PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

ANA CARAIANI, MATTHEW EMERTON, TOBY GEE, DAVID GERAGHTY, VYTAUTAS PAŠKŪNAS, AND SUG WOO SHIN

Abstract. We use the patching method of Taylor–Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GL_{n}(F), F a finite extension of ℚ_{p}. We use our construction to prove many new cases of the Breuil–Schneider conjecture.

Contents

1. Introduction  1
2. The patching argument  7
3. Hecke algebras and types  19
4. Local-global compatibility  30
5. The Breuil–Schneider conjecture  42
6. Relationship with a hypothetical p-adic local Langlands correspondence 45
References  46

1. Introduction

Our goal in this paper is to use global methods (specifically, the Taylor–Wiles– Kisin patching method) to construct a candidate for the p-adic local Langlands correspondence for GL_{n}(F), where F is an arbitrary finite extension of ℚ_{p}, and p ∤ 2n. At present, the existence of such a correspondence is only known for GL_{1}(F) (where it is given by local class field theory), and for GL_{2}(ℚ_{p}) (cf. [Col10, Paš13]). We do not prove that our construction gives a purely local correspondence (and it would perhaps be premature to conjecture that it should), but we are able to say enough about our construction to prove many new cases of the Breuil–Schneider conjecture, and to reduce the general case of the Breuil–Schneider conjecture (under some mild technical hypotheses) to standard conjectures related to automorphy lifting theorems.

2 A. CARAIANI, M. EMERTON, T. GEE, D. GERAGHTY, V. PAŠKŪNAS, AND S. W. SHIN

The idea that global methods could be used to construct the correspondence is a natural one; the only proof at present of the classical local Langlands correspondence [HT01, Hen00] is by global means, and indeed the first proofs of local class field theory were global. The basic idea is to embed a local situation into a global one, apply a global correspondence (for example, the association of Galois representations to certain automorphic forms), and then to prove that the construction is independent of the choice of global situation. In this paper, we carry out the first half of this idea (although, in contrast to the constructions of [HT01], the direction of the correspondence we construct is from representations of the local Galois group G_{F} to representations of GL_{n}(F)). We intend to return to the second half (investigating the question of independence of the global situation) in subsequent work. (See Section 6 for a discussion of the relationship of our construction to conjectural extensions of the existing p-adic Langlands correspondence for GL_{2}(ℚ_{p}).)

A.C. was partially supported by the NSF Postdoctoral Fellowship DMS-1204465. M.E. was partially supported by NSF grants DMS-1003339, DMS-1249548, and DMS-1303450. T.G. was partially supported by a Marie Curie Career Integration Grant, and by an ERC Starting Grant.

D.G. was partially supported by NSF grants DMS-1200304 and DMS-1128155. V.P. was partially supported by the DFG, SFB/TR45. S.W.S. was partially supported by NSF grant DMS-1162250 and a Sloan Fellowship.

1.1. The Breuil–Schneider conjecture. Before giving an overview of our construction, we discuss an application of it to the Breuil–Schneider conjecture [BS07]; this conjecture predicts that locally algebraic representations of GL_{n}(F) admit invariant norms (and thus nonzero completions to unitary Banach representations) if and only if they arise from regular de Rham Galois representations by applying (a generic version of) the classical local Langlands correspondence to the corresponding Weil–Deligne representations. (This conjecture is motivated by the case of GL_{2}(ℚ_{p}), where it is an immediate consequence of known properties of the p-adic local Langlands correspondence.) In one direction, Hu [Hu09] showed that if such a norm exists, the locally algebraic representation necessarily comes from a regular de Rham representation.

The converse direction is largely open. We recall the conjecture in more detail in Section 5 below, to which the reader should refer for any unfamiliar notation or terminology. Given a de Rham representation r : G_{F} → GL_{n}(\overline{ℚ}_{p}) of regular weight, in [BS07] there is associated to r a locally algebraic \overline{ℚ}_{p}-representation BS(r) of GL_{n}(F). The following is [BS07, Conjecture 4.3] (in the open direction).

A. Conjecture. If r : G_{F} → GL_{n}(\overline{ℚ}_{p}) is de Rham and has regular weight, then BS(r) admits a nonzero unitary Banach completion.

In fact, the known properties of the p-adic local Langlands correspondence for GL_{2}(ℚ_{p}) suggest that there should even be a nonzero admissible completion. Conjecture A was proved in the case that π_{sm}(r) is supercuspidal in [BS07, Theorem 5.2], and in the more general case that WD(r) is indecomposable in [Sor13]. The argument of [Sor13] is global. It makes use of a strategy of one of us (M.E.) who observed that if r arises as the local Galois representation coming from an automorphic representation, then one can obtain an admissible completion from the completed cohomology of [Eme06b], cf. Proposition 4.6 of [Eme05] (and also [Sor12]). However, as there are only countably many automorphic representations, it is not possible to say anything about most principal series representations in this way; indeed, as was already remarked in [BS07] (see the discussion before Remark 5.7), the principal series case seems to be the deepest case of the conjecture.

Other than for GL_{2}(ℚ_{p}), the only previous results in the general principal series case that we are aware of are those of [AKdS13, Ies12, KdS12, Vig08], which prove the conjecture for certain principal series cases for GL_{2}(F), under additional restrictions on the Hodge filtration of r. The methods of these papers do not seem to shed any light on the stronger question of the existence of admissible completions. Under the assumption that p ∤ 2n, which we make from now on, the construction explained below associates an admissible unitary Banach representation V (r) to any continuous representation r : G_{F} → GL_{n}(\overline{ℚ}_{p}). In order to prove Conjecture A, it would be enough to establish that V (r) contains a copy of BS(r) when r is de Rham of regular weight. We expect this to be true in general, and we are able to show that it is equivalent to proving a certain automorphy lifting theorem. The following is our main result in this direction. (See sections 2 and 5 for any unfamiliar terminology; note in particular that the hypothesis that r lies on an automorphic component does not imply that r arises from the Galois representation associated to an automorphic representation, but is rather the much weaker condition that it lies on the same component of a local deformation ring as some such representation. It is a folklore conjecture (closely related to the problem of deducing the Fontaine–Mazur conjecture from generalisations of Serre’s conjecture via automorphy lifting theorems) that every de Rham representation of regular weight satisfies this condition.)

B. Theorem (Theorem 5.3 and Remark 4.19). Suppose that p ∤ 2n, that r : G_{F} → GL_{n}(\overline{ℚ}_{p}) is potentially crystalline of regular weight, and that r is generic. Suppose that r lies on an automorphic component of the corresponding potentially crystalline deformation ring. Then BS(r) admits a nonzero unitary admissible Banach completion.

By taking known automorphy lifting theorems, in particular those of [BLGGT13], we are able to deduce new cases of Conjecture A. In particular, we deduce the following result (Corollary 5.5).

C. Theorem. Suppose that p > 2, that r : G_{F} → GL_{n}(\overline{ℚ}_{p}) is de Rham of regular weight, and that r is generic. Suppose further that either

(1) n = 2, and r is potentially Barsotti–Tate, or

(2) F/ℚ_{p} is unramified and r is crystalline with Hodge–Tate weights in the

extended Fontaine–Laffaille range, and n ≠ p. Then BS(r) admits a nonzero unitary admissible Banach completion.

Actually, we prove a more general result (Corollary 5.4) which establishes the conjecture for potentially diagonalizable representations; conjecturally, every potentially crystalline representation is potentially diagonalizable. We remark that while we expect these results to extend to potentially semistable (rather than just potentially crystalline) representations, and to non-generic representations, we have restricted to the potentially crystalline case for two reasons: we can use the main theorems of [BLGGT13] without modification, and we do not have to consider issues related to the possible reducibility of BS(r).

1.2. An overview of our construction. In the proof of the classical local Langlands correspondence [HT01, Hen00], the globalisation argument uses a reduction to the supercuspidal case (via the classification of irreducible smooth representations of GL_{n}(F) given in [BZ77, Zel80]), and then uses trace formula methods to realise supercuspidal representations as the local components of cuspidal automorphic representations. No such argument is possible in our setting; there is at present no analogue of the theory of [BZ77, Zel80] in the setting of admissible padic Banach space representations, and even in the supercuspidal case, there are uncountably many p-adic Galois representations up to twist, but only countably many automorphic representations.

It is natural to hope that in the p-adic setting, one could carry out an analogous globalisation using “p-adic automorphic representations”, such as those arising from the completed cohomology of [Eme06b]. However, since the locally algebraic vectors in completed cohomology are computed by classical automorphic representations, one cannot expect to see any regular de Rham Galois representations in completed cohomology other than those arising from classical automorphic representations.

The globalisation argument in the proof of classical local Langlands is effectively a result showing the Zariski-density of automorphic points in the Bernstein spectrum; the analogous result for p-adic local Langlands (or rather, for the part of it pertaining to regular de Rham representations) would be a Zariski-density result for automorphic points in the corresponding local Galois deformation rings. This is not known in general, but strong results in this direction follow from the Taylor–Wiles– Kisin patching method, which provides a Zariski-density result for a non-empty collection of components of a local deformation ring (and in general shows that each component either contains no automorphic points, or a Zariski-dense set of points; as mentioned above, the problem of showing that each component contains an automorphic point is closely related to the problem of deducing the Fontaine–Mazur conjecture from generalisations of Serre’s conjecture, cf. Remark 5.5.3 of [EG13]).

The Taylor–Wiles–Kisin method patches together spaces of automorphic forms with varying tame level. Usually, the weight and the p-part of the level of these forms is fixed, and one obtains a patched module for a certain universal local deformation ring corresponding to de Rham representations of fixed Hodge–Tate weights, and a fixed inertial type. Instead, we vary, in addition, over all weights and levels at p, obtaining a module over the unrestricted local deformation ring. By construction, this module naturally has an action of GL_{n}(O_{F}); by keeping track of the action of the Hecke operators at p, we are able to promote this to an action of GL_{n}(F). Taking the fiber of this patched module at the point corresponding to a particular Galois representation r, and inverting p, gives the unitary admissible Banach representation V (r) that we seek. The condition that p ∤ 2n is needed to employ the Taylor–Wiles–Kisin method (for example, this condition is necessary in order to be able to appeal to various results from [BLGGT13]), but we suspect that it is not ultimately needed to carry out variants of these constructions.

We do not know whether it is reasonable to expect that our construction is purely local, and thus defines a p-adic local Langlands correspondence; this amounts to the problem of showing that the patched modules that we construct are purely local objects. For some weak evidence in this direction, see [EGS13], which proves a related result for lattices corresponding to certain 2-dimensional tamely potentially Barsotti–Tate representations. It can also be shown that our construction recovers the known correspondence for GL_{2}(ℚ_{p}); we will report on this in a future paper. We refer to Section 6 below for a slightly more detailed discussion of how the construction of this paper might relate to a hypothetical p-adic local Langlands correspondence.

1.3. Outline of the paper. In Section 2, we carry out our patching construction. Section 3 contains an introduction to the results of Bernstein–Zelevinsky, Bushnell– Kutzko, Schneider–Zink and Dat on types and the local Langlands correspondence for GL_{n}. We then prove some refinements of some of these results, and explain how to descend them from algebraically closed coefficient fields to finite extensions of ℚ_{p}. In Section 4 we begin by proving a result of independent interest, showing that the classical local Langlands correspondence interpolates over a (potentially semistable) local deformation ring, and then apply this to show local-global compatibility for our patched modules. Finally, in Section 5 we combine this local-global compatibility result with automorphy lifting theorems to prove our results on the Breuil–Schneider conjecture.

1.4. Acknowledgements. The constructions made in this paper originate in work carried out at a focused research group on “The p-adic Langlands program for nonsplit groups” at the Banff Centre in August 2012; we would like to thank BIRS for providing an excellent working atmosphere, and for its financial support. We would also like to thank AIM for providing financial support and an excellent working atmosphere towards the end of this project. The idea that patching could be used to prove the Breuil–Schneider conjecture goes back to conversations between two of us (M.E. and T.G.) at the Harvard Eigen-semester in 2006, and we would like to thank the mathematics department of Harvard University for its hospitality. Finally, we thank Brian Conrad and Florian Herzig for helpful remarks on an earlier draft of this paper.

1.5. Notation. We fix a prime p, and an algebraic closure \overline{ℚ}_{p} of ℚ_{p}. Throughout the paper we work with a finite extension E/ℚ_{p} in \overline{ℚ}_{p}, which will be our coefficient field. We write O = O_{E} for the ring of integers in E, ϖ = ϖ_{E} for a uniformiser, and 𝔽 := O/ϖ for the residue field. At any particular moment E is fixed, but we allow ourselves to modify E (typically via an extension of scalars) during the course of our arguments. Furthermore, we will often assume without further comment that E and 𝔽 are sufficiently large, and in particular that if we are working with representations of the absolute Galois group of a p-adic field F, then the images of all embeddings F ↪ \overline{E} are contained in E.

If F is a field, we let G_{F} denote its absolute Galois group. Let ε denote the p-adic cyclotomic character, and \overline{ε} the mod p cyclotomic character. If F is a finite extension of ℚ_{p} for some p, we write I_{F} for the inertia subgroup of G_{F}, and ϖ_{F} for a uniformiser of the ring of integers O_{F} of F. If F̃ is a number field and v is a finite place of F̃ then we let Frob_{v} denote a geometric Frobenius element of GF̃_{v}.

If F is a p-adic field, W is a de Rham representation of G_{F} over E, and κ : F ↪ E, then we will write HT_{κ}(W) for the multiset of Hodge–Tate weights of W with respect to κ. By definition, the multiset HT_{κ}(W) contains i with multiplicity dim_{E}(W ⊗_{κ,F} ˆ \overline{F}(i))^{G_{F}}. Thus for example HT_{κ}(ε) = \{−1\}.

We say that W has regular Hodge–Tate weights if for each κ, the elements of HT_{κ}(W) are pairwise distinct. Let ℤ_{+}^{n} denote the set of tuples (ξ_{1}, …, ξ_{n}) of integers with ξ_{1} ≥ ξ_{2} ≥ ⋯ ≥ ξ_{n}. Then if W has regular Hodge–Tate weights, there is a ξ = (ξ_{κ,i}) ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (F,E)} such that for each κ : F ↪ E,

HT_{κ}(W) = {ξ_{κ,1} + n − 1, ξ_{κ,2} + n − 2, …, ξ_{κ,n}},

and we say that W is regular of weight ξ. For any ξ ∈ ℤ_{+}^{n}, view ξ as a dominant weight (with respect to the upper triangular Borel subgroup) of the algebraic group GL_{n} in the usual way, and let M′_{ξ} be the algebraic O_{F} -representation of GL_{n} given by

M′_{ξ} := Ind_{B_{n}}^{GL_{n}}(w_{0}ξ)_{/O_{F}}

where B_{n} is the Borel subgroup of upper-triangular matrices of GL_{n}, and w_{0} is the longest element of the Weyl group (see [Jan03] for more details of these notions, and note that M′_{ξ} has highest weight ξ). Write M_{ξ} for the O_{F} -representation of GL_{n}(O_{F}) obtained by evaluating M′_{ξ} on O_{F}. For any ξ ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (F,E)} we write L_{ξ} for the O-representation of GL_{n}(O_{F}) defined by

L_{ξ} := ⊗_{κ:F ↪E}M_{ξ_{κ}} ⊗_{O_{F} ,κ} O.

If F is a p-adic field, then an inertial type is a representation τ : I_{F} → GL_{n}(\overline{ℚ}_{p}) with open kernel which extends to the Weil group W_{F}.

Then we say that a de Rham representation ρ : G_{F} → GL_{n}(E) has inertial type τ if the restriction to I_{F} of the Weil–Deligne representation WD(ρ) associated to ρ is equivalent to τ. Given an inertial type τ, there is a finite-dimensional smooth irreducible \overline{ℚ}_{p}-representation σ(τ) of GL_{n}(O_{F}) associated to τ by the “inertial local Langlands correspondence”; see Theorem 3.7 below. (Note that by the results of Section 3.12 below, we will be able to replace σ(τ) by a model defined over a finite extension of E in our main arguments.)

Let F be a finite extension of ℚ_{p}, and let rec denote the local Langlands correspondence from isomorphism classes of irreducible smooth representations of GL_{n}(F) over ℂ to isomorphism classes of n-dimensional Frobenius semisimple Weil– Deligne representations of W_{F} as in the introduction to [HT01]. Fix once and for all an isomorphism ı : \overline{ℚ}_{p} ⟶^{∼} ℂ. We define the local Langlands correspondence rec_{p} over \overline{ℚ}_{p} by ı ∘ rec_{p} = rec ∘ ı. This depends only on ı^{−1}(\sqrt{p}), and if we define r_{p}(π) := rec_{p}(π ⊗ | det |^{(1−n)/2}), then r_{p} is independent of the choice of ı. Furthermore, if V is a Frobenius semisimple Weil–Deligne representation of W_{F} over E, then r_{p}^{−1}(V) is also defined over E by [Clo90, Prop 3.2] and the fact that r_{p} commutes with automorphisms of ℂ, cf. Section 4 of [BS07]

If A is a linear topological O-module, we write A^{∨} for its Pontrjagin dual Hom_{O}^{cont}(A, E/O), where E/O has the discrete topology, and we give A^{∨} the compact open topology.

By the proof of Theorem 1.2 of [ST02], the functor A ↦ A^{d} := Hom_{O}^{cont}(A, O) induces an anti-equivalence of categories between the category of compact, Otorsion-free linear-topological O-modules A and the category of ϖ-adically complete and separated O-torsion-free O-modules. A quasi-inverse is given by B ↦ B^{d} := Hom_{O}(B, O), where the target is given the weak topology of pointwise convergence. We refer to this duality as Schikhof duality. Note that if A is an O-torsion free profinite linear-topological O-module, then A^{d} is the unit ball in the E-Banach space Hom_{O}(A, E).

If r : G_{F} → GL_{n}(E) is de Rham of regular weight a, then we write π_{alg}(r) := L_{a}^{d} ⊗_{O} E, and π_{sm}(r) := r_{p}^{−1}(WD(r)^{F −ss}), both of which are E-representations of GL_{n}(F). (The GL_{n}(O_{F})-action on L_{a}^{d} extends linearly to a GL_{n}(F)-action on π_{alg}(r).) As the names suggest, π_{alg}(r) is an algebraic representation, and π_{sm}(r) is a smooth representation. Note that π_{alg}(r) = L_{ξ} ⊗_{O} E for ξ_{κ,i} := −a_{κ,n+1−i}.

