Emerton’s Jacquet Functors for

Non-Borel Parabolic Subgroups

Richard Hill and David Loeffler

Received: May 30, 2010 Revised: October 18, 2010

Communicated by Peter Schneider

Abstract. This paper studies Emerton’s Jacquet module functor for locally analytic representations of p-adic reductive groups, introduced in [Eme06a]. When P is a parabolic subgroup whose Levi factor M is not commutative, we show that passing to an isotypical subspace for the derived subgroup of M gives rise to essentially admissible locally analytic representations of the torus Z(M), which have a natural interpretation in terms of rigid geometry. We use this to extend the construction in of eigenvarieties in [Eme06b] by constructing eigenvarieties interpolating automorphic representations whose local components at p are not necessarily principal series.

2010 Mathematics Subject Classification: 11F75, 22E50, 11F70 Keywords and Phrases: Eigenvarieties, p-adic automorphic forms, completed cohomology

1 Introduction

1.1 Background

Let 𝔊 be a reductive group over a number field F. The automorphic representations of the group 𝔊(𝔸), where 𝔸 is the adèle ring of F, are central objects of study in number theory. In many cases, it is expected that the set Π(𝔊) of automorphic representations contains a distinguished subset Π(𝔊)^{arith} of representations which are (in some sense) “definable over \overline{ℚ}”. The subject of this paper is the p-adic interpolation properties of these representations (and their associated Hecke eigenvalues). Following the pioneering work of Coleman and Coleman-Mazur [Col96, Col97, CM98] for the automorphic representations attached to modular forms with nonzero Hecke eigenvalue at p, it is expected that these Hecke eigenvalues should be parametrised by p-adic rigid spaces (eigenvarieties).

0The second author is grateful for the support of EPSRC Postdoctoral Fellowship EP/F04304X/2.

Documenta Mathematica 16 (2011) 1–31

A very general construction of eigenvarieties is provided by the work of Emerton [Eme06b], using the cohomology of arithmetic quotients of 𝔊. For any fixed open compact subgroup K_{f} ⊆ 𝔊(𝔸_{f}) (where 𝔸_{f} is the finite adèles of F), and K_{∞}^{∘} the identity component of a maximal compact subgroup of 𝔊(F ⊗ ℝ), the quotients Y (K_{f}) = 𝔊(F)\𝔊(𝔸)/K_{f}K_{∞}^{∘} are real manifolds, equipped with natural local systems V_{X} for each algebraic representation X of 𝔊. The cohomology groups H^{i}(Y (K_{f}), V_{X}) are finite-dimensional, and passing to the direct limit over K_{f} gives an admissible smooth representation H^{i}(V_{X}) of 𝔊(𝔸_{f}). Every irreducible subquotient of H^{i}(V_{X}) is the finite part of an automorphic representation; we say that the representations arising in this way are cohomological (in degree i).

Emerton’s construction proceeds in two major steps. Fix a prime 𝔭 above p and an open compact subgroup K^{(𝔭)} ⊆ 𝔊(𝔸_{f}^{(𝔭)}) (a “tame level”). Firstly, from the spaces H^{i}(Y (K^{(𝔭)}K_{𝔭}), V_{X}) for various open compact subgroups K_{𝔭} ⊆ G = 𝔊(F_{𝔭}), Emerton constructs Banach space representations H̃^{i}(K^{(𝔭)}) of G. For any complete subfield L of F_{𝔭}, the spaces H̃^{i}(K^{(𝔭)})_{la} of locally L-analytic vectors are locally L-analytic representations of G, and there are natural maps

H^{i}(V_{X})^{K^{(𝔭)}} → Hom_{𝔤}(X′, H̃^{i}(K^{(𝔭)})_{la})  (1.1)

where 𝔤 = Lie G. In many cases, these maps are known to be isomorphisms; if this holds, the automorphic representations which are cohomological in degree i are exactly those which appear as subquotients of Hom_{𝔤}(X′, H̃^{i}(K^{(𝔭)})_{la}) for some X and tame level K^{(𝔭)}.

The second step in the construction is to extract the desired information from the space H̃^{i}(K^{(𝔭)})_{la}. This is carried out by applying the Jacquet module functor of [Eme06a], for a Borel subgroup B ⊆ G. This then produces an essentially admissible locally analytic representation of the Levi factor M of B, which is a torus. There is an anti-equivalence of categories between essentially admissible locally analytic representations of M and coherent sheaves on the rigidanalytic space M̂ parametrising characters of M. The eigenvariety E(i, K^{(𝔭)}) is then constructed from this sheaf by passing to the relative spectrum of the unramified Hecke algebra H^{sph} of K^{(𝔭)}; points of this variety correspond to characters (κ, λ) ∈ M̂×Spec H^{sph} such that the (M = κ, H^{sph} = λ)-eigenspace of J_{B}(H̃^{i}(K^{(𝔭)})_{la}) is nonzero. Hence if the map (1.1) above is an isomorphism, there is a point of E(i, K^{(𝔭)}) for each automorphic representation π = ⨂_{v} π_{v} which is cohomological in degree i with (π_{f}^{(𝔭)})^{K^{(𝔭)}} ⊗ J_{B}(π_{𝔭}) ≠ 0.

1.2 Statement of the main result

In this paper, we consider the situation where B is replaced by a general parabolic subgroup P of G. This extends the scope of the theory in two ways: firstly, it may happen that no Borel subgroup exists (G may not be quasi-split); and even if a Borel subgroup exists, there will usually be automorphic representations for which J_{B}(π_{𝔭}) = 0, which do not appear in Emerton’s eigenvariety. As above, we choose a number field F, a connected reductive group 𝔊 over F, and a prime 𝔭 of F above the rational prime p. Let G = 𝔊 ×_{F} F_{𝔭}, a reductive group over F_{𝔭}, and G = G(F_{𝔭}). Let us choose a parabolic subgroup P of G (not necessarily arising from a parabolic subgroup of 𝔊), with unipotent radical N; and let M be a Levi factor of P, with centre Z and derived subgroup D. We write G = G(F_{𝔭}), and similarly for P, M, D, Z. We choose a complete extension L of ℚ_{p} contained in F_{𝔭}, so G, P, M, D, Z are locally L-analytic groups. Let Γ = D × 𝔊(𝔸_{f}^{𝔭}) × π_{0}, where π_{0} is the component group of 𝔊(F ⊗ ℝ). Let us choose an open compact subgroup U ⊆ Γ (this is the most natural notion of a “tame level” in this context), and a finite-dimensional irreducible algebraic representation W of M. As we will explain below, the Hecke algebra H(Γ//U) can be written as a tensor product H^{ram} ⊗ H^{sph}, where H^{sph} is commutative, and H^{ram} is finitely-generated (and supported at a finite set of places S).

Theorem (Theorem 6.3). There exists a rigid-analytic subvariety E(i, P, W, U) of Ẑ × Spec H^{sph}, endowed with a coherent sheaf \overline{F}(i, P, W, U) with a right action of H^{ram}, such that:

1. The natural projection E(i, P, W, U) → 𝔷′ has discrete fibres. In particular, the dimension of E(i, P, W, U) is at most equal to the dimension of Z.

2. The point (χ, λ) ∈ Ẑ × Spec H^{sph} lies in E(i, P, W, U) if and only if the (Z = χ, H^{sph} = λ)-eigenspace of Hom_{U} (W, J_{P} (H̃^{i})_{la}) is nonzero. If

this is so, the fibre of \overline{F}(i, P, W, U) at (χ, λ) is isomorphic as a right H^{ram}-module to the dual of that eigenspace.

3. If there is a compact open subgroup G_{0} ⊆ G such that (H̃_{la}^{i})^{U^{(𝔭)}} is isomorphic as a G_{0}-representation to a finite direct sum of copies of C^{la}(G_{0}) (where U^{(𝔭)} = U ∩ 𝔊(𝔸_{f}^{𝔭})), then E(i, P, W, U) is equidimensional, of dimension equal to the rank of Z.

Now let us suppose that W is absolutely irreducible, and write Π(i, P, W, U) for the set of irreducible smooth 𝔊(𝔸_{f}) × π_{0}-representations π_{f} such that J_{P} (π_{f})^{U} ≠ 0, and π_{f} appears as a subquotient of the cohomology space H^{i}(V_{X}) for some irreducible algebraic representation X of G such that (X′)^{N} ≅ W ⊗ χ for a character χ. To any such π_{f}, we may associate the point (θχ, λ) ∈ Ẑ × Spec H^{sph}, where θ is the smooth character by which Z acts on J_{P} (π_{p}), and λ the character by which H^{sph} acts on J_{P} (π_{f})^{U}. Let E(i, P, W, U)_{cl} denote the set of points of Ẑ × Spec H^{sph} obtained in this way from representations π_{f} ∈ Π(i, P, W, U).

Corollary (Corollary 6.4). If the map (1.1) is an isomorphism in degree i for all irreducible algebraic representations X such that (X′)^{N} is a twist of W, then E(i, P, W, U)_{cl} ⊆ E(i, P, W, U). In particular, the Zariski closure of E(i, P, W, U)_{cl} has dimension at most dim Z.

In the special case when 𝔊(F ⊗ ℝ) is compact modulo centre, a related statement has been proved (by very different methods) by the second author [Loe11]. If P_{1} and P_{2} are two different choices of parabolic, with P_{1} ⊇ P_{2}, we have a relation between the eigenvarieties attached to P_{1} and P_{2} under a mild additional hypothesis, namely that the tame level be of the form U^{(𝔭)} × U_{𝔭}, with U^{(𝔭)} an open compact subgroup away from 𝔭 and U_{𝔭} an open compact subgroup of D_{1} = [M_{1}, M_{1}] which admits a certain decomposition with respect to the parabolic P_{2} ∩D_{1} (see §5.2 below). In this situation, we have the following:

Theorem (Theorem 6.5). If U is of the above type, then the space E(i, P_{1}, W, U) is equal to the union of two subvarieties E(i, P_{1}, W, U)_{P_{2}−fs} and E(i, P_{1}, W, U)_{P_{2}−null}, which are respectively endowed with sheaves of H^{ram}-modules \overline{F}(i, P, W, U)_{P_{2}−fs} and \overline{F}(i, P, W, U)_{P_{2}−null} whose direct sum is \overline{F}(i, P, W, U). If π_{f} ∈ Π(i, P, W, U) and π_{f} is not annihilated by the map (1.1), then the point of E(i, P_{1}, W, U) corresponding to π_{f} lies in the former subvariety if J_{P_{2}}(π_{𝔭}) ≠ 0, and in the latter if J_{P_{2}}(π_{𝔭}) = 0. Moreover, there is a closed subvariety of E(i, P_{2}, W^{N_{12}}, U ∩ D_{2}) whose image in Ẑ_{1} × Spec H^{sph} is E(i, P_{1}, W, U)_{P_{2}−fs}.

