes the cyclotomic character (thought of as taking values in A): concretely,

ε : ℚ_{p}^{×} = p^{ℤ} × ℤ_{p}^{×} → ℤ_{p}^{×} → A^{×},

where the first arrow is given by projection onto the second factor, and the second arrow arises from the fact that A is a ℤ_{p}-algebra.

4.2.5. Proposition. If U is a locally admissible smooth M-representation over A, then there are isomorphisms:

(4.2.6)  Ord_{P} (U ⊗ C_{c}^{sm}(N, A)) ⟶^{∼} U,

(4.2.7)  Ord_{P} (U) = 0,

(4.2.8)  H^{1}(N_{0}, U ⊗ C_{c}^{sm}(N, A)) = 0,

and

(4.2.9)  H^{1}(N_{0}, U) ⟶^{∼} (α^{u})^{−1} ⊗ U,

Proof. The isomorphism (4.2.6) is constructed in [10, Prop. 4.2.7]. The isomorphism (4.2.7) follows from the fact that if m ∈ M^{+} is chosen so that mN_{0}m^{−1} is strictly contained in N_{0}, then the operator h_{N_{0},m} is locally nilpotent on U. The isomorphism (4.2.8) follows from the isomorphism of N_{0}-representations

C_{c}^{sm}(N, A) ⟶^{∼} ⨁_{n∈N/N_{0}} C^{sm}(N_{0}, A),

while the isomorphism (4.2.9) follows from (3.6.5). □

4.2.10. Corollary. For any locally admissible smooth representation U of M over A, there are isomorphisms Ord_{P} (IndG ; \overline{P} U) ⟶^{∼} U and H^{1}Ord_{P} (IndG ; \overline{P} U) ⟶^{∼} (α^{u})^{−1} ⊗ U^{w}.

Proof. Let G = \overline{P} N ∐ \overline{P} w denote the Bruhat decomposition of G. (Here w denotes some representative of the non-trivial element of the Weyl group of M in G, e.g. one can take w = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ).) Evaluation of elements of IndG ; \overline{P} U at w yields a P -equivariant surjection

IndG ; \overline{P} U → U^{w},

where P acts on U^{w} via the natural quotient map P → M. The kernel of this surjection is precisely (IndG ; \overline{P} U)(N), the P -invariant submodule of IndG ; \overline{P} U consisting of functions whose support lies in \overline{P} N. As noted in [10, §4.1], there is a P -equivariant isomorphism

C_{c}^{sm}(N, A) ⊗_{A} U ⟶^{∼} (IndG ; \overline{P} U)(N).

Altogether, we find that as a P -representation, IndG ; \overline{P} U sits in a short exact sequence

(4.2.11)  0 → C_{c}^{sm}(N, A) ⊗_{A} U → IndG ; \overline{P} U → U^{w} → 0.

Applying Ord_{P}, and taking into account its left exactness, together with the isomorphisms (4.2.6) and (4.2.7), we find that

Ord_{P} (IndG ; \overline{P} U) ⟶^{∼} U,

while forming the long exact sequence of continuous N_{0}-cohomology, and taking into account the isomorphisms (4.2.8) and (4.2.9), we find that H^{1}(N_{0}, IndG ; \overline{P} U) ⟶^{∼} (α^{u})^{−1} ⊗ U^{w}, and thus also that

H^{1}Ord_{P} (IndG ; \overline{P} U) ⟶^{∼} (α^{u})^{−1} ⊗ U^{w}.

This proves the corollary. □

4.2.12. Theorem. Let V be an absolutely irreducible admissible representation of G over k. (Recall that these are classified by Proposition 4.1.10.)

(1) If V ⟶^{∼} IndG ; \overline{P} χ_{1} ⊗ χ_{2}, for a pair of smooth characters χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×},

then Ord_{P} (V) ⟶^{∼} χ_{1} ⊗ χ_{2}, while H^{1}Ord_{P} (V) ⟶^{∼} χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε}.

(2) If V ⟶^{∼} (χ ∘ det) ⊗ St, for some smooth character χ : ℚ_{p}^{×} → k^{×}, then Ord_{P} (V) ⟶^{∼} χ ⊗ χ, while H^{1}Ord_{P} (V) = 0.

(3) If V ⟶^{∼} χ ∘ det for some smooth character χ : ℚ_{p}^{×} → k^{×}, then Ord_{P} (V) = 0, while H^{1}Ord_{P} (V) = χ ⊗ χ.

(4) If V is a supersingular absolutely irreducible admissible smooth representation of G, then Ord_{P} (V) = H^{1}Ord_{P} (V) = 0.

Proof. Claim (1) is a special case of Corollary 4.2.10. By definition, the representation (χ ∘ det) ⊗ St sits in the short exact sequence

0 → χ ∘ det → IndG ; \overline{P} χ ⊗ χ → (χ ∘ det) ⊗ St → 0.

Comparing this with the short exact sequence (4.2.11) (in the case when U = χ⊗χ), we find that St ⟶^{∼} (χ ⊗ χ) ⊗_{A} C_{c}^{sm}(N, k) as P -representations. Thus claim (2) follows from Proposition 4.2.5. As a P -representation, the representation χ ∘ det is isomorphic to the M-representation χ ⊗ χ. Thus claim (3) also follows from Proposition 4.2.5. Claim (4) follows from Proposition 3.6.2, and the adjointness properties of V ↦ Ord_{P} (V) and V ↦ V_{N}. (Recall from Proposition 4.1.10 (4) that if V is supersingular, then it is neither a subrepresentation of nor a quotient of a representation induced from P or \overline{P}.) □

4.2.13. Remark. If V is supersingular, then, using the description of V in terms of (φ, Γ)-modules provided by [2, Thm. 2.2.1], one can show that in fact H^{1}(N_{0}, V) (which up to a twist is isomorphic to the space of N_{0}-coinvariants V_{N_{0}}) vanishes for any compact open subgroup N_{0} of N.

Thus we see that for any absolutely irreducible object V of Mod_{G}^{adm}(A) the natural surjection V_{N_{0}} → V_{N} is in fact an isomorphism. It then follows that this surjection is an isomorphism for any object V of Mod_{G}^{l.adm}(A). Indeed, beginning with the absolutely irreducible case, one uses Lemma 4.1.9 to extend to the irreducible case, and then proceeds by induction on length to prove it for any finite length object. Since any object Mod_{G}^{l.adm}(A) is the inductive limit of finitely generated admissible – and hence finite length [10, Thm. 2.3.8] – objects, and since the formation of coinvariants commutes with the formation of inductive limits, we conclude that it holds for arbitrary such V. We don’t know whether or not the analogous result holds for other groups G.

We conclude this subsection by presenting an application of the preceding computations.

4.2.14. Proposition. Let V be an object of Mod_{G}^{adm}(A) which is faithful as an A-module, and for which k ⊗_{A} V ⟶^{∼} IndG ; \overline{P} \overline{χ}_{1} ⊗ \overline{χ}_{2}, where \overline{χ}_{1}, \overline{χ}_{2} : ℚ_{p}^{×} → k^{×} are smooth characters for which \overline{χ}_{1}\overline{χ}_{2}^{−1} ≠ \underline{1}, \overline{ε}. Then V ⟶^{∼} IndG ; \overline{P} χ_{1} ⊗ χ_{2}, where χ_{1} and χ_{2} are lifts of \overline{χ}_{1} and \overline{χ}_{2} to smooth characters ℚ_{p}^{×} → A^{×}.

Proof. We prove this by induction on the length of A. Let 𝔪 denote the maximal ideal of A, let A[𝔪] denote the ideal in A consisting of elements annihilated by 𝔪 (note that is it non-zero, since A is Artinian), and let I ⊂ A be the ideal generated by some non-zero element a ∈ A[𝔪]. By our inductive hypothesis, there is an isomorphism V/IV ⟶^{∼} IndG ; \overline{P} χ′_{1} ⊗ χ′_{2}, for some lifts χ′_{1}, χ′_{2} of \overline{χ}_{1} and \overline{χ}_{2} to (A/I)^{×}-valued characters. Since V is a faithful A-module, IV ≠ 0, and thus IV ⟶^{∼} IndG ; \overline{P} \overline{χ}_{1} ⊗ \overline{χ}_{2}. (The latter representation is irreducible, since \overline{χ}_{1} ≠ \overline{χ}_{2} by assumption.) Thus V sits in the short exact sequence

(4.2.15)  0 → IndG ; \overline{P} \overline{χ}_{1} ⊗ \overline{χ}_{2} → V → IndG ; \overline{P} χ′_{1} ⊗ χ′_{2} → 0.

Applying H^{•}Ord_{P} to (4.2.15), and taking into account Corollary 4.2.10, we obtain the exact sequence

(4.2.16)  0 → \overline{χ}_{1} ⊗ \overline{χ}_{2} → Ord_{P} (V) → χ′_{1} ⊗ χ′_{2} → χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε}^{−1}.

Since χ_{1} ≠ χ_{2}\overline{ε}^{−1} by assumption, we find that the final arrow in (4.2.16) vanishes, and thus that Ord_{P} (V) is an extension of χ′_{1} ⊗ χ′_{2} by \overline{χ}_{1} ⊗ \overline{χ}_{2}. Now consider the commutative diagram

[figure: 0; IndG ; \overline{P} \overline{χ}_{1} ⊗ \overline{χ}_{2}; IndG ; \overline{P} Ord_{P} (V); IndG ; \overline{P} \overline{χ}_{1} ⊗ \overline{χ}_{2}; 0; 0; IndG ; \overline{P} \overline{χ}_{1} ⊗ \overline{χ}_{2}; V; IndG ; \overline{P} χ′_{1} ⊗ χ′_{2}; 0,]

induced by the adjointness between Ord_{P} (V) and Ind\overline{P}. Since the outer two vertical arrows are isomorphisms, so is the middle isomorphism. We conclude that Ord_{P} (V) is given by an A^{×}-valued character χ_{1} ⊗ χ_{2}, and that IndG ; \overline{P} χ_{1} ⊗ χ_{2} ⟶^{∼} V. □

4.2.17. Remark. The preceding result shows that if \overline{χ}_{1}, \overline{χ}_{2} : ℚ_{p}^{×} → k^{×} are smooth characters for which \overline{χ}_{1}\overline{χ}_{2}^{−1} ≠ \underline{1}, \overline{ε}^{−1}, then the map from the deformation functor of \overline{χ}_{1}⊗\overline{χ}_{2} to the deformation functor of IndG ; \overline{P} \overline{χ}_{1}⊗\overline{χ}_{2} induced by IndG ; \overline{P} is an equivalence.