We let Art_{F} : F^{×} ⟶^{∼} W_{F}^{ab} be the isomorphism provided by local class field theory, which we normalise so that uniformisers correspond to geometric Frobenius elements.

We write all matrix transposes on the left; so ^{t}g is the transpose of g. We let G_{n} denote the group scheme over ℤ defined to be the semidirect product of GL_{n} × GL_{1} by the group {1, j}, which acts on GL_{n} × GL_{1} by

j(g, μ)j^{−1} = (μ ·^{t}g^{−1}, μ).

We have a homomorphism ν : G_{n} → GL_{1}, sending (g, μ) to μ and j to −1.

Further notation is introduced in the course of our arguments; we mention just some of it here, for the reader’s convenience.

From Subsection 2.6 on, we will have fixed a particular finite extension F of ℚ_{p}, with ring of integers O_{F} and uniformiser ϖ_{F}. To ease notation we will typically write G := GL_{n}(F), K := GL_{n}(O_{F}), and Z := Z(G). For each m ≥ 0, we write Γ_{m} = GL_{n}(O_{F} /ϖ_{F}^{m}) and K_{m} := ker( GL_{n}(O_{F}) → GL_{n}(O_{F} /ϖ_{F}^{m})), so that

K/K_{m} ⟶^{∼} Γ_{m}.

Furthermore, throughout Section 2, a large amount of notation is introduced related to automorphic forms on a definite unitary group, and Taylor–Wiles–Kisin patching. Here we merely signal that the main construction of this section, and our major object of study in the paper, is a patched G-representation that we denote by M_{∞}. Beginning in Section 4, we will also write

M_{∞}(σ^{∘}) := (Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘})^{d}))^{d} ,

when σ^{∘} is a K-invariant O_{E}-lattice in a K-representation σ of finite dimension over E.

We use Ind to denote induction, and c-Ind to denote induction with compact supports. In Sections 3 and 4, we use i_{P}^{G} to denote normalised parabolic induction.

If σ is a representation of K, then we will write H(σ) := End(c-Ind_{K}^{G}σ) to denote the Hecke algebra of G with respect to σ. Sometimes, when it is helpful to emphasise the role of G, we will write H(G, σ) instead. We also use obvious variants with G and K replaced by another p-adic group and compact open subgroup.

2. The patching argument

In this section we will carry out our patching argument on definite unitary groups. The key difference between the construction presented here and previous patching constructions is that the object we end up with is not simply a module over a certain Galois deformation ring, but rather a GL_{n}(F)-representation over that ring; we refer to the discussion at the beginning of Subsection 2.6 below for a more detailed discussion of this difference.

Our construction uses the same general framework as in section 5 of [EG13] (which in turn is based on the approach taken in [CHT08], [BLGG11] and [Tho12]); we recall the key elements of this framework in the first several subsections that follow. We follow the notation of [EG13] as closely as possible, and we indicate explicitly where we deviate from it.

The construction itself is the subject of Subsection 2.6, and the key fact that it actually produces a GL_{n}(F)-representation is verified in Proposition 2.8. In Subsection 2.10 we explain how our patched GL_{n}(F)-representation gives rise to admissible unitary Banach representations attached to local Galois representations.

2.1. Globalisation. Let F/ℚ_{p} be a finite extension, and fix a continuous representation r̄: G_{F} → GL_{n}(𝔽). Our goal in this subsection is to give a criterion for r̄ to be obtained as the restriction of a global Galois representation that is automorphic, in a suitable sense (and satisfies some additional convenient properties).

We will assume that the following hypotheses are satisfied:

• p ∤ 2n, and • r̄ admits a potentially crystalline lift of regular weight, which is potentially

diagonalizable in the sense of [BLGGT13]. Conjecturally, the second hypothesis is always satisfied; this is Conjecture A.3 of [EG13]. In this direction, we note the following result.

2.2. Lemma. After possibly making a finite extension of scalars, the second hypothesis is satisfied if either n = 2 or r̄ is semisimple.

Proof. If n = 2, this is Remark A.4 of [EG13]. If r̄ is semisimple, then after extending scalars, we may write it as a sum of inductions of characters, and it is easy to see that by lifting these characters to crystalline characters, we can find a potentially crystalline lift which has regular weight, and is a sum of inductions of characters. Such a lift is obviously potentially diagonalizable (indeed, after restriction to some finite extension, it is a sum of crystalline characters). □

Having assumed these hypotheses, Corollary A.7 of [EG13] (with K our F, and F our F̃) provides us with an imaginary CM field F̃ with maximal totally real subfield F̃^{+}, and a continuous irreducible representation \overline{ρ} : GF̃^{+} → G_{n}(𝔽) such that \overline{ρ} is a suitable globalisation of r̄ in the sense of Section 5.1 of [EG13]. In particular, \overline{ρ} is automorphic, it is unramified outside of p, and it globalises r̄ in the sense that:

• each place v | p of F̃^{+} splits in F̃, and has F̃_{v}^{+} ≅ F; we fix a choice of such isomorphisms.

• For each place v | p of F̃^{+}, there is a place ṽ of F̃ lying over v with \overline{ρ}|_{GF̃ṽ}

isomorphic to r̄. In fact, in order to arrange that our patched modules have a certain multiplicity one property, we will also demand that:

• \overline{ρ}(GF̃) = GL_{n}(𝔽′) for some subfield 𝔽′ ⊆ 𝔽 with #𝔽′ > 3n, and

• \overline{F̃}^{ker ad \overline{ρ}|_{GF̃}} does not contain F̃(ζ_{p}).

The second of these properties is already part of the definition of a suitable globalisation. To see that we can arrange that the first property holds, note that the proof of Proposition A.2 of [EG13] allows us to arrange that \overline{ρ}(GF̃) = GL_{n}(𝔽_{p^{m}}) for any sufficiently large m.

2.3. Unitary groups. We now use the globalisation \overline{ρ} of our local Galois representation r̄ to carry out the Taylor–Wiles–Kisin patching argument as in Section 5 of [EG13]. The definitions of Hecke algebras, the choices of auxiliary primes and so on are essentially identical to the arguments made in [EG13], and rather than repeating them verbatim, we often refer the reader to [EG13] for the details of these definitions, indicating only the differences in our construction.

As in Sections 5.2 and 5.3 of [EG13], we fix a certain definite unitary group G̃/F̃^{+} together with a model (which we will also denote by G̃) over OF̃^{+}. (The group G̃ is denoted G in [EG13], but we will later use G to denote GL_{n}(F).) This model has the property that for each place v of F̃^{+} which splits as ww^{c} in F̃, there is an isomorphism ι_{w} : G̃(OF̃_{v}^{+}) ⟶^{∼} GL_{n}(OF̃_{w}); we fix a choice of such isomorphisms. We also choose a finite place v_{1} of F̃^{+} which is prime to p, with the properties that

• v_{1} splits in F̃, say as v_{1} = ṽ_{1}v_{1}^{c},

• v_{1} does not split completely in F̃(ζ_{p}), and • \overline{ρ}(FrobF̃ṽ_{1} ) has distinct 𝔽-rational eigenvalues, no two of which have ratio

(Nv_{1})^{±1}.

(It is possible to find such a place v_{1} by our assumptions that \overline{ρ}(GF̃) = GL_{n}(𝔽′)

with #𝔽′ > 3n, and that \overline{F̃}^{ker ad \overline{ρ}|_{GF̃}} does not contain F̃(ζ_{p}). Note that this differs slightly from the choice of place v_{1} in the first paragraph of Section 5.3 of [EG13], but that it is still the case that any deformation of \overline{ρ}|_{GF̃ṽ_{1}} is unramified. We have made this choice in order to be able to arrange that our patched modules satisfy multiplicity one.)

Let S_{p} denote the set of primes of F̃^{+} dividing p. We now fix a place 𝔭 | p of F̃^{+}, and for each integer m ≥ 0 we consider the compact open subgroup U_{m} = ∏_{v} U_{m,v}

of G̃(𝔸∞ ; F̃^{+}), where

• U_{m,v} = G̃(OF̃_{v}^{+}) for all v which split in F̃ other than v_{1} and 𝔭;

• U_{m,v_{1}} is the preimage of the upper triangular matrices under

G̃(OF̃_{v_{1}}^{+} ) → G̃(k_{v_{1}}) ⟶^{∼}_{ιṽ_{1}} GL_{n}(k_{v_{1}})

• U_{m,𝔭} is the kernel of the map G̃(OF̃_{𝔭}^{+}) → G̃(OF̃_{𝔭}^{+}/ϖm ; F̃_{𝔭}^{+});

• U_{m,v} is a hyperspecial maximal compact subgroup of G̃(F̃_{v}^{+}) if v is inert in F̃.

By the choice of v_{1} and U_{m,v_{1}} we see that U_{m} is sufficiently small. Write U := U_{0}. In order to make the patching argument, we will need to consider certain compact open subgroups of the U_{m} corresponding to choices of sets of auxiliary primes Q_{N} that will be introduced in Section 2.5. Specifically, for each integer N ≥ 1, we will have a finite set of primes Q_{N} of F̃^{+} disjoint from S_{p} ∪ {v_{1}} as well as open compact subgroups U_{i}(Q_{N})_{v} of G̃(OF̃_{v}^{+}) for each v ∈ Q_{N} and i = 0, 1. We then define subgroups U_{i}(Q_{N})_{m} = ∏_{v} U_{i}(Q_{N})_{m,v} ⊂ U_{m}, for i = 0, 1 by setting U_{0}(Q_{N})_{m,v} = U_{1}(Q_{N})_{m,v} = U_{m,v} for v ∉ Q_{N}, and U_{i}(Q_{N})_{m,v} = U_{i}(Q_{N})_{v} for v ∈ Q_{N}.

By assumption, r̄ has a potentially diagonalizable lift of regular weight, say r_{pot.diag} : G_{F} → GL_{n}(O). Suppose that r_{pot.diag} has weight ξ and inertial type τ (in the sense of Section 1.5). Extending E if necessary, we may assume that the GL_{n}(O_{F})-representation σ(τ) is defined over E. Then we have two representations L_{ξ} and L_{τ} of GL_{n}(O_{F}) on finite free O-modules in the following way: the representation L_{ξ} is the one defined in the notation section, and L_{τ} is a choice of GL_{n}(O_{F})-stable lattice in σ(τ). Set L_{ξ,τ} := L_{τ} ⊗_{O} L_{ξ}, a finite free O-module with an action of GL_{n}(O_{F}).

Returning to our global situation, let W_{ξ,τ} denote the finite free O-module with an action of ∏_{v∈S_{p}\{𝔭}} U_{m,v} given by W_{ξ,τ} = ⊗_{v∈S_{p}\{𝔭},O}L_{ξ,τ} where U_{m,v} acts on

the factor corresponding to v via U_{m,v} = G̃(OF̃_{v}^{+}) ⟶^{∼}_{ιṽ} GL_{n}(OF̃ṽ) ⟶^{∼} GL_{n}(O_{F}). In

order to avoid duplication of definition, we allow Q_{N} = ∅ in the definitions we now make. For any finite O-module V with a continuous action of U_{m,𝔭}, we have spaces of algebraic modular forms S_{ξ,τ} (U_{i}(Q_{N})_{m}, V); these are just the functions

f : G̃(F̃^{+})\G̃(𝔸∞ ; F̃^{+}) → W_{ξ,τ} ⊗_{O} V

with the property that if g ∈ G̃(𝔸∞ ; F̃^{+}) and u ∈ U_{i}(Q_{N})_{m} then f(gu) = u^{−1}f(g), where U_{i}(Q_{N})_{m} acts on W_{ξ,τ} ⊗_{O} V via projection to ∏_{v∈S_{p}} U_{m,v}. (For example: when V = O is the trivial representation, then after extending scalars from O to ℂ via O ⊂ \overline{ℚ}_{p} ⟶^{∼}_{ı} ℂ, this space corresponds to classical automorphic forms of fixed type σ(τ) at the places in S_{p} \ {𝔭}, full level 𝔭^{m} at 𝔭, and whose weight (via our fixed isomorphism ı : \overline{ℚ}_{p} → ℂ) is 0 at places above 𝔭, and given by ξ at each of the places in S_{p} \ {𝔭}.)

We let 𝕋^{S_{p}∪Q_{N} ,univ} be the commutative O-polynomial algebra generated by formal variables T_{w}^{(j)} for all 1 ≤ j ≤ n, w a place of F̃ lying over a place v of F̃^{+} which splits in F̃ and is not contained in S_{p} ∪ Q_{N} ∪ {v_{1}}, together with formal variables T (j) ; ṽ_{1} for 1 ≤ j ≤ n. The algebra 𝕋^{S_{p}∪Q_{N} ,univ} acts on S_{ξ,τ} (U_{i}(Q_{N})_{m}, V) via the Hecke operators

T_{w}^{(j)} := [U_{m,w} ι_{w}^{−1} (\begin{matrix}ϖ_{w}1_{j} & 0 \\ 0 & 1_{n−j} \end{matrix}) U_{m,w}]

where ϖ_{w} is a fixed uniformiser in OF̃_{w}.

Choose an ordering δ_{1}, …, δ_{n} of the (distinct) eigenvalues of \overline{ρ}(Frobṽ_{1}). Since \overline{ρ} is a suitable globalisation of r̄, it is in particular automorphic in the sense of Definition 5.3.1 of [EG13], and we let 𝔪_{Q_{N}} be the maximal ideal of 𝕋^{S_{p}∪Q_{N} ,univ} corresponding

to \overline{ρ}, and containing (T (j) ; ṽ_{1} − (Nv_{1})^{j(1−j)/2}(δ_{1} ⋯ δ_{j})) for each 1 ≤ j ≤ n.

We will write 𝔪 for 𝔪_{∅}.

2.4. Galois deformations. Let S be a set of places of F̃^{+} which split in F̃, with S_{p} ⊆ S. As in [CHT08], we will write F̃(S) for the maximal extension of F̃ unramified outside S, and from now on we will write G_{F̃^{+},S} for Gal(F̃(S)/F̃^{+}). We will freely make use of the terminology (of liftings, framed liftings etc.) of Section 2 of [CHT08].

Let T = S_{p} ∪{v_{1}}. For each v ∈ S_{p}, we let ṽ be a choice of a place of F̃ lying over v, with the property that \overline{ρ}|_{GF̃ṽ} ≅ r̄. (Such a choice is possible by our assumption

that \overline{ρ} is a suitable globalisation of r̄.) We let T̃ denote the set of places ṽ, v ∈ T. For each v ∈ T, we let R□ ; ṽ denote the maximal reduced and p-torsion free quotient of the universal O-lifting ring of \overline{ρ}|_{GF̃ṽ}. For each v ∈ S_{p}\\{𝔭\}, we write R□,ξ,τ ; ṽ for the

reduced and p-torsion free quotient of R□ ; ṽ corresponding to potentially crystalline lifts of weight ξ and inertial type τ.

Consider (in the terminology of [CHT08]) the deformation problem

S := (F̃/F̃^{+}, T, T̃, O, \overline{ρ}, ε^{1−n}δ_{F̃/F̃^{+}}^{n}, \{R□ ; ṽ_{1}\} ∪ \{R□ ; 𝔭̃\} ∪ \{R□,ξ,τ ; ṽ \}_{v∈S_{p}\\{𝔭\}}).

There is a corresponding universal deformation ρ_{S}^{univ} : G_{F̃^{+},T} → G_{n}(R_{S}^{univ}) of \overline{ρ}. In addition, there is a universal T -framed deformation ring R_{S}^{□_{T}} in the sense of Definition 1.2.1 of [CHT08], which parameterises deformations of \overline{ρ} of type S together with particular local liftings for each ṽ ∈ T̃.

2.5. Auxiliary primes. As in Section 5.5 of [EG13], the version of the Taylor– Wiles patching argument given in [Tho12] allows us to choose an integer q ≥ [F̃^{+} : ℚ]n(n − 1)/2 and for each N ≥ 1 sets of primes Q_{N}, Q̃_{N} with the following properties:

• Q_{N} is a finite set of finite places of F̃^{+} of cardinality q which is disjoint from T and consists of places which split in F̃;

• Q̃_{N} consists of a single place ṽ of F̃ above each place v of Q_{N};

• Nv ≡ 1 mod p^{N} for v ∈ Q_{N};

• for each v ∈ Q_{N}, \overline{ρ}|_{GF̃ṽ} ≅ s̄ṽ ⊕ \overline{ψ}ṽ where \overline{ψ}ṽ is an eigenspace of Frobenius on which Frobenius acts semisimply.

For each N, v ∈ Q_{N} and i = 0, 1, we let U_{i}(Q_{N})_{v} ⊂ G̃(OF̃_{v}^{+}) denote the parahoric open compact subgroups defined in Section 5.5 of [EG13], following [Tho12].

For each v ∈ Q_{N}, a quotient R\overline{ψ}ṽ ; ṽ of R□ ; ṽ is defined in section 5.5 of [EG13] (following [Tho12]). We let S_{Q_{N}} denote the deformation problem

S_{Q_{N}} := (F̃/F̃^{+}, T ∪ Q_{N}, T̃ ∪ Q̃_{N}, O, \overline{ρ}, ε^{1−n}δ_{F̃/F̃^{+}}^{n},

\{R□ ; ṽ_{1}\} ∪ \{R□ ; 𝔭̃\} ∪ \{R□,ξ,τ ; ṽ \}_{v∈S_{p}\\{𝔭\}} ∪ \{R\overline{ψ}ṽ ; ṽ\}_{v∈Q_{N}} ).

We let R_{S_{Q_{N}}}^{univ} denote the corresponding universal deformation ring, and we let R_{S_{Q_{N}}}^{□_{T}} denote the corresponding universal T -framed deformation ring. We define

R^{loc} := R□ ; 𝔭̃ ˆ⊗ (ˆ⊗_{v∈S_{p}\{𝔭}}R□,ξ,τ ; ṽ ) ˆ⊗R□ ; ṽ_{1}

where all completed tensor products are taken over O. By the choice of the sets of primes Q_{N}, we also know that

• the ring R_{S_{Q_{N}}}^{□_{T}} can be topologically generated over R^{loc} by

q − [F̃^{+} : ℚ]n(n − 1)/2 elements.