2 Preliminaries

2.1 Notation and definitions

Let p be a prime. Let K ⊇ ℚ_{p} be a complete discretely valued field, which will be the coefficient field for all the representations we consider, and L a finite extension of ℚ_{p} contained in K. If V is a locally convex K-vector space, we let V′ denote the continuous dual of V. We write V′_{b} for V′ endowed with the strong topology (which is the only topology on V′ we shall consider). Let S be an abstract semigroup. A topological representation of S is a locally convex Hausdorff topological K-vector space V endowed with a left action of S by continuous operators. If S has a topology, we say that the representation is separately continuous if the orbit map of each v ∈ V is a continuous map S → V, and continuous if the map S × V → V is continuous. In particular, this applies when S is a topological K-algebra and V is an S-module, in which case we shall refer to V as a separately continuous or continuous topological S-module. If G is a locally compact topological group and V is a continuous representation of G, then V′ is a module over the algebra D(G) of measures on G [Eme04, 5.1.7], defined as C(G)′ where C(G) is the space of continuous K-valued functions on G. If G is a locally p-adic analytic group, then for any open compact subgroup H ⊆ G, the subalgebra D(H) is Noetherian, and we say V is admissible continuous [ST02a, Lemma 3.4] if V is a Banach space and V′ is finitely generated over D(H) for one (and hence every) open compact H. If G is a locally L-analytic group, in the sense of [ST02b], then we say the representation V is locally analytic if it is a continuous G-representation on a space of compact type, and the orbit maps are locally L-analytic functions G → V. This implies [Eme04, 5.1.9] that V′_{b} is a separately continuous topological module over the topological K-algebra D^{la}(G) of distributions on G, defined as C^{la}(G)′_{b} where C^{la}(G) is the space of locally L-analytic K-valued functions on G. For H an open compact subgroup, the subalgebra D^{la}(H) is a Fréchet-Stein algebra [ST03, 5.1], so the category of coadmissible D^{la}(H)-modules is defined [ST03, §3]; we say V is admissible locally analytic if V′_{b} is coadmissible as a module over D^{la}(H) for one (and hence every) open compact H. Finally, if G is a locally L-analytic group for which Z = Z(G) is topologically finitely generated, we say the representation V is Z-tempered if it is locally analytic and can be written as an increasing union of Z-invariant BH-spaces. This implies that for any open compact subgroup H ⊆ G, V′_{b} is a jointly continuous topological module over the algebra D^{ess}(H, Z(G)) = D^{la}(H) ˆ⊗_{D^{la}(Z∩H)} C^{an}(Ẑ), where Ẑ is the rigid space^{1} parametrising characters of Z. The algebra D^{ess}(H, Z(G)) is also a Fréchet-Stein algebra [Eme04, 5.3.22], and we say V is essentially admissible locally analytic if V′_{b} is coadmissible as a module over D^{ess}(H, Z(G)) for one (and hence every) open compact H. We write Rep_{top}(G) for the category of topological representations of G, with morphisms being G-equivariant continuous linear maps. We consider the following full subcategories:

• Rep_{cts}(G): continuous representations

• Rep_{cts,ad}(G): admissible continuous representations

• Rep_{top,c}(G): topological representations on compact type spaces

• Rep_{la,c}(G): locally analytic representations

• Rep_{la,c}^{z}(G): Z-tempered representations

• Rep_{la,ad}(G): admissible locally analytic representations

• Rep_{ess}(G): essentially admissible locally analytic representations

• Rep_{cts,fd}(G): finite-dimensional continuous representations

• Rep_{la,fd}(G): finite-dimensional locally analytic representations Each of these categories is stable under passing to closed G-invariant submodules. The categories Rep_{cts,ad}(G), Rep_{la,ad}(G) and Rep_{ess}(G) have the additional property that all morphisms are strict, with closed image. The definition of Rep_{top} and Rep_{top,c} makes sense if G is only assumed to be a semigroup. We will need one more category of representations of semigroups: if S is a semigroup which contains a locally L-analytic subgroup S_{0}, we define Rep_{la,c}^{z}(S) to be the full subcategory of Rep_{top,c}(S) of representations which are locally analytic as representations of S_{0}, and can be written as an increasing union of Z(S)-invariant BH-subspaces. We will, in fact, only use this when either S is a group (in which case the definition reduces to the definition of Rep_{la,c}^{z} above) or S is commutative.

1The space Ẑ is in fact defined over L, but we shall always consider it as a rigid space over K by base extension.

Documenta Mathematica 16 (2011) 1–31

Remark. If V ∈ Rep_{top}(G), V′ naturally carries a right action of G. Hence we follow the conventions of [Eme04, §5.1] by defining the algebra structures on D(G) and its cousins in such a way that the Dirac distributions satisfy δ_{g} ⋆ δ_{h} = δ_{hg}, so all of our modules are left modules. The alternative is to consider the contragredient action on V′, which is the convention followed in [ST02b, ST03]; we do not adopt this approach here as we will occasionally wish to consider semigroups rather than groups.

2.2 Smooth and locally isotypical vectors

We now present a slight generalisation of the theory of [Eme04, §7]. Let G be a locally compact topological group and H ⊴ G closed. We suppose that G admits a countable basis of neighbourhoods of the identity consisting of open compact subgroups; this is automatic if G is locally p-adic analytic, for instance. The action of any g ∈ G on H by conjugation gives a homeomorphism from H to itself, so the conjugation action of G preserves the set of open compact subgroups of H.

Definition 2.1. Let V be an (abstract) K-vector space with an action of G. We say a vector v ∈ V is H-smooth if there is an an open compact subgroup U of H such that Uv = v.

Our assumptions imply that the space V_{H−sm} of H-smooth vectors is Ginvariant.

Definition 2.2 ([Eme04, 7.1.1]). Suppose V ∈ Rep_{top}(G). We define

V_{H−st.sm} = \underrightarrow{lim}_{\begin{matrix}U⊆H \\ U open \end{matrix}} V^{U},

equipped with the locally convex inductive limit topology.

Clearly V_{H−st.sm} can be identified with V_{H−sm} as an abstract K-vector space, but the inductive limit topology on the former is generally finer than the subspace topology on the latter. It is clear that the action of G on V induces a topological action on V_{H−st.sm}, so (−)_{H−st.sm} is a functor from Rep_{top}(G) to itself, and the natural injection V_{H−st.sm} ↪ V is G-equivariant. We say V is strictly H-smooth if this map is a topological isomorphism.

Proposition 2.3.

(i) If V ∈ Rep_{cts}(G), then V_{H−st.sm} ∈ Rep_{cts}(G).

(ii) If V ∈ Rep_{top,c}(G), then V_{H−st.sm} is of compact type and the natural map V_{H−st.sm} → V is a closed embedding.

Proof. To show (i), we argue as in [Eme04, 7.1.10]. We let G_{0} be an open compact subgroup of G and (H_{i})_{i≥0} a decreasing sequence of open compact subgroups of H satisfying ⋂_{i} H_{i} = {1} and with each H_{i} normal in G_{0}; it is clear that we may do this, by our assumption on G. We set H_{i} = G_{i} ∩H. Then V^{H_{i}} is a G_{0}-invariant closed subspace of V, and letting V_{i} denote the kernel of the “averaging” map V^{H_{i}} → V^{H_{i−1}}, we have V^{H−st.sm} = ⨁_{i} V_{i}. Since each V_{i} is in Rep_{cts}(G_{0}), V_{H−st.sm} ∈ Rep_{cts}(G_{0}), which implies it is in Rep_{cts}(G). Statement (ii) depends only on V as an H-representation, so we are reduced to the case of [Eme04, 7.1.3].

It follows from (ii) that for V ∈ Rep_{top,c}(G) we do not need to distinguish between V_{H−st.sm} and V_{H−sm}. Moreover, we see that if V ∈ Rep_{la,c}(G) or any of the subcategories of admissible representations introduced above, V_{H−st.sm} has the same property.

Definition 2.4. Let V, W be abstract K-vector spaces with an action of G. We say a vector v ∈ V is locally (H, W)-isotypic if there is an integer n, an open compact subgroup U of H, and a U-equivariant linear map W^{n} → V whose image contains v.

The locally (H, W)-isotypic vectors clearly form a G-invariant subspace of V, since H is normal in G. By construction, this is the image of the evaluation map Hom_{H−sm}(W, V) ⊗_{K} W → V, where Hom_{H−sm}(W, V) denotes the subspace of H-smooth vectors in Hom_{K}(W, V) = W′ ⊗_{K} V with its diagonal G-action. If V and W are in Rep_{top}(G), with W finite-dimensional, then Hom_{K}(W, V) has a natural topology (as a direct sum of finitely many copies of V) and we write Hom_{H−st.sm}(W, V) for Hom_{K}(W, V)_{H−st.sm}, with its inductive limit topology as above. Then Hom_{H−st.sm}(W, V) ⊗_{K} W is an object of Rep_{top}(G) with a natural morphism to V. We let V_{(H,W)−liso} denote the image of Hom_{H−st.sm}(W, V)⊗_{K} W in V, endowed with the quotient topology from the source (which is generally finer than the subspace topology on the target). We say V is strictly locally (H, W)-isotypical if the map V_{(H,W)−liso} → V is a topological isomorphism.

Definition 2.5. We say W is H-good if W is finite-dimensional, and for any open compact subgroup U ⊆ H, End_{U}(W) = End_{H}(W) = End_{G}(W).

Proposition 2.6. Suppose W is H-good, with B = End_{G}(W). Then for any representation V of G on an abstract K-vector space, the natural map

Hom_{K}(W, V)_{H−sm} ⊗_{B} W → V

is a G-equivariant injection. Dually, for any abstract right B-module X with a B-linear G-action which is smooth restricted to H, the natural map

X → Hom_{K}(W, X ⊗_{B} W)_{H−sm}

is an isomorphism.

Proof. If G = H, the first statement is [Eme04, 4.2.4] (the assumption in op.cit. that W be algebraic is only used to show that W is H-good). For the general case, the map exists and is injective at the level of H-representations, so it suffices to note that the assumption on W implies that the left-hand side has a well-defined G-action, for which the map is G-equivariant. For the second part, it suffices to show that the map restricts to an isomorphism X^{U} → Hom_{U}(W, X ⊗_{B} W) for any open U ⊆ H. Since W is faithful as a Bmodule by construction, the natural map is an injection. Since X is smooth as an H-representation, any vector in the left-hand side is in Hom_{U}(W, X^{U′} ⊗_{B}W) for some U′, which we may assume to be normal in U. However, we have

Hom_{U}(W, X^{U′} ⊗_{B} W) ⊆ Hom_{U′}(W, X^{U′} ⊗_{B} W) = X^{U′} ⊗_{B} Hom_{U′}(W, W).

and since W is H-good, we have Hom_{U′}(W, W) = B, so Hom_{U′}(W, X^{U′} ⊗_{B} W) = X^{U′}. Passing to U/U′-invariants gives the result.

Combining the preceding results shows that for W an H-good representation, the two functors

Hom_{H−st.sm}(W, −) and − ⊗_{B}W

are mutually inverse equivalences between the categories of strictly locally (H, W)-isotypical representations of G and strictly H-smooth Grepresentations on right B-modules.

Proposition 2.7. If H is a locally L-analytic group, and V is in Rep_{top}(G) ∩ Rep_{la,c}(H), then there is a topological isomorphism V_{H−st.sm} ≅ V^{𝔥}, where 𝔥 is the Lie algebra of H. More generally, if W is an H-good locally analytic representation of G, V_{(H,W)−liso} ≅ Hom_{𝔥}(W, V) ⊗_{B} W.