4.3. Extension computations. In this subsection we let k denote a finite field of characteristic p. Our goal is to compute the spaces Ext_{G}^{1}(V_{1}, V_{2}), where V_{1} and V_{2} are absolutely irreducible objects of Mod_{G}^{adm}(k). We do this for every possible choice of V_{1} and V_{2} other than for the cases when V_{1} and V_{2} are both supersingular, when V_{1} is a character and V_{2} is a twist of Steinberg or supersingular (although we do treat the extensions of a character by Steinberg in one particular case, originally considered by Colmez – see the second isomorphism of claim (2) of Proposition 4.3.22), and when V_{1} is a supersingular and V_{2} is a twist of Steinberg. We also omit certain cases when V_{1} is a twist of Steinberg and V_{2} is a character when p = 2 or 3. Breuil and Paškūnas, and Colmez, have each found a different approach to many of the Ext calculations that we make; see Remark 4.3.35 below.

Our basic tool will be the exact sequence (3.7.5), together with the computations of the preceding subsection, and of course Corollary 3.12 of [11], verifying that Conjecture 3.7.2 holds in our context, which we will apply without further mention. We also require some additional technical computations which we have relegated to an appendix.^{1} We will also exploit the short exact sequence

(4.3.1)  0 → χ ∘ det → IndG ; \overline{P} χ ⊗ χ → (χ ∘ det) ⊗ St → 0,

which gives rise to an exact sequence

(4.3.2) Hom_{G}(χ ∘ det, V) → Ext_{G}^{1}((χ ∘ det) ⊗ St, V)

→ Ext_{G}^{1}(IndG ; \overline{P} χ ⊗ χ, V) → Ext_{G}^{1}(χ ∘ det, V),

for any object V of Mod_{G}^{adm}(k). Finally, we will utilize the following lemmas:

4.3.3. Lemma. If V is an object of Mod_{G}^{sm}(k), and W is an object of Modsm ; \overline{P}(k), then there are natural isomorphisms

Ext• ; \overline{P} (V, W) ⟶^{∼} Ext_{G}^{•}(V, IndG ; \overline{P} W).

(Here the Ext spaces are computed in Modsm ; \overline{P}(k) and Mod_{G}^{sm}(k), respectively.)

Proof. If W ↪ I^{•} is an injective resolution in Modsm ; \overline{P}(k), then IndG ; \overline{P} W ↪ IndG ; \overline{P} I^{•} is an injective resolution in Mod_{G}^{sm}(k). (Indeed, the functor IndG ; \overline{P} is right adjoint to the exact forgetful functor Mod_{G}^{sm}(k) → Modsm ; \overline{P}(k), and hence preserves injectives, and is furthermore exact [10, Prop. 4.1.5].) The adjunction property of IndG ; \overline{P} gives an isomorphism Hom_{P} (V, I^{•}) ⟶^{∼} Hom_{G}(V, IndG ; \overline{P} I^{•}). Passing to cohomology, and taking into account the preceding discussion, we obtain the isomorphism in the statement of the lemma. □

4.3.4. Lemma. If V_{1} and V_{2} are smooth P -representation over k, with V_{1} being finite-dimensional over k, then restricting from P to M induces an isomorphism Hom_{P} (V_{1}, V_{2}) ⟶^{∼} Hom_{M}(V_{1}, V_{2}), and consequently an embedding Ext_{P}^{1} (V_{1}, V_{2}) ↪ Ext_{M}^{1}(V_{1}, V_{2}). (Here the Ext^{1} spaces are computed in the categories of smooth representations over k.)

1In an earlier version of this paper, these computations were quite extensive, and were largely necessitated by the fact that we didn’t know Conjecture 3.7.2, and so were forced to rely on the weaker exact sequence (3.7.6), rather than (3.7.5). With the results of [11] available, this complication has been obviated, and the computations of the appendix have become correspondingly shorter and comparatively less technical.

Proof. Let ϕ ∈ Hom_{M}(V_{1}, V_{2}). Since V_{1} and ϕ(V_{1}) are both finite-dimensional smooth representations of P, we may find a compact open subgroup N_{1} of N which acts trivially on each of V_{1} and ϕ(V_{1}). The map ϕ is then both M-equivariant and N_{1}-equivariant. Since M and N_{1} together generate P, the map ϕ is in fact P - equivariant. This proves the first statement of the lemma. The claim regarding Ext spaces follows, either by a general argument with δ-functors, or more directly by noting that any M-equivariant section to an extension of V_{1} by V_{2} is then necessarily P -equivariant. □

4.3.5. Lemma. If V_{1} and V_{2} are two objects of Mod_{G}^{sm}(k), and if V_{1} is finitedimensional over k, then the restriction map Hom_{G}(V_{1}, V_{2}) → Hom_{P} (V_{1}, V_{2}) is an isomorphism, and consequently, the restriction map Ext_{G}^{1}(V_{1}, V_{2}) → Ext_{P}^{1} (V_{1}, V_{2}) (in which the Hom and Ext spaces are computed in the categories Mod_{G}^{sm}(k) and Mod_{P}^{sm}(k) respectively) is injective.

Proof. This can be proved in a similar manner to the preceding lemma. (In particular, the claim regarding Ext^{1} spaces follows formally from the claim regarding Hom spaces.) Namely, let ϕ ∈ Hom_{P} (V_{1}, V_{2}). Since V_{1} is finite-dimensional over k, so is ϕ(V_{1}), and so there is some open subgroup G_{0} of G which fixes both V_{1} and ϕ(V_{1}) elementwise. The map ϕ is then both G_{0} and P -equivariant. Since G_{0} and P generate G as a group, we see that ϕ is in fact G-equivariant.

Alternatively, one can argue directly from (an obvious variant, with P replaced by \overline{P}, of) the preceding lemma: if ϕ is P -equivariant, then it is in particular Mequivariant, and hence also \overline{P} -equivariant. Since P and \overline{P} generated G, we see that ϕ is G-equivariant, as claimed. □

4.3.6. Remark. The main theorem of [14] shows that the analogue of the preceding result holds for any absolutely irreducible object V_{1} of Mod_{G}^{adm}(k) which is not a twist of St.

We also have the following variant of Lemma 4.3.4.

4.3.7. Lemma. If W_{1} and W_{2} are two objects of Mod_{M}^{sm}(k) that are finite dimensional over k, then restriction from P to M induces isomorphisms

Ext_{P}^{i} (W_{1}, W_{2}) ⟶^{∼} Ext_{M}^{i}(W_{1}, W_{2}).

for 0 ≤ i ≤ 2. (In the source, the Ext space is computed in the category Mod_{P}^{sm}(k), and W_{1} and W_{2} are regarded as P -representations via the surjection P → M, while in the target, the Ext space is computed in Mod_{M}^{sm}(k).)

Proof. When i = 0 this is a special case of Lemma 4.3.4. To treat the case when i = 1, note that Lemma 4.3.8 below shows that N acts trivially on W_{1} and W_{2}, and on any extension of W_{1} by W_{2} in the category Mod_{P}^{sm}(k). Thus computing extensions of W_{1} by W_{2} as P -representations is the same as computing extensions of W_{1} by W_{2} as M = P/N-representations.

The argument in the case when i = 2 is slightly more involved. (We also note that this case is required only in Section A.1 of the appendix.) We begin by showing that if V is a finite-dimensional representation of M over k, regarded as a P -representation by having N act trivially, then the inflation map H^{i}(M, V) → H^{i}(P, V) is an isomorphism. Indeed, by the Hochschild–Serre spectral sequence for continuous cohomology, it suffices to verify that H^{i}(N, V) = 0 for i > 0 when V is regarded as a trivial N-representation.

Of course there is an isomorphism ℚ_{p} ⟶^{∼} N; for each j ≥ 0, write N^{j} to denote the image of p^{−j}ℤ_{p} under this isomorphism. We then have that N = ⋃ N^{j}, and for each i ≥ 1 there is a short exact sequence

0 → R^{1} lim_{\overleftarrow{j}} H^{i−1}(N_{j}, V) → H^{i}(N, V) → lim_{\overleftarrow{j}} H^{i}(N_{j}, V) → 0.

Since each H^{i−1}(N_{j}, V) is finite-dimensional (as V is), the projective system of H^{i−1}(N_{j}, V) satisfies the Mittag–Leffler condition, and so R^{1} lim_{\overleftarrow{j}} H^{i−1}(N_{j}, V) = 0.

Thus in fact H^{i}(N, V) ⟶^{∼} lim_{\overleftarrow{j}} H^{i}(N_{j}, V). Now H^{i}(N_{j}, V) = 0 if i > 1, and thus

H^{i}(N, V) = 0 for such i. On the other hand, H^{1}(N_{j}, V) = Hom_{cont}(N_{j}, V) ⟶^{∼} V (since N, and so N_{j}, acts trivially on V). As [N^{j+1} : N^{j}] = p, we see that the transition map H^{1}(N_{j+1}, V) → H^{1}(N_{j}, V) vanishes (since V is a k-vector space, and k has characteristic p); thus in fact H^{1}(N, V) = 0 as well.

We now return to proving that the map Ext_{P}^{2} (W_{1}, W_{2}) → Ext_{M}^{2}(W_{1}, W_{2}) is an isomorphism. Since W_{1} is finite-dimensional, we see that if W_{1}^{∨} denotes the contragredient representation to W_{1}, then there is a commutative diagram

\xymatrix{Ext_{P}^{1} (W_{1}, W_{2}) \ar[d] \ar[r] & Ext_{M}^{1}(W_{1}, W_{2}) \ar[d] \\ Ext_{P}^{1} (\underline{1} ⊗ \underline{1}, W_{1}^{∨} ⊗ W_{2}) \ar[r] & Ext_{M}^{1}(\underline{1} ⊗ \underline{1}, W_{1}^{∨} ⊗ W_{2}).}

Thus it suffices to consider the case when W_{1} is the trivial representation. Now consider the diagram

\xymatrix{Ext_{M}^{2}(\underline{1}, W_{2}) \ar[d] \ar[r] & Ext_{P}^{2} (\underline{1}, W_{2}) \ar[d] \ar[r] & Ext_{M}^{2}(\underline{1}, W_{2}) \ar[d] \\ H^{2}(M, W_{2}) \ar[r] & H^{2}(P, W_{2}) \ar[r] & H^{2}(M, W_{2}),}

where the first horizontal arrows are given by inflation from M to P, the second horizontal arrows are given by restriction from P to M, and the vertical arrows are given by (2.2.1). Composing either the upper or lower horizontal arrows yields the identity, and we showed above that the first of the lower horizontal arrows is an isomorphism. Thus both lower horizontal arrows are isomorphisms. On the other hand, it follows from Corollary 2.2.4 that the vertical arrows are injective. From these facts we conclude that the upper horizontal arrows are injective; since they compose to give the identity, they must be isomorphisms. This completes the proof of the lemma for i = 2. □

The following lemma is proved in [6, Lem. III.1.3]. For the reader’s convenience, we give a proof here.