For each v ∈ Q_{N} we choose a uniformiser ϖṽ ∈ OF̃ṽ, so that we have the projection operator pr_{ϖṽ} ∈ End_{O}(S_{ξ,τ} (U_{i}(Q_{N})_{m}, O/ϖ^{r})_{𝔪_{Q_{N}}}) defined as in Proposition 5.9 of [Tho12]. We define pr to be the composite of the projections pr_{ϖṽ}. (These projections commute among themselves, and so it doesn’t matter in which order we compose them. Whenever we use pr it will be clear from the context what the underlying set Q_{N} is.) Then, as in Section 5.5 of [EG13]:

(1) The map

pr : S_{ξ,τ} (U_{m}, O/ϖ^{r})_{𝔪} → pr (S_{ξ,τ} (U_{0}(Q_{N})_{m}, O/ϖ^{r})_{𝔪_{Q_{N}}} )

is an isomorphism. Moreover, since pr is defined using Hecke operators at places in Q_{N}, it commutes with the action of G̃(F̃_{𝔭}^{+}) on the spaces of algebraic automorphic forms. More precisely, if g ∈ G̃(F̃_{𝔭}^{+}) satisfies g^{−1}U_{m′,𝔭}g ⊆ U_{m,𝔭} for some positive integers m ≤ m′, then we have a commutative diagram

S_{ξ,τ} (U_{m}, O/ϖ^{r})_{𝔪} ⟶^{pr} pr (S_{ξ,τ} (U_{0}(Q_{N})_{m}, O/ϖ^{r})_{𝔪_{Q_{N}}} )
↓^{g}  ↓^{g}
S_{ξ,τ} (U_{m′}, O/ϖ^{r})_{𝔪} ⟶^{pr} pr (S_{ξ,τ} (U_{0}(Q_{N})_{m′}, O/ϖ^{r})_{𝔪_{Q_{N}}} )

where both vertical arrows are induced by the action of g ∈ G̃(F̃_{𝔭}^{+}) (namely: (gf)(g′) = f(g′g)).

(2) Let Γ_{m} = GL_{n}(O_{F} /ϖ_{F}^{m}) ≅ U/U_{m} and Δ_{Q_{N}} = ∏_{v∈Q_{N}} U_{0}(Q_{N})_{v}/U_{1}(Q_{N})_{v}. Then U_{0}(Q_{N})_{0} acts on S_{ξ,τ} (U_{1}(Q_{N})_{m}, O/ϖ^{r}) via (gf)(g′) = g_{p}f(g′g), and this action factors through Δ_{Q_{N}} × Γ_{m}. With respect to this action, pr (S_{ξ,τ} (U_{1}(Q_{N})_{m}, O/ϖ^{r})_{𝔪_{Q_{N}}} ) is a projective (O/ϖ^{r})[Δ_{Q_{N}}][Γ_{m}]-module, and there is a natural Γ_{m}-equivariant isomorphism

pr (S_{ξ,τ} (U_{1}(Q_{N})_{m}, O/ϖ^{r})_{𝔪_{Q_{N}}} )^{Δ_{Q_{N}}} ⟶^{∼} S_{ξ,τ} (U_{m}, O/ϖ^{r})_{𝔪}.

(The projectivity follows from the proof of Lemma 3.3.1 of [CHT08]. The fact that there is a Γ_{m}-equivariant isomorphism follows immediately from point (1) and the definitions. We shall not need the analogous statement about coinvariants which is recalled in [EG13] and proved in [Tho12]; see Remark 2.7 below for an indication as to why not.)

(3) Let 𝕋_{ξ,τ}^{S_{p}∪Q_{N}} (U_{i}(Q_{N})_{m}, O/ϖ^{r}) be the image of 𝕋^{S_{p}∪Q_{N} ,univ} in the ring End_{O} (pr (S_{ξ,τ} (U_{i}(Q_{N})_{m}, O/ϖ^{r}))). Then there exists a deformation

G_{F̃^{+},T ∪Q_{N}} → G_{n} (𝕋_{ξ,τ}^{S_{p}∪Q_{N}} (U_{1}(Q_{N})_{m}, O/ϖ^{r})_{𝔪_{Q_{N}}} )

of \overline{ρ} which is of type S_{Q_{N}}. In particular, pr (S_{ξ,τ} (U_{i}(Q_{N})_{m}, O/ϖ^{r})_{𝔪_{Q_{N}}} )

is a finite R_{S_{Q_{N}}}^{univ} -module.

As in Section 5.5 of [EG13], there is a homomorphism Δ_{Q_{N}} → (R_{S_{Q_{N}}}^{univ} )^{×} obtained by identifying Δ_{Q_{N}} with the product of the inertia subgroups in the maximal abelian p-power order quotient of ∏_{v∈Q_{N}} GF̃ṽ, and thus a homomorphism

O[Δ_{Q_{N}}] → R_{S_{Q_{N}}}^{univ} . The R_{S_{Q_{N}}}^{univ} -module structure on pr (S_{ξ,τ} (U_{1}(Q_{N})_{m}, O/ϖ^{r})_{𝔪_{Q_{N}}} )

thus induces an action of O[Δ_{Q_{N}}] on pr (S_{ξ,τ} (U_{1}(Q_{N})_{m}, O/ϖ^{r})_{𝔪_{Q_{N}}} ), which agrees with the one in (2) above.

2.6. Patching. We now make our patching construction, by applying the Taylor– Wiles–Kisin method. Before doing so, we provide a brief comparison and contrast with the patching constructions in some previous papers, such as [EG13] and [EGS13]. In the latter paper, we employ Taylor–Wiles–Kisin patching to construct what we call patching functors, which are (essentially) certain exact functors from the category of continuous GL_{n}(O_{F})-representations on finitely generated Omodules to the category of coherent sheaves on an appropriate deformation space of local Galois representations (perhaps with some auxiliary patching variables added). Although this is not discussed in [EGS13], such a functor can be (pro-)represented by an object M_{∞}, which is a continuous GL_{n}(O_{F})-representation over the local deformation ring (again, perhaps with patching variables added). More precisely, in terms of such a GL_{n}(O_{F})-representation M_{∞}, the patching functor can be defined as Hom_{O[[GL_{n}(O_{F})]]}^{cont}(M_{∞}, V^{∨})^{∨}, if V is a continuous representation of GL_{n}(O_{F}) on a finitely generated O-module. The exactness of the patching functor can be encoded in the requirement that M_{∞} be a projective O[[GL_{n}(O_{F})]]-module.

In this paper our approach is to construct the representing object M_{∞} directly, and (most importantly) to promote it from being merely a GL_{n}(O_{F})-representation to being a representation of the full p-adic group GL_{n}(F). (In terms of patching functors, one can somewhat loosely think of this as extending the patching functor from the category of GL_{n}(O_{F})-representations to a category that we might call the Hecke category, whose objects are the same, but in which the morphisms between any two GL_{n}(O_{F})-representations U and V are defined to be Hom_{GL_{n}(F)}(c-Ind_{GL_{n}(O_{F})}^{GL_{n}(F)}U, c-Ind_{GL_{n}(O_{F})}^{GL_{n}(F)}V ).)

From now on, to ease notation we write K = GL_{n}(OF̃𝔭̃) = GL_{n}(O_{F}), G =

GL_{n}(F̃𝔭̃) = GL_{n}(F), and Z = Z(G). For each integer N ≥ 0, we set K_{N} := ker(GL_{n}(O_{F}) → GL_{n}(O_{F} /ϖ_{F}^{N})), so that K/K_{N} ⟶^{∼} Γ_{N}. We have the Cartan decomposition G = KAK, where A is the set of diagonal matrices whose diagonal entries are powers of the uniformiser ϖ_{F}, and we let A_{N} be the subset of A consisting of matrices with the property that the ratio of any two diagonal entries is of the form ϖ_{F}^{r} with |r| ≤ N, and set G_{N} = KA_{N}K. Note that G_{N} is not a subgroup of G unless N = 0, but that each K\G_{N}/KZ is finite, and G = ∪_{N≥0}G_{N}.

If (σ, W) is a representation of KZ, then we write Ind_{KZ}^{G_{N}} σ for the space of functions f : G_{N} → W with f(kg) = σ(k)f(g) for all g ∈ G_{N}, k ∈ KZ; this is naturally a KZ-representation via (kf)(g) := f(gk). We define Ind_{KZ}^{G} σ in the same way; then Ind_{KZ}^{G} σ is a representation of G via (gf)(g′) := f(g′g).

For each N, we set

M_{i,Q_{N}} := pr(S_{ξ,τ} (U_{i}(Q_{N})_{2N}, O/ϖ^{N})_{𝔪_{Q_{N}}} )^{∨}.

Note that M_{i,Q_{N}} depends on the integer N as well as on the set of primes Q_{N} (it could happen that Q_{M} = Q_{N} for M ≠ N), but we will only include Q_{N} in the notation for the sake of simplicity. Note also that we could have equivalently defined

M_{i,Q_{N}} := pr^{∨}(S_{ξ,τ} (U_{i}(Q_{N})_{2N}, O/ϖ^{N})_{𝔪_{Q_{N}}}^{∨} ),

since pr is an endomorphism of S_{ξ,τ} (U_{i}(Q_{N})_{2N}, O/ϖ^{N})_{𝔪_{Q_{N}}} and Pontrjagin duality is an exact contravariant functor.

Let Δ_{Q_{N}} be as above; it is of p-power order by the definitions of the U_{i}(Q_{N})_{2N}. It follows from point (2) in the previous section that M_{1,Q_{N}} is a finite projective (O/ϖ^{N})[Δ_{Q_{N}}][Γ_{2N}]-module. Since Z centralises U_{1}(Q_{N})_{2N}, there is also a natural action of Z on M_{1,Q_{N}}.

2.7. Remark. The reason for including a Pontrjagin dual in the definition of M_{i,Q_{N}} is that S_{ξ,τ} (U_{i}(Q_{N})_{2N}, O/ϖ^{N}) is a space of automorphic forms, and so is most naturally thought of as being contravariant in the level, while patching is a process that involves passing to a projective limit over the level (rather than a direct limit). Now since S_{ξ,τ} (U_{i}(Q_{N})_{2N}, O/ϖ^{N}) is a space of automorphic forms on the definite unitary group G̃, it is a space of functions on a finite set, and so has a natural self-duality. Thus, by exploiting this self-duality to convert its contravariant functoriality into a covariant functoriality, we could omit the Pontrjagin dual in the preceding definition, and indeed it is traditionally omitted (see e.g. [CHT08], [BLGG11], [Tho12], and [EG13]). However, we have found it conceptually clearer to include this duality in our definitions and constructions.

We now define a KZ-equivariant map

α_{N} : M_{1,Q_{N}} → Ind_{KZ}^{G_{N}} ((M_{1,Q_{N}})_{K_{N}})

(where (M_{1,Q_{N}})_{K_{N}} denotes the K_{N}-coinvariants in M_{1,Q_{N}}) in the following way. Note firstly that there is a natural identification

(M_{1,Q_{N}})_{K_{N}} = pr^{∨}(S_{ξ,τ} (U_{1}(Q_{N})_{N}, O/ϖ^{N})_{𝔪_{Q_{N}}}^{∨} ),

so it suffices to define a KZ-equivariant map

α_{N} : S_{ξ,τ} (U_{1}(Q_{N})_{2N}, O/ϖ^{N})^{∨} → Ind_{KZ}^{G_{N}} S_{ξ,τ} (U_{1}(Q_{N})_{N}, O/ϖ^{N})^{∨}.

Now, given g ∈ G_{N}, we have g^{−1}K_{2N}g ⊆ K_{N}, so that there is a natural map

g^{∗} : S_{ξ,τ} (U_{1}(Q_{N})_{N}, O/ϖ^{N}) → S_{ξ,τ} (U_{1}(Q_{N})_{2N}, O/ϖ^{N})

given by (g^{∗} · f)(x) := f(xg), and a map

g_{∗} := ((g^{−1})^{∗})^{∨} : S_{ξ,τ} (U_{1}(Q_{N})_{2N}, O/ϖ^{N})^{∨} → S_{ξ,τ} (U_{1}(Q_{N})_{N}, O/ϖ^{N})^{∨}.

(The latter is well defined since G_{N} is stable under taking inverses. We note that g^{∗} (resp. g_{∗}) may be interpreted as the natural pullback (resp. pushforward) map on cohomology (resp. homology) under the natural right (resp. left) action of G̃(𝔸^{∞}) on the tower of arithmetic quotients of G̃.) We have (gh)^{∗} = g^{∗} ∘ h^{∗} and hence (gh)_{∗} = g_{∗} ∘ h_{∗}, whenever all are defined. Then we define α_{N} by

(α_{N}(x))(g) := g_{∗}(x).

In order to check that this is KZ-equivariant, we must check that for all k ∈ KZ we have (α_{N}(kx))(g) = (kα_{N}(x))(g); this is equivalent to checking that g_{∗}(kx) = (gk)_{∗}(x), which is immediate from the definition.

Now set M_{1,Q_{N}}^{□} := M_{1,Q_{N}} ⊗_{R_{S_{Q_{N}}}^{univ}} R_{S_{Q_{N}}}^{□_{T}}. We have an induced KZ-equivariant

map α_{N} : M_{1,Q_{N}}^{□} → Ind_{KZ}^{G_{N}}(M_{1,Q_{N}}^{□})_{K_{N}}. Define

R_{∞} := R^{loc}[[x_{1}, …, x_{q−[F̃^{+}:ℚ]n(n−1)/2}]],
S_{∞} := O[[z_{1}, …, z_{n^{2}#T} , y_{1}, …, y_{q}]],

for formal variables x_{1}, …, x_{q−[F̃^{+}:ℚ]n(n−1)/2}, y_{1}, …, y_{q} and z_{1}, …, z_{n^{2}#T}. For each N, we fix a surjection R_{∞} ↠ R_{S_{Q_{N}}}^{□_{T}} of R^{loc}-algebras (which, as recalled in Section 2.5, is possible by the choice of the sets Q_{N}). These choices allow us to regard each M_{1,Q_{N}}^{□} as an R_{∞}-module. Also, we fix choices of lifts representing the universal deformations over R_{S}^{univ} and each R_{S_{Q_{N}}}^{univ} such that our chosen lift over each R_{S_{Q_{N}}}^{univ} reduces to our chosen lift over R_{S}^{univ}. These choices give rise to

isomorphisms R_{S_{Q_{N}}}^{□_{T}} ⟶^{∼} R_{S_{Q_{N}}}^{univ} ˆ⊗_{O}O[[z_{1}, …, z_{n^{2}#T}]] compatible with a fixed iso-

morphism R_{S}^{□_{T}} ⟶^{∼} R_{S}^{univ} ˆ⊗_{O}O[[z_{1}, …, z_{n^{2}#T}]]; they also allow us to regard each M_{1,Q_{N}}^{□} as an O[[z_{1}, …, z_{n^{2}#T}]]-module. Finally, for each N, choose a surjection O[[y_{1}, …, y_{q}]] ↠ O[Δ_{Q_{N}}]. This gives each M_{1,Q_{N}}^{□} the structure of an S_{∞}-module and hence the structure of an S_{∞}[[K]]-module (where the action of K factors through Γ_{2N}).

We now apply the Taylor–Wiles method in the usual way to pass to a subsequence, and patch the modules M_{1,Q_{N}}^{□} together with the maps α_{N} : M_{1,Q_{N}}^{□} → Ind_{KZ}^{G_{N}}(M_{1,Q_{N}}^{□})_{K_{N}}. More precisely, for each N ≥ 1, let 𝔟_{N} denote the ideal of S_{∞} generated by ϖ^{N}, z_{i}^{N} and (1 + y_{i})^{p^{N}} − 1. Let 𝔞 denote the ideal of S_{∞} generated by the z_{i} and the y_{i}. Fix a sequence (𝔡_{N})_{N≥1} of open ideals of R_{S}^{univ} such that

• ϖ^{N}R_{S}^{univ} ⊂ 𝔡_{N} ⊂ Ann_{R_{S}^{univ}}(S_{ξ,τ} (U_{2N}, O/ϖ^{N})_{𝔪}^{∨});

• 𝔡_{N} ⊃ 𝔡_{N+1} for all N;
• ∩_{N≥1}𝔡_{N} = (0).

(For example, we may take 𝔡_{N} = 𝔪_{R_{S}^{univ}}^{N} ∩ Ann_{R_{S}^{univ}}(S_{ξ,τ} (U_{2N}, O/ϖ^{N})_{𝔪}^{∨}).)

At level N, we consider tuples consisting of • a surjective homomorphism of R^{loc}-algebras ϕ : R_{∞} ↠ R_{S}^{univ}/𝔡_{N};

• a finite projective (S_{∞}/𝔟_{N})[Γ_{2N}]-module M^{□} which carries a commuting action of R_{∞} ⊗_{O} O[Z] such that the action of S_{∞} can be factored through that of R_{∞};

• a Γ_{2N} and Z-equivariant isomorphism ψ : (M^{□}/𝔞M^{□}) ⟶^{∼} S_{ξ,τ} (U_{2N}, O/ϖ^{N})_{𝔪}^{∨} compatible with ϕ;

• a KZ-equivariant and R_{∞} ⊗_{O} S_{∞}-linear map α : M^{□} → Ind_{KZ}^{G_{N}}[(M^{□})_{K_{N}}], whose reduction modulo 𝔞 is compatible with the map

S_{ξ,τ} (U_{2N}, O/ϖ^{N})_{𝔪}^{∨} → Ind_{KZ}^{G_{N}}[S_{ξ,τ} (U_{N}, O/ϖ^{N})_{𝔪}^{∨}]

induced by the action of G_{N} on the spaces of algebraic modular forms. We consider two such tuples (ϕ, M^{□}, ψ, α) and (ϕ′, M^{□′}, ψ′, α′) to be equivalent if ϕ = ϕ′ and if there is an isomorphism of (S_{∞}/𝔟_{N})[Γ_{2N}] ⊗_{O} R_{∞} ⊗_{O} O[Z]-modules M^{□} ⟶^{∼} M^{□′} which identifies ψ with ψ′ and α with α′. Note that there are at most finitely many equivalence classes of such tuples. (For a given M^{□} there are only finitely many KZ-equivariant homomorphisms α : M^{□} → Ind_{KZ}^{G_{N}}[(M^{□})_{K_{N}}], because M^{□} and K\G_{N}/KZ are finite.) Note also that a tuple (ϕ, M^{□}, ψ, α) of level N gives rise to a tuple (ϕ′, M^{□′}, ψ′, α′) of level (N − 1) by setting ϕ′ := ϕ mod 𝔡_{N−1}, M^{□′} := (M^{□}/𝔟_{N−1})_{K_{2(N−1)}} and ψ′ := ψ mod ϖ^{N−1}. The map α′ is defined by the formula α′(\overline{m})(g) = \overline{α(m)(g)}. Here \overline{m} denotes the image of m ∈ M^{□} in M^{□′} and \overline{α(m)(g)} denotes the image of α(m)(g) in (M^{□′})_{K_{N−1}}. Note that for any m ∈ M^{□}, γ ∈ K_{2(N−1)} and g ∈ G_{N−1}, we have

α((γ − 1)m)(g) = α(m)(g(γ − 1)) = α(m)((gγg^{−1} − 1)g) = (gγg^{−1} − 1)α(m)(g),

by the K-equivariance of α. Using the fact that gγg^{−1} ∈ K_{N−1} it is straightforward to see that α′ is well-defined.