Proof. Clear from proposition 2.3(i), since a vector v ∈ V is in V_{H−sm} if and only if it is 𝔥-invariant.

3 Preservation of admissibility

3.1 Spaces of invariants

In this section we consider a group G and a normal subgroup H, and consider the functor of H-invariants V ↦ V^{H} : Rep_{top}(G) → Rep_{top}(G/H). Our aim is to show that this preserves the various subcategories of admissible representations introduced in the previous section.

Proposition 3.1. If V is an admissible Banach representation of a locally p-adic analytic group G, and H ⊴ G is a closed normal subgroup, then V^{H} is an admissible Banach representation of G/H.

Proof. Suppose first G is compact, so D(G) is Noetherian. Since H is normal and acts continuously on V, V^{H} is a G-invariant closed subspace; so (V^{H})′ is a D(G)-module quotient of a finitely-generated D(G)-module, and hence is a finitely-generated D(G)-module. However, the closed embedding C(G/H) ↪ C(G) dualises to a surjection D(G) → D(G/H), and it is clear that the D(G)-action on (V^{H})′ factors through this surjection. Hence (V^{H})′ is finitely-generated over D(G/H). In the general case, let G_{0} be a compact open subgroup of G and H_{0} = G_{0} ∩H. Then G_{0}/H_{0} is an open compact subgroup of G/H. By the above, V^{H_{0}} is an admissible continuous G_{0}/H_{0}-representation. Since V^{H} is a closed G_{0}/H_{0}-invariant subspace of V^{H_{0}} it is also admissible continuous as a representation of G_{0}/H_{0} and hence of G/H.

We now suppose G is a locally L-analytic group. We write H ⊴_{L} G to mean that H is a closed normal subgroup of G and the ℚ_{p}-subspace Lie(H) ⊆ Lie(G) is in fact an L-subspace, so H and G/H also inherit locally L-analytic structures.

Proposition 3.2. If V is an admissible locally analytic representation of G, and H ⊴_{L} G. Then V^{H} is an admissible locally analytic representation of G/H.

Proof. As above, we may assume G is compact. As in the Banach case, we note that V^{H} is a closed G-invariant subspace of V, so it is an admissible locally analytic G-representation [ST03, 6.4(ii)] on which the action of G factors through G/H. Hence the action of D^{la}(G) on (V^{H})′ factors through D^{la}(G/H). Since the natural map C^{la}(G/H) → C^{la}(G) is a closed embedding, D^{la}(G/H) is a Hausdorff quotient of D^{la}(G) and hence a coadmissible D^{la}(G)-module, and so by [ST03, 3.8] we see that (V^{H})′_{b} is coadmissible as a D(G/H)-module as required.

We now assume that G is a locally L-analytic group with Z(G) topologically finitely generated, and H ⊴_{L} G. In this case Z(G/H) may be much larger than Z(G)/(Z(G) ∩ H), as in the case of ℚ_{p}^{×} ⋉ ℚ_{p}; so an element of Rep_{la,c}^{z}(G) on which H acts trivially need not lie in Rep_{la,c}^{z}(G/H). Moreover, it is not obvious that Z(G/H) need be topologically finitely generated if Z(G) is so. We shall therefore assume that G is a direct product H × J, with H, J ⊴_{L} G, and Z(H) and Z(J) are both topologically finitely generated.

Proposition 3.3. In the above situation, for any essentially admissible locally analytic G-representation V, the space V^{H} is an essentially admissible locally analytic representation of J.

Proof. By [Eme04, 6.4.11], any closed invariant subspace of an essentially admissible representation is essentially admissible; so it suffices to assume that V = V^{H}. Let J_{0} ⊆ J and H_{0} ⊆ H be open compact subgroups. Then G_{0} = J_{0}×H_{0} is an open compact subgroup of G. We have Z(G) = Z(H)×Z(J),

and hence { \atop Z(G)} = { \atop Z(H)} × { \atop Z(J)}.

We now unravel the tensor products to find that the algebra

D^{ess}(G_{0}, Z(G)) = D^{la}(G_{0}) ˆ⊗_{D^{la}(G_{0}∩Z(G))} C^{an}({ \atop Z(G)})

decomposes as

(D^{la}(H_{0}) ˆ⊗_{K} D^{la}(J_{0}))  ˆ⊗_{D^{la}(H_{0}∩Z(H)) ˆ⊗_{K} D^{la}(J_{0}∩Z(J))}(C^{an}({ \atop Z(H)}) ˆ⊗_{K} C^{an}({ \atop Z(J)}))

= (D^{la}(H_{0}) ˆ⊗_{D^{la}(H_{0}∩Z(H))} C^{an}({ \atop Z(H)})) ˆ⊗_{K} (D^{la}(J_{0}) ˆ⊗_{D^{la}(J_{0}∩Z(J))} C^{an}({ \atop Z(J)}))

= D^{ess}(H_{0}, Z(H)) ˆ⊗_{K} D^{ess}(J_{0}, Z(J)).

By assumption, the action of D^{ess}(H_{0}, Z(H)) on V′_{b} factors through the augmentation map to K; so the action of D^{ess}(G_{0}, Z(G)) factors through D^{ess}(J_{0}, Z(J)). Since D^{ess}(J_{0}, Z(J)) is a Hausdorff quotient of D^{ess}(G_{0}, Z(G)), it is a coadmissible D^{ess}(G_{0}, Z(G))-algebra, and thus V′_{b} is a coadmissible D^{ess}(J_{0}, Z(J))-module as required.

3.2 Admissible representations of product groups

In this section, we’ll recall the theory presented in [Eme04, §7] of representations of groups of the form G × Γ, where G is a locally L-analytic group and Γ an arbitrary locally profinite (locally compact and totally disconnected) topological group. This will allow us to give more “global” formulations of the results of the previous section. Let ∗ denote one of the set {“admissible Banach”, “admissible locally analytic”, “essentially admissible locally analytic”}, so we shall speak of “∗-admissible representations”. Whenever we consider essentially admissible representations we will assume that the groups concerned have topologically finitely generated centre, so the concept is well-defined.

Definition 3.4 ([Eme04, 7.2.1]). A ∗-admissible representation of (G, Γ) is a locally convex K-vector space V with an action of G × Γ such that

• For each open compact subgroup U ⊆ Γ, V^{U} has property ∗ as a representation of G (in the subspace topology);

• V is a strictly smooth Γ-representation in the sense of definition 2.1.

Remark. Our terminology is slightly different from that of [Eme04], where such representations are described as ∗-admissible representations of G × Γ. We adopt the formulation above in order to avoid ambiguity when Γ is also a locally analytic group.

The results of the preceding section can be combined to prove:

Proposition 3.5. If G and H are locally L-analytic groups, V is a ∗- representation of G × H, and Z(H) is compact if ∗ = “essentially admissible locally analytic”, then the space

V_{H−st.sm} = \underrightarrow{lim}_{U⊆H open compact} V^{U}

is a ∗-admissible representation of (G, H).

Proof. Since the natural maps V^{U} ↪ V^{U′} for U′ ⊆ U are closed embeddings, the map V^{U} ↪ V_{H−st.sm} is also a closed embedding [Bou87, page II.32]; and its image is clearly (V_{H−st.sm})^{U}, so it suffices to check that V^{U} has property ∗ for each U. In the admissible Banach case, this is clear from proposition 3.1. In the admissible locally analytic case, it likewise follows from proposition 3.2. In the essentially admissible case, it suffices to note that the assumption on Z(H) implies that V is essentially admissible as a representation of G×H if and only if it is essentially admissible as a representation of G×U for any open compact U ⊆ H; so we are in the situation of proposition 3.3.

A slightly more general version of this applies to groups of the form G×H ×J, where G and H are locally L-analytic and J is an arbitrary locally compact topological group.

Theorem 3.6. Let V be a ∗-admissible representation of (G × H, J), where Z(H) is compact in the essentially admissible case. Then V_{H−st.sm} is a ∗- admissible representation of (G, H × J).

Proof. We have

V_{H−st.sm} = (V_{J−st.sm})_{H−st.sm} = \underrightarrow{lim}_{U⊆H,U′⊆J} V^{U×U′},

which is clearly a strict inductive limit; and V^{U×U′} is the U-invariants in the ∗-admissible G × H-representation V^{U′}, and hence an admissible Grepresentation. The open compact subgroups of H × J of the form U × U′ are cofinal in the family of all open compact subgroups, so V_{H−st.sm} is a ∗- admissible (G, H × J)-representation as required.

We write Rep_{cts,ad}(G, Γ) for the category of admissible continuous (G, Γ)-rep-resentations, and similarly for the other admissibility conditions.

3.3 Ordinary parts and Jacquet modules

Let G be a connected reductive algebraic group over L, and P a parabolic subgroup of G with Levi factor M. We write Z = Z(M), D = M^{ss}. We use Roman letters G, P, M, Z, D for the L-points of these, which are locally L-analytic groups. Note that the multiplication map Z × D → M has finite kernel and cokernel, and hence a representation of M has property ∗ if and only if it has the corresponding property as a representation of Z × D. Suppose that V ∈ Rep_{cts,adm}(G). We say V is unitary if the topology of V can be defined by a G-invariant norm (or equivalently if V contains a G-invariant separated open lattice); this is automatic if G is compact, but not otherwise. The category Rep_{u,adm}(G) of unitary admissible Banach representations of G over K is equivalent to Mod_{G}^{ϖ−adm}(O_{K})_{ℚ}, where Mod_{G}^{ϖ−adm}(O_{K}) is the category considered in [Eme10, 2.4.5] and the subscript ℚ denotes the category with the same objects but all Hom-spaces tensored with ℚ. In [Eme10, §3], Emerton constructs the ordinary part functor

Ord_{P} : Mod_{G}^{ϖ−adm}(O_{K}) → Mod_{M}^{ϖ−adm}(O_{K}).

This functor is additive, so it extends to a functor

Ord_{P} : Rep_{u,adm}(G) → Rep_{u,adm}(M).

It is easy to extend this to representations of product groups of the type considered above. Let Γ be a locally profinite topological group, and V a unitary admissible Banach (G, Γ)-representation (i.e. admitting a G × Γ-invariant norm). We define

Ord_{P} (V) = \underrightarrow{lim}_{\begin{matrix}U⊆Γ \\ open \end{matrix}} Ord_{P} (V^{U}).

Given any subgroups U′ ⊆ U, there is an “averaging” map π : V^{U′} → V^{U}; and we may write V^{U′} as a locally convex direct sum V^{U′} = V^{U} ⊕ V^{π}, where V^{π} denotes the kernel of π. Since the ordinary part functor commutes with direct sums, we find that Ord_{P} (V^{U′}) = Ord_{P} (V^{U}) ⊕ Ord_{P} (V^{π}); thus the natural map Ord_{P} (V^{U}) → Ord_{P} (V^{U′}) is a closed embedding, and if U′ ⊴ U, we have Ord_{P} (V^{U′})^{U} = Ord_{P} (V^{U}). Passing to the direct limit, we have Ord_{P} (V)^{U} = Ord_{P} (V^{U}), and Ord_{P} (V) is an admissible Banach (M, Γ)-representation. An identical argument applies to the Jacquet module functor J_{P} : Rep_{ess}(G) → Rep_{ess}(M) of [Eme06a] (and indeed to any functor which preserves direct sums). Combining this with theorem 3.6 above, we have:

Proposition 3.7.