4.3.8. Lemma. Any smooth N-action on a finite-dimensional k-vector space is trivial.

Proof. Suppose that W is finite-dimensional over k and equipped with a smooth N-action. Write N = ⋃ N^{i}, where each N^{i} is a compact open subgroup. Since N^{i} is in particular pro-p, we see that W^{N^{i}} ≠ 0. Since W is finite-dimensional, we find that W^{N} = ⋂ W^{N^{i}} ≠ 0. (A nested intersection of non-zero subspaces of a finite-dimensional space is non-zero.) Thus, proceeding by induction, we find that the Jordan–Hölder factors of W, as N-representation, are copies of the trivial representation. The lemma then follows from the fact that

Ext_{N}^{1}(\underline{1}, \underline{1}) ⟶^{∼} H^{1}(N, \underline{1}) = Hom_{cont}(N, k) = 0.

(Here the isomorphism is provided by Proposition 2.2.2, the first equality is the standard description of H^{1} with coefficients in the trivial representations as a space of homomorphisms, and the second equality holds since N is a divisible group, while k is of characteristic p.) □

We also have the following closely related lemma.

4.3.9. Lemma. Any smooth SL_{2}(ℚ_{p})-action on a finite-dimensional k-vector space is trivial.

Proof. If V is a finite-dimensional k-vector space equipped with a smooth SL_{2}(ℚ_{p})- action, then there exists an open subgroup H of SL_{2}(ℚ_{p}) which acts trivially on V. Since by the preceding lemma N also acts trivially on V, and N and H together generate SL_{2}(ℚ_{p}), the present lemma follows. (Alternatively, one can combine the preceding lemma with its obvious analogoue for \overline{N} to conclude that both N and \overline{N} act trivially on V, and then note that N and \overline{N} together generate SL_{2}(ℚ_{p}).) □

4.3.10. Lemma. If α, α′, β, β′ : ℚ_{p}^{×} → k^{×} are smooth characters, then Ext_{M}^{i}(α ⊗ β, α′ ⊗β′) = 0 for all i ≥ 0, unless α = α′ and β = β′. (Here the Ext^{i} are computed in the category Mod_{M}^{sm}(k).)

Proof. Since M is commutative, for any two smooth M-representations U and V, there is a natural M-action on the space Hom_{M}(U, V) (induced by the M-action on either U or V), and hence (by passing to derived functors) a natural M-action on Ext_{M}^{i}(U, V), for each i ≥ 0. If now we take U = α ⊗ β and V = α′ ⊗ β′, we see that this M-action is simultaneously given by the character α ⊗ β and by the character α′ ⊗β′. Thus, unless these characters coincide, the space Ext_{M}^{i}(α ⊗β, α′ ⊗β′) must vanish. □

4.3.11. Remark. We also remark that if α, β : ℚ_{p}^{×} → k^{×} are smooth characters, so that α ⊗ β is a smooth character of M, then

Ext_{M}^{1}(α ⊗ β, α ⊗ β) ⟶^{∼} Ext_{M}^{1}(\underline{1}, \underline{1}) ⟶^{∼} H^{1}(M, \underline{1}) ⟶^{∼} Hom_{cont}(M, \underline{1}),

by Proposition 2.2.2.

We are now ready to begin our calculation of Ext spaces for G-representations. We present the calculations in a series of propositions. We begin with the case of extensions by a principal series.

4.3.12. Proposition. Let χ : ℚ_{p}^{×} → k^{×} be a smooth character.

(1) If χ_{1} ⊗ χ_{2} ≠ χ ⊗ χ, then Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, (χ ∘ det) ⊗ St) = 0.

(2) There is a natural isomorphism

Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ) ⟶^{∼} Ext_{G}^{1}(IndG ; \overline{P} χ ⊗ χ, (χ ∘ det) ⊗ St).

Proof. This is a direct application of (3.7.5), setting U = χ_{1} ⊗ χ_{2} and V = (χ ∘ det) ⊗ St, and taking into account part (2) of Theorem 4.2.12. □

4.3.13. Proposition. Let χ : ℚ_{p}^{×} → k^{×} be a smooth character.

(1) If χ_{1} ⊗ χ_{2} ≠ χ\overline{ε}^{−1} ⊗ χ\overline{ε}, then Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, (χ ∘ det)) = 0.

(2) The space Ext_{G}^{1}(IndG ; \overline{P} χ\overline{ε}^{−1} ⊗ χ\overline{ε}, (χ ∘ det)) is one-dimensional.

Proof. These claims follow directly from (3.7.5), together with part (3) of Theorem 4.2.12. □

4.3.14. Proposition. The space Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, V) vanishes for any pair of smooth characters χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×}, and any supersingular absolutely irreducible admissible smooth representation V of G over k.

Proof. This follows directly from (3.7.5) (taking U = χ_{1}⊗χ_{2}), together with part (4) of Theorem 4.2.12. □

4.3.15. Proposition. Let χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×} be a pair of smooth characters. (1) If χ′_{1}, χ′_{2} : ℚ_{p}^{×} → k^{×} is a second pair of smooth characters, then

Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, IndG ; \overline{P} χ′_{1} ⊗ χ′_{2})

vanishes unless either χ_{1} = χ′_{1} and χ_{2} = χ′_{2}, or else χ_{1} = χ′_{2}\overline{ε}^{−1} and

χ_{2} = χ′_{1}\overline{ε}.
(2) If χ_{1} ≠ χ_{2}\overline{ε}^{−1}, then

Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, IndG ; \overline{P} χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε})

is one-dimensional over k.

(3) If either χ_{1} ≠ χ_{2}\overline{ε}^{−1} or p ≠ 2, then the natural map

Ext_{M}^{1}(χ_{1} ⊗ χ_{2}, χ_{1} ⊗ χ_{2}) → Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, IndG ; \overline{P} χ_{1} ⊗ χ_{2})

given by applying IndG ; \overline{P} is an isomorphism.

(4) If p = 2 and χ_{1} = χ_{2} (= χ_{2}\overline{ε}^{−1}), then the natural map

Ext_{M}^{1}(χ_{1} ⊗ χ_{2}, χ_{1} ⊗ χ_{2}) → Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, IndG ; \overline{P} χ_{1} ⊗ χ_{2})

given by applying IndG ; \overline{P} is injective, with a one-dimensional cokernel.

Proof. We apply the exact sequence (3.7.5), taking U to be χ_{1} ⊗ χ_{2}, and V to be IndG ; \overline{P} χ′_{1} ⊗χ′_{2}. Taking into account Theorem 4.2.12 (1), we obtain an exact sequence

(4.3.16) 0 → Ext_{M}^{1}(χ_{1} ⊗ χ_{2}, χ′_{1} ⊗ χ′_{2}) → Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, IndG ; \overline{P} χ′_{1} ⊗ χ′_{2})

→ Hom_{M}(χ_{1} ⊗ χ_{2}, χ′_{2}\overline{ε}^{−1} ⊗ χ′_{1}\overline{ε}) → Ext_{M}^{2}(χ_{1} ⊗ χ_{2}, χ′_{1} ⊗ χ′_{2})

→ Ext_{G}^{2}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, IndG ; \overline{P} χ′_{1} ⊗ χ′_{2}).

Claim (1) is now seen to follow immediately from the exactness of (4.3.16) together with Lemma 4.3.10.

If we set χ′_{1} = χ_{2}\overline{ε}^{−1} and χ′_{2} = χ_{1}\overline{ε}, and assume that χ′_{1} ≠ χ_{1} (or equivalently, that χ′_{2} ≠ χ_{2}), then the exact sequence (4.3.16), together with Lemma 4.3.10, gives rise to an isomorphism

Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, IndG ; \overline{P} χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε}) ⟶^{∼} Hom_{M}(χ_{1} ⊗ χ_{2}, χ_{1} ⊗ χ_{2}),

whose target is one-dimensional. This establishes claim (2).

If we set χ′_{1} = χ_{1} and χ′_{2} = χ_{2}, then the exact sequence (4.3.16) includes the subsequence

0 → Ext_{M}^{1}(χ_{1} ⊗ χ_{2}, χ_{1} ⊗ χ_{2}) → Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, IndG ; \overline{P} χ_{1} ⊗ χ_{2})

→ Hom_{M}(χ_{1} ⊗ χ_{2}, χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε}).

If in addition χ_{1} ≠ χ_{2}\overline{ε}^{−1} (or equivalently, χ_{2} ≠ χ_{1}\overline{ε}), we find that the last term in this exact sequence vanishes, and so claim (3) follows in this case. To complete the proof of (3), suppose that p ≠ 2 and that χ_{1} = χ_{2}\overline{ε}^{−1}. In this case the exact sequence (4.3.16) takes the form

(4.3.17) 0 → Ext_{M}^{1}(χ ⊗ χ\overline{ε}, χ ⊗ χ\overline{ε}) → Ext_{G}^{1}(IndG ; \overline{P} χ ⊗ χ\overline{ε}, IndG ; \overline{P} χ ⊗ χ\overline{ε})

→ Hom_{M}(χ ⊗ χ\overline{ε}, R^{1}Ord_{P} (IndG ; \overline{P} χ ⊗ χ\overline{ε})) → Ext_{M}^{2}(χ ⊗ χ\overline{ε}, χ ⊗ χ\overline{ε})

→ Ext_{G}^{2}(IndG ; \overline{P} χ ⊗ χ\overline{ε}, IndG ; \overline{P} χ ⊗ χ\overline{ε}).

We show in Subsection A.1 of the appendix that the final arrow in this exact sequence is not injective. On the other hand, we have shown in Theorem 4.2.12 that R^{1}Ord_{P} (IndG ; \overline{P} χ ⊗ χ\overline{ε}) is one-dimensional, and so we conclude that the first non-trivial arrow in (4.3.17) is an isomorphism, as required.

We now turn to claim (4). Thus we assume that p = 2, and that χ_{1} = χ_{2} = χ. To simplify the notation, we apply, as we may, a twist by χ^{−1} ∘ det, and thus in fact assume that χ = \underline{1}. The short exact sequence (4.3.1) gives rise to a long exact sequence

0 → Hom_{G}(IndG ; \overline{P} \underline{1}⊗\underline{1}, \underline{1}) → Hom_{G}(IndG ; \overline{P} \underline{1}⊗\underline{1}, IndG ; \overline{P} \underline{1}⊗\underline{1}) → Hom_{G}(IndG ; \overline{P} \underline{1}⊗\underline{1}, St)

→ Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, \underline{1}) → Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, IndG ; \overline{P} \underline{1} ⊗ \underline{1}) → Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, St).

Since (4.3.1) is non-split, this long exact sequence induces an exact sequence

(4.3.18) 0 → Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, \underline{1}) → Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, IndG ; \overline{P} \underline{1} ⊗ \underline{1})

→ Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, St).