For each pair of integers N′ ≥ N ≥ 1, we define a tuple (ϕ, M^{□}, ψ, α) of level N as follows: we set ϕ equal to R_{∞} ↠ R_{S_{Q_{N′}}}^{□_{T}} ↠ R_{S}^{univ}/𝔡_{N} and we set M^{□} =

(M_{1,Q_{N′}}^{□} ⊗_{S_{∞}} S_{∞}/𝔟_{N})_{K_{2N}}. The map ψ comes from points (1) and (2) in the previous section. The map α comes from α_{N′} defined above (in the same way that α′ is defined in terms of α in the previous paragraph) and the compatibility it is required to satisfy comes from the commutative diagram in point (1) of the previous section. Since there are only finitely many isomorphism classes of tuples at each level N, but N′ is allowed to be arbitrarily large, we can apply a diagonal argument to find a subsequence of pairs (N′(N), N)_{N≥1} indexed by N such that for each N ≥ 2, the tuple indexed by N reduced to level (N − 1) is isomorphic to the tuple indexed by (N − 1). For each N ≥ 2, we fix a choice of such an isomorphism.

We now define

M_{∞} := \underleftarrow{lim}_{N}(M_{1,Q_{N′(N)}}^{□} ⊗_{S_{∞}} S_{∞}/𝔟_{N})_{K_{2N}} ,

where the transition maps are induced by the isomorphisms fixed in the previous paragraph. (We drop the square from the notation here in order to avoid notational overload in later sections.) Each of the terms in the projective limit is a (literally) finite O-module, endowed with commuting actions of S_{∞}[[K]] and R_{∞} ⊗_{O} O[Z], and by construction the transition maps in the projective limit respect these actions. Thus M_{∞} is naturally a profinite topological S_{∞}[[K]]-module which carries a commuting action of R_{∞} ⊗_{O} O[Z], the topology on M_{∞} being the projective limit topology (where each of the terms in the projective limit is endowed with the discrete topology). Moreover, the action of S_{∞} on M_{∞} can be factored through a map S_{∞} → R_{∞} (since the analogous statement holds at each level N).

The module M_{∞} is the key construction of the paper; the remainder of this section is devoted to recording some additional properties that it enjoys. Firstly, since the transition maps in the projective limit are given simply by reducing from level N to level N − 1, it is easily verified that the natural map induces an isomorphism

(M_{∞}/𝔟_{N})_{K_{2N}} ⟶^{∼} (M_{1,Q_{N′(N)}}^{□}/𝔟_{N})_{K_{2N}} .

Next, from this, it follows from the topological form of Nakayama’s lemma that M_{∞} is in fact a finite S_{∞}[[K]]-module. It follows that the topology on M_{∞} coincides with the quotient topology obtained by writing it as a quotient of S_{∞}[[K]]^{r}, where S_{∞}[[K]] is endowed with its natural profinite topology. Crucially, there is a KZequivariant and R_{∞}-linear map

α_{∞} : M_{∞} → Ind_{KZ}^{G} M_{∞}

by taking the projective limit of the maps

(M_{1,Q_{N′(N)}}^{□} ⊗_{S_{∞}} S_{∞}/𝔟_{N})_{K_{2N}} → Ind_{KZ}^{G_{N}} ((M_{1,Q_{N′(N)}}^{□} ⊗_{S_{∞}} S_{∞}/𝔟_{N})_{K_{N}} ) .

induced by α_{N′(N)} : M_{1,Q_{N′(N)}} → Ind_{KZ}^{G_{N′(N)}} ((M_{1,Q_{N′(N)}})_{K_{N′(N)}}). We denote this induced map by \overline{α}_{N′(N)}.

The following proposition establishes the additional key properties of the patched module M_{∞} that we will need.

2.8. Proposition. M_{∞} is projective over S_{∞}[[K]]. Furthermore, α_{∞} is injective, and its image is G-stable, so that α_{∞} induces an action of G on M_{∞}.

Proof. Since M_{∞} is finite over S_{∞}[[K]], we may choose a presentation S_{∞}[[K]]^{r} ↠ M_{∞} for some r ≥ 1. In order to check that M_{∞} is a projective S_{∞}[[K]]-module, it is enough to check that this surjection splits. Since each M_{1,Q_{N}} is a projective (O/ϖ^{N})[Δ_{Q_{N}}][Γ_{2N}]-module, we see that

(M_{∞}/𝔟_{N})_{K_{2N}} ≅ (M_{1,Q_{N′(N)}}^{□}/𝔟_{N})_{K_{2N}}

is a projective S_{∞}/𝔟_{N}[Γ_{2N}]-module, so we have a cofinal system of ideals (namely, the ideals generated by 𝔟_{N} + ker(O[[K]] → O[Γ_{2N}])) defining the topology of S_{∞}[[K]] modulo which the surjection splits. The sets of possible splittings at these finite levels then give us a projective system of non-empty finite sets, and an element of the projective limit of this projective system gives the required splitting.

We now check that α_{∞} is injective. Note that by definition, for each α_{N} : M_{1,Q_{N}} → Ind_{KZ}^{G_{N}}((M_{1,Q_{N}})_{K_{N}} ) we have (α_{N}(m))(1) = 1_{∗}(m) = \overline{m}, where \overline{m} denotes the image of m in (M_{1,Q_{N}})_{K_{N}}. From this (with N replaced by N′(N)) we deduce that (\overline{α}_{N′(N)})(m))(1) = \overline{m} where \overline{m} is the image of m ∈ (M_{1,Q_{N′(N)}}^{□}/𝔟_{N})_{K_{2N}}

in (M_{1,Q_{N′(N)}}^{□}/𝔟_{N})_{K_{N}}. This then implies that (α_{∞}(m))(1) = m for each m ∈ M_{∞}, and α_{∞} is certainly injective.

In order to show that the image of α_{∞} is G-stable, we will show that for all g ∈ G, m ∈ M_{∞} we have

g(α_{∞}(m)) = α_{∞}((α_{∞}(m))(g)).

In other words, we will show that for all g, h ∈ G and m ∈ M_{∞}, we have

(α_{∞}(m))(hg) = (α_{∞}((α_{∞}(m))(g)))(h).

Let m be an element of M_{∞} and let N be any integer large enough so that g, h, gh ∈ G_{N}. This certainly means that g, h, gh ∈ G_{2N} as well. Since N can be arbitrarily large, it is enough to show that both sides of the equation above become equal in (M_{∞}/𝔟_{N})_{K_{N}} and we do this by explicit computation.

We let π_{N} : M_{∞} → (M_{∞}/𝔟_{N})_{K_{2N}} and σ_{N} : M_{∞} → (M_{∞}/𝔟_{N})_{K_{N}} denote the projection maps. Then by definition, we have

σ_{N}(α_{∞}(m)(hg)) = \overline{α}_{N′(N)}(π_{N}(m))(hg)

and

σ_{N}(α_{∞}(α_{∞}(m)(g))(h)) = \overline{α}_{N′(N)}(π_{N}(α_{∞}(m)(g)))(h)
= \overline{α}_{N′(N)}(σ_{2N}(α_{∞}(m)(g)) mod 𝔟_{N})(h)
= \overline{α}_{N′(N)}(\overline{α}_{N′(2N)}(π_{2N}(m))(g) mod 𝔟_{N})(h).

Now, for integers N, Ñ, N′′ ≥ 1 with Ñ ≤ N′(N), we let U_{1}(Q_{N′(N)}, Ñ)_{N′′} be the open compact subgroup intermediate to U_{1}(Q_{N′(N)})_{N′′} and U_{0}(Q_{N′(N)})_{N′′} such that U_{0}(Q_{N′(N)})_{N′′}/U_{1}(Q_{N′(N)}, Ñ)_{N′′} ≅ (ℤ/pÑℤ)^{q}. Then we have a commutative diagram

S_{ξ,τ} (U_{1}(Q_{N′(2N)}, 2N)_{4N}, O/ϖ^{2N})^{∨} ⟶^{g_{∗}} S_{ξ,τ} (U_{1}(Q_{N′(2N)}, 2N)_{2N}, O/ϖ^{2N})^{∨}
∥  ↓^{nat}
S_{ξ,τ} (U_{1}(Q_{N′(2N)}, 2N)_{4N}, O/ϖ^{2N})^{∨}  S_{ξ,τ} (U_{1}(Q_{N′(2N)}, N)_{2N}, O/ϖ^{N})^{∨}
↓^{nat}  ↓^{h_{∗}}
S_{ξ,τ} (U_{1}(Q_{N′(2N)}, N)_{2N}, O/ϖ^{N})^{∨} ⟶^{(hg)_{∗}} S_{ξ,τ} (U_{1}(Q_{N′(2N)}, N)_{N}, O/ϖ^{N})^{∨}.

Localising at 𝔪_{Q_{N′(2N)}}, applying the projector pr^{∨}, tensoring over R_{S_{Q_{N′(N)}}}^{univ} with

R_{S_{Q_{N′(N)}}}^{□_{T}} and reducing modulo 𝔟_{2N} or 𝔟_{N} as appropriate, we obtain a commutative

diagram:

(M_{∞}/𝔟_{2N})_{K_{4N}} ⟶^{g_{∗}} (M_{∞}/𝔟_{2N})_{K_{2N}}
∥  ↓^{nat}
(M_{∞}/𝔟_{2N})_{K_{4N}}  (M_{∞}/𝔟_{N})_{K_{2N}}
↓^{nat}  ↓^{h_{∗}}
(M_{∞}/𝔟_{N})_{K_{2N}} ⟶^{(hg)_{∗}} (M_{∞}/𝔟_{N})_{K_{N}} .

Now, on the one hand, the element π_{2N}(m) lies in the space in the upper left corner of this diagram. By definition, we have

\overline{α}_{N′(N)}(\overline{α}_{N′(2N)}(π_{2N}(m))(g) mod 𝔟_{N})(h) = h_{∗} ∘ nat ∘ g_{∗}(π_{2N}(m))

On the other hand, π_{N}(m) lies in the space in the lower left corner of the diagram, and we have

\overline{α}_{N′(N)}(π_{N}(m))(hg) = (hg)_{∗}(π_{N}(m)) = (hg)_{∗} ∘ nat(π_{2N}(m)).

The desired equality now follows from the commutativity of the above diagrams. □

Note that for each positive integer m, the compact open subgroups U_{m} have the same level away from 𝔭. Let U^{𝔭} ⊂ G̃(𝔸∞,𝔭 ; F̃^{+}) denote that common level. Define the ϖ-adically completed cohomology space

S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪} := \underleftarrow{lim}_{s}(\underrightarrow{lim}_{m} S_{ξ,τ} (U_{m}, O/ϖ^{s})_{𝔪}).

The space S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪} is equipped with a natural G-action, induced from the action of G on algebraic automorphic forms. Moreover, the space S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪} is equipped with a natural action of the Hecke algebra

𝕋_{ξ,τ}^{S_{p}} (U^{𝔭}, O)_{𝔪} := \underleftarrow{lim}_{m} 𝕋_{ξ,τ}^{S_{p}} (U_{m}, O)_{𝔪}.

By taking the inverse limit of the 𝕋_{ξ,τ}^{S_{p}} (U_{m}, O)_{𝔪}-valued deformations of ρ̄ of type S we obtain a map R_{S}^{univ} → 𝕋_{ξ,τ}^{S_{p}} (U^{𝔭}, O)_{𝔪}. Therefore, S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪} is also equipped with an action of the local deformation ring R□ ; 𝔭̃ via the composition R□ ; 𝔭̃ → R_{S}^{univ} →

𝕋_{ξ,τ}^{S_{p}} (U^{𝔭}, O)_{𝔪}.

2.9. Corollary. There is a G-equivariant isomorphism

(M_{∞}/𝔞M_{∞}) ⟶^{∼} S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪}^{d},

which is compatible with the R□ ; 𝔭̃-action on both sides.

Proof. We have a KZ-equivariant isomorphism ψ_{∞} : (M_{∞}/𝔞M_{∞}) ⟶^{∼} S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪}^{d} obtained by taking the projective limit of the Γ_{2N}- and Z-equivariant isomorphisms ψ at level N.

We first show that ψ_{∞} is compatible with the R□ ; 𝔭̃-action on both sides. The R□ ; 𝔭̃-action on the left hand side is via the map R□ ; 𝔭̃ → R_{∞} and each ψ at level N is compatible with ϕ : R_{∞} ↠ R_{S}^{univ}/𝔡_{N}. On the other hand, the action of R□ ; 𝔭̃ on the right hand side is via the map R□ ; 𝔭̃ → R_{S}^{univ} and ∩_{N≥1}𝔡_{N} = (0). The desired compatibility follows.

Moreover, ψ_{∞} fits into a commutative diagram

(M_{∞}/𝔞M_{∞}) ⟶^{ψ_{∞}}  S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪}^{d}
↓^{g}  ↓^{g}
Ind_{KZ}^{G}(M_{∞}/𝔞M_{∞}) ⟶^{ψ_{∞}} Ind_{KZ}^{G}(S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪}^{d})

where both vertical maps are induced by the action of g ∈ G. The G-equivariance of ψ_{∞} now follows from Proposition 2.8. □

2.10. Admissible unitary Banach representations. Fix a lifting r : G_{F} → GL_{n}(O) of r̄. We now explain how M_{∞} allows us to associate an admissible unitary Banach representation V (r) of GL_{n}(F) to r.

As above, we identify F with F̃𝔭̃. By definition, r comes from a homomorphism of O-algebras x : R□ ; 𝔭̃ → O. We extend this to a homomorphism of O-algebras

x′ : R□ ; 𝔭̃ ˆ⊗ (ˆ⊗_{v∈S_{p}\{𝔭}}R□,ξ,τ ; ṽ ) → O

by using the homomorphisms R□,ξ,τ ; ṽ → O corresponding to our given potentially diagonalizable representation r_{pot.diag}, and then extend x′ arbitrarily to a homomorphism of O-algebras y : R_{∞} → O. We set V (r) := (M_{∞} ⊗_{R_{∞},y} O)^{d}[1/p].

2.11. Proposition. The representation V (r) is an admissible unitary Banach representation of GL_{n}(F).

2.12. Remark. Note that we do not know if V (r) is independent of either the global setting or the choice of y. We also do not know that V (r) is necessarily nonzero, although we will prove that V (r) ≠ 0 for many regular de Rham representations r in Section 5 below, as a consequence of the stronger result that (for the particular choice of r under consideration) the subspace of locally algebraic vectors in V (r) is nonzero.

Proof of Proposition 2.11. The image of (M_{∞}⊗_{R_{∞},y}O)^{d} in V (r) is a GL_{n}(F)-stable unit ball, so V (r) is a unitary representation.

In order to see that V (r) is admissible, we must show that for each N ≥ 0, the 𝔽-vector space ((M_{∞} ⊗_{R_{∞},y} O)^{d} ⊗ 𝔽)^{K_{N}} is finite-dimensional. Writing \overline{y} for the composite y : R_{∞} → O → 𝔽, we must check that (M_{∞} ⊗_{R_{∞},\overline{y}} 𝔽)_{K_{N}} is finitedimensional. Since M_{∞} is a finite S_{∞}[[K]]-module, and ker \overline{y} induces an open ideal of S_{∞}, this is immediate. □

2.13. Remark. While we have assumed throughout this section that r̄ has a potentially diagonalizable lift with regular Hodge–Tate weights, this hypothesis is not needed for our main results, which concern representations r : G_{F} → GL_{n}(E). Indeed, possibly after making a ramified extension of E, it is easy to see that any such representation can be conjugated to a representation r′ which factors through GL_{n}(O) and whose reduction r̄′ is semisimple, so that r̄′ has a lift of the required kind (possibly after further extending E) by Lemma 2.2.

3. Hecke algebras and types

In the following two sections we will use the local Langlands correspondence and the theory of types to establish local-global compatibility results for our patched modules. In particular, we will make use of the results of [BK99] and [SZ99] in order to study spaces of automorphic forms which correspond to fixed inertial types. As explained in the introduction, in some of our results we will for simplicity restrict ourselves to the case of Weil–Deligne representations with N = 0; this means that we will limit ourselves to considering potentially crystalline Galois representations. However, some of our results are more naturally expressed in the more general context of representations with arbitrary monodromy, so we will make it clear when we impose this restriction. We begin by collecting and explaining various results from the literature that we will need.

Let F/ℚ_{p} be a finite extension. Recall that we write K = GL_{n}(O_{F}), G = GL_{n}(F), and that W_{F} is the Weil group of F. Although several of our references work over ℂ, we work consistently over \overline{ℚ}_{p} (except in Subsection 3.12, where we fix a single Bernstein component, and work over a finite extension of ℚ_{p}). The various results over ℂ are transferred to our context over \overline{ℚ}_{p} via the fixed choice of ı : \overline{ℚ}_{p} →^{∼} ℂ everywhere so this is harmless. (The transfer depends on ı since an automorphism of ℂ does not fix isomorphism classes of representations in general.)