(i) If V ∈ Rep_{u,ad}(G, Γ) and W ∈ Rep_{cts,fd}(M), then

Hom_{D−st.sm}(W, Ord_{P} V) ∈ Rep_{cts,ad}(Z, D × Γ).

Moreover, Hom_{D−st.sm}(W, Ord_{P} V) is unitary if W is.

(ii) If V ∈ Rep_{ess}(G, Γ) and W ∈ Rep_{la,fd}(M), and 𝔡 = Lie D, then

Hom_{D−st.sm}(W, J_{P} V) = Hom_{𝔡}(W, J_{P} V) ∈ Rep_{ess}(Z, D × Γ).

4 Jacquet modules of admissible representations

As in section 3.3 above, let G be the L-points of a connected reductive algebraic group over L, and P a parabolic subgroup with Levi subgroup M. Proposition 3.7(ii) gives us a copious supply of essentially admissible locally analytic representations of the torus Z = Z(M): for any V ∈ Rep_{ess}(G), any open compact U ⊆ D = M^{ss}, and any finite-dimensional M-representation W, Hom_{U}(W, J_{P} V) = (W′ ⊗_{K} J_{P} V)^{U} ∈ Rep_{ess}(Z). These correspond, by the equivalence of categories of [Eme06b, 2.3.2], to coherent sheaves on the rigid space Ẑ. For V ∈ Rep_{ess}(Z), we will write Exp V for the support of the sheaf corresponding to V, a reduced rigid subspace of Ẑ. In this section, we’ll prove two results describing the geometry of the rigid spaces Exp Hom_{U}(W, J_{P} V), for U ⊆ D open compact, under additional assumptions on V. These generalise the corresponding results in [Eme06a] when P is a Borel subgroup.

4.1 Compact maps

We begin by generalising some results from [Eme06a, §2.3] on compact endomorphisms of topological modules. Recall that a topological K-algebra is said to be of compact type if it can be written as an inductive limit of Banach algebras, with injective transition maps that are both algebra homomorphisms and compact as maps of topological K-vector spaces. If A is such an algebra, then a topological A-module is said to be of compact type if it is of compact type as a topological K-vector space. In this situation, we have the following definition of a compact morphism (op.cit., def. 2.3.3):

Definition 4.1. A continuous A-linear morphism ϕ : M → N between compact type topological A-modules is said to be A-compact if there is a commutative diagram

\xymatrix{M \ar[rr]^{ϕ} \ar[dr]^{α} &  & N \\  & N_{1} \ar[ur]^{β} \\ V \ar[ur]^{γ} \ar@{-->}[uurr]}  (4.1)

where N_{1} is a compact type topological A-module, α and β are continuous A-linear maps, V is a compact type K-vector space, and γ is a continuous Klinear map for which A ˆ⊗_{K} V → N_{1} is surjective, and the composite dashed arrow is compact as a map of compact type K-vector spaces.

Lemma 4.2. If M is a compact type module over a compact type topological K-algebra A; ϕ : M → M is an A-compact map; N is a finitely-generated module over a finite-dimensional K-algebra B; and ψ : N → N is K-linear, then the map ϕ ⊗ ψ : M ⊗_{K} N → M ⊗_{K} N is (A ⊗_{K} B)-compact.

Proof. We may assume without loss of generality that ψ is the identity, by [Eme06a, 2.3.4(i)]. This case follows immediately by tensoring each of the spaces in the diagram with N.

Lemma 4.3. Let σ : A → A′ be a finite morphism of compact type topological K-algebras, and ϕ : M → N a morphism of topological A′-modules which is A′-compact. Then ϕ is A-compact.

Proof. By assumption, we have a diagram as in lemma 4.1, where the map A′ ˆ⊗_{K} V → N_{1} is surjective. Let a_{1}, …, a_{k} be a set of elements generating A′ as an A-module, let V′ = V^{k}, and define the map γ′ : V′ → N_{1} by (v_{1}, …, v_{k}) ↦ ∑ a_{i}γ(v_{i}). Then it is clear that 1 ˆ⊗ γ′ gives a surjection A ˆ⊗_{K} V^{k} → N_{1}. Furthermore, the composite ϕ ∘ γ′ : V′ → N is the map (v_{1}, …, v_{k}) ↦ ∑ β(a_{i}γ(v)). As β is a morphism of A′-modules, this equals ∑ a_{i}(β ∘γ)(v), which is clearly compact (since β∘γ is). So the map γ′ : V′ → N_{1} witnesses ϕ as an A-compact map.

4.2 Twisted distribution algebras

Let L be a finite extension of ℚ_{p}, and G a locally L-analytic group. Let (H_{n})_{n≥0} be a decreasing sequence of good L-analytic open subgroups of G, in the sense of [Eme04, §5.2], such that

• the subgroups H_{n} form a basis of neighbourhoods of the identity in G;

• H_{n} is normal in H_{0} for all n;

• the inclusion H_{n+1} ↪ H_{n} extends to a morphism of rigid spaces between the underlying affinoid rigid analytic groups ℍ_{n+1} ↪ ℍ_{n}, which is relatively compact.

Such a sequence certainly always exists, since the choice of H_{0} determines a Lie O_{L}-lattice 𝔥 in the Lie algebra of G, and we may take H_{n} to be the subgroup attached to the sublattice π^{n}𝔥. We may use this sequence to write the topological K-algebra A := D^{la}(H_{0}) = C^{la}(H_{0})′_{b} as an inverse limit of the spaces A_{n} := D(ℍ_{n}^{∘}, H_{0}) = [C(H_{0})_{ℍ_{n}^{∘}−an}]′_{b}. For all n, A_{n} is a compact type topological K-algebra, and the sequence (A_{n})_{n≥0} is a weak Fréchet-Stein structure on A. We begin with a construction related to the “untwisting isomorphism” of [Eme04, 3.2.4]. Let (ρ, W) be any finite-dimensional K-representation of H_{0}, and let E = End_{K} W. We consider the following commutative diagram of K-vector spaces:

\xymatrix{ & K[H_{0}] ⊗_{K} E \ar[dd]^{\begin{matrix}g⊗m↦ \\ g⊗ρ(g)m \end{matrix}}_{γ} \\ K[H_{0}] \ar@{-->}[ur]^{g↦g⊗1}_{α} \ar[dr]_{g↦g⊗ρ(g)}^{β} \\  & K[H_{0}] ⊗_{K} E}  (4.2)

Here α and β are ring homomorphisms, and although γ is not a ring homomorphism, it satisfies the relation γ(α(x)y) = β(x)γ(y), so it intertwines the two K[H_{0}]-module structures on K[H_{0}] ⊗_{K} E given by α and β. Furthermore γ is clearly invertible. We now assume that (ρ, W) is locally analytic (when W is equipped with its unique Hausdorff locally convex topology).^{2} Hence there is an integer n(ρ) such that W_{ℍ_{n}^{∘}−an} = W for all n ≥ n(ρ).

Proposition 4.4. Let n ≥ n(ρ). Then there exist unique continuous maps

α_{n}, β_{n} : A_{n} → A_{n} ⊗_{K} End(W) and γ_{n} : A_{n} ⊗_{K} End(W) →^{∼} A_{n} ⊗_{K} End W

extending the maps α, β, γ above.

Proof. Taking the (algebraic) K-dual of the diagram (4.2), we have a diagram

\xymatrix{ & F(H_{0}, E′) \ar[dl]^{α′} \\ F(H_{0}, K) \\  & F(H_{0}, E′) \ar[uu]^{γ′} \ar@{-->}[ul]_{β′}}

where for K-vector space V, F(H_{0}, V) indicates the K-vector space of arbitrary functions H_{0} → V. One finds that for a function f : H_{0} → E′, we have α′(f)(m) = f(m)(1) and β′(f)(m) = f(m)(ρ(m)), while γ′(f)(m) = x ↦ f(ρ(m)x). All of these maps manifestly preserve the subspaces of ℍ_{n}^{∘}-analytic functions for n ≥ n(ρ), and are continuous for the natural topologies of these subspaces; so there are corresponding maps between the duals of these subspaces, as required.

Corollary 4.5. For each n ≥ n(ρ), the map β_{n} makes B_{n} = A_{n} ⊗_{K} End W a finitely-generated topological A_{n}-module, and the natural map B_{n+1} → B_{n} induces an isomorphism A_{n} ˆ⊗_{A_{n+1}} B_{n+1} →^{∼} B_{n}.

2If L = ℚ_{p} this is equivalent to the (a priori weaker) assumption that (ρ, W) is continuous.

This follows from the p-adic analogue of Cartan’s theorem, which states that any continuous homomorphism between two ℚ_{p}-analytic groups is locally analytic; see [Ser92, Part II, §V.9].

Documenta Mathematica 16 (2011) 1–31

Proof. This is clearly true for the A_{n}-module structure on B_{n} given by α_{n}, so it follows for the β_{n}-structure (since the untwisting isomorphisms γ_{n} and γ_{n+1} are compatible with the map B_{n+1} → B_{n}).

Proposition 4.6. Let n ≥ n(ρ) and let X be a compact type topological A_{n}- module. Then the diagonal H_{0}-action on X ⊗_{K} W extends to a topological A_{n}-module structure. Moreover, if n ≥ n(ρ) + 1, we have an isomorphism of topological A_{n−1}-modules

A_{n−1} ˆ⊗_{A_{n}} (X ⊗_{K} W) →^{∼} (A_{n−1} ˆ⊗_{A_{n}} X) ⊗_{K} W.

Proof. We clearly have commuting, K-linear, continuous actions of A_{n} and End W on X ⊗_{K} W, so we obtain an action of A_{n} ⊗_{K} End W. Pulling back via the map β_{n}, we obtain an A_{n}-module structure, which clearly restricts to the diagonal action of H_{0}. The isomorphism follows from the last statement of the preceding corollary via the associativity of the tensor product, since

A_{n−1} ˆ⊗_{A_{n}} (X ⊗_{K} W)

=(A_{n−1} ˆ⊗_{A_{n}} B_{n}) ˆ⊗_{B_{n}} (X ⊗_{K} W)
=B_{n−1} ˆ⊗_{B_{n}} (X ⊗_{K} W)
=(A_{n−1} ˆ⊗_{A_{n}} X) ⊗_{K} W.