We claim that the last arrow is surjective; indeed, it is easily checked that the composite

Ext_{M}^{1}(\underline{1} ⊗ \underline{1}, \underline{1} ⊗ \underline{1}) → Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, IndG ; \overline{P} \underline{1} ⊗ \underline{1}) → Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, St),

with the first arrow being induced by the functor IndG ; \overline{P}, coincides with the isomorphism of Proposition 4.3.12 (2). Thus not only have we shown that (4.3.18) extends to a short exact sequence, but in fact we have provided a splitting, and thus have established an isomorphism

Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, \underline{1}) ⊕ Ext_{M}^{1}(\underline{1} ⊗ \underline{1}, \underline{1} ⊗ \underline{1}) ⟶^{∼} Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \underline{1}, IndG ; \overline{P} \underline{1} ⊗ \underline{1}).

Since Proposition 4.3.13 (2) shows that the first summand is one-dimensional (recall that p = 2, so that \overline{ε} = \underline{1}), claim (4) follows. □

We now consider the case of extensions by Steinberg and by characters.

4.3.19. Proposition. Let χ, χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×} be smooth characters.

(1) If χ_{1} ⊗ χ_{2} ≠ χ ⊗ χ or χ\overline{ε}^{−1} ⊗ χ\overline{ε}, then Ext_{G}^{1}((χ ∘ det) ⊗ St, IndG ; \overline{P} χ_{1} ⊗ χ_{2})

vanishes.

(2) If p ≠ 2, then Ext_{G}^{1}((χ ∘ det) ⊗ St, IndG ; \overline{P} χ\overline{ε}^{−1} ⊗ χ\overline{ε}) is one-dimensional.

Proof. Claim (1) follows directly from a consideration of (4.3.2), together with Proposition 4.3.15. Claim (2) follows in a similar manner, since

Ext_{G}^{1}(χ ∘ det, IndG ; \overline{P} χ\overline{ε}^{−1} ⊗ χ\overline{ε}) ⟶^{∼} Ext_{M}^{1}(χ ⊗ χ, χ\overline{ε} ⊗ χ\overline{ε}^{−1})

by Lemma 4.3.3 and (the analogue for \overline{P} of) Lemma 4.3.7, and the latter Ext space vanishes, by Lemma 4.3.10, since χ ⊗ χ and χ\overline{ε} ⊗ χ\overline{ε}^{−1} are distinct characters of M (as p ≠ 2). □

Recall that in Subsection 4.2 we defined the action of the Weyl group element w on the category of smooth M-representations. Clearly, if χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×} are smooth characters, then (χ_{1} ⊗ χ_{2})^{w} = χ_{2} ⊗ χ_{1}. In particular, if χ : ℚ_{p}^{×} → k^{×} is a smooth character, then (χ ⊗ χ)^{w} = χ ⊗ χ, and hence the action of w on Mrepresentations induces an involution of Ext_{M}^{1}(χ⊗χ, χ⊗χ). We let Ext_{M}^{1}(χ⊗χ, χ⊗ χ)^{w} (resp. Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w=−1}) denote the fixed space under this involution (resp. the anti-fixed space of the involution w).

4.3.20. Lemma. (1) The subspace of Ext_{M}^{1}(χ⊗χ, χ⊗χ)^{w} of Ext_{M}^{1}(χ⊗χ, χ⊗χ)

consists of extensions which are trivial when restricted to M ⋂ SL_{2}(ℚ_{p}).

(2) The subspace Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w=−1} of Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ) consists of extensions which are trivial when restricted to the centre Z_{G} of G.

Proof. Under the composition of the isomorphisms

Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ) ⟶^{∼} Ext_{M}^{1}(\underline{1}, \underline{1}) ⟶^{∼} Hom_{cont}(M, \underline{1})

discussed in Remark 4.3.11, the action of w on the source corresponds to the waction ϕ ↦ (ϕ^{w} : m ↦ ϕ(w(m))) on the target. Also, an element of the source is trivial when restricted to M ⋂ SL_{2}(ℚ_{p}) (resp. Z_{G}) precisely when the corresponding homomorphism ϕ is trivial on M ⋂ SL_{2}(ℚ_{p}) (resp. Z_{G}). An elementary computation shows that this latter condition holds if and only if ϕ^{w} = ϕ (resp. ϕ^{w} = −ϕ), and so the lemma follows. □

We note that Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w=−1} is linearly independent of (resp. coincides with) Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w} when p ≠ 2 (resp. p = 2).

4.3.21. Proposition. Let χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×} be smooth characters.

(1) If χ_{1} ≠ χ_{2}, then Ext_{G}^{1}(χ_{1} ∘ det, χ_{2} ∘ det) vanishes.

(2) If χ_{1} = χ_{2} = χ (say), then restricting to M induces an isomorphism

Ext_{G}^{1}(χ ∘ det, χ ∘ det) ⟶^{∼} Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w}.

Proof. Note that Lemma 4.3.9 shows that any extension of χ_{1} ∘ det by χ_{2} ∘ det is trivial on SL_{2}(ℚ_{p}), and so factors through det. Thus the natural map

Ext_{ℚ_{p}^{×}}^{1}(χ_{1}, χ_{2}) → Ext_{G}^{1}(χ_{1} ∘ det, χ_{2} ∘ det),

given by pulling back ℚ_{p}^{×} representations along det : G → ℚ_{p}^{×}, is an isomorphism. Now det also induces an isomorphism M/(M ⋂ SL_{2}(ℚ_{p})) ⟶^{∼} ℚ_{p}^{×}, and so we see that restriction to M induces an isomorphism between Ext_{G}^{1}(χ_{1} ∘ det, χ_{2} ∘ det) and the subspace of extensions of Ext_{M}^{1}(χ_{1} ⊗ χ_{1}, χ_{2} ⊗ χ_{2}) which are trivial when restricted to M ⋂ SL_{2}(ℚ_{p}). Claim (1) now follows from Lemma 4.3.10 (which shows that Ext_{M}^{1}(χ_{1} ⊗ χ_{1}, χ_{2} ⊗ χ_{2}) = 0 when χ_{1} ≠ χ_{2}), while claim (2) follows from Lemma 4.3.20 (1). □

4.3.22. Proposition. Let χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×} be smooth characters.

(1) If χ_{1} ≠ χ_{2}, then Ext_{G}^{1}((χ_{1} ∘ det) ⊗ St, (χ_{2} ∘ det) ⊗ St) vanishes.

(2) If χ_{1} = χ_{2} = χ (say), then there are natural isomorphisms

(4.3.23)  Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w} ⟶^{∼} Ext_{G}^{1}((χ ∘ det) ⊗ St, (χ ∘ det) ⊗ St)

and

(4.3.24)  Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w=−1} ⟶^{∼} Ext_{G}^{1}(χ ∘ det, (χ ∘ det) ⊗ St).

Proof. Claim (1) follows from (4.3.2) and the vanishing result of Proposition 4.3.12. To prove claim (2), we again consider the exact sequence (4.3.2) (taking V = (χ ∘ det) ⊗ St), and place it into the following commutative diagram, as the lefthand column:

\xymatrix{0 \ar[d] \\ Ext_{G}^{1}((χ ∘ det) ⊗ St, (χ ∘ det) ⊗ St) \ar[d] \\ Ext_{G}^{1}(IndG ; \overline{P} χ ⊗ χ, (χ ∘ det) ⊗ St) \ar[d] & Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ) \ar[l]_{∼} \ar[d] \\ Ext_{G}^{1}(χ ∘ det, (χ ∘ det) ⊗ St) \ar[r] & Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w=−1},}

in which the upper horizontal arrow is given by part (2) of Proposition 4.3.12, the lower horizontal arrow will be specified below, and the right-hand vertical arrow is defined to make the diagram commute.

Let us now specify the lower horizontal arrow. There is a P -equivariant isomorphism

St ⟶^{∼} C_{c}^{sm}(N, k) ⊗_{k} (χ ⊗ χ).

Evaluation at the identity of N induces an M-equivariant surjection

C_{c}^{sm}(N, k) ⊗_{k} (χ ⊗ χ) → χ ⊗ χ.

Composing these two, we obtain an M-equivariant surjection

(χ ⊗ χ) ⊗ St → χ ⊗ χ,

which induces a map

(4.3.25)  Ext_{P}^{1} (χ ∘ det, (χ ∘ det) ⊗ St) → Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w=−1}.

(Note that it does indeed land in Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w=−1}, by Lemma 4.3.20 (2), because any P -equivariant extension of χ ∘ det by (χ ∘ det) ⊗ St must have a central character, there being no P -equivariant maps from the former representation to the latter.) We take the lower horizontal arrow to be the map

(4.3.26)  Ext_{P}^{1} (χ ∘ det, (χ ∘ det) ⊗ St) → Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w=−1}.

obtained by composing of (4.3.25) with the restriction map

(4.3.27)  Ext_{G}^{1}(χ ∘ det, (χ ∘ det) ⊗ St) → Ext_{P}^{1} (χ ∘ det, (χ ∘ det) ⊗ St).

The map (4.3.26) was first constructed by Colmez, who proved that it is an isomorphism [6, Thm. VII.4.19]. We will give another proof of this fact in the course of the present argument.

Corollary A.2.4 of the appendix shows that (4.3.25) is injective. Lemma 4.3.5 shows that (4.3.27) in also injective. Thus their composite (4.3.26) is again injective. A computation shows that (up to a sign) the right-hand vertical arrow in our diagram is given by the formula ϕ ↦ ϕ − ϕ^{w}. Thus this arrow is surjective. Since the lower horizontal arrow is injective, we discover that it is in fact an isomorphism. Thus we obtain the isomorphism (4.3.24). Also, restricting the upper horizontal arrow to Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w}, we obtain the isomorphism (4.3.23). □

4.3.28. Remark. The isomorphism (4.3.23) has a concrete interpretation. Namely, if U is an extension of χ⊗χ by itself, corresponding to an element of Ext_{M}^{1}(χ⊗χ, χ⊗ χ)^{w}, then Proposition 4.3.21 shows that U extends uniquely to a G-representation. The induction IndG ; \overline{P} U then contains a copy of U as a subrepresentation, and the corresponding quotient is an extension V of (χ∘det)⊗St by itself. This map U ↦ V gives rise to the isomorphism (4.3.23) (up to a possible sign).

4.3.29. Remark. As was observed above, Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w=−1} is linearly independent of (resp. coincides with) Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ)^{w} when p ≠ 2 (resp. p = 2). Thus if p ≠ 2, then we find that none of the non-trivial extensions of (χ ∘ det) ⊗ St by (χ∘det)⊗St constructed in the preceding proposition admit a central character, while if p = 2, then they all do.