As recalled in Section 1.5, the local Langlands correspondence rec_{p} gives a bijection between the isomorphism classes of irreducible smooth representations of G over \overline{ℚ}_{p}, and the n-dimensional Frobenius semisimple Weil–Deligne representations of W_{F}.

3.1. Bernstein–Zelevinsky theory. We now recall some details of the local Langlands correspondence and its relationship to Bernstein–Zelevinsky theory, following [Rod82].

Given a partition n = n_{1} + ⋯ + n_{r}, let P = MN be the corresponding standard parabolic subgroup of G with Levi subgroup M and unipotent radical N (standard with respect to the Borel subgroup of upper triangular matrices), so that M ≅ GL_{n_{1}}(F) × ⋯ × GL_{n_{r}}(F). For any smooth representations π_{i} of GL_{n_{i}}(F), the tensor product π_{1} ⊗ ⋯ ⊗ π_{r} is naturally a representation of M and thus of P, and we denote by π_{1} × ⋯ × π_{r} the normalised induction i_{P}^{G} π_{1} ⊗ ⋯ ⊗ π_{r}. If π_{1}, …, π_{r} are irreducible supercuspidal representations then π_{1} × ⋯ × π_{r} has finite length by [Rod82, Proposition 4].

Any smooth irreducible representation π of G is necessarily a subrepresentation of some π_{1} ×⋯×π_{r} for some P and some irreducible supercuspidal representations π_{1}, …, π_{r}. By [Rod82, Proposition 5], π determines the multiset {π_{1}, …, π_{r}} (and thus the corresponding partition of n, up to reordering) uniquely, and we refer to it as the cuspidal support of π. For any representation π_{i} of GL_{n_{i}}(F), and any integer s, we write π_{i}(s) := π_{i} ⊗ | det |^{s}. Then by [Rod82, Théorème 1] (and the remark which follows it), π_{1} × ⋯ × π_{r} is reducible if and only if there are some i, j with n_{i} = n_{j} and π_{j} ≅ π_{i}(1).

We define a segment to be a set of isomorphism classes of irreducible cuspidal representations GL_{n_{i}}(F) of the form Δ = {π_{i}, π_{i}(1), …, π_{i}(r − 1)} for some r ≥ 1, and we write Δ = [π_{i}, π_{i}(r − 1)]. We say that two segments Δ_{1}, Δ_{2} are linked if neither contains the other, and Δ_{1} ∪ Δ_{2} is also a segment. If Δ_{1} = [π_{i}, π′_{i}] and Δ_{2} = [π′′_{i}, π′′′_{i}] are linked, we say that Δ_{1} precedes Δ_{2} if π′′_{i} = π_{i}(r) for some r ≥ 0.

If Δ = [π_{i}, π_{i}(r−1)] is a segment, then π_{i}×⋯×π_{i}(r−1) has a unique irreducible subrepresentation Z(Δ) and a unique irreducible quotient L(Δ). By [Rod82, Théorème 2], if Δ_{1}, …, Δ_{r} are segments such that if i < j, then Δ_{i} does not precede Δ_{j}, then the representation Z(Δ_{1})×⋯×Z(Δ_{r}) has a unique irreducible subrepresentation, which we denote Z(Δ_{1} ×⋯×Δ_{r}). Every irreducible smooth representation of G is of the form Z(Δ_{1} ×⋯×Δ_{r}) for some segments Δ_{1}, …, Δ_{r}, which are uniquely determined up to reordering. By [Rod82, Théorème 3], the analogous statement also holds for the L(Δ_{i}). By [Rod82, Théorème 5], Z(Δ_{1}, …, Δ_{r}) occurs with multiplicity one as a subquotient of Z(Δ_{1}) × ⋯ × Z(Δ_{r}).

The link with the local Langlands correspondence is as follows (see [Rod82, §4.4]). The representation rec_{p}(π) is irreducible if and only if π is supercuspidal. For each integer r ≥ 1 there is an explicitly defined r-dimensional Weil–Deligne representation Sp(r) whose restriction to the Weil group is unramified and whose monodromy operator has rank r−1. Then if Δ = [π_{i}, π_{i}(r−1)] is a segment, we have rec_{p}(L(Δ)) = rec_{p}(π_{i}) ⊗ Sp(r), and more generally we have rec_{p}(L(Δ_{1}, …, Δ_{r})) = ⊕_{i=1}^{r}rec_{p}(L(Δ_{i})).

3.2. The Bernstein Centre. We now briefly recall some of the results of [Ber84] (in the special case that the reductive group under consideration there is GL_{n}) in a fashion adapted to our needs. The Bernstein spectrum is the set of G-orbits of pairs (M, ω), where M is a Levi subgroup of G, ω is an irreducible supercuspidal representation of M, and the action of G is via conjugation; note that up to conjugacy, (M, ω) is of the form (GL_{n_{1}}(F) × ⋯ × GL_{n_{r}}(F), π_{1} ⊗ ⋯ ⊗ π_{r}), as in the preceding section. As we will explain below, the Bernstein spectrum naturally has the structure of an algebraic variety over \overline{ℚ}_{p} (with infinitely many connected components), the Bernstein variety. Given an irreducible smooth G-representation π, we obtain a point of the Bernstein spectrum by passing to the cuspidal support of π. This map is surjective; any point (M, ω) of the Bernstein spectrum is equal to the cuspidal support of π for any Jordan–Hölder factor π of i_{P}^{G} ω (and indeed these are precisely the π for which (M, ω) arises as the cuspidal support; here P is any parabolic subgroup of G admitting M as a Levi quotient — the collection of Jordan–Hölder factors of i_{P}^{G} ω is independent of the choice of P).

The connected components of the Bernstein variety are as follows. Fix a pair (M, ω) as above; then the component of the Bernstein variety containing (M, ω) is the union of the G-orbits of the pairs (M, αω), where α is an unramified quasicharacter of M. We say that two pairs (M, ω) and (M′, ω′) are inertially equivalent if they are in the same Bernstein component, and write [M, ω] for the equivalence class. Fixing one such pair (M, ω), it is easy to see that there is a natural algebraic structure on the inertial equivalence class, because the set of unramified quasicharacters of M has a natural algebraic structure, and thus so does any quotient of it by a finite group; this gives the structure of the Bernstein variety. Given an irreducible smooth G-representation π, we will refer to the inertial equivalence class of its cuspidal support as the inertial support of π.

For any connected component Ω of the Bernstein variety, we have a corresponding full subcategory of the category of smooth G-representations, whose objects are the smooth representations all of whose irreducible subquotients have cuspidal support in Ω. Such a subcategory is called a Bernstein component of the category of smooth G-representations, and in fact the category of smooth G-representations is a direct product of the Bernstein components. Given a Bernstein component Ω, the centre ℨ_{Ω} of Ω is the centre of the corresponding Bernstein component (that is, the endomorphism ring of the identity functor), so that ℨ_{Ω} acts naturally on each irreducible smooth representation π ∈ Ω. Since π is irreducible, each element of ℨ_{Ω} will act on π through a scalar. In fact this scalar depends only on the cuspidal support of π, and in this way ℨ_{Ω} is identified with the ring of regular functions on the connected component Ω of the Bernstein variety.

The above notions extend in an obvious way to products of groups GL_{n_{i}}(F), and we will make use of this extension below without further comment (in order to compare representations of G with representations of a Levi subgroup).

Finally, we note that from the link between the local Langlands correspondence and Bernstein–Zelevinsky theory explained in Section 3.1, it is immediate that two irreducible smooth representations π, π′ of G lie in the same Bernstein component if and only if rec_{p}(π)|_{I_{F}} ≅ rec_{p}(π′)|_{I_{F}} (where we ignore the monodromy operators).

3.3. Bushnell–Kutzko theory. Fix a Bernstein component Ω. In [BK99], there is the definition of a semisimple Bushnell–Kutzko type (J, λ) for Ω, where J ⊆ K is a compact open subgroup, and λ is a smooth irreducible \overline{ℚ}_{p}-representation of J. The pair (J, λ) has the property that if π is an irreducible smooth representation of G, then Hom_{J}(λ, π) ≠ 0 if and only if π ∈ Ω, and in fact the functor Hom_{J}(λ, ∗) induces an equivalence of categories between Ω and the category of left modules of the Hecke algebra H(G, λ) := H(G, J; λ) := End_{G}(c-Ind_{J}^{G}λ), with the inverse functor being given by c-Ind_{J}^{G}λ ⊗_{H(G,λ)} (∗).

Let π be an irreducible element of Ω, and let (M, ω) be a representative for the inertial support of π. We can and do suppose that M is a standard Levi subgroup ∏_{i=1}^{r} ∏_{j=1}^{d_{i}} GL_{e_{i}}(F), and that ω = ⊗_{i=1}^{r}π_{i}^{⊗d_{i}}, where π_{i} and π_{i′} are not inertially equivalent (that is, do not differ by a twist by an unramified quasicharacter) if i ≠ i′. Then by the construction of (J, λ) in [BK99], there is a pair (J ∩ M, λ_{M}) which is a semisimple Bushnell–Kutzko type for the Bernstein component Ω_{M} of M determined by ω, in the sense that if π_{M} is an irreducible smooth representation of M, then Hom_{J∩M}(λ_{M}, π_{M}) ≠ 0 if and only if π_{M} ∈ Ω_{M}, and the functor Hom_{J}(λ_{M}, ∗) induces an equivalence of categories between Ω_{M} and the category of left modules of H(M, λ_{M}) in the same way as above.

Let P be a parabolic subgroup of G with Levi factor M. Then the normalised induction i_{P}^{G} restricts to a functor from Ω_{M} to Ω, and there is a unique injective algebra homomorphism t_{P} : H(M, λ_{M}) → H(G, λ) (which is denoted j_{Q} on page 55 of [BK99]) such that under the equivalences of categories explained above, i_{P}^{G} corresponds to the pushforward along t_{P}.

It will be useful to us to have a somewhat more explicit description of the pair (J ∩ M, λ_{M}). Recall that we may write M = ∏_{i=1}^{r} ∏_{j=1}^{d_{i}} GL_{e_{i}}(F), and that ω = ⊗_{i=1}^{r}π_{i}^{⊗d_{i}}. Then as is explained in [BK99, §1.4-5] (see also the paragraph after Lemma 7.6.3 of [BK93]), we may write J ∩ M = ∏_{i=1}^{r} ∏_{j=1}^{d_{i}} J_{i}, λ_{M} = ⨂_{i=1}^{r} λ_{i}^{⊗d_{i}}, where (J_{i}, λ_{i}) is a maximal simple type occurring in π_{i} in the sense of [BK98]. There is a corresponding isomorphism H(M, λ_{M}) ≅ ⊗_{i=1}^{r}H(GL_{e_{i}}(F), λ_{i})^{⊗d_{i}}, and each H(GL_{e_{i}}(F), λ_{i}) is commutative, so that H(M, λ_{M}) is commutative. The following somewhat technical lemma will be useful to us in Section 4. We remind the reader that the algebra H(M, λ_{M}) is naturally isomorphic to the convolution algebra of compactly supported functions f : M → End_{\overline{ℚ}_{p}}(λ_{M}) such that f(jmj′) = j ∘f(m)∘ j′ for all m ∈ M, j, j′ ∈ J ∩ M (see [BL94, §2.2]).

3.4. Lemma. There is an integer e ≥ 1 and a \overline{ℚ}_{p}-basis for H(M, λ_{M}) with the property that if ν is an element of this basis, then the e-fold convolution of ν with itself is supported on t_{ν}(J ∩ M) for some t_{ν} ∈ Z(M).

Proof. By the above remarks, we need only prove the corresponding result for the Hecke algebra of a maximal simple type H(GL_{e_{i}}(F), λ_{i}). In this case, the proof of Theorem 7.6.1 of [BK93] shows that we can take a basis given by Hecke operators supported on cosets of the form tJ_{i} where t ∈ D(𝔅) in the notation of [BK93]. By Lemma 7.6.3 of [BK93], it suffices to show that there is a positive integer e such that if t ∈ D(𝔅) then the e-fold composition of a Hecke operator ψ_{t} supported on tJ_{i} with itself is supported on sJ_{i}, where s is a scalar matrix. But D(𝔅) is a cyclic group, generated by a uniformiser ϖ_{E} of some finite extension of fields E/F inside GL_{e_{i}}(F), so we can just take e to be the ramification degree e(E : F). In that case, the e-fold composition of a Hecke operator supported on ϖ_{E}J_{i} with itself is supported on ϖ_{F} J_{i}, as remarked on the bottom of page 201 of [BK93]. □

3.5. Remark. If λ_{i} is a maximal simple type for GL_{n_{i}}(F), then the ramification degree e(E : F) is equal to n_{i}/f_{i}, where f_{i} is the number of unramified characters of F^{×} which preserve a supercuspidal GL_{n_{i}}(F)-representation containing λ_{i}. This follows from Lemma 6.2.5 of [BK93]. Therefore, det(ϖ_{E}) ∈ F^{×} has valuation f_{i}.

3.6. Results of Schneider–Zink and Dat. We will need a slight refinement of the Bernstein centre and of the theory of Bushnell–Kutzko, which is constructed in the paper [SZ99]. Note that there is not quite a bijection between irreducible smooth representations of G and characters of the Bernstein centre; as explained in Section 3.2, any two Jordan–Hölder constituents of a parabolic induction from an irreducible cuspidal representation correspond to the same character of the centre, so that for example the trivial representation and the Steinberg representation correspond to the same character. Furthermore, as recalled in Section 3.2, two irreducible representations lie in the same Bernstein component if and only if the corresponding Weil–Deligne representations agree on inertia, but have possibly differing monodromy operators, and it can be useful to have a finer decomposition. In particular, we wish to be able to consider only the representations with N = 0.

Let Ω be a Bernstein component with corresponding semisimple Bushnell–Kutzko type (J, λ). Let Irr(Ω)_{0} denote the irreducible elements π of Ω with the property that N = 0 on rec_{p}(π). In [SZ99, §2], the material recalled in Section 3.1 above is recast in terms of certain partition valued functions which are relevant for Ω in the terminology of [BS07]. A partial ordering is defined on these functions, and there is a unique maximal element for this ordering, which we will denote by P from now on. In [SZ99, §6] an irreducible direct summand σ_{P}(λ) of Ind_{J}^{K} λ is constructed, with the property that if π is an irreducible smooth G-representation, then Hom_{K}(σ_{P}(λ), π) ≠ 0 if and only if π ∈ Irr(Ω)_{0}, in which case σ_{P}(λ) occurs in π|_{K} with multiplicity one. (This follows immediately from Proposition 6.2 of [SZ99], using the relationship between the partial ordering on partition valued functions in [SZ99] and the monodromy operators, which is explained in the proof of Proposition 6.5.3 of [BC09].)

If τ is the inertial type corresponding to Ω, then we write σ(τ) for σ_{P}(λ). As remarked on page 201 of [SZ99], since P is maximal, σ(τ) coincides with the representation ρ_{𝔰} = e_{K} Ind_{J}^{K} λ constructed in Theorem 4.1 of [Dat99]. Here e_{K} is a certain idempotent in H(G, λ) which is constructed in [Dat99, §4.2].

Let ℨ_{Ω} be the centre of Ω. By Theorem 4.1 of [Dat99], the action of ℨ_{Ω} on c-Ind_{K}^{G}σ(τ) induces an isomorphism

ℨ_{Ω} ≅ End_{G}(c-Ind_{K}^{G}σ(τ)) =: H(G, σ(τ)),

so in particular H(G, σ(τ)) is commutative.

Let W_{[M,ω]} := N_{G}([M, ω])/M be the relative normaliser of the inertial equivalence class of (M, ω), with (M, ω) as in Section 3.2. Then by Proposition 2.1 of [Dat99], the natural algebra homomorphism t_{P} : H(M, λ_{M}) → H(G, λ) induces an isomorphism t_{P} : H(M, λ_{M})^{W_{[M,ω]}} ⟶^{∼} Z(H(G, λ)). It follows from the discussion in [Dat99, §4.2] that there is an algebra homomorphism s_{P} : H(G, λ) → H(G, σ(τ)), which makes H(G, σ(τ)) a direct summand of H(G, λ), and the composite s_{P} ∘ t_{P} induces an isomorphism H(M, λ_{M})^{W_{[M,ω]}} ⟶^{∼} H(G, σ(τ)). (The map s_{P} is just given by h ↦ e_{K} ∗ h ∗ e_{K}, where e_{K} is the idempotent mentioned above.)

We summarise much of the preceding discussion as the following theorem.

3.7. Theorem. If τ is an inertial type, then there is a finite-dimensional smooth irreducible \overline{ℚ}_{p}-representation σ(τ) of K such that if π is any irreducible smooth \overline{ℚ}_{p}-representation of G, then the restriction of π to K contains (an isomorphic copy of) σ(τ) as a subrepresentation if and only if rec_{p}(π)|_{I_{F}} ∼ τ and N = 0 on rec_{p}(π). Furthermore, in this case the restriction of π to K contains a unique copy of σ(τ). The ring H(G, σ(τ)) := End_{G}(c-Ind_{K}^{G}σ(τ)) is commutative.

3.8. Proposition. Let π_{i} be an irreducible supercuspidal representation of GL_{n_{i}}(F) for 1 ≤ i ≤ r, and n = n_{1} + … + n_{r}. If π_{j} ≅ π̸_{i}(1) (the condition being empty if n_{i} ≠ n_{j}) for every i < j then the G-socle and the G-cosocle of π_{1} × … × π_{r} are irreducible. Moreover, the socle occurs as a subquotient with multiplicity one and is the only generic subquotient of π_{1} × ⋯ × π_{r}.

Proof. This is an exercise in Bernstein–Zelevinsky theory [BZ77, Zel80]. We will make use of the material recalled from [Rod82] at the beginning of this section. If we let Δ_{1} = {π_{1}}, …, Δ_{r} = {π_{r}}, then by assumption the Δ_{i} are segments such that Δ_{i} does not precede Δ_{j} for i < j. Since the segments are of length one, we have L(Δ_{i}) ≅ Z(Δ_{i}) ≅ π_{i} by definition, so that as recalled above π_{1} × ⋯ × π_{r} has a unique irreducible subrepresentation Z(Δ_{1}, …, Δ_{r}) and a unique irreducible quotient L(Δ_{1}, …, Δ_{r}). In addition, Z(Δ_{1}, …, Δ_{r}) occurs as a subquotient with multiplicity one, so we only need to show that it is the unique generic subquotient.