4.3 Twisted Jacquet modules

We now return to the situation considered above, so G is the group of L-points of a reductive algebraic group G over L as above, with P a parabolic subgroup, M a Levi subgroup of P, N the unipotent radical, and Z = Z(M). We choose a sequence (H_{n})_{n≥0} of good L-analytic open subgroups of G admitting rigid analytic Iwahori decompositions ℍ_{n} = \overline{ℕ}_{n} × 𝕄_{n} × ℕ_{n}, as in [Eme06a, 4.1.6]. We also impose the additional condition that 𝕄_{n} = ℤ_{n} × 𝔻_{n} where ℤ_{n} and 𝔻_{n} are the affinoid subgroups underlying good analytic open subgroups of Z and of D = M^{ss}; it is clear that we can always do this (by exactly the same method as in Emerton’s case). We let Z^{+} be the submonoid {z ∈ Z(M) : zN_{0}z^{−1} ⊆ N_{0}} of Z. Our starting point is the following, which is part of the proof of [Eme06a, 4.2.23]:

Proposition 4.7. Let V be an admissible locally analytic representation of G. Then for all n ≥ 0, the action of M_{0} × Z^{+} on the space

U_{n} = (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0})} V′_{b})_{N_{0}}

extends to an A_{n}[Z^{+}]-module structure. Moreover, the transition map A_{n} ˆ⊗_{A_{n+1}} U_{n+1} → U_{n} is A_{n}-compact and Z^{+}-equivariant, and there is some z ∈ Z^{+} (independent of n) such that there exists a map α : U_{n} → A_{n} ˆ⊗_{A_{n+1}} U_{n+1} making the following diagram commute:

\xymatrix{A_{n} ˆ⊗_{A_{n+1}} U_{n+1} \ar[r] \ar[d]^{id ˆ⊗ z} & U_{n} \ar[d]^{z} \ar@{-->}[dl]^{α} \\ A_{n} ˆ⊗_{A_{n+1}} U_{n+1} \ar[r] & U_{n}.}  (4.3)

We now let Ũ_{n} = U_{n} ⊗_{K} W, where (W, ρ) is a fixed, finite-dimensional, continuous representation of M. By the last proposition of the preceeding section (taking the groups there denoted by G and H_{i} to be those we are now calling M and M_{i}), we have a diagonal A_{n}-module structure on Ũ_{n}, and there is also a diagonal action of Z^{+} on Ũ_{n} commuting with the M_{0}-action.

Proposition 4.8. For any n ≥ n(ρ) the following holds:

• Ũ_{n} is a compact type topological A_{n}-module, and the action of Z^{+} is A_{n}-linear.

• There is an A_{n+1}[Z^{+}]-linear map U_{n+1} → U_{n} such that the induced map A_{n} ˆ⊗_{A_{n+1}} Ũ_{n+1} → Ũ_{n} is A_{n}-compact.

• For any good z ∈ Z^{+}, we can find a map α̃ : U_{n} → A_{n} ˆ⊗_{A_{n+1}} Ũ_{n+1} such that the diagram corresponding to (4.3) commutes.

Also, the direct limit \underleftarrow{lim} U_{n} (with respect to the transition maps above) is isomorphic as a topological A[Z^{+}]-module to (V^{N_{0}} ⊗ W′)′_{b}.

Proof. Since Ũ_{n} is isomorphic to (U_{n})^{⊕ dim W} as a topological K-vector space, it is certainly of compact type, and we have already observed that it is a topological A_{n}-module for all n ≥ n(ρ). Furthermore the Z^{+}-action commutes with the M_{0}-action, and thus it must be A_{n}-linear by continuity. Moreover, we have an A_{n}-compact map A_{n} ˆ⊗_{A_{n+1}} U_{n+1} → U_{n}. Tensoring with the identity map gives a morphism of A_{n} ⊗ End W -modules (A_{n} ˆ⊗_{A_{n+1}} U_{n+1}) ⊗_{K} W → U_{n} ⊗_{K} W, which is A_{n} ⊗_{K} End W -compact by lemma 4.2. But the map β : A_{n} → A_{n} ⊗_{K} End W is a finite morphism, so by lemma 4.3, this map is A_{n}-compact. Finally, we know that there exists a map α : U_{n} → A_{n} ˆ⊗_{A_{n+1}} U_{n+1} through which z factors, and it is clear that if we define α̃ to be the map α ⊗ ρ(z) then the diagram corresponding to (4.3) commutes.

The preceding proposition asserts precisely that the hypotheses of [Eme06a, 3.2.24] are satisfied, and that proposition (and its proof) give us the following:

Corollary 4.9. The space X = [(V^{N_{0}} ⊗_{K} W′)_{fs}]′_{b} is a coadmissible C^{an}(Ẑ) ˆ⊗_{K} A-module, where (−)_{fs} denotes the finite-slope-part functor Rep_{top,c}(Z^{+}) → Rep_{la,c}^{z}(Z) of [Eme06a, 3.2.1]. Moreover, if (Y_{n})_{n≥0} is any increasing sequence of affinoid subdomains of Ẑ whose union is the entire space, then for any n ≥ n(ρ) we have

(C^{an}(Y_{n})^{†} ˆ⊗_{K} A_{n}) ˆ⊗_{(C^{an}(Ẑ) ˆ⊗_{K} A)} X = C^{an}(Y_{n})^{†} ˆ⊗_{K[Z^{+}]} Ũ_{n}.

By [Eme04, 3.2.9] we have X = [(V^{N_{0}} ⊗_{K} W′)_{fs}]′_{b} = [(V^{N_{0}})_{fs} ⊗_{K} W′]′_{b} = [J_{P} (V) ⊗_{K} W′]′_{b}, so the above corollary gives us a description of the strong dual of the W -twisted Jacquet module. We can now prove the first of the two main theorems of this section. Proposition 3.7(ii) above shows that for any V ∈ Rep_{ess}(G), (J_{P} (V) ⊗_{K} W′) ∈ Rep_{ess}(Z, D). Equivalently, for any open compact subgroup Γ ⊆ D, the space (J_{P} (V) ⊗_{K} W′)^{Γ} is an essentially admissible locally analytic Z-representation, and hence corresponds to a coherent sheaf on Ẑ. The previous corollary allows us to describe the support of this sheaf when V is admissible:

Theorem 4.10. Suppose V is an admissible locally analytic G-representation, W is a finite-dimensional locally analytic representation of M, and Γ is an open compact subgroup of D. Let E ⊆ Ẑ be the support of the coherent sheaf on Ẑ corresponding to (J_{P} (V) ⊗_{K} W′)^{Γ}. Then the natural map E → (Lie Z)′ (induced by the differentiation map Ẑ → (Lie Z)′) has discrete fibres.

Proof. Since we are free to replace the sequence (H_{n}) of subgroups of G with a cofinal subsequence, we may assume that Γ ⊇ D_{0}. So it suffices to prove the result for Γ = D_{0}. Furthermore, since the differentiation map Ẑ_{0} → (Lie Z)′ has discrete fibres, it suffices to show that for any character χ of Z_{0}, the rigid space

Exp (J_{P} (V) ⊗_{K} W′)^{D_{0},Z_{0}=χ} ⊆ Ẑ

is discrete. If χ does not extend to a character of M, then this space is clearly empty, so there is nothing to prove; otherwise, let us fix such an extension, which gives us an isomorphism (J_{P} (V) ⊗_{K} W′)^{D_{0},Z_{0}=χ} = [J_{P} (V) ⊗_{K} (W ⊗_{K} χ)′]^{M_{0}}. So we may assume without loss of generality that χ is the trivial character, and it suffices to show that

C^{an}(Y_{n})^{†} ˆ⊗_{C^{an}(Ẑ)} [(J_{P} (V) ⊗_{K} W′)^{M_{0}}]′_{b}

is finite-dimensional over K for all n, or (equivalently) all sufficiently large n. If we take the completed tensor product of both sides of the formula in corollary 4.9 with C^{an}(Y_{n})^{†}, regarded as a C^{an}(Y_{n})^{†} ˆ⊗_{K} A_{n}-algebra via the augmentation map A_{n} → K, we have

C^{an}(Y_{n})^{†}  ˆ⊗_{(C^{an}(Y_{n})^{†} ˆ⊗_{K} A_{n})}(C^{an}(Y_{n})^{†} ˆ⊗_{K} A_{n})  ˆ⊗_{(C^{an}(Ẑ) ˆ⊗_{K} A)}[J_{P} (V) ⊗_{K} W′]′_{b}

= C^{an}(Y_{n})^{†} ˆ⊗_{(C^{an}(Y_{n})^{†} ˆ⊗_{K} A_{n})} (C^{an}(Y_{n})^{†} ˆ⊗_{K[Z^{+}]} Ũ_{n}) . (4.4)

The left-hand side of (4.4) simplifies as

C^{an}(Y_{n})^{†} ˆ⊗_{(C^{an}(Y_{n})^{†} ˆ⊗_{K} A_{n})} (C^{an}(Y_{n})^{†} ˆ⊗_{K} A_{n}) ˆ⊗_{(C^{an}(Ẑ) ˆ⊗_{K} A)} [J_{P} (V) ⊗_{K} W′]′_{b}
= C^{an}(Y_{n})^{†} ˆ⊗_{(C^{an}(Ẑ) ˆ⊗_{K} A)} [J_{P} (V) ⊗_{K} W′]′_{b}

= C^{an}(Y_{n})^{†} ˆ⊗_{C^{an}(Ẑ)} (K ˆ⊗_{A} [J_{P} (V) ⊗_{K} W′]′_{b})

= C^{an}(Y_{n})^{†} ˆ⊗_{C^{an}(Ẑ)} [(J_{P} (V) ⊗_{K} W′)^{M_{0}}]′_{b} .

Meanwhile, the right-hand side of (4.4) is

C^{an}(Y_{n})^{†} ˆ⊗_{(C^{an}(Y_{n})^{†} ˆ⊗_{K} A_{n})} (C^{an}(Y_{n})^{†} ˆ⊗_{K[Z^{+}]} Ũ_{n})

= C^{an}(Y_{n})^{†} ˆ⊗_{K[Z^{+}]} (K ˆ⊗_{A_{n}} Ũ_{n}) .

Any z ∈ Z^{+} that induces an A_{n}-compact endomorphism of Ũ_{n} will induce a K-compact endomorphism of K ˆ⊗_{A_{n}} Ũ_{n}, by [Eme06a, 2.3.4(ii)]. Such a z does exist, by hypothesis. Hence C^{an}(Y_{n})^{†} ˆ⊗_{K[Z^{+}]} (K ˆ⊗_{A_{n}} Ũ_{n}) is finitedimensional over K, by [Eme06a, 2.3.6]. Comparing the two sides of (4.4), we are done.

We also have a version of [Eme06a, 4.2.36] in this context.

Theorem 4.11. If V is an admissible locally analytic representation of G such that there is an isomorphism of H-representations V →^{∼} C^{la}(H)^{r}, for some open compact H ⊆ G and some r ∈ ℕ, then for any W and Γ, E = Exp (J_{P} (V) ⊗_{K} W′)^{Γ} is equidimensional of dimension d, where d is the dimension of Z.