4.3.30. Proposition. Let χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×} be smooth characters.

(1) If χ_{1} ≠ χ_{2}, and furthermore either χ_{1} ≠ χ_{2}\overline{ε} or else p ≠ 3, then Ext_{G}^{1}((χ_{1} ∘ det) ⊗ St, χ_{2} ∘ det) vanishes.

(2) If χ_{1} = χ_{2} = χ (say), and if p ≠ 2, then Ext_{G}^{1}((χ ∘ det) ⊗ St, χ ∘ det) is one-dimensional.

Proof. Both claims follow directly from a consideration of (4.3.2), together with Proposition 4.3.13.

□

4.3.31. Proposition. If χ : ℚ_{p}^{×} → k^{×} is smooth and V is a supersingular absolutely irreducible object of Mod_{G}^{adm}(k), then Ext_{G}^{1}((χ ∘ det) ⊗ St, V ) vanishes.

Proof. This follows directly from a consideration of (4.3.2), together with the vanishing result of Proposition 4.3.14 (in the case χ_{1} = χ_{2} = χ). □

4.3.32. Proposition. Let χ, χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×} be smooth characters.

(1) If χ_{1} ⊗ χ_{2} ≠ χ ⊗ χ, then Ext_{G}^{1}(χ ∘ det, IndG ; \overline{P} χ_{1} ⊗ χ_{2}) vanishes.

(2) There is a natural isomorphism

Ext_{G}^{1}(χ ∘ det, IndG ; \overline{P} χ ⊗ χ) ⟶^{∼} Ext_{M}^{1}(χ ⊗ χ, χ ⊗ χ).

Proof. This follows directly from Lemmas 4.3.3 and 4.3.7 (or rather, the analogue of the latter for \overline{P}). □

4.3.33. Proposition. Let χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×} be smooth characters, with χ_{1} ≠ χ_{2}. If V is a supersingular absolutely irreducible object of Mod_{G}^{adm}(k), then the space Ext_{G}^{1}(V, IndG ; \overline{P} χ_{1} ⊗ χ_{2}) vanishes.

Proof. Let 0 → IndG ; \overline{P} χ_{1} ⊗ χ_{2} → E → V → 0 be an extension. Applying H^{•}Ord_{P}, and taking into account Theorem 4.2.12, we find that

H^{1}Ord_{P} (E) ⟶^{∼} H^{1}Ord_{P} (IndG ; \overline{P} χ_{1} ⊗ χ_{2}) ⟶^{∼} χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε},

and hence, by Proposition 3.6.2, we obtain an isomorphism of Jacquet modules

E_{N} ⟶^{∼} χ_{2} ⊗ χ_{1}.

From the adjointness between Jacquet modules and induction, we find that there exists a map E → IndG ; \overline{P} χ_{1} ⊗ χ_{2} extending the identity map from IndG ; \overline{P} χ_{1} ⊗ χ_{2} to itself. Thus E is a split extension, as claimed. □

4.3.34. Proposition. Let χ : ℚ_{p}^{×} → k^{×} be a smooth character. If V is a supersingular absolutely irreducible object of Mod_{G}^{adm}(k), then Ext_{G}^{1}(V, χ ∘ det) vanishes.

Proof. This is proved in an analogous manner to the preceding proposition. Namely, if 0 → χ ∘ det → E → V → 0 is an extension, then applying H^{•}Ord_{P}, and taking into account Theorem 4.2.12 and Proposition 3.6.2, we obtain an isomorphism of Jacquet modules E_{N} ⟶^{∼} χ ⊗ χ. Thus, from the adjointness between Jacquet modules and induction, we obtain a map E → IndG ; \overline{P} χ ⊗ χ that extends the embedding χ ∘ det ↪ IndG ; \overline{P} χ ⊗ χ Thus E is a split extension, as claimed. □

4.3.35. Remark. In his paper [6], Colmez constructs a functor from the full subcategory of Mod_{G}^{adm}(k) consisting of finitely generated objects to the category of continuous G_{ℚ_{p}}-representations on k-vector spaces. Using this functor, Colmez has found another approach to many of the preceding computations, and some of the above computations were first made by him. In particular, Claim (3) of the Proposition 4.3.15 (under the additional assumption that χ_{1} ≠ χ_{2}) is originally due to Colmez [6, Prop. VII.4.15], as is the isomorphism (4.3.24) [6, §VII.4.3]. In [6, §VII.4.2] Colmez gives another proof of (most of) Proposition 4.3.15, and in [6, Prop. 7.5.4], he gives alternative proofs of Propositions 4.3.14, 4.3.31, 4.3.33, and 4.3.34.

In the paper [4], Breuil and Paškūnas study the functor of passage to pro-p-Iwahori invariants (in the context of representations of GL_{2}(F), for a finite extension F of ℚ_{p}), as well as its first derived functor, and in this way obtain results regarding extensions by principal series representations. In particular they recover (most of) Proposition 4.3.15 [4, Cor. 8.2, Cor. 8.3], as well as Proposition 4.3.33 [4, Cor. 8.4].

Appendix A

In this appendix we present some technical calculations needed to complete the computations of Subsection 4.3. We have placed these calculations in an appendix, since they are of a more ad hoc nature than the computations using the functors H^{•}Ord_{P} that we have presented in the main body of the text. Throughout the appendix, G, P, etc. have the same meanings as in Section 4 (i.e. G = GL_{2}(ℚ_{p}), P is the Borel subgroup of upper triangular matrices, etc.).

A.1. We maintain the notation of the preceding subsection. Our goal now is to consider those aspects of the situation when χ_{1} = χ_{2}\overline{ε}^{−1} needed to complete the proof of part (3) of Proposition 4.3.15. To simplify the notation, we make a twist by χ_{1}^{−1} ∘ det, and assume that in fact χ_{1} = \underline{1} from now on (so that χ_{2} = \overline{ε}). We also assume that p ≠ 2. We must prove that the map

(A.1.1)  Ext_{M}^{2}(\underline{1} ⊗ \overline{ε}, \underline{1} ⊗ \overline{ε}) → Ext_{G}^{2}(IndG ; \overline{P} \underline{1} ⊗ \overline{ε}, IndG ; \overline{P} \underline{1} ⊗ \overline{ε})

induced by applying IndG ; \overline{P} is not an injection. Here, the Ext^{2} spaces are understood to be computed in the space of locally admissible representations.

In fact, we will prove that a specific line in the source lies in the kernel. To describe this line, we will have to describe some of the Ext spaces under discussion more explicitly. To begin with, note that we have isomorphisms

(A.1.2)  Ext_{M}^{1}(\underline{1} ⊗ \overline{ε}, \underline{1} ⊗ \overline{ε}) ⟶^{∼} Ext_{M}^{1}(\underline{1} ⊗ \underline{1}, \underline{1} ⊗ \underline{1}) ⟶^{∼} H^{1}(M, \underline{1} ⊗ \underline{1}),

the first being given by twisting, and the second by Proposition 2.2.2, together with the fact that Ext_{M}^{1} between admissible smooth representations is the same whether computed in the category Mod_{M}^{l.adm}(k) or Mod_{M}^{sm}(k).

Now write M′ = M/Z_{G} ⟶^{∼} ℚ_{p}^{×}. Since p ≠ 2, the group admits the following description:

M′ ≅ ℚ_{p}^{×} ≅ Δ × Γ × A,

where Δ has order p−1, Γ = 1+ℤ_{p} ≅ ℤ_{p}, and A = p^{ℤ} is the cyclic group generated by p. Since Δ has prime-to-p order, it has no cohomology above degree 0; of course H^{0}(Δ, \underline{1}) is one-dimensional. Simple computations in continuous cohomology show that H^{0}(Γ, \underline{1}), H^{1}(Γ, \underline{1}), H^{0}(A, \underline{1}), and H^{1}(A, \underline{1}) are all one-dimensional, and that the continuous cohomology of Γ and A in degrees > 1 vanishes. Each of the tensor products H^{0}(Δ, \underline{1})⊗_{k} H^{0}(Γ, \underline{1})⊗_{k} H^{0}(A, \underline{1}) and H^{0}(Δ, \underline{1})⊗_{k} H^{1}(Γ, \underline{1})⊗_{k} H^{0}(A, \underline{1}) is thus one-dimensional; let α and β respectively denote basis vectors. The Künneth formula then shows that H^{1}(M′, \underline{1}) is two-dimensional, spanned by α and β, and that H^{2}(M′, \underline{1}) is one-dimensional, spanned by the Yoneda product (which in this context coincides with the cup product) α∪β. Another application of the Künneth formula (applied to the evident isomorphism M ⟶^{∼} Z_{G} × M′) shows that the pullback maps H^{i}(M′, \underline{1}) → H^{i}(M, \underline{1} ⊗ \underline{1}) are injective, for all i. We may thus regard α and β as elements of H^{1}(M, \underline{1} ⊗ \underline{1}), and hence they give rise to corresponding elements α′ and β′ of Ext_{M}^{1}(\underline{1} ⊗ \underline{1}, \underline{1} ⊗ \underline{1}) and α′′ and β′′ of Ext_{M}^{1}(\underline{1} ⊗ \overline{ε}, \underline{1} ⊗ \overline{ε}), via the isomorphisms (A.1.2).

A.1.3. Lemma. The Yoneda product α′′ ∪ β′′ ∈ Ext_{M}^{2}(\underline{1} ⊗ \overline{ε}, \underline{1} ⊗ \overline{ε}) is non-zero.

Proof. By twisting, it is equivalent to show that Yoneda product α′∪β′ ∈ Ext_{M}^{2}(\underline{1}⊗ \underline{1}, \underline{1} ⊗ \underline{1}) is non-zero. Passing from the category Mod_{M}^{l.adm}(k) to the category Mod_{M}^{sm}(k) induces a corresponding map on Ext^{2} spaces, which we may then compose with the map to H^{2}(M, \underline{1} ⊗ \underline{1}) given by (2.2.1). Under this composite, α′ ∪ β′ maps to α ∪ β, which is non-zero by the Künneth formula for continuous cohomology. □

We will show that the Yoneda product α′′ ∪ β′′ ∈ Ext_{M}^{2}(\underline{1} ⊗ \overline{ε}, \underline{1} ⊗ \overline{ε}), which by the preceding lemma is non-zero, has zero image under (A.1.1). The first step in the argument is to show that it suffices to check the corresponding fact with Ext^{2} computed in the category of locally admissible representations replaced by Ext^{2} computed in the category of smooth representations. To see this, we first note that applying IndG ; \overline{P} to the extension classes α′ and β′ gives rise to extension classes X and Y lying in Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \overline{ε}, IndG ; \overline{P} \underline{1} ⊗ \overline{ε}), and the image of α′ ∪ β′ under (A.1.1) coincides with X ∪ Y. The following Lemma then shows that the cup product of X and Y vanishes when computed in Ext_{G}^{2}(IndG ; \overline{P} \underline{1} ⊗ \overline{ε}, IndG ; \overline{P} \underline{1} ⊗ \overline{ε}) for the category Mod_{G}^{l.adm}(k) if and only if it vanishes when computed in Ext_{G}^{2}(IndG ; \overline{P} \underline{1}⊗\overline{ε}, IndG ; \overline{P} \underline{1}⊗\overline{ε}) for the category Mod_{G}^{sm}(k).