Let U_{n} be the subgroup of unipotent upper-triangular matrices in GL_{n}(F), and let θ_{n} : U_{n} → \overline{ℚ}_{p}^{×} be the character (u_{ij}) ↦ ψ(∑_{i=1}^{n−1} u_{i,i+1}), where ψ : F → \overline{ℚ}_{p}^{×} is a fixed smooth non-trivial character. If π is a representation of G, we let π_{θ_{n}} be the largest quotient of π on which U_{n} acts by θ_{n}. If π is an irreducible representation of G then the dimension of π_{θ_{n}} is at most one, and is equal to one if and only if π is generic. Since U_{n} is equal to the union of its compact open subgroups, the functor π ↦ π_{θ_{n}} is exact. Thus it is enough to show that (π_{1} × ⋯ × π_{r})_{θ_{n}} is one dimensional and Z(Δ_{1}, …, Δ_{r})_{θ_{n}} is non-zero.

In [BZ77, Zel80] the authors define a family of exact functors π ↦ π^{(i)} for 0 ≤ i ≤ n from the category of smooth representations of GL_{n}(F) to the category of smooth representations of GL_{n−i}(F). The representation π^{(i)} is called the i-th derivative of π. We have π^{(0)} = π and π^{(n)} = π_{θ_{n}}. If π is irreducible and supercuspidal then π^{(i)} = 0 for 0 < i < n and π^{(n)} is one dimensional, by [BZ77, Theorem 4.4]. By [BZ77, Corollary 4.6] we have that (π_{1} × ⋯ × π_{r})^{(n)} ≅ (π_{1})^{(n_{1})} ⊗ … ⊗ (π_{r})^{(n_{r})}, which is one dimensional, since the π_{i} are supercuspidal representations of GL_{n_{i}}(F). It follows from [Zel80, Theorem 6.2] that Z(Δ_{1}, …, Δ_{r})^{(n)} ≠ 0, as required. □

If π in Ω is irreducible, then the action of ℨ_{Ω} on π defines a maximal ideal χ_{π} : ℨ_{Ω} → End_{G}(π) ≅ \overline{ℚ}_{p}. The following is a strengthening of Theorem 4.1 of [Dat99].

3.9. Proposition. Let π be an irreducible smooth \overline{ℚ}_{p}-representation of G, such that rec_{p}(π)|_{I_{F}} ∼ τ and N = 0 on rec_{p}(π). Then

c-Ind_{K}^{G}σ(τ) ⊗_{ℨ_{Ω},χ_{π}} \overline{ℚ}_{p} ≅ π_{1} × ⋯ × π_{r},

where π_{i} is a supercuspidal representation of GL_{n_{i}}(F), such that if i < j then π_{j} ≅ π̸_{i}(1) (the condition being empty if n_{i} ≠ n_{j}), and rec_{p}(π) ≅ ⊕_{i=1}^{r}rec_{p}(π_{i}). Moreover, π is the unique irreducible quotient of c-Ind_{K}^{G}σ(τ) ⊗_{ℨ_{Ω},χ_{π}} \overline{ℚ}_{p}.

Proof. Since N = 0 on rec_{p}(π), we may write it as a direct sum of irreducible representations of the Weil group W_{F}, and there is a partition n = n_{1} + ⋯ + n_{r} and supercuspidal representations π_{i} of GL_{n_{i}}(F) such that rec_{p}(π) ≅ ⊕_{i=1}^{r}rec_{p}(π_{i}). After reordering we may assume that if i < j then π_{j} ≅ π̸_{i}(1). It follows from Proposition 3.8 that π_{1} × ⋯ × π_{r} has a unique irreducible quotient π′.

Then rec_{p}(π′) ≅ ⊕_{i=1}^{r}rec_{p}(π_{i}) as representations of W_{F}, and N = 0 on rec_{p}(π′) since all the segments have length one, as in the proof of Proposition 3.8. Thus rec_{p}(π′) ≅ rec_{p}(π), which implies that π ≅ π′. Since the socle of π_{1} × ⋯ × π_{r} is irreducible and occurs as a subquotient with multiplicity one, the action of ℨ_{Ω} on π_{1} × ⋯ × π_{r} factors through a maximal ideal χ : ℨ_{Ω} → \overline{ℚ}_{p}. Since the cosocle is isomorphic to π, we deduce that χ = χ_{π}.

Theorem 3.7 implies that π is a quotient of c-Ind_{K}^{G}σ(τ). Since c-Ind_{K}^{G}σ(τ) is projective there exists a G-equivariant map φ : c-Ind_{K}^{G}σ(τ) → π_{1} × ⋯ × π_{r} such that the composition with π_{1} × ⋯ × π_{r} ↠ π is surjective. The G-cosocle of the cokernel of φ is zero. Since π_{1} × ⋯ × π_{r} is of finite length, so is the cokernel of φ, and we deduce that φ is surjective. Since ℨ_{Ω} acts naturally on everything, φ induces a surjection φ̄ : c-Ind_{K}^{G}σ(τ) ⊗_{ℨ_{Ω},χ_{π}} \overline{ℚ}_{p} ↠ π_{1} × ⋯ × π_{r}.

Then [Dat99, Theorem 4.1] implies that the semisimplification of c-Ind_{K}^{G}σ(τ)⊗_{ℨ_{Ω}} χ_{π} is isomorphic to the semisimplification of π′_{1}×⋯×π′_{s}, where π′_{i} are supercuspidal representations of GL_{n′_{i}}(F) for some integers n′_{i}, such that n = n′_{1}+…+n′_{s}. Since π is an irreducible subquotient of both π′_{1} ×…×π′_{s} and π_{1} ×⋯×π_{r}, [BZ77, Theorem 2.9] implies that π′_{1} × … × π′_{s} and π_{1} × ⋯ × π_{r} have the same semisimplification. Thus c-Ind_{K}^{G}σ(τ) ⊗_{ℨ_{Ω},χ_{π}} \overline{ℚ}_{p} and π_{1} × ⋯ × π_{r} have the same semisimplification, which implies that φ̄ is an isomorphism, as required. □

3.10. Corollary. Let Ω be a Bernstein component corresponding to an inertial type τ and let ℨ_{Ω} be the centre of Ω. Let χ : ℨ_{Ω} → \overline{ℚ}_{p} be a maximal ideal of ℨ_{Ω}. Then the G-socle of c-Ind_{K}^{G}σ(τ) ⊗_{ℨ_{Ω},χ} \overline{ℚ}_{p} is irreducible and generic, and all the other irreducible subquotients are not generic.

Conversely, if an irreducible representation π in Ω is generic then π is isomorphic to the G-socle of c-Ind_{K}^{G}σ(τ) ⊗_{ℨ_{Ω},χ_{π}} \overline{ℚ}_{p}.

Proof. The first part follows from Propositions 3.8 and 3.9. The converse may be seen as follows. There exist supercuspidal representations π_{i} of GL_{n_{i}}(F) such that n = n_{1} + … + n_{r} and π is a subquotient of π_{1} × ⋯ × π_{r}. If w is a permutation of {1, …, r} then π_{1} ×⋯×π_{r} and π_{w(1)} ×⋯×π_{w(r)} have the same semisimplification by [BZ77, Theorem 2.9], so we may assume that the π_{i} satisfy the conditions of Proposition 3.8. Since the socle of π_{1} × ⋯ × π_{r} is irreducible and occurs as a subquotient with multiplicity one, the action of ℨ_{Ω} on π_{1} × ⋯ × π_{r} factors through a maximal ideal, which is equal to χ_{π}, as π occurs as a subquotient. If we let π′ be the G-cosocle of π_{1} × ⋯ × π_{r} then π′ satisfies the conditions of Proposition 3.9, and we have χ_{π′} = χ_{π}. The assertion follows from Propositions 3.8 and 3.9. □

3.11. Corollary. Let π be an irreducible smooth generic \overline{ℚ}_{p}-representation of G, such that rec_{p}(π)|_{I_{F}} ∼ τ and N = 0 on rec_{p}(π). Then we have a natural isomorphism c-Ind_{K}^{G}σ(τ) ⊗_{ℨ_{Ω},χ_{π}} \overline{ℚ}_{p} ≅ π.

Proof. By Corollary 3.10, we see that the G-socle of c-Ind_{K}^{G}σ(τ)⊗_{ℨ_{Ω},χ_{π}} \overline{ℚ}_{p} is isomorphic to π and occurs with multiplicity one as a subquotient. Theorem 3.7 implies that π is a quotient of c-Ind_{K}^{G}σ(τ) ⊗_{ℨ_{Ω},χ_{π}} \overline{ℚ}_{p}. This implies the assertion. □

3.12. Rationality. As a preparation for the next section we explain in this subsection how the results above remain true with a finite extension E of ℚ_{p} as coefficient field, as long as E is sufficiently large (depending on the Bernstein component Ω). We do this by following various parts of the construction of the Bernstein centre in [Ren10], working over E rather than over \overline{ℚ}_{p}, and we then deduce the results from those over \overline{ℚ}_{p} by faithfully flat descent. Let (M, ω) be the supercuspidal support of some irreducible representation in Ω, let X (M) be the group of unramified characters χ : M → \overline{ℚ}_{p}^{×}, and let

X (M)(ω) := {χ ∈ X (M) : ω ≅ ω ⊗ χ}.

Let M^{0} be the intersection of the kernels of the characters χ ∈ X (M), and let T be the intersection of the kernels of the χ ∈ X (M)(ω). The restriction of ω to M^{0} is a finite direct sum of irreducible representations, see [Ren10, VI.3.2]. We fix one irreducible summand ρ. It follows from Lemme VI.4.4 of [Ren10] and its proof that ρ extends to a representation ρ_{T} of T, such that Ind_{T}^{M} ρ_{T} is isomorphic to a finite direct sum of copies of ω. Thus χ ∈ X (M) lies in X (M)(ω) if and only if the restriction of χ to T is trivial. Thus the restriction to T induces a bijection

(3.13)  X (M)/X (M)(ω) →^{≅} X (T),

where X (T) is the group of characters from T to \overline{ℚ}_{p}^{×}, which are trivial on M^{0}.

Let D be the Bernstein component (for M) containing ω, let ℨ_{D} be the centre of D and let Π(D) := c-Ind_{M^{0}}^{M}ρ. It is shown in [Ren10, VI.4.1] that Π(D) is a projective generator for D. Thus we may identify ℨ_{D} with the centre of the ring End_{M}(Π(D)). Since ρ is irreducible, Π(D) is a finitely generated \overline{ℚ}_{p}[M]-module.

Let ℨ_{Ω} be the Bernstein centre of Ω and let Π(Ω) := i_{P}^{G} Π(D), where P is any parabolic subgroup with Levi subgroup M. It is shown in [Ren10, Thm.VI.10.1] that Π(Ω) is a projective generator of Ω, which is a finitely generated \overline{ℚ}_{p}[G]-module. Thus we may identify ℨ_{Ω} with the centre of the ring End_{G}(Π(Ω)).

It follows from Bushnell–Kutzko theory that ω ≅ c-Ind_{J}^{M}Λ, and ρ ≅ c-Ind_{J}^{M^{0}}λ, where J is an open compact-mod-centre subgroup of M, J is an open compact subgroup of M, and Λ, λ are (necessarily) finite-dimensional irreducible representations. We may realise both Λ and λ over a finite extension E of ℚ_{p}. By compactly inducing these realisations, we deduce that both ω and ρ can be realised over E. We denote these representations by ω_{E} and ρ_{E}, respectively. It is shown in [Ren10, Lemme V.2.7] that X (M)(ω) is a finite group. Let W (D) be the subgroup of N_{G}(M)/M stabilising D. For each w ∈ W (D) there are precisely |X (M)(ω)| unramified characters ξ such that ω^{w} ≅ ω ⊗ ξ. Since the group M/M^{0} is finitely generated, by replacing E with a finite extension, we may assume that all the characters ξ are E-valued. By further enlarging E we may assume that \sqrt{q}, where q is the number of elements in the residue field of F, is contained in E. Then the modulus character of P is defined over E.

Let Π(D)_{E} := c-Ind_{M^{0}}^{M}ρ_{E} and let ℨ_{D,E} be the centre of End_{M}(c-Ind_{M^{0}}^{M}ρ_{E}). Since ρ_{E} ⊗_{E} \overline{ℚ}_{p} ≅ ρ, we have Π(D)_{E} ⊗_{E} \overline{ℚ}_{p} ≅ Π(D). We may express the generators of Π(D) (as a \overline{ℚ}_{p}[M]-module) as a finite \overline{ℚ}_{p}-linear combination of elements of Π(D)_{E}.

The E[M]-submodule of Π(D)_{E} generated by these elements has to equal to Π(D)_{E}, as the quotient is zero once we extend the scalars to \overline{ℚ}_{p}. In particular, Π(D)_{E} is a finitely generated E[M]-module.

Let Π(Ω)_{E} := i_{P}^{G} Π(D)_{E} and let ℨ_{Ω,E} be the centre of End_{G}(Π(Ω)_{E}). The smooth parabolic induction commutes with ⊗_{E}\overline{ℚ}_{p}, as the set P \G/H is finite for every open subgroup H of G and tensor products commute with inductive limits, so Π(Ω)_{E} ⊗_{E} \overline{ℚ}_{p} ≅ Π(Ω). Since Π(Ω) is a finitely generated \overline{ℚ}_{p}[G]-module, arguing as in the previous paragraph we deduce that Π(Ω)_{E} is a finitely generated E[G]- module.

The following observation (see for example Lemma 5.1 of [Paš13]) is very useful. If π and π′ are representations of some group G on E-vector spaces, such that π is a finitely generated E[G]-module, then

(3.14)  Hom_{E[G]}(π, π′) ⊗_{E} \overline{ℚ}_{p} ≅ Hom_{\overline{ℚ}_{p}[G]}(π ⊗_{E} \overline{ℚ}_{p}, π′ ⊗_{E} \overline{ℚ}_{p}).

It follows from (3.14) that

(3.15)  End_{M}(Π(D)_{E}) ⊗_{E} \overline{ℚ}_{p} ≅ End_{M}(Π(D)),
(3.16)  End_{G}(Π(Ω)_{E}) ⊗_{E} \overline{ℚ}_{p} ≅ End_{G}(Π(Ω)).

Let D_{E} be the full subcategory of smooth representation ω′ of M on E-vector spaces, such that ω′⊗_{E} \overline{ℚ}_{p} is in D. It follows from (3.14) that Hom_{M}(Π(D)_{E}, ω′)⊗_{E} \overline{ℚ}_{p} ≅ Hom_{M}(Π(D), ω′ ⊗_{E} \overline{ℚ}_{p}). Since Π(D) is a projective generator of D, we deduce that Π(D)_{E} is a projective generator of D_{E}. (This follows from the fact that \overline{ℚ}_{p} is faithfully flat over E; we will repeatedly use this fact below without further comment.) In particular, ℨ_{D,E} is naturally isomorphic to the centre of the category D_{E}.

Similarly we let Ω_{E} be the full subcategory of smooth representations π′ of G on E-vector spaces, such that π′ ⊗_{E} \overline{ℚ}_{p} is in Ω. The same argument as above gives that Π(Ω)_{E} is a projective generator of Ω_{E} and ℨ_{Ω,E} is naturally isomorphic to the centre of Ω_{E}.

3.17. Lemma. Let A be an E-algebra and let Z be an E-subalgebra of A. If Z⊗_{E} \overline{ℚ}_{p} is the centre of A ⊗_{E} \overline{ℚ}_{p} then Z is the centre of A.

Proof. For each z ∈ Z, we define an E-linear map ϕ_{z} : A → A, a ↦ az − za. Since z⊗1 is central in A⊗_{E} \overline{ℚ}_{p} by assumption, we deduce that (Im ϕ_{z})⊗_{E} \overline{ℚ}_{p} = 0, which implies that Im ϕ_{z} = 0. We deduce that Z is contained in Z(A), the centre of A. If z ∈ Z(A) then z ⊗ 1 is contained in the centre of A ⊗_{E} \overline{ℚ}_{p} and thus in Z ⊗_{E} \overline{ℚ}_{p} by assumption. Hence (Z(A)/Z) ⊗_{E} \overline{ℚ}_{p} = 0, which implies that Z = Z(A). □

3.18. Lemma. The isomorphism (3.15) induces an isomorphism ℨ_{D,E} ⊗_{E} \overline{ℚ}_{p} ≅ ℨ_{D}.

Proof. Since Π(D) ≅ Ind_{T}^{M}(c-Ind_{M^{0}}^{T}ρ) and induction is a functor, we have an inclusion End_{T} (c-Ind_{M^{0}}^{T}ρ) ⊂ End_{M}(Π(D)). It follows from [Ren10, Thm.VI.4.4] that this inclusion identifies End_{T} (c-Ind_{M^{0}}^{T}ρ) with the centre of End_{M}(Π(D)). The assertion follows from Lemma 3.17 applied to Z = End_{T} (c-Ind_{M^{0}}^{T}ρ_{E}) and A = End_{M}(Π(D)_{E}). □

Let Irr(D) be the set of irreducible representations in D. Every such irreducible representation is of the form ω ⊗ χ for some χ ∈ X (M). We thus have a bijection X (M)/X (M)(ω) →^{≅} Irr(D), χ ↦ ω ⊗ χ. Composing this bijection with (3.13) we obtain a bijection

(3.19)  Irr(D) →^{≅} X (T).

Now X (T) is naturally isomorphic to the set of homomorphisms of \overline{ℚ}_{p}-algebras from \overline{ℚ}_{p}[T/M^{0}] to \overline{ℚ}_{p}. It is explained in [Ren10, VI.4.4] that we have identifications

ℨ_{D} ≅ End_{T} (Ind_{M^{0}}^{T} ρ) ≅ \overline{ℚ}_{p}[T/M^{0}];

see Théorème VI.4.4 for the first isomorphism and Proposition VI.4.4 for the second, so that (3.19) induces a natural bijection between Irr(D) and MaxSpec ℨ_{D}.