Proof. As in [Eme06a], we may assume (by replacing the sequence (G_{n})_{n≥0} with a cofinal subsequence if necessary) that H = H_{0} and Γ ⊇ D_{0}. But then we can identify (J_{P} (V) ⊗_{K} W′)^{Γ} with a direct summand of (J_{P} (V) ⊗_{K} W′)^{D_{0}}; this identifies Exp (J_{P} (V) ⊗_{K} W′)^{Γ} with a union of irreducible components of Exp (J_{P} (V) ⊗_{K} W′)^{D_{0}}. We may therefore assume that in fact Γ = D_{0}. As a final reduction, letting U_{n} = (D(ℍ_{n}^{∘}, H_{0}) ˆ⊗_{D^{la}(H_{0})} V′_{b})_{N_{0}} as before,

we note that the untwisting isomorphism U_{n} →^{∼} D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0})^{r} ˆ⊗_{K} A_{n} (equation 4.2.39 in [Eme04]) can be extended to an isomorphism U_{n} ⊗_{K} W → D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0})^{r dim W} ˆ⊗_{K} A_{n}. We thus assume that W is the trivial representation. Following Emerton, we choose Banach spaces W_{n} such that the map D(\overline{ℕ}_{n+1}^{∘}, \overline{N}_{0})^{r} → D(\overline{ℕ}_{n}^{∘}, \overline{N}_{0})^{r} factors through W_{n}, and (exactly as in the Borel case) for a suitable z ∈ Z^{+} we have

J_{P} (V)′_{b} →^{∼} \underleftarrow{lim}_{n} K\{\{z, z^{−1}\}\} ˆ⊗_{K[z]} (W_{n} ˆ⊗_{K} A_{n}),

for some A_{n}-linear action of z on W_{n} ˆ⊗_{K} A_{n} which factors through D(\overline{ℕ}_{n+1}^{∘}, \overline{N}_{0})^{r} ˆ⊗_{K} A_{n}. Taking the completed tensor product with the map A_{n} → D(ℤ_{n}^{∘}, Z_{0}) given by the augmentation map of D_{0}, we have

[J_{P} (V)^{D_{0}}]′_{b} →^{∼} \underleftarrow{lim}_{n} K\{\{z, z^{−1}\}\} ˆ⊗_{K[z]} W_{n} ˆ⊗_{K} D(ℤ_{n}^{∘}, Z_{0}).

Let us write Ẑ_{0} as an increasing union of affinoid subdomains (X_{n})_{n≥0}, such that the natural map D^{la}(Z_{0}) →^{∼} C^{an}(Ẑ_{0}) → C^{an}(X_{n}) factors through D(ℤ_{n}^{∘}, Z_{0}). Extending scalars from D(ℤ_{n}^{∘}, Z_{0}) to C^{an}(Ẑ) via this map, the above formula becomes

[J_{P} (V)^{D_{0}}]′_{b} = \underleftarrow{lim}_{n} K\{\{z, z^{−1}\}\} ˆ⊗_{K[z]} W_{n} ˆ⊗_{K} C^{an}(X_{n}).

The action of z on W_{n} ˆ⊗_{K} C^{an}(X_{n}) is a C^{an}(X_{n})-compact morphism of an orthonormalizable C^{an}(X_{n})-Banach module, so the result follows by the methods of [Buz07].

5 Change of parabolic

We now consider the problem of relating the geometric objects arising from the above construction for two distinct parabolic subgroups.

5.1 Transitivity of Jacquet functors

Let us recall the definition of the finite-slope-part functor, which we have already seen in the previous section. We let Z be a topologically finitely generated abelian locally L-analytic group, and Z^{+} an open submonoid of Z which generates Z as a group. Then we have the following functor Rep_{top,c}(Z^{+}) → Rep_{la,c}^{z}(Z):

Definition 5.1 ([Eme06a, 3.2.1]). For any object V ∈ Rep_{top,c}(Z^{+}), we define

V_{fs} = L_{b,Z^{+}}(C^{an}(Ẑ), V),

endowed with the action of Z on the first factor.

Lemma 5.2. Let Z be a topologically finitely generated abelian group and Y a closed subgroup, and suppose Y^{+} and Z^{+} are submonoids of Y and Z satisfying the conditions above, with Y^{+} ⊆ Y ∩ Z^{+}. Then for all V ∈ Rep_{top,c}(Z^{+}), the natural map V_{Y −fs} → V induces an isomorphism

(V_{Y −fs})_{Z−fs} →^{∼} V_{Z−fs}.

Proof. Consider the canonical Z^{+}-equivariant map V_{Z−fs} → V. We note that V_{Z−fs} is in Rep_{la.c}^{z}(Z), and hence a fortiori in Rep_{la.c}^{z}(Y). Hence by the universal property of [Eme06a, 3.2.4(ii)], the above map factors through V_{Y −fs}. The factored map is still Z^{+}-equivariant, so by a second application of the universal property it factors through (V_{Y −fs})_{Z−fs}. This gives a continuous Z-equivariant map V_{Z−fs} → (V_{Y −fs})_{Z−fs}, which is clearly inverse to the map in the statement of the proposition.

Now suppose P_{1} and P_{2} are parabolic subgroups of the reductive group G over L, with P_{1} ⊇ P_{2}. We let N_{1}, N_{2} be their unipotent radicals, so we have a chain of inclusions G ⊇ P_{1} ⊇ P_{2} ⊇ N_{2} ⊇ N_{1}. Let us choose a Levi subgroup M_{2} of P_{2}, so P_{2} = N_{2} ⋊ M_{2}. There is a unique Levi subgroup M_{1} of P_{1} containing M_{2}; and P_{12} = P_{2} ∩ M_{1} is a parabolic subgroup of M_{1} of which M_{2} is also a Levi factor. We write Z_{1}, Z_{2} for the centres of M_{1} and M_{2}. All of the above are algebraic groups over L, so their L-points are locally Lanalytic groups; we denote these groups of points by the corresponding Roman letters.

Theorem 5.3.

1. For any unitary admissible continuous G-representation V, there is a unique isomorphism of admissible continuous M_{2}-representations

Ord_{P_{12}} (Ord_{P_{1}} V) = Ord_{P_{2}} V

commuting with the canonical lifting maps from both sides into V^{N_{2}}.

2. For any essentially admissible locally analytic G-representation V, there is a unique isomorphism of essentially admissible locally analytic M_{2}- representations

J_{P_{12}} (J_{P_{1}}V) = J_{P_{2}}V

commuting with the canonical lifting maps.

Proof. We begin by proving the second statement. We have N_{2} = N_{1} ⋊ N_{12}, where N_{12} = N_{2} ∩ M_{1} is the unipotent radical of P_{12}. Let N_{2,0} be an open compact subgroup of N_{2} which has the form N_{1,0}⋊N_{12,0}, for open compact subgroups of the two factors; such subgroups certainly exist, since the conjugation action of N_{1} on N_{12} is continuous.

For i = 1, 2 we write M_{i}^{+} for the submonoid of elements m ∈ M_{i} for which mN_{i,0}m^{−1} ⊆ N_{i,0} and m^{−1}\overline{N}_{i,0}m ⊆ \overline{N}_{i,0}, and Z_{i} = M_{i}^{+} ∩ Z_{i}. Then we have M_{2}^{+} ⊆ M_{1}^{+}, and in particular Z_{1}^{+} ⊆ Z_{2}^{+}. We have

J_{P_{1}}V = L_{b,Z_{1}^{+}} (C^{an}(Ẑ_{1}), V^{N_{1,0}})

endowed with the action of M_{1} = Z_{1} ×_{Z_{1}^{+}} M_{1}^{+} determined by the actions of Z_{1}

on C^{an}(Ẑ_{1}) and M_{1}^{+} on V^{N_{1,0}}. The restriction of this action to N_{12,0} is simply the action on the right factor (since N_{12,0} ⊆ M_{1,0} ⊆ M_{1}^{+}) and hence

(J_{P_{1}}V)^{N_{12,0}} = L_{b,Z_{1}^{+}} (C^{an}(Ẑ_{1}), (V^{N_{1,0}})^{N_{12,0}}) = L_{b,Z_{1}^{+}} (C^{an}(Ẑ_{1}), V^{N_{2,0}}) .

The Hecke operator construction of [Eme06a, §3.4] gives two actions of M_{2}^{+} on V^{N_{2,0}}, given respectively by m∘v = π_{N_{2,0}}mv and m∘v = π_{N_{12,0}}π_{N_{1,0}}mv, where the operators π_{N_{i,0}} are the averaging operators with respect to Haar measure on the subgroups N_{i,0}. Since N_{2,0} = N_{12,0} ⋉N_{1,0}, and the Haar measure on the product is the product of the Haar measures on the factors, these two actions coincide. Applying the preceding lemma with Z = Z_{2} and Y = Z_{1} gives the result. The statement for the ordinary part functor can be proved along similar lines, but it is easier to note that the functor Ord_{P} is right adjoint to the parabolic induction functor IndG ; \overline{P} [Eme10, 4.4.6], for \overline{P} an opposite parabolic to P. Since a composition of adjunctions is an adjunction, it suffices to check instead that Ind_{\overline{P}_{1}}^{G} Ind_{\overline{P}_{12}}^{M_{1}} U = Ind_{\overline{P}_{2}}^{G} U for any U ∈ Rep_{u,adm}(M_{2}). We may identify C(G, C(M_{1}, U)) with C(G × M_{1}, U). Evaluating at 1 ∈ M_{1} gives a map to C(G, U), and it is easy to check that this restricts to an isomorphism between the subspaces realising the two induced representations.

5.2 Hecke algebras and the canonical lifting

We now turn to studying the Jacquet functor in a special case; later we will combine this with the transitivity result above to deduce a general statement. As before, let G be a reductive algebraic group over L, and let H = [G, G], a semisimple group. There is a bijection between parabolics of G and H, given by P ↦ P′ = P ∩ H and P′ ↦ P = N_{G}(P′). We also choose an opposite parabolic \overline{P}, determining a Levi subgroup M of P, and also a Levi M′ of P′ in the obvious way. Write Z_{M}, Z_{M′} and Z_{G} for the centres of these subgroups, so Z_{M} is isogenous to Z_{M′} × Z_{G}. As before, we use Roman letters for the L-points of these algebraic groups. Let H_{0} be an open compact subgroup of H. We say H_{0} is decomposed with respect to P′ and \overline{P}′ if the product of the subgroups M′_{0} = H_{0} ∩ M′, N_{0} = H_{0} ∩ N and \overline{N}_{0} = H_{0} ∩ \overline{N} is H_{0}, for any ordering of the factors. We say an element m ∈ M is positive (for H_{0}) if mN_{0}m^{−1} ⊆ N_{0} and m^{−1}\overline{N}_{0}m ⊆ \overline{N}_{0} (see [Bus01, §3.1]). Let M^{⊕} ⊆ M be the monoid of positive elements, and Z_{M}^{⊕} its intersection with Z_{M}; and let H^{⊕}(M′_{0}) denote the subalgebra of the Hecke algebra H(M′_{0}) supported on M^{′+} = M^{+} ∩ M′. Note that M^{⊕} is contained in the monoid M^{+} of the previous section, and clearly has finite index therein. We say an element z ∈ Z_{M} is strongly positive if the sequences z^{n}N_{0}z^{−n} and z^{−n}\overline{N}_{0}z^{n} tend monotonically to \{1\}; if this holds, then z^{−1} and M^{⊕} together generate M. Such elements exist in abundance; any element whose pairing with the roots corresponding to P has sufficiently large valuation will suffice. In particular, there exist strongly positive elements in Z_{M′}.