A.1.4. Lemma. Let U, V, and W be admissible smooth G-representations over k. If we are given extensions 0 → U → X → V → 0 and 0 → V → Y → W → 0, then the resulting 2-extension 0 → U → X → Y → W → 0 given by forming the Yoneda product of X with Y is trivial in the category Mod_{G}^{l.adm}(k) if and only if it is trivial in the category Mod_{G}^{sm}(k).

Proof. Let C denote either of the categories Mod_{G}^{l.adm}(k) or Mod_{G}^{sm}(k). As is wellknown, the Yoneda product of X with Y is trivial in the category C if and only there exists an extension 0 → X → Z → W → 0, in the category C, such that the extension 0 → V → Z′ → W → 0 obtained by pushing forward Z along the surjection X → V is isomorphic (as an extension of W by V) to Y. However, such an object Z, being a successive extension of admissible representations, is necessarily admissible [10, Prop. 2.2.13]. Thus trivializing the Yoneda product of X with Y in C is equivalent to trivializing it in the category Mod_{G}^{adm}(k), for either choice of C. □

From now on we compute all Ext^{2} spaces in the appropriate category of smooth representations (as opposed to the category of locally admissible representations). As already noted, by the preceding lemma, we are reduced to showing that α′′∪β′′ ∈ Ext_{M}^{2}(\underline{1}⊗\overline{ε}, \underline{1}⊗\overline{ε}) has vanishing image in Ext_{G}^{2}(IndG ; \overline{P} \underline{1}⊗\overline{ε}, IndG ; \overline{P} \underline{1}⊗\overline{ε}). Our strategy for proving this will be to exhibit an explicit element of Ext_{M}^{2}(\underline{1} ⊗ \overline{ε}, \underline{1} ⊗ \overline{ε}) which does vanish when mapped to Ext_{G}^{2}(IndG ; \overline{P} \underline{1} ⊗ \overline{ε}, IndG ; \overline{P} \underline{1} ⊗ \overline{ε}), and then to show that this element is equal to a non-zero scalar multiple of α′′ ∪ β′′.

We begin by writing down the diagram

(A.1.5)  \xymatrix{Ext_{M}^{1}(\overline{ε} ⊗ \underline{1}, \overline{ε} ⊗ \underline{1}) \ar[d]_{U↦IndG ; \overline{P} U^{w}} \ar[r]^{∼} & Ext_{P}^{1} (\overline{ε} ⊗ \underline{1}, \overline{ε} ⊗ \underline{1}) \ar[d] \\ Ext_{G}^{1}(IndG ; \overline{P} \underline{1} ⊗ \overline{ε}, IndG ; \overline{P} \underline{1} ⊗ \overline{ε}) \ar[r]^{∼} & Ext_{P}^{1} (IndG ; \overline{P} \underline{1} ⊗ \overline{ε}, \overline{ε} ⊗ \underline{1}) \ar[d] \\  & Ext_{P}^{1} (C_{c}^{sm}(N, k) ⊗ (\underline{1} ⊗ \overline{ε}), \overline{ε} ⊗ \underline{1}) \ar[d]^{δ} \\ Ext_{M}^{2}(\overline{ε} ⊗ \underline{1}, \overline{ε} ⊗ \underline{1}) \ar[d]_{U↦IndG ; \overline{P} U^{w}} & Ext_{P}^{2} (\overline{ε} ⊗ \underline{1}, \overline{ε} ⊗ \underline{1}) \ar[d] \ar[l]_{∼} \\ Ext_{G}^{2}(IndG ; \overline{P} \underline{1} ⊗ \overline{ε}, IndG ; \overline{P} \underline{1} ⊗ \overline{ε}) \ar[r]^{∼} & Ext_{P}^{2} (IndG ; \overline{P} \underline{1} ⊗ \overline{ε}, \overline{ε} ⊗ \underline{1}),}

whose right-hand column is obtained by applying Hom_{P} (–, χ_{1}\overline{ε}⊗χ_{2}\overline{ε}^{−1}) to (4.2.11) (taking U = \underline{1} ⊗ \overline{ε}), and whose horizontal arrows are provided by Lemmas 4.3.3 and 4.3.7. Both squares commute (possibly up to a sign). (Of course, all the Ext spaces in this diagram are computed in the category of smooth representations.)

Note that we have labelled by δ the boundary map

δ : Ext_{P}^{1} (C_{c}^{sm}(N, k) ⊗ (\underline{1} ⊗ \overline{ε}), \overline{ε} ⊗ \underline{1}) → Ext_{P}^{2} (\overline{ε} ⊗ \underline{1}, \overline{ε} ⊗ \underline{1}).

A.1.6. Lemma. The space Ext_{P}^{1} (C_{c}^{sm}(N, k) ⊗ (\underline{1} ⊗ \overline{ε}), \overline{ε} ⊗ \underline{1}) is non-zero.

Proof. A simple descent argument shows that the formation of this space is compatible with extension of scalars, and thus we may enlarge k if necessary so that we can choose smooth characters χ_{1}, χ_{2} : ℚ_{p}^{×} → k^{×} such that χ_{1} ≠ χ_{2}\overline{ε}^{−1}. Proposition 4.3.15 (2) and its proof then shows that there is an isomorphism

Ext_{G}^{1}(IndG ; \overline{P} χ_{1} ⊗ χ_{2}, IndG ; \overline{P} χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε}) → Hom_{M}(χ_{1} ⊗ χ_{2}, χ_{1} ⊗ χ_{2}),

which, according to the discussion of Remark 3.7.7, is given by mapping an extension

(A.1.7)  0 → IndG ; \overline{P} χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε} → V → IndG ; \overline{P} χ_{1} ⊗ χ_{2} → 0

to the corresponding boundary map

δ_{E} : Ord_{P} (IndG ; \overline{P} χ_{1} ⊗ χ_{2}) → H^{1}Ord_{P} (IndG ; \overline{P} χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε}),

obtained by applying H^{•}Ord_{P} to (A.1.7).

As is explained in the proof of Corollary 4.2.10, there is a P -equivariant embedding

(A.1.8)  C_{c}^{sm}(N, k) ⊗ (χ_{1} ⊗ χ_{2}) ↪ IndG ; \overline{P} χ_{1} ⊗ χ_{2}

and a P -equivariant surjection

(A.1.9)  IndG ; \overline{P} χ_{2}\overline{ε}^{−1} ⊗ χ_{1}\overline{ε} → χ_{1}\overline{ε} ⊗ χ_{2}\overline{ε}^{−1}.

If we pull E back along the first of these maps, and then push the resulting extension forward along the second, we obtain an extension

(A.1.10)  0 → χ_{1}\overline{ε} ⊗ χ_{2}\overline{ε}^{−1} → E′ → C_{c}^{sm}(N, k) ⊗ (χ_{1} ⊗ χ_{2}) → 0.

We may apply H^{•}Ord_{P} to this short exact sequence to obtain a complex of the form (3.3.4). In particular, we obtain a map

(A.1.11)  Ord_{P} (C_{c}^{sm}(N, k) ⊗ (χ_{1} ⊗ χ_{2})) → H^{1}Ord_{P} (χ_{1}\overline{ε} ⊗ χ_{2}\overline{ε}^{−1}).

It is shown in the course of the proof of Corollary 4.2.10 that (A.1.8) induces an isomorphism upon applying Ord_{P}, and that (A.1.9) induces an isomorphism upon applying H^{1}Ord_{P}. It follows from this that the map (A.1.11) coincides with the map δ_{E} (after identifying their respective domains and codomains via these isomorphisms).

Combining the results of the preceding two paragraphs, we find that if E is non-trivial, so that δ_{E} is non-trivial, then the map (A.1.11) is non-trivial, and hence (A.1.10) is non-split. Twisting this extension by χ_{1}^{−1} ⊗ χ_{2}^{−1}\overline{ε}, we obtain a non-trivial extension of the form required to establish the lemma. □

If c ∈ Ext_{P}^{1} (C_{c}^{sm}(N, k) ⊗ (\underline{1} ⊗ \overline{ε}), \overline{ε} ⊗ \underline{1}) is non-zero, then applying first the third horizontal isomorphism in (A.1.5), and then the automorphism w, to δ(c), we see from a consideration of (A.1.5) (in particular, using the fact that its right-hand column is exact) that δ(c) gives rise to an element γ ∈ Ext_{M}^{2}(\underline{1} ⊗ \overline{ε}, \underline{1} ⊗ \overline{ε}) whose image under IndG ; \overline{P} vanishes. We will be done once we prove the following result.

A.1.12. Proposition. The element γ is a non-zero scalar multiple of α′′ ∪ β′′ in Ext_{M}^{2}(\underline{1} ⊗ \overline{ε}, \underline{1} ⊗ \overline{ε}).

Proof. If we let d ∈ Ext_{P}^{1} (\overline{ε} ⊗ \underline{1}, C_{c}^{sm}(N, k) ⊗ (\underline{1} ⊗ \overline{ε})) be the extension class corresponding to the short exact sequence (4.2.11) (with U taken to be \underline{1} ⊗ \overline{ε}), then δ(c) = ±c ∪ d (we don’t concern ourselves with the precise value of the sign), and so to prove the lemma, we have to show that c_{|M} ∪ d_{|M} = (c ∪ d)_{|M} is a non-zero scalar multiple of α′′ ∪ β′′. (Here the subscript “|M” denotes the restriction of a class from P to M.)

We identify N with ℚ_{p} in the usual fashion, via ℚ_{p} ∋ x ↦ ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ∈ N, and we identify M′ = M/Z_{G} both with ℚ_{p}^{×} and with a subgroup of M via ℚ_{p}^{×} ∋ a ↦ ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) ∈ M′ ⊂ M. With these identifications we obtain isomorphisms C_{c}^{sm}(N, k) ⟶^{∼} C_{c}^{sm}(ℚ_{p}, k), and M′ = ℚ_{p}^{×} acts via the formula

(⟨a⟩f)(x) = f(a^{−1}x),

if a ∈ ℚ_{p}^{×} and x ∈ ℚ_{p}, where we have written ⟨a⟩ to denote a thought of as an element of M′ (to avoid any possible confusion with the element a thought of as a scalar).