The group W (D) acts on Irr(D) by conjugation. For each w ∈ W (D) let ξ ∈ X (M) be any character such that ω^{w} ≅ ω ⊗ ξ, and let ξ_{w} be the restriction of ξ to T. Then ξ_{w} depends only on w and not on the choice of ξ. If χ ∈ X (M) then (ω ⊗ χ)^{w} ≅ ω ⊗ χ^{w}ξ. Thus the action of W (D) on X (T) via (3.19) is given by w.χ = χ^{w}ξ_{w}. It is immediate that this action is induced by the action of W (D) on \overline{ℚ}_{p}[T/M^{0}] given by w.(tM^{0}) = ξ_{w}^{−1}(t)t^{w}M^{0}.

3.20. Lemma. The action of W (D) on ℨ_{D} preserves ℨ_{D,E}.

Proof. Since ω and ρ can both be defined over E, so can the representation ρ_{H} defined at the beginning of [Ren10, VI.4.4], and in particular so can ρ_{T}, its restriction to T. Hence, if we identify ℨ_{D} with \overline{ℚ}_{p}[T/M^{0}] as in [Ren10, Prop.VI.4.4] then ℨ_{D,E} is identified with E[T/M^{0}]. Since the characters ξ_{w} are E-valued by the choice of E, we get the assertion. □

3.21. Lemma. ℨ_{Ω,E} = ℨ_{D,E}^{W (D)}.

Proof. Since Π(Ω) = i_{P}^{G} Π(D) and parabolic induction is a functor, we have an inclusion ℨ_{D} ⊂ End_{G}(Π(Ω)). It follows from the discussion immediately preceding the proof of Theorem VI.10.4 of [Ren10] that this inclusion identifies ℨ_{D}^{W (D)} with the centre of End_{G}(Π(Ω)). The assertion follows from Lemma 3.17 applied to Z = ℨ_{D,E}^{W (D)} and A = End_{G}(Π(Ω)_{E}). □

3.22. Proposition. The isomorphism (3.16) induces an isomorphism ℨ_{Ω,E}⊗_{E} \overline{ℚ}_{p} ≅ ℨ_{Ω}.

Proof. Using Lemmas 3.18 and 3.21 we obtain

ℨ_{Ω,E} ⊗_{E} \overline{ℚ}_{p} ≅ ℨ_{D,E}^{W (D)} ⊗_{E} \overline{ℚ}_{p} ≅ (ℨ_{D,E} ⊗_{E} \overline{ℚ}_{p})^{W (D)} ≅ ℨ_{D}^{W (D)} ≅ ℨ_{Ω},

the last isomorphism following from [Ren10, VI.10.4] as in the proof of Lemma 3.21. □

3.23. Lemma. ℨ_{Ω,E} coincides with the ring E[B] constructed in [Che09, Prop. 3.11].

Proof. Let Δ be the subgroup of X (M) ⋊ W (D) consisting of pairs (ξ, w), such that ω^{w} ≅ ω ⊗ ξ. This subgroup acts naturally on E[M/M^{0}]. The map ξ ↦ (ξ, 1) identifies X (M)(ω) with a normal subgroup of Δ and the quotient is isomorphic to W (D). We have

\overline{ℚ}_{p}[M/M^{0}]^{Δ} ≅ (\overline{ℚ}_{p}[M/M^{0}]^{X (M)(ω)})^{W (D)} ≅ \overline{ℚ}_{p}[T/M^{0}]^{W (D)} ≅ ℨ_{D}^{W (D)},

see [Ren10, Rem.VI.4.4] for the second isomorphism. Chenevier defines E[B] to be E[M/M^{0}]^{Δ}. This subring gets identified with E[T/M^{0}]^{W (D)} inside \overline{ℚ}_{p}[T/M^{0}]^{W (D)}, and with ℨ_{D,E}^{W (D)} inside ℨ_{D}^{W (D)}, see the proof of Lemma 3.20. The assertion follows from Lemma 3.21. □

Let σ(τ) be the representation of K given by Theorem 3.7. After replacing E by a finite extension we may assume that there exists a representation σ(τ)_{E} of K on an E-vector space, such that σ(τ)_{E} ⊗_{E} \overline{ℚ}_{p} ≅ σ(τ). Then c-Ind_{K}^{G}σ(τ)_{E} is an object in Ω_{E}. Since ℨ_{Ω,E} is the centre of Ω_{E} it acts on c-Ind_{K}^{G}σ(τ)_{E}, thus inducing a homomorphism ℨ_{Ω,E} → End_{G}(c-Ind_{K}^{G}σ(τ)_{E}).

3.24. Lemma. The map ℨ_{Ω,E} → End_{G}(c-Ind_{K}^{G}σ(τ)_{E}) is an isomorphism.

Proof. It follows from Theorem 4.1 of [Dat99] and Proposition 3.22 above that the map is an isomorphism once we extend scalars to \overline{ℚ}_{p}. This implies the assertion. □

Let R := End_{G}(Π(Ω)). Since Π(Ω) is a projective generator the functors M ↦ M ⊗_{R} Π(Ω) and π ↦ Hom_{G}(Π(Ω), π) induce an equivalence of categories between the category of right R-modules and Ω. If π is irreducible, then the action of ℨ_{Ω} on π factors through χ_{π} : ℨ_{Ω} → \overline{ℚ}_{p}. It follows from [Ren10, Lem.VI.10.4] that R is a finitely generated ℨ_{Ω}-module , which implies that the module corresponding to π is a finite dimensional \overline{ℚ}_{p}-vector space. Since ℨ_{Ω,E} is a finitely generated algebra over E, E(χ_{π}) := χ_{π}(ℨ_{Ω,E}) is a finite extension of E.

In the above E was only required to be sufficiently large. Thus if E′ is a subfield of \overline{ℚ}_{p} containing E, then we let Ω_{E′}, Π(Ω)_{E′} be the corresponding objects defined over E′ instead of E. Then Π(Ω)_{E′} is a projective generator of Ω_{E′} and the functors M ↦ M ⊗_{R_{E′}} Π(Ω)_{E′} and π ↦ Hom_{G}(Π(Ω)_{E′}, π) induce an equivalence of categories between the category of right R_{E′}-modules and Ω_{E′}, where R_{E′} := End_{G}(Π(Ω)_{E′}).

3.25. Lemma. Every irreducible generic π ∈ Ω can be realised over E(χ_{π}).

Proof. To ease the notation we let E′ := E(χ_{π}) and let π′ := c-Ind_{K}^{G}σ(τ)_{E} ⊗_{ℨ_{Ω,E}} E′. Then

π′ ⊗_{E′} \overline{ℚ}_{p} ≅ c-Ind_{K}^{G}σ(τ) ⊗_{ℨ_{Ω},χ_{π}} \overline{ℚ}_{p} ≅ π_{1} × … × π_{r},

where the last isomorphism is given by Proposition 3.9. Hence, π′ ⊗_{E′} \overline{ℚ}_{p} is of finite length, which implies that Hom_{G}(Π(Ω), π′ ⊗_{E′} \overline{ℚ}_{p}) is a finite dimensional \overline{ℚ}_{p}-vector space, which implies that M′ := Hom_{G}(Π(Ω)_{E}, π′) is a finite dimensional E′-vector space.

If π′′ is an irreducible E′-subrepresentation of π′, and if we define M′′ := Hom_{G}(Π(Ω)_{E′}, π′′), then M′′ is an irreducible R_{E′}-module which is finite dimensional over E′. It follows from [Bou12, Cor.12.7.1a)] that M′′⊗_{E′} \overline{ℚ}_{p} is a semi-simple R-module. Hence, π′′ ⊗_{E′} \overline{ℚ}_{p} is a semi-simple G-representation. Proposition 3.8 implies that the G-socle of π′′ ⊗_{E′} \overline{ℚ}_{p} is irreducible and is isomorphic to π. Thus π′′ ⊗_{E′} \overline{ℚ}_{p} ≅ π. □

Henceforth for each Bernstein component Ω we will fix a sufficiently large E as above and work with it. Agreeing on this, we will omit E from the notation when there is no danger of confusion. For instance we will write ℨ_{Ω}, σ(τ), and so on, in place of ℨ_{Ω,E}, σ(τ)_{E} and so on. Note that we fixed a choice of E in Section 2; however, it is harmless to replace our patched module M_{∞} with its base extension to the ring of integers in any larger choice of E, and we will do so without further comment.

4. Local-global compatibility

The goal of this section is to prove that the patched module M_{∞} satisfies localglobal compatibility, in the following sense: the G-action on M_{∞} (obtained by patching global objects) will induce a tautological Hecke action on certain patched modules for particular K-types. On the other hand, we will define a second Hecke action via an interpolation of the classical local Langlands correspondence. We will then prove that these two Hecke actions coincide. The details are made explicit below.

Note that it is plausible that M_{∞} should satisfy local-global compatibility, since it is patched together from spaces of algebraic modular forms; the difficulty in proving this is that the modules at finite level are all p-power torsion, while localglobal compatibility is usually defined after inverting p, so that we need to establish some integral control on the compatibility. Some of our arguments were inspired by the treatment of the two-dimensional crystalline case in [Kis07, §3.6], and somewhat related considerations in the arguments of [Kis09a].

Let σ be a locally algebraic type for G = GL_{n}(F) defined over E. Then by definition σ is an absolutely irreducible representation of K = GL_{n}(O_{F}) over E of the form σ_{sm} ⊗ σ_{alg}, where σ_{sm} is a smooth type for K (i.e. σ_{sm} = σ(τ) for some inertial type τ) and σ_{alg} is the restriction to K of an irreducible algebraic representation of Res_{F/ℚ_{p}}G; we will sometimes also write σ_{alg} for the corresponding G-representation. (So, all of our locally algebraic types are “potentially crystalline”, in the sense that they detect representations for which N = 0.) Set H(σ) := End_{G}(c-Ind_{K}^{G}σ).

We say that a continuous representation r : G_{F} → GL_{n}(E) is potentially crystalline of type σ if it is potentially crystalline with inertial type τ and Hodge–Tate weights prescribed by σ_{alg}. We will also say a global representation has type σ if it restricts to such an r. Let R□ ; 𝔭̃(σ) be the local framed universal deformation ring of type σ at 𝔭̃ (i.e. the unique reduced and p-torsion free quotient of R□ ; 𝔭̃ corresponding to potentially crystalline lifts of type σ).

Let X = Spf R□ ; 𝔭̃(σ), with ideal of definition taken to be the maximal ideal, and let X^{rig} denote its rigid generic fiber (as constructed in [dJ95, §7]). Note that X^{rig} = ∪_{j}U_{j} is an increasing union of affinoids, and in fact is a quasi-Stein rigid space, since it is a closed subspace of an open polydisc, which is an increasing union of closed polydiscs. By a standard abuse of notation, we will write O_{X^{rig}} for the ring of rigid-analytic functions on X^{rig}. Then O_{X^{rig}} = \underleftarrow{lim}_{j} Γ(U_{j}, O_{U_{j}}) and we equip it with the inverse limit topology. We note that by [dJ95, Lemma 7.1.9], there is a bijection between the points of X^{rig} and the closed points of Spec R□ ; 𝔭̃(σ)[1/p]. The universal lift over R□ ; 𝔭̃(σ) gives rise to a continuous family of representations ρ^{rig} : G_{F} → GL_{n}(O_{X^{rig}}). (The continuity of ρ^{rig} is equivalent to that of each of the representations G_{F} → GL_{n}(Γ(U_{j}, O_{U_{j}})) obtained by restricting elements of GL_{n}(O_{X^{rig}}) to U_{j}.) If x is a point of X^{rig} with residue field E_{x}, we denote by ρ_{x} : G_{F} → GL_{n}(E_{x}) the specialisation of ρ^{rig} at x. We define the locally algebraic G-representation π_{l.alg,x} := π_{sm}(ρ_{x})⊗_{E} σ_{alg} = π_{sm}(ρ_{x})⊗_{E_{x}} π_{alg}(ρ_{x}). (Recall the notation π_{sm}(ρ_{x}) and π_{alg}(ρ_{x}) from §1.5; in particular, π_{sm}(ρ_{x}) = r_{p}^{−1}(WD(ρ_{x})^{F −ss}).) Note that H(σ) acts via a character on the space Hom_{K}(σ, π_{l.alg,x}), the latter being one-dimensional (by Theorem 3.7 together with the argument of [ST06, Lemma 1.4]).

The following theorem, which may be of independent interest, gives our interpolation of the local Langlands correspondence. Its proof will occupy much of this section.

4.1. Theorem. There is an E-algebra homomorphism η : H(σ) → R□ ; 𝔭̃(σ)[1/p] which interpolates the local Langlands correspondence r_{p}. More precisely, for any closed point x of Spec R□ ; 𝔭̃(σ)[1/p], the action of H(σ) on Hom_{K}(σ, π_{l.alg,x}) factors as η composed with the evaluation map R□ ; 𝔭̃(σ)[1/p] → E_{x}.

We begin by proving the following weaker result, showing the existence of a rigid analytic local Langlands map.

4.2. Proposition. There is an E-algebra homomorphism η : H(σ) → O_{X^{rig}} which interpolates the local Langlands correspondence r_{p}. More precisely, for any point x ∈ X^{rig}, the action of H(σ) on Hom_{K}(σ, π_{l.alg,x}) factors as η composed with the evaluation map O_{X^{rig}} → E_{x}.

Let ρ_{WD}^{rig} : W_{F} → GL_{n}(O_{X^{rig}}) be the family of Weil–Deligne representations associated to the family of representations ρ^{rig} by Corollaire 3.19 of [Che09]. More precisely, this result of Chenevier generalises Théorème C of [BC08] to the case of an arbitrary reduced quasi-compact and quasi-separated rigid space. Applying it to each U_{j}, we obtain a compatible family of Weil–Deligne representations ρ_{WD}^{rig} : W_{F} → GL_{n}(Γ(U_{j}, O_{U_{j}})) and thus a Weil–Deligne representation ρ_{WD}^{rig} : W_{F} → GL_{n}(O_{X^{rig}}). Note that ρ_{WD}^{rig} has N = 0.

For a point x of X^{rig}, we denote by ρ_{WD,x} the specialisation of ρ_{WD}^{rig} at x. Then ρ_{WD,x}|_{I_{F}} ≃ τ for all points x of X^{rig}. Recall that ℨ_{Ω} is the Bernstein centre for the Bernstein component Ω corresponding to σ(τ).

4.3. Proposition. There exists a unique E-algebra map I : ℨ_{Ω} → O_{X^{rig}} such that for any point x of X^{rig} with residue field E_{x}, the smooth G-representation π_{x} corresponding to ρ_{WD,x} via the local Langlands correspondence rec_{p} determines via specialisation the map x ∘ I : ℨ_{Ω} → E_{x}.

Proof. Consider the following map, obtained by specialisation:

γ_{G} : ℨ_{Ω} → ∏_{x∈X^{rig}} E′_{x},

where γ_{G} is defined on the factor corresponding to x by evaluating ℨ_{Ω} at the closed point in the Bernstein component Ω determined via local Langlands by x, and E′_{x}/E_{x} is a sufficiently large finite extension.

Consider as well the following map, also obtained by specialisation:

γ_{WD} : O_{X^{rig}} → ∏_{x∈X^{rig}} E′_{x}.

This is an injection since X^{rig} is reduced and since each Γ(U_{j}, O_{U_{j}}) is Jacobson. (The Jacobson property is true of any affinoid algebra. To see that X^{rig} is reduced, it is enough to check it on completed local rings at closed points, but these are the same as the completed local rings of R□ ; 𝔭̃(σ) [\frac{1}{p}] by [dJ95, Lemma 7.1.9]. The

latter is reduced (being a localization of R□ ; 𝔭̃(σ), which is reduced by definition) and excellent, since R□ ; 𝔭̃(σ) is a complete, local, noetherian ring (and thus excellent by [Gro65, Scholie 7.8.3(iii)]). The reducedness of the completed local rings now follows from [Gro65, Scholie 7.8.3(v)].)

In order to define our map I, it suffices to show that the image of ℨ_{Ω} under γ_{G} is contained in the image of γ_{WD}. Let T : W_{F} → ℨ_{Ω} be the pseudo-representation constructed in Proposition 3.11 of [Che09]. (Note that Chenevier’s E[B] is our ℨ_{Ω} by Lemma 3.23.) By the construction of T, we have γ_{G} ∘ T = γ_{WD} ∘ tr(ρ_{WD}^{rig}). Therefore the proof of the proposition is reduced to Lemma 4.5 below. □

Write v : W_{F} ↠ ℤ for the valuation map assigning 1 to any lift of the geometric Frobenius. Let ϕ ∈ W_{F} be an element of valuation 1. For w ∈ W_{F} and any I_{F} -representation r_{0}, let r_{0}^{w} be the twist r_{0}^{w}(γ) := r_{0}(w^{−1}γw).

4.4. Lemma. Let r be an irreducible continuous representation of W_{F} over \overline{ℚ}_{p}.

(1) The restriction r|_{I_{F}} decomposes as a direct sum of non-isomorphic irreducible I_{F} -representations ⊕_{i=1}^{f}r_{1}^{ϕ^{i}} for some integer f ≥ 1. If t ∈ ℤ then r(ϕ^{t}) respects the decomposition (i.e. r(ϕ^{t}) sends r_{1}^{ϕ^{i}} into itself for 1 ≤ i ≤ f) exactly when f | t.

(2) We have tr(r(w)) ≠ 0 for some w ∈ W_{F} of valuation t if and only if f | t.

(3) The unramified characters χ of W_{F} satisfying r ⊗ χ ≃ r are exactly the characters of order dividing f.