Proposition 5.4. For any essentially admissible G-representation V, we have J_{P} (V) = (V^{N_{0}})_{Y −fs}, where Y is any closed subgroup of M that contains a strongly positive element. In particular, J_{P} (V) = J_{P′}(V).

Proof. For any open compact N_{0} ⊆ N, [Eme06a, lemma 3.2.29] and the discussion following it shows that V^{N_{0}} is in the category denoted therein by Rep_{la,c}^{z}(Z_{M}^{+}); thus the hypotheses of [Eme06a, prop 3.2.28] are satisfied for the subgroup Y = Z_{M′}. The conclusion of that proposition then states that J_{P} (V) = (V^{N_{0}})_{Z_{M} −fs} = (V^{N_{0}})_{Y −fs}.

We now lighten the notation somewhat by writing superscript + for ⊕, since the proposition shows that the distinction between M^{+} and M^{⊕} is unimportant from the perspective of Jacquet modules.

Proposition 5.5. Let j be the morphism H^{+}(M′_{0}) → H(H_{0}) constructed in [Bus01, §3.3]. Then the natural inclusion V^{H_{0}} ↪ V^{M′_{0}N_{0}} is H^{+}(M′_{0})- equivariant, where H^{+}(M′_{0}) acts via j on the first space and via its inclusion into H(M′_{0}) on the second.

Proof. Easy check.

Proposition 5.6. For any essentially admissible locally analytic G-represen-tation V which is smooth as an H-representation, the above inclusion induces an isomorphism

(V^{H_{0}})_{Z_{M′} −fs} →^{∼} (V^{M′_{0}N_{0}})_{Z_{M′} −fs} = J_{P} (V)^{M′_{0}}.

Moreover, there exists a direct sum decomposition

V^{H_{0}} = (V^{H_{0}})_{Z_{M′} −fs} ⊕ (V^{H_{0}})_{Z_{M′} −null}

where the summands are closed subspaces of V^{H_{0}}, stable under the action of Z_{G} and H(M′_{0}).

Proof. Let Q = V^{M′_{0}N_{0}}/V^{H_{0}}. By the left-exactness of the finite slope part functor [Eme06a, 3.2.6(ii)], there is a closed embedding

(V^{M′_{0}N_{0}})_{Z_{M′} −fs}/(V^{H_{0}})_{Z_{M′} −fs} ↪ Q_{Z_{M′} −fs}.

But since V is smooth as an H-representation, every element v ∈ V^{M′_{0}N_{0}} is in fact in V^{UM′_{0}N_{0}} for some open U ⊆ \overline{N}; any such U contains a Z_{M′}^{+}-conjugate of \overline{N}_{0}, so there is some z ∈ Z^{+} such that zv ∈ V^{\overline{N}_{0}M′_{0}}. Our hypothesis that H_{0} is decomposed implies that the averaging operator π_{N_{0}} : V^{𝔫} → V^{N_{0}} preserves V^{\overline{N}_{0}M′_{0}}, so z∘v = π_{N_{0}}(zv) ∈ V^{H_{0}}. Therefore Q is Z_{M′}^{+}-torsion, and thus clearly Q_{Z−fs} = 0. For the second statement, let z be any strongly positive element of Z_{M′}. By [Bus01, Theorem 1], there exists an integer n (depending only on P, H_{0} and z) such that for any smooth H-representation V, the action of z on V^{H_{0}} via j satisfies

V^{H_{0}} = z^{n}V^{H_{0}} ⊕ Ker(z^{n} | V^{H_{0}}),

with z invertible on the subspace z^{n}V^{H_{0}}. For representations V as in the statement, the subspace Ker(z^{n} | V^{H_{0}}) is clearly closed, and moreover z^{n} gives a continuous map from the essentially admissible Z_{G}-representation V^{H_{0}} to itself, so its image is also closed.

In particular, since V^{H_{0}} is an essentially admissible Z_{G}-representation, J_{P} (V)^{M′_{0}} is essentially admissible as a Z_{G}-representation, not just as a representation of the larger group Z_{G} × Z_{M′}/(Z_{M′} ∩ H_{0}). Remark. If H_{0} satisfies the stronger conditions of [Bus01, §1.2], we obtain a finer decomposition of V^{H_{0}} into a direct sum of closed Z_{G}-subrepresentations corresponding to Bernstein components of H.

5.3 Jacquet modules of locally isotypical representations

We now extend the results on H-smooth representations above to certain locally H-isotypical representations.

Proposition 5.7. If W is a twist of an absolutely irreducible algebraic representation of G, and P = MN is a parabolic subgroup of G with [M, M] = D, then End_{𝔡}(W^{N}) = K, so in particular the M-representation W^{N} is D-good.

Proof. The twist is of no consequence, so suppose that W is algebraic. Let us choose a maximal torus T in M, and a field K′ ⊃ K over which M is split; then there is a Borel subgroup B ⊆ P defined over K′ with Levi factor T. The theory of highest weights then shows that W is absolutely irreducible if and only if the highest weight space of W is 1-dimensional; applying this condition to W and to the M-representation W^{N}, we deduce that W^{N} is absolutely irreducible as an M-representation. Since M is isogenous to D × Z(M) and all absolutely irreducible representations of Z(M) are one-dimensional, it follows that W^{N} is in fact absolutely irreducible as a D-representation.

Proposition 5.8. If W ∈ Rep_{la,fd}(G) is H-good, with B = End_{𝔥}(W) = End_{G} W, and furthermore W^{𝔫} = W^{N}, then for any X ∈ Rep_{la,c}^{z}(G) which is smooth as an H-representation and has a right action of B, we have

J_{P} (X ⊗_{B} W) = J_{P} (X) ⊗_{B} W^{N}.

Proof. Compare [Eme06a, 4.3.4]. Since X is smooth as an H-representation it is certainly smooth as an N-representation. Arguing as in the proof of proposition 2.6, we have (X ⊗_{B} W)^{N_{0}} = X^{N_{0}} ⊗_{B} W^{N_{0}}, which by assumption equals X^{N_{0}} ⊗_{B} W^{N}. Passing to finite-slope parts now yields the result.

The condition W^{𝔫} = W^{N} is certainly satisfied for any W that is algebraic as a representation of N (since any open subgroup of N is Zariski-dense).

Proposition 5.9. Let W be a twist of an absolutely irreducible algebraic representation of G, and let V ∈ Rep_{la,c}^{z}(P) be locally (H, W)-isotypical. Then J_{P} (V) is locally (D, W^{N})-isotypical, and

Hom_{𝔡}(W^{N}, J_{P} (V)) = J_{P} (Hom_{𝔥}(W, V)).

Proof. Let X = Hom_{𝔥}(W, V). By proposition 2.6, we have V = X ⊗_{K} W; so by proposition 5.8 and the remark following, J_{P} (V) = J_{P} (X) ⊗_{K} W^{N}. Since W^{N} is D-good, we can apply the converse implication of proposition 2.6 to deduce that J_{P} (X) = Hom_{𝔡}(W^{N}, J_{P} (V)) as required.

5.4 Combining the constructions

We now summarize the results of the above analysis.

Theorem 5.10. For any V ∈ Rep_{ess}(G), we have:

1. For any parabolic subgroup P ⊆ G with Levi subgroup M, any finitedimensional W ∈ Rep_{la,c}(M), and any open compact subgroup U ⊆ D =

[M, M], there is a coherent sheaf F(V, P, W, U) on { \atop Z(M)} with a right action of H(U), whose fibre at a character χ ∈ { \atop Z(M)} is isomorphic (as a right H(U)-module) to the dual of the space Hom_{U}(W, J_{P} V)^{Z(M)=χ}. In particular, a character χ lies in the subvariety S(V, P, W, U) = support F(V, P, W, U) if and only if this eigenspace is nonzero.

2. If V ∈ Rep_{la,ad}(G), then the projection S(V, P, W, U) → (Lie Z)′ has discrete fibres.

3. If V is isomorphic as an H-representation to C^{la}(H)^{m} for some m and some open compact H ⊆ G, then S(V, P, W, U) is equidimensional of dimension dim Z.

4. If P_{1}, P_{2} are parabolics with P_{1} ⊇ P_{2} as above, W is an absolutely irreducible algebraic representation of M_{1}, and U is an open compact subgroup of D_{1} which is decomposed with respect to the parabolic P_{2} ∩ D_{1}, then there is a decomposition

F(V, P_{1}, U, W) = F(V, P_{1}, U, W)_{Z_{2}−null} ⊕ F(V, P_{1}, U, W)_{Z_{2}−fs},

where the latter factor is isomorphic to a quotient of the pushforward to Z_{1} of the sheaf

F(V, P_{2}, W^{N}, U ∩ D_{2})

on Z_{2}.

Proof. The only statement still requiring proof is the last one. Let Y = (J_{P_{1}}V)_{D_{1},W −liso}. The closed embedding Y ↪ J_{P_{1}}(V) induces by functoriality a closed embedding J_{P_{12}}Y ↪ J_{P_{12}}(J_{P_{1}}V). The right-hand side is simply J_{P_{2}}V, by theorem 5.3. Thus we have a closed embedding

Hom_{𝔡_{2}}(W^{N}, J_{P_{12}}Y) ↪ Hom_{𝔡_{2}}(W^{N}, J_{P_{2}}V).

The left-hand side is isomorphic, by proposition 5.9, to J_{P_{12}} [Hom_{𝔡_{1}}(W, Y)]. We may now apply proposition 5.6 to the M_{1}-representation Hom_{𝔡_{1}}(W, Y) = Hom_{𝔡_{1}}(W, J_{P_{1}}V), to deduce that there is a direct sum decomposition

Hom_{U}(W, J_{P_{1}}V) = Hom_{U}(W, J_{P_{1}}V)_{Z_{2}−fs} ⊕ Hom_{U}(W, J_{P_{1}}V)_{Z_{2}−null}

and the first direct summand is isomorphic as a Z_{2}-representation to a closed subspace of

Hom_{U∩M′_{2}}(W^{N}, J_{P_{12}}Y) ⊆ Hom_{U∩D_{2}}(W^{N}, J_{P_{2}}V).

Dualising, we obtain the stated relation between the sheaves F(…).

6 Application to completed cohomology

6.1 Construction of eigenvarieties

Let us now fix a number field F, a connected reductive group 𝔊 over F, and a prime 𝔭 of F above p. Let G = 𝔊 ×_{F} F_{𝔭}, a reductive group over F_{𝔭}, and G = G(F_{𝔭}). Let us choose a parabolic subgroup P of G (not necessarily arising from a parabolic subgroup of 𝔊), and set P = P(F_{𝔭}), and similarly for M, N, D, Z as above. We suppose our base field L is a subfield of F_{𝔭}, so G, P, M, N, D, Z are locally L-analytic groups. We recall from [Eme04, 2.2.16] the construction of the completed cohomology spaces H̃^{i} for each cohomological degree i ≥ 0, which are unitary admissible Banach representations of (G, 𝔊(𝔸_{f}^{𝔭})×π_{0}), where π_{0} is the group of components of 𝔊(F ⊗_{ℚ} ℝ). The following is immediate from the above:

Proposition 6.1. Let Γ = D × 𝔊(𝔸_{f}^{𝔭}) × π_{0}. For any i ≥ 0, we have:

1. For any W ∈ Rep_{cts,fd}(M), the space

Hom_{D−st.sm}(W, Ord_{P} H̃^{i})

is an admissible continuous (Z, Γ)-representation.