As already observed above, M′ × Z_{G} ⟶^{∼} M. Since c and d are both trivial when restricted to Z_{G}, so is their Yoneda product. Thus it suffices to verify that (c ∪ d)_{|M′} is a non-zero scalar multiple of (α′′ ∪ β′′)_{|M′}, and this is what we will do.

Evaluation at 0 ∈ ℚ_{p} gives an M′-equivariant short exact sequence

(A.1.13)  0 → C_{c}^{sm}(ℚ_{p}^{×}, k) → C_{c}^{sm}(ℚ_{p}, k) → \underline{1} → 0.

Now (\overline{ε} ⊗ \underline{1})_{|M′} = \overline{ε}, while (\underline{1} ⊗ \overline{ε})_{|M′} = \underline{1}. Thus c_{|M′} ∈ Ext_{M′}^{1}(\overline{ε}, C_{c}^{sm}(ℚ_{p}, k)). Since p ≠ 2, the character \overline{ε} is non-trivial, and thus the long exact sequences of Exts arising from the short exact sequence (A.1.13) gives rise to an isomorphism

Ext_{M′}^{1}(C_{c}^{sm}(ℚ_{p}, \underline{1}), \overline{ε}) ⟶^{∼} Ext_{M′}^{1}(C_{c}^{sm}(ℚ_{p}^{×}, \underline{1}), \overline{ε}).

We let c′ ∈ Ext_{M′}^{1}(C_{c}^{sm}(ℚ_{p}^{×}, \underline{1}), \overline{ε}) denote the element which corresponds to c_{|M′} via this isomorphism. Similarly, there is an isomorphism

Ext_{M′}^{1}(\overline{ε}, C_{c}^{sm}(ℚ_{p}^{×}, k)) ⟶^{∼} Ext_{M′}^{1}(\overline{ε}, C_{c}^{sm}(ℚ_{p}, k)),

and we let d′ ∈ Ext_{M′}^{1}(\overline{ε}, C_{c}^{sm}(ℚ_{p}^{×}, k)) denote the element which corresponds to d_{|M′} via this isomorphism. There are equalities c′ ∪ d′ = c_{|M′} ∪ d_{|M′} = (c ∪ d)_{|M′}.

As above, write A = p^{ℤ} ⊂ ℚ_{p}^{×}, so that ℚ_{p}^{×} = ℤ_{p}^{×} × A. There is an isomorphism of M′-representations C_{c}^{sm}(ℚ_{p}^{×}, k) ⟶^{∼} c−Ind_{A}^{M′} C^{sm}(ℤ_{p}^{×}, k), and hence an isomorphism

Ext_{M′}^{1}(C_{c}^{sm}(ℚ_{p}^{×}, k), \overline{ε}) ⟶^{∼} Ext_{ℤ_{p}^{×}}^{1}(C^{sm}(ℤ_{p}^{×}, k), \overline{ε}).

We let c′′ ∈ Ext_{ℤ_{p}^{×}}^{1}(C^{sm}(ℤ_{p}^{×}, k), \overline{ε}) denote the element which corresponds to c′ via this isomorphism. If c′′ is given by the extension

0 → \overline{ε} → E → C^{sm}(ℤ_{p}^{×}, k) → 0,

then we in fact regard c′′ as an extension of M′-representations, by having A act trivially on all the terms of this extension. The extension class c′ is then obtained by first tensoring E with C_{c}^{sm}(A, k) (regarded as an M′-representation by having ℤ_{p}^{×} act trivially) to obtain an extension

0 → C_{c}^{sm}(A, k) ⊗_{k} \overline{ε} → C_{c}^{sm}(A, k) ⊗_{k} E → C_{c}^{sm}(ℚ_{p}^{×}, k) → 0,

and then pushing forward via the map C_{c}^{sm}(A) ⊗_{k} \overline{ε} → \overline{ε}, obtained by tensoring the M′-equivariant map I : C_{c}^{sm}(A, k) → \underline{1} (given by “integrating with respect to Haar measure” on the discrete group A) with the character \overline{ε}.

Now tensoring I with the identity automorphism of C^{sm}(ℤ_{p}^{×}, k) (the latter being regarded as an M′-representation by having A act trivially) similarly induces an M′-equivariant map

J : C_{c}^{sm}(ℚ_{p}^{×}, k) → C^{sm}(ℤ_{p}^{×}, k),

and we write d′′ := J_{∗}d (the pushforward of d with respect to J), an element of Ext_{M′}^{1}(C^{sm}(ℤ_{p}^{×}, k), \overline{ε}). It follows from the description of the relationship between c′ and c′′ in the preceding paragraph that c′ ∪ d′ = c′′ ∪ d′′.

The inclusion of the scalar multiples of the character \overline{ε}_{|ℤ_{p}^{×}} in the space of all locally constant functions gives an embedding ı : \overline{ε} ↪ C^{sm}(Γ, k), and Lemma A.1.14 below shows that d′′ = ı_{∗}d′′′ for some class d′′′ ∈ Ext_{M′}^{1}(\overline{ε}, \overline{ε}) which is non-trivial when restricted to A. We then compute that

c′′ ∪ d′′ = c′′ ∪ ı_{∗}d′′′ = (ı^{∗}c′′) ∪ d′′′.

Lemma A.1.20 below shows that ı^{∗}c′′ ∈ Ext_{M′}^{1}(\overline{ε}, \overline{ε}) is non-trivial when restricted to ℤ_{p}^{×} (or equivalently, when restricted to Γ). It follows that indeed (ı^{∗}c′′) ∪ d′′′ is a non-zero multiple of α′′_{|M′} ∪ β′′_{|M′}, and the proposition is proved. □

A.1.14. Lemma. If ı : \overline{ε} ↪ C^{sm}(ℤ_{p}^{×}, k) denotes the inclusion of the scalar multiples of the character \overline{ε} into the space of locally constant functions on ℤ_{p}^{×}, then there exists a class d′′′ ∈ Ext_{M′′}^{1}(\overline{ε}, \overline{ε}), whose restriction to A is non-trivial, and such that d′′ = ı_{∗}d′′′.

Proof. To see this, we begin by recalling the construction of d′′, beginning with the class d, which corresponds to the extension

(A.1.15)  0 → C_{c}^{sm}(ℚ_{p}, k) → IndG ; \overline{P} \underline{1} ⊗ \overline{ε} → \overline{ε} ⊗ \underline{1} → 0.

We extend the identification of N with ℚ_{p} to an identification of \overline{P} \G with ℙ^{1}(ℚ_{p}) = ℚ_{p} ∪ \{∞\}. Any element f of IndG ; \overline{P} \underline{1} ⊗ \overline{ε} has a support in G, which is invariant under left translation by \overline{P}, and so projects to a compact open subset of \overline{P} \G, which as in [10, §4] we again refer to as the support of f. If we write

Ω = ℙ^{1}(ℚ_{p}) \ \{0\} ⊂ ℙ^{1}(ℚ_{p}) ⟶^{∼} \overline{P} \ G,

and let (IndG ; \overline{P} \underline{1} ⊗ \overline{ε})(Ω) denote the subspace of IndG ; \overline{P} \underline{1} ⊗ \overline{ε} consisting of elements whose support is contained in Ω, then the class d′ corresponds to (the restriction to M′ of) the extension of M-representations

(A.1.16)  0 → C_{c}^{sm}(ℚ_{p}^{×}, k) → (IndG ; \overline{P} \underline{1} ⊗ \overline{ε})(Ω) → \overline{ε} ⊗ \underline{1} → 0

induced by (A.1.15). The class d′′ := I_{∗}d′ then corresponds to the extension

(A.1.17)  0 → C^{sm}(ℤ_{p}^{×}, k) → F → \overline{ε} ⊗ \underline{1} → 0

obtained by pushing (A.1.16) forward via the map J : C_{c}^{sm}(ℚ_{p}^{×}, k) → C^{sm}(ℤ_{p}^{×}, k) given by “integrating along the fibres” (the fibres of the projection ℚ_{p}^{×} ⟶^{∼} A × ℤ_{p}^{×} → ℤ_{p}^{×} being cosets of A, and hence discrete).

Let

U := (IndG ; \overline{P} \underline{1} ⊗ \overline{ε})(Ω)^{ℤ_{p}^{×}=\overline{ε}} ⊂ (IndG ; \overline{P} \underline{1} ⊗ \overline{ε})(Ω)

denote the subspace consisting of elements which transform under ℤ_{p}^{×} via the character \overline{ε}. This is clearly an M-invariant subspace of (IndG ; \overline{P} \underline{1} ⊗ \overline{ε})(Ω). Certainly, the image of U ⋂ C_{c}^{sm}(ℚ_{p}^{×}, k) under J is one-dimensional. Thus there are two possibilities for the image V of U under the projection

(A.1.18)  (IndG ; \overline{P} \underline{1} ⊗ \overline{ε})(Ω) → F,

namely: either V is one-dimensional (or equivalently, U is contained in C_{c}^{sm}(ℚ_{p}^{×}, k)), or else V is two-dimensional, and thus gives a subextension

(A.1.19)  0 → \overline{ε} ⊗ \underline{1} → V → \overline{ε} ⊗ \underline{1} → 0

of (A.1.17).

Let f denote the characteristic function of ℤ_{p}, thought of as an element of C_{c}^{sm}(ℚ_{p}, k) ⊗ (\underline{1} ⊗ \overline{ε}). Then f is an element of IndG ; \overline{P} \underline{1} ⊗ \overline{ε}, and one easily verifies that wf ∈ U (where w = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )), and that wf does not lie in C_{c}^{sm}(ℚ_{p}^{×}, k), and hence has a non-trivial projection onto \overline{ε} ⊗ \underline{1}. Thus in fact V is two-dimensional, and gives rise to the subextension (A.1.19) of (A.1.17). If we let d′′′ denote the class of this subextension, then indeed d′′ = ı_{∗}d′′′.

It remains to be shown that d′′′ is non-trivial when restricted to A. To see this, one explicitly computes J(⟨p⟩(wf) − wf), and checks that the result is non-zero. We leave this straightforward computation to the sufficiently interested reader. □

A.1.20. Lemma. If ı : \overline{ε} ↪ C^{sm}(ℤ_{p}^{×}, k) is as in the statement of Lemma A.1.14, and if e ∈ Ext_{ℤ_{p}^{×}}^{1}(C_{c}^{sm}(ℤ_{p}^{×}, k), \overline{ε}) is a non-trivial class, then the same is true of ı^{∗}e.

Proof. Recall that ℤ_{p}^{×} = Δ × Γ, where Δ has order prime to p. We may thus verify the assertion of the lemma after passing to \overline{ε}-eigenspaces for the action of Δ in all the representations and extensions considered, and so reduce to checking the analogous statement for ȷ^{∗}e, where e ∈ Ext_{Γ}^{1}(C_{c}^{sm}(Γ, k), \underline{1}) is non-trivial, and ȷ : \underline{1} ↪ C^{sm}(Γ, k). We do this by passing to k-duals, under which admissible smooth Γ-representations become finitely generated modules for the completed group ring k[[Γ]].

We must then verify that if e′ ∈ Ext_{k[[Γ]]}^{1}(\underline{1}, k[[Γ]]) is non-trivial, and if π : k[[Γ]] → \underline{1} is the canonical surjection given by taking the quotient by the augmentation ideal, then the pushforward π_{∗}e′ is again non-trivial. This is easily verified by the reader, using the well-known isomorphism k[[Γ]] ⟶^{∼} k[[t]] (whose target is a power series ring in one variable). □

A.2. Our goal in this subsection is to prove that the map (4.3.25) is injective. We begin by recalling that P = MN acts on C_{c}^{sm}(N, k) via the formula

(mnf)(n′) = f(m^{−1}n′mn),

for m ∈ M, n, n′ ∈ N, and f ∈ C_{c}^{sm}(N, k). The same formula defines an action of P on the space C^{sm}(N, k) of all smooth (i.e. locally constant) k-valued functions on N. The P -action on this space is not smooth, and we denote by C_{u}^{sm}(N, k) the subspace of P -smooth vectors in C^{sm}(N, k). (The subscript u is for uniform; a function is P - smooth if and only if it is N-smooth, and so the elements of C_{u}^{sm}(N, k) are precisely the uniformly locally constant functions on N.) Evaluation at the identity of N gives a T -equivariant map

C_{u}^{sm}(N, k) → \underline{1},

which by Frobenius reciprocity induces an isomorphism

(A.2.1)  C_{u}^{sm}(N, k) ⟶^{∼} Ind_{M}^{P} \underline{1}.

By construction the embedding C_{c}^{sm}(N, k) ⊂ C_{u}^{sm}(N, k) is P -equivariant. Thus if ψ : ℚ_{p}^{×} → k is a smooth character, then we obtain a long exact sequence

0 → Hom_{P} (ψ ⊗ ψ^{−1}, C_{c}^{sm}(N, k)) → Hom_{P} (ψ ⊗ ψ^{−1}, C_{u}^{sm}(N, k))

→ Hom_{P} (ψ ⊗ ψ^{−1}, C_{u}^{sm}(N, k)/ C_{c}^{sm}(N, k)) → Ext_{P}^{1} (ψ ⊗ ψ^{−1}, C_{c}^{sm}(N, k))

→ Ext_{P}^{1} (ψ ⊗ ψ^{−1}, C_{u}^{sm}(N, k)).

(Here and below, all Ext^{1} spaces are computed in the category of smooth representations.) Clearly Hom_{P} (ψ ⊗ ψ^{−1}, C_{c}^{sm}(N, k)) must vanish, while Frobenius reciprocity together with the isomorphism (A.2.1) yields isomorphisms

Hom_{P} (ψ ⊗ ψ^{−1}, C_{u}^{sm}(N, k)) ⟶^{∼} Hom_{M}(ψ ⊗ ψ^{−1}, \underline{1})

and

Ext_{P}^{1} (ψ ⊗ ψ^{−1}, C_{u}^{sm}(N, k)) ⟶^{∼} Ext_{M}^{1}(ψ ⊗ ψ^{−1}, \underline{1}).

(The latter isomorphism is proved in a manner identical to Lemma 4.3.3.) Finally, Lemma 4.3.4 yields an isomorphism

Hom_{P} (ψ⊗ψ^{−1}, C_{u}^{sm}(N, k)/ C_{c}^{sm}(N, k)) ⟶^{∼} Hom_{M}(ψ⊗ψ^{−1}, C_{u}^{sm}(N, k)/ C_{c}^{sm}(N, k)).

Thus we may rewrite the above long-exact sequence as

(A.2.2) 0 → Hom_{M}(ψ ⊗ ψ^{−1}, \underline{1}) → Hom_{M}(ψ ⊗ ψ^{−1}, C_{u}^{sm}(N, k)/ C_{c}^{sm}(N, k))

→ Ext_{P}^{1} (ψ ⊗ ψ^{−1}, C_{c}^{sm}(N, k)) → Ext_{M}^{1}(ψ ⊗ ψ^{−1}, \underline{1}).

A.2.3. Proposition. Hom_{M}(ψ ⊗ ψ^{−1}, C_{u}^{sm}(N, k)/ C_{c}^{sm}(N, k)) is one-dimensional.

Proof. We identify N with ℚ_{p}, and so identify C_{u}^{sm}(N, k) with C_{u}^{sm}(ℚ_{p}, k). We similarly identify ℚ_{p}^{×} with a subgroup of M, and observe that since both ψ ⊗ ψ^{−1} and C_{u}^{sm}(ℚ_{p}, k) are acted on trivially by Z, restricting from M-representations to ℚ_{p}^{×}-representations induces an isomorphism

Hom_{M}(ψ ⊗ ψ^{−1}, C_{u}^{sm}(N, k)/ C_{c}^{sm}(N, k)) ⟶^{∼} Hom_{ℚ_{p}^{×}}(ψ, C_{u}^{sm}(N, k)/ C_{c}^{sm}(N, k)).

If a ∈ ℚ_{p}^{×} and f ∈ C_{u}^{sm}(ℚ_{p}, k) then we will write ⟨a⟩f to denote the action of a on f. (This should prevent any possible confusion with the element a thought of as a scalar.) Explicitly, the action of ℚ_{p}^{×} on C_{u}^{sm}(ℚ_{p}, k) is given by the formula

(⟨a⟩f)(x) := f(a^{−1}x),

for a ∈ ℚ_{p}^{×}, x ∈ ℚ_{p}, and f ∈ C_{u}^{sm}(ℚ_{p}, k).

If f ∈ C_{u}^{sm}(ℚ_{p}, k), then corresponding to the decomposition

ℚ_{p} = ℤ_{p} ⊔ ∐_{n=1}^{∞} p^{−n}ℤ_{p}^{×},

we may write f = f_{0} + ∑_{i=1}^{∞}⟨p^{−i}⟩f_{i}, where f_{i} is a smooth function supported on ℤ_{p}^{×}, so that ⟨p^{−i}⟩f_{i} is supported on p^{−i}ℤ_{p}^{×}. Since f is uniformly smooth, there is an r > 0 such that every f_{i} is constant on the cosets of 1 + p^{r}ℤ_{p} in ℤ_{p}^{×}.

Suppose that f mod C_{c}^{sm}(ℚ_{p}, k) transforms via the character ψ with respect to the action of ℚ_{p}^{×}. Let {a_{j}} denote a set of coset representatives for 1 + p^{r}ℤ_{p} in ℤ_{p}^{×}. By assumption ⟨a_{j}⟩f − ψ(a_{j})f is compactly supported. We compute that

⟨a_{j}⟩f − ψ(a_{j})f = ⟨a_{j}⟩f_{0} − ψ(a_{j})f_{0} + ∑_{i=1}^{∞}⟨p^{−i}⟩(⟨a_{j}⟩f_{i} − ψ(a_{j})f_{i}).

Thus for each j, if i is sufficiently large, we find that ⟨a_{j}⟩f_{i} = ψ(a_{j})f_{i}. Since the number of a_{j} is finite, we may choose i_{0} so that if i > i_{0} then ⟨a_{j}⟩f_{i} = ψ(a_{j})f_{i} for all j. Since f_{i} is constant on 1+p^{r}ℤ_{p} cosets, we see that then in fact ⟨a⟩f_{i} = ψ(a)f_{i} for all a ∈ ℤ_{p}^{×}, and thus that f_{i} = c_{i}ψ_{|ℤ_{p}^{×}}^{−1}, for some scalar c_{i} ∈ k, if i > i_{0}. Thus

we may write f = ∑_{i=0}^{i_{0}}⟨p^{−i}⟩f_{i} + ∑_{i>i_{0}} c_{i}⟨p^{−i}⟩ψ_{|ℤ_{p}^{×}}^{−1}.

By assumption we also have that ⟨p⟩f − ψ(p)f is compactly supported. Since

⟨p⟩f − ψ(p)f

= ⟨p⟩f_{0} + ∑_{i=0}^{i_{0}−1}(⟨p^{−i}⟩f_{i+1} − ψ(p)⟨p^{−i}⟩f_{i}) − ψ(p)⟨p^{−i_{0}}⟩f_{i_{0}} + c_{i_{0}+1}⟨p^{−i_{0}}⟩ψ_{|ℤ_{p}^{×}}

+ ∑_{i>i_{0}}(c_{i+1} − ψ(p)c_{i})⟨p^{−i}⟩ψ_{|ℤ_{p}^{×}},

we find that c_{i+1} − ψ(p)c_{i} = 0 for sufficiently large i, and hence, enlarging i_{0} if necessary, we may write

f = ∑_{i=0}^{i_{0}}⟨p^{−i}⟩f_{i} + c ∑_{i>i_{0}} ψ(p)^{i}⟨p^{−i}⟩ψ_{|ℤ_{p}^{×}}^{−1} = ∑_{i=0}^{i_{0}}⟨p^{−i}⟩f_{i} + c ∑_{i>i_{0}} ψ_{|p^{−i}ℤ_{p}^{×}}^{−1},

for some c ∈ k. Thus we see that, away from a compact subset ℚ_{p}, the function f coincides with a scalar multiple of ψ^{−1}. On the other hand, any function f with this property certainly lies in C_{u}^{sm}(ℚ_{p}, k), and f mod C_{c}^{sm}(ℚ_{p}, k) transforms via ψ with respect to the action of ℚ_{p}^{×}. The space of C_{c}^{sm}(ℚ_{p}, k)-cosets of such f is clearly one-dimensional, and so the proposition follows. □

A.2.4. Corollary. The map (4.3.25) is injective.

Proof. Take ψ to be trivial. The map (4.3.25) is then given by the final arrow in the exact sequence (A.2.2). Clearly Hom_{M}(\underline{1} ⊗ \underline{1}, \underline{1} ⊗ \underline{1}) is one-dimensional. It thus follows from Proposition A.2.3 that the second arrow of (A.2.2) is an isomorphism. Thus the final arrow is an embedding, as claimed. □

A.2.5. Remark. If ψ is non-trivial, then it follows from a Proposition A.2.3 and a consideration of (A.2.2) that Ext_{P}^{1} (ψ ⊗ ψ^{−1}, C_{c}^{sm}(N, k)) is one-dimensional. A non-trivial such extension may obtained from the restriction to P of IndG ; \overline{P} \underline{1} ⊗ ψ by applying a twist by \underline{1} ⊗ ψ^{−1}.

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Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208

E-mail address: emerton@math.northwestern.edu