2. For any W ∈ Rep_{la,fd}(M), the space

Hom_{𝔡}(W, J_{P} H̃_{la}^{i})

is an essentially admissible locally L-analytic (Z, Γ)-representation.

Let us fix an open compact subgroup U ⊆ Γ (this is the most natural notion of a “tame level” in this context). Then we can use the above result to define an eigenvariety of tame level U, closely following [Eme06b, §2.3]. Let v be a (finite or infinite) prime of S. We set

Γ_{v} = \begin{cases}𝔊(F_{v}) & if v ∤ ∞ and v ≠ 𝔭 \\ D & if v = 𝔭 \\ π_{0}(𝔊(F_{v})) & if v | ∞. \end{cases}

Then Γ = ∏′_{v} Γ_{v}. Let us set U_{v} = U ∩ Γ_{v}. We say v is unramified (for U) if v is finite, v ≠ 𝔭, and U_{v} is a hyperspecial maximal compact subgroup of Γ_{v}. Let S be the (clearly finite) set of ramified primes, and Γ^{S} = ∏_{v∉S} Γ_{v}, Γ_{S} = ∏_{v∈S} Γ_{v}. It is easy to see that U = U_{S} × U^{S}, where U^{S} = U ∩ Γ^{S} and similarly U_{S} = U ∩ Γ_{S}, and hence we have a tensor product decomposition of Hecke algebras

H(Γ//U) = H(Γ_{S}//U_{S}) ⊗ H(Γ^{S}//U^{S}) =: H^{ram} ⊗ H^{sph}.

As is well known, the algebra H^{sph} is commutative (but not finitely generated over K), while H^{ram} is finitely generated (but not commutative in general). By construction, H(Γ//U) acts on the essentially admissible Z-representation Hom_{U}(W, J_{P} H̃_{la}^{i}), and hence it also acts on the corresponding sheaf F(i, P, W, U) on Ẑ.

Definition 6.2. Let E(i, P, W, U) be the relative spectrum \underline{Spec} A, where A is the OẐ-subsheaf of \underline{End} F(i, P, W, U) generated by the image of H^{sph}.

For the definition of the relative spectrum, see [Con06, Thm 2.2.5]. By definition E(i, P, W, U) is a rigid space over K, endowed with a finite morphism π : E(i, P, W, U) → Ẑ and an isomorphism of sheaves of OẐ-algebras A ≅ π_{∗}O_{E(i,P,W,U)}. Consequently, F(i, P, W, U) lifts to a sheaf \overline{F}(i, P, W, U) on E(i, P, W, U). We can regard E(i, P, W, U) as a subvariety of Ẑ_{K} × Spec H^{sph} (although the latter will not be a rigid space if 𝔊 is not the trivial group); in particular, a K-point of E(i, P, W, U) gives rise to a homomorphism λ : H^{sph} → K. We record the following properties of this construction, which are precisely analogous to [Eme06b, 2.3.3]:

Theorem 6.3.

1. The natural projection E(i, P, W, U) → 𝔷′ has discrete fibres. In particular, the dimension of E(i, P, W, U) is at most equal to the dimension of Z.

2. The action of H^{ram} on F(i, P, W, U) lifts to an action on \overline{F}(i, P, W, U), and the fibre of \overline{F}(i, P, W, U) at a point (χ, λ) ∈ Ẑ × Spec H^{sph} is isomorphic as a right H^{ram}-module to the dual of the (Z = χ, H^{sph} = λ)- eigenspace of Hom_{U}(W, J_{P} H̃_{la}^{i}). In particular, the point (χ, λ) lies in E(i, P, W, U) if and only if this eigenspace is non-zero.

3. If there is a compact open subgroup G_{0} ⊆ G such that (H̃_{la}^{i})^{U^{(𝔭)}} is isomorphic as a G_{0}-representation to a finite direct sum of copies of C^{la}(G_{0}) (where U^{(𝔭)} = U ∩ 𝔊(𝔸_{f}^{𝔭})), then E(i, P, W, U) is equidimensional, of dimension equal to the rank of Z.

Remark. The hypothesis in the last point above is always satisfied when i = 0 and 𝔊(F ⊗ℝ) is compact, since for any open compact subgroup U^{(𝔭)} ⊆ 𝔊(𝔸_{f}^{𝔭}), the image of G(F) ∩ U^{(𝔭)} in G is a discrete cocompact subgroup Λ, and the U^{(𝔭)}-invariants H̃^{0}(U^{(𝔭)}) are isomorphic as a representation of G and as a H(U^{(𝔭)})-module to C(Λ\G). This case is considered extensively in an earlier publication of the second author [Loe11]. Now let us suppose G is split over K, and fix an irreducible (and therefore absolutely irreducible) algebraic representation W of M. We let Π(P, W, U) denote the set of irreducible smooth 𝔊(𝔸_{f}) × π_{0}-representations π_{f} such that J_{P} (π_{f})^{U} ≠ 0, and π_{f} appears as a subquotient of the cohomology space H^{i}(V_{X}) of [Eme06b, §2.2] for some irreducible algebraic representation X of G with (X′)^{N} ≅ W ⊗ χ for some character χ. To any such π_{f}, we may associate the point (θχ, λ) ∈ Ẑ × Spec H^{sph}, where θ is the smooth character by which Z acts on J_{P} (π_{p}), and λ the character by which H^{sph} acts on J_{P} (π_{f})^{U}. Let E(i, P, W, U)_{cl} denote the set of points of Ẑ × Spec H^{sph} obtained in this way from representations π_{f} ∈ Π(i, P, W, U).

Corollary 6.4. If the map (1.1) is an isomorphism for all irreducible algebraic representations X such that (X′)^{N} is a twist of W, then E(i, P, W, U)_{cl} ⊆ E(i, P, W, U). In particular, the Zariski closure of E(i, P, W, U)_{cl} has dimension at most dim Z.

Proof. Let π_{f} ∈ Π(i, P, W, U). Then the locally algebraic (G, 𝔊(𝔸_{f}^{(𝔭)}) × π_{0})- representation π_{f} ⊗ X′ appears in H^{i}(V_{X}) ⊗_{K} X′. By assumption, the latter embeds as a closed subrepresentation of H̃_{la}^{i}. The Jacquet functor is exact restricted to locally X′-algebraic representations (since this is so for smooth representations). Moreover, the functor Hom_{𝔡}(W, −) is exact restricted to locally W -algebraic representations, so Hom_{𝔡} [W, J_{P} (π_{f}) ⊗_{K} (X′)^{N}] appears as

a subquotient of Hom_{𝔡} [W, J_{P} (H̃_{la}^{i})]. Since (X′)^{N} = W ⊗ χ, the former space is simply J_{P} (π_{f}) ⊗_{K} χ, so the point (θχ, λ) appears in E(i, P, W, U) as required.

Remarks. 1. The entire construction can also be carried out with the spaces H̃^{i} replaced by the compactly supported versions H̃_{c}^{i} or the parabolic versions H̃_{par}^{i}; we then obtain analogues of the above proposition for the compactly supported or parabolic cohomology of the arithmetic quotients.

2. It suffices to check that the map (1.1) is an isomorphism for L = ℚ_{p}. This is known to hold in many cases, e.g. in degree i = 0 for any 𝔊, and in degree 1 for GL_{2}(ℚ) (as shown in [Eme06b]) or for a semisimple and simply connected group (as shown by the first author in [Hil07]). The “weak Emerton criterion” of [Hil07, defn. 2] suffices to prove corollary 6.4 when W is not a character; this is known in many more cases, e.g. when i = 2 and the congruence kernel of 𝔊 is finite. When W is a character χ : M → 𝔾_{m}, the weak Emerton criterion implies that the points E(i, P, W, U)_{cl} are contained in the union of E(i, P, W, U) and the single point (χ^{−1}, 1).

Theorem 6.5. Suppose P_{1} ⊇ P_{2} are two parabolics, and U = U^{(𝔭)} × U_{𝔭}, where U^{(𝔭)} ⊆ 𝔊(𝔸_{f}^{(𝔭)}) × π_{0} and U_{𝔭} ⊆ D_{1} is decomposed with respect to P_{2} ∩ D_{1}. Then E(i, P_{1}, W, U) is equal to a union of two closed subvarieties

E(i, P_{1}, W, U)_{P_{2}−fs} ∩ E(i, P_{1}, W, U)_{P_{2}−null},

which are respectively equipped with sheaves of H^{ram}-modules \overline{F}(i, P, W, U)_{P_{2}−fs} and \overline{F}(i, P, W, U)_{P_{2}−null} whose direct sum is \overline{F}(i, P, W, U). The element of H^{ram} corresponding to any strictly positive element of Z_{2} acts invertibly on \overline{F}(i, P, W, U)_{P_{2}−fs} and nilpotently on \overline{F}(i, P, W, U)_{P_{2}−null}; and there is a subvariety of E(i, P_{2}, W^{N_{12}}, U ∩ D_{2}) whose image in Ẑ_{1} × Spec H^{sph} coincides with E(i, P_{1}, W, U)_{P_{2}−fs}.

Proof. By theorem 5.10, we may decompose F(i, P_{1}, W, U) as a direct sum of a null part and a finite slope part; this decomposition is clearly functorial, and hence it is preserved by the action of the Hecke algebra H^{sph}, so we may define the spaces E(i, P_{1}, W, U)_{P_{2}−fs} and E(i, P_{1}, W, U)_{P_{2}−null} to be the relative spectra of the Hecke algebra acting on the two summands. For the final statement, we note that there is a quotient Q of F(i, P_{2}, W^{N_{12}}, U ∩ D_{2}), corresponding to the Z_{2}-subrepresentation

J_{P_{12}} (Hom_{𝔡_{1}}(W, J_{P_{1}} H̃_{la}^{i}))^{U∩M_{2}} ⊆ Hom_{𝔡_{2}}(W^{N_{12}}, J_{P_{2}} H̃_{la}^{i})^{U∩D_{2}}

such that the pushforward of Q to Ẑ_{1} is isomorphic to F(i, P_{1}, W, U)_{P_{2}−fs}. This isomorphism clearly commutes with the action of H^{sph} on both sides, from which the result follows.

7 Acknowledgements

The second author would like to thank Kevin Buzzard and Matthew Emerton for illuminating conversations; the participants in the autum 2009 “P -adic Analysis” study group at DPMMS, University of Cambridge, for their assistance in learning the necessary theory; and Sarah for her love and support.

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Richard Hill Department of Mathematics University College London Gower Street London WC1E 6BT, UK rih@math.ucl.ac.uk David Loeffler Warwick Mathematics Institute Zeeman Building University of Warwick Coventry CV4 7AL, UK d.loeffler.01@cantab.net 32

Documenta Mathematica 16 (2011)