Proof. (1) The representation r|_{I_{F}} factors through a finite quotient I_{F} /H, so it decomposes as a direct sum of irreducible I_{F} -representations ⊕_{i=1}^{f}r_{i}, for some integer f ≥ 1. The fact that r is irreducible as a W_{F} -representation implies that r(ϕ) acts transitively on (the representation spaces of) the r_{i}. Up to reordering the r_{i}, we may assume that it sends r_{i} to r_{i+1}, where r_{f+1} := r_{1}. Moreover, we also deduce that r_{i+1} ≃ r_{i}^{ϕ} and that r_{1} ≃ r_{1}^{ϕ^{f}}. Finally, all the representations r_{i} are non-isomorphic, since if there was an isomorphism between them, we could define a proper W_{F} -subrepresentation of r and thus contradict the irreducibility of r. (More precisely, if we had an isomorphism r_{1} ≃ r_{1+s} for some 1 ≤ s < f, then we could assume that f = sf′ for some integer f′ and get I_{F} -isomorphisms

α_{sk} : r_{1} ⊕ ⋯ ⊕ r_{s} ⟶^{∼} r_{1+sk} ⊕ ⋯ ⊕ r_{s(1+k)}

for each 1 ≤ k < f′. In that case, we could take the I_{F} -subrepresentation of r generated by v + α_{s}(v) + ⋯ + α_{s(f′−1)}(v) with v ∈ r_{1} ⊕ ⋯ ⊕ r_{s}; it is easy to check that this space is also stable under ϕ if we choose the α_{sk} appropriately.) The fact that r(ϕ) induces a cyclic permutation of the f irreducible constituents implies the statement about r(ϕ^{t}).

(2) Since r(w) is not supported on the diagonal unless f | t we get the only if part. For the if part, assume that f | t. By part 1, the matrix r(ϕ^{t}) has the same block decomposition as r|_{I_{F}}. Note that the group algebra of I_{F} /H surjects onto ⊕_{i=1}^{f}End_{\overline{ℚ}_{p}}(r_{i}), since the r_{i} are non-isomorphic irreducible representations of the finite group I_{F} /H. Therefore, there is some linear combination of matrices ∑_{h∈I_{F}} c_{h} · r(h) which has non-zero trace against the non-zero matrix r(ϕ^{t}). This implies that tr(r(h · ϕ^{t})) ≠ 0 for some h ∈ I_{F}.

(3) Observe that r ⊗ χ ≃ r if and only if χ(w)tr r(w) = tr r(w), ∀w ∈ W. The latter condition is equivalent via part 2 to the condition that χ(w) = 1 for all w ∈ W_{F} such that f|v(w), or equivalently that χ^{f} = 1. Hence part 3 is verified. □

4.5. Lemma. The image of T generates ℨ_{Ω} as an E-algebra.

Proof. It suffices (by the faithful flatness of the extension \overline{ℚ}_{p}/E) to prove the result after replacing E with \overline{ℚ}_{p}. Since the inertial type τ factors through a finite quotient I_{F} /H, it decomposes as a direct sum ⊕_{i=1}^{r}(τ_{i})_{i}^{d}, where the τ_{i} are nonisomorphic inertial types such that σ(τ_{i}) is cuspidal. As in the proof of Proposition 3.11 of [Che09], the Bernstein component Ω decomposes as Ω_{1} × ⋯ × Ω_{r} and ℨ_{Ω} = ⊗_{i=1}^{r}ℨ_{Ω_{i}}, where each Ω_{i} corresponds to the simple type σ((τ_{i})^{d_{i}}). If we let T_{i} : W_{F} → ℨ_{Ω_{i}} be the pseudo-representation associated to Ω_{i} by Proposition 3.11 of [Che09], then by definition T (g) := ∑_{i=1}^{r} T_{i}(g). It suffices to show that the image of T generates each ℨ_{Ω_{i}} for i = 1, …, r, where ℨ_{Ω_{i}} is regarded as a subalgebra of ℨ_{Ω}.

Let r_{i} be an irreducible W_{F} -representation such that r_{i}|_{I_{F}} ≃ τ_{i}, and let f_{i} be the integer associated to r_{i} by Lemma 4.4. By choosing rec_{p}^{−1}(r_{i})^{⊗d_{i}} as a base point, each closed point of Spec ℨ_{Ω_{i}} may be represented by an unramified character χ_{i} = (χ_{i,1}, …, χ_{i,d_{i}}) (or more precisely by ⊗_{j=1}^{d_{i}}(rec_{p}^{−1}(r_{i}) ⊗ χ_{i,j}) up to a permutation of factors), where the χ_{i,j} are unramified characters of F^{×}. Then each T_{i}(g) is defined by

T_{i}(g)(χ_{i}) := tr(r_{i})(g) ∑_{j=1}^{d_{i}} χ_{i,j}(Art_{F} (g)).

Consider elements g of the form h · ϕ^{t_{i}}, with h ∈ I_{F} and t_{i} a multiple of f_{i}. By Lemma 4.4(1), the matrix r_{i}(ϕ^{t_{i}}) is non-zero and consists of f_{i} blocks which match the block decomposition of τ_{i}. Because the constituents of τ_{i} are non-isomorphic for different i’s, we may choose the c_{h} such that

∑_{h∈I_{F} /H} c_{h} · tr(r_{i}(h) · r_{i}(ϕ^{t_{i}})) ≠ 0

and ∑_{h∈I_{F} /H} c_{h} · r_{i′}(h) = 0 for i′ ≠ i. In particular, this means that ∑_{h∈I_{F} /H} c_{h} · T (hϕ^{t_{i}}) ∈ ℨ_{Ω_{i}}.

Now notice that Art_{F} (ϕ^{t_{i}}) ∈ F^{×} has valuation t_{i}, which by Lemma 4.4(3) and Remark 3.5 coincides with the valuation of det(π_{E_{i}})^{t_{i}/f_{i}}, where E_{i}/F is the extension in Lemma 3.4 for the cuspidal type σ(τ_{i}). Therefore, ∑_{j=1}^{d_{i}} χ_{i,j}(Art_{F} (g)) as

a regular function on the Bernstein component ℨ_{Ω_{i}} is exactly ∑_{j=1}^{d_{i}} X_{ij}^{t_{i}/f_{i}}, where X_{i1} is a generator of the Bernstein centre for σ(τ_{i}) and the X_{ij} are obtained from X_{i1} from the Weyl group action. This implies that for our choice of coefficients c_{h}, ∑_{h∈I_{F} /H} c_{h} · T (hϕ^{t_{i}}) is equal to ∑_{j=1}^{d_{i}} X_{ij}^{t_{i}/f_{i}} ∈ ℨ_{Ω_{i}}, up to a non-zero scalar.

Note that we can ensure that t_{i}/f_{i} is any integer. Therefore, we can generate all elements in ℨ_{Ω_{i}} which are sums of k-th powers of the d_{i} generators of the Bernstein centre for a single supercuspidal, for any integer k. However, since ℨ_{Ω_{i}} is obtained by taking Weyl-group invariants, it is generated as a \overline{ℚ}_{p}-algebra by the elementary symmetric polynomials in those d_{i} variables together with the product of the inverses of the variables. Over \overline{ℚ}_{p}, which is a field of characteristic 0, we may take the sums of powers of d_{i} variables as generators for the elementary symmetric polynomials in those variables. We may also generate the product of the inverses of the variables from sums of powers with negative exponents. □

4.6. Remark. While the proof of Lemma 4.5 is slightly technical, the lemma itself is rather natural; it expresses the idea that local Langlands should make sense in families, and hence that the family of G-representations parameterised by ℨ_{Ω} — and thus the parameter ring ℨ_{Ω} itself — should be completely determined by the corresponding family of Weil group representations, which are encoded by the ℨ_{Ω}-valued pseudo-representation T.

If we let 𝔄_{Ω} denote the E-subalgebra of ℨ_{Ω} generated by the image of T, then this is a finite type E-algebra, and we have a morphism Spec ℨ_{Ω} → Spec 𝔄_{Ω}. It is not hard to see (e.g. by applying local Langlands over the fraction field of 𝔄_{Ω}) that this is a birational map, which is in fact a bijection on points (as one sees by applying local Langlands at the closed points). Unfortunately, we were unable to find a completely conceptual proof in general that this morphism is an isomorphism of varieties over E.

In the case when ℨ_{Ω} parameterises supercuspidal representations, one can see this as follows: it suffices to check that one obtains an isomorphism after passing to the formal completion at each closed point x ∈ Spec ℨ_{Ω}. Let π_{x} be the supercuspidal G-representation corresponding to x, and let T_{x} : W_{F} → E_{x} the specialisation of T to the image of x in Spec 𝔄_{Ω}. Let R_{x} be the universal formal deformation ring of T_{x}, so that we have morphisms

Spf ℨ̂_{Ω}_{x} → Spf 𝔄̂_{Ω} → Spf R_{x},

the second being induced by T. Let r_{x} : W_{F} → GL_{n}(E′_{x}) denote the (absolutely) irreducible representation attached to π_{x} via local Langlands, where E′_{x}/E_{x} is a finite extension, and let T′_{x} denote the composite T_{x} : W_{F} → E_{x} → E′_{x}. Then T′_{x} is the pseudo-representation attached to r_{x}. Since r_{x} is irreducible, the universal formal deformation rings of r_{x} and T′_{x} coincide ([Nys96, Théorème 3], [Rou96, Corollaire 6.2]), and are thus both given by R_{x}⊗_{E_{x}} E′_{x}. A direct analysis, using that the source and target are both obtained simply by forming unramified twists, and that local Langlands gives a bijection on isomorphism classes that is compatible with twisting, shows that the composite of the base change to E′_{x} of the above morphisms is an isomorphism. Since the first of them is dominant, it is also an isomorphism. Thus the morphism Spec ℨ_{Ω} → Spec 𝔄_{Ω} is a bijection on closed points and induces isomorphisms after completing at each closed point. From the latter, we see that it is étale and radiciel, hence an open immersion by [Gro67, Théorème 17.9.1] and, since it is also surjective, we see that it is in fact an isomorphism.

One could use a variant of the argument in first paragraph of the proof of Lemma 4.5 to reduce the general case of the lemma to the case when ℨ_{Ω} parameterises a family of supercuspidal representations, where the preceding argument then applies. In this way, one could give a slightly more conceptual proof of the Lemma.

Proof of Proposition 4.2. We adopt the notation of §3.12. In particular M is the Levi subgroup in the supercuspidal support of some (thus any) irreducible representation in Ω, and X (M) is the group of unramified characters of M(F). The group automorphism X (M) ⟶^{∼} X (M) given by χ_{M} ↦ χ_{M}| det |^{\frac{1−n}{2}} gives rise to an E-isomorphism Spec ℨ_{D} ⟶^{∼} Spec ℨ_{D}. The latter map is invariant under the W (D)- action (the point is that | det | is invariant under G-conjugation) so it descends to an E-isomorphism Spec ℨ_{Ω} ⟶^{∼} Spec ℨ_{Ω} in view of Lemma 3.21. Let tw : ℨ_{Ω} → ℨ_{Ω} denote the induced isomorphism.

We have a natural isomorphism ι_{σ} : H(σ_{sm}) →^{∼} H(σ); viewing Hecke algebras as endomorphism-valued functions on G, this is given by ψ ↦ ψ · σ_{alg} . (This is a priori only an injection, but in fact is an isomorphism by the proof of Lemma 1.4 of [ST06].) Now we construct η as the following composite map

H(σ) ≃^{ι_{σ}^{−1}} H(σ_{sm}) ≃ ℨ_{Ω} →^{tw} ℨ_{Ω} →^{I} O_{X^{rig}},

where the second map comes from Lemma 3.24. Note that η is already defined over E. To verify the desired interpolation property of η, we let x : O_{X^{rig}} → E_{x} be an E-algebra map. Then x ∘ I ∘ tw : ℨ_{Ω} → E_{x} gives the supercuspidal support of π_{sm}(ρ_{x}) = r_{p}^{−1}(WD(ρ_{x})^{F −ss}) by Proposition 4.3; indeed, since I interpolates rec_{p} by that proposition, I ∘ tw interpolates r_{p}. In order to complete the proof, we can and do base change to \overline{ℚ}_{p}. Then Proposition 3.9 shows us that ℨ_{Ω} acts on Hom_{K}(σ_{sm}, π_{sm}(ρ_{x})) through x ∘ I ∘ tw.

To conclude, it is enough to observe that the action of H(σ_{sm}) on the space Hom_{K}(σ_{sm}, π_{sm}(ρ_{x})) is compatible with the ℨ_{Ω}-action on the same space via the isomorphism H(σ_{sm}) ≃ ℨ_{Ω}, and also with the H(σ)-action on Hom_{K}(σ, π_{l.alg,x}) via the canonical isomorphisms between the algebras (via ι_{σ}) and the modules. These are readily checked. □

In order to deduce Theorem 4.1 from Proposition 4.2, we will now use the results of [Hu09] to show that the image of η is bounded, in the sense that for any h ∈ H(σ), the valuation of η(h) at each point of X^{rig} is uniformly bounded.

Recall that σ = σ_{alg} ⊗ σ_{sm}, where σ_{sm} = σ(τ) for some inertial type τ : I_{F} → GL_{n}(E). Let x : R□ ; 𝔭̃(σ)[1/p] → E_{x} be a closed point, so that E_{x} is a finite extension field of E. Then x defines a local Galois representation ρ_{x} : G_{F} → GL_{n}(E_{x}) which is potentially crystalline, and has Hodge–Tate weights determined by the highest weight ξ of σ_{alg}. Set π_{x} := π_{sm}(ρ_{x}). Recall that π_{l.alg,x} is the locally algebraic representation defined over E_{x} corresponding to ρ_{x} (see §1.5), whose smooth part is π_{x} and which determines the character χ_{π_{x}} ∘ ι_{σ}^{−1} : H(σ) → E_{x} (via the action of H(σ) on Hom_{K}(σ, π_{l.alg,x}).

Let P = MN be a parabolic subgroup of G, with Levi M and unipotent radical N. Let Z(M) be the centre of M, let N_{0} ⊂ N be a compact open subgroup and define Z(M)^{+} := {t ∈ Z(M)|tN_{0}t^{−1} ⊂ N_{0}}. When P is a standard (upper) parabolic, the subgroup Z(M)^{+} of Z(M) consists of elements with non-decreasing p-adic valuations on the diagonal. Then [Eme06a] defines a Jacquet module functor J_{P} on locally analytic representations of G.

We will consider the following condition on a locally analytic representation V of G.

4.7. Condition. For every parabolic subgroup P = MN as above, with modulus character δ_{P}, every χ : Z(M) → E^{×} such that Hom_{Z(M)}(χ, J_{P} (V)) ≠ 0, and every t ∈ Z(M)^{+}, we have |χ(t)δ_{P} (t)^{−1}|_{p} ≤ 1.

As above, we write τ = ⊕_{i=1}^{r}(τ_{i})^{d_{i}}, where the τ_{i} are pairwise non-isomorphic I_{F} -representations corresponding via the inertial local Langlands correspondence to cuspidal types σ(τ_{i}) of GL_{e_{i}}(O_{F}). Let M = ∏_{i=1}^{r} GL_{e_{i}}(F)^{d_{i}} be a standard Levi of G, with corresponding standard parabolic P = MN.

From now on until the end of this section, we will replace π_{x} (as well as χ_{π_{x}}, σ_{sm}, and so on) by its base extension to \overline{ℚ}_{p}. Then as recalled in Section 3.1, π_{x} is the unique irreducible quotient of a normalised parabolic induction π̃_{x} := i_{P}^{G} π_{x,M}, where π_{x,M} = ⊗_{i=1}^{r}(⊗_{j=1}^{d_{i}}π_{x,i,j}) such that each π_{x,i,j} is a supercuspidal representation of GL_{e_{i}}(F) containing the type σ(τ_{i}) and where for each i, we have π_{x,i,j} ≄ π_{x,i,j′}(1) for 1 ≤ j′ < j ≤ d_{i}.

By Proposition 3.9 we have a G-equivariant homomorphism φ : c-Ind_{K}^{G}σ_{sm} → π̃_{x}, which identifies π̃_{x} with c-Ind_{K}^{G}σ_{sm} ⊗_{H(σ_{sm}),χ_{πx}} \overline{ℚ}_{p}. We identify W_{[M,π_{x,M}]} with ∏_{i=1}^{r} S_{d_{i}} in the obvious way, where S_{d_{i}} is the symmetric group on {1, …, d_{i}}. Note that W_{[M,π_{x,M}]} and the identification are independent of x.

4.8. Lemma. Let χ(π_{x,i,j}) denote the central character of π_{x,i,j}. For an element w = \{w_{i}\}_{i=1}^{r} ∈ W_{[M,π_{x,M}]}, define characters χ_{x,w} : Z(M) → \overline{ℚ}_{p}^{×} by χ_{x,w} = ⨂_{i=1}^{r} ⨂_{j=1}^{d_{i}} χ(π_{x,i,w_{i}(j)}).

For every t ∈ Z(M)^{+}, there exists a constant C_{t} such that |χ_{x,w}(t)|_{p} ≤ C_{t} for all points x of X^{rig}.

Proof. We know that σ_{alg} ⊗π̃_{x}, after twisting by a unitary character (det | det |)^{1−n} (this twist is discussed at the beginning of §5 below), corresponds to the potentially crystalline Galois representation ρ_{x} with Hodge–Tate weights determined by σ_{alg}, in the sense that π̃_{x}| det |^{1−n} ↔ WD(ρ_{x})^{F −ss} via the modified local Langlands correspondence as in Section 4 of [BS07]. Moreover, note that by Lemma 4.2 of Section 4 of [BS07], σ_{alg} ⊗ π̃_{x} actually has a model over a sufficiently large finite extension of ℚ_{p}, so the characters χ_{x,w} then take values in some sufficiently large finite extension E′_{x}/ℚ_{p}.

By the equivalence between parts (ii) and (iv) of Theorem 1.2 of [Hu09] (where our coefficient field is taken to be E′_{x}), we know that σ_{alg}(det)^{1−n} ⊗π̃_{x}| det |^{1−n} has a unitary central character and satisfies Condition 4.7 [Eme06a]. Therefore so does σ_{alg} ⊗ π̃_{x}.

Note that by Proposition 4.3.6 of [Eme06a] we have J_{P} (σ_{alg} ⊗ π̃_{x}) →^{∼} σ_{alg}^{N} ⊗

r_{P}^{G}(π̃_{x})δ_{P}^{1/2}, where r_{P}^{G} is the normalised Jacquet functor for smooth representations. Putting this formula together with Condition 4.7, we see that then we have |χ(t)|_{p} ≤ |σ_{alg}^{N}(t) · δ_{P} (t)^{−1/2}|^{−1} for every χ occurring in r_{P}^{G}(π̃_{x}).

Now, Proposition 3.2(2) of [Hu09] computes r_{P}^{G}(π̃_{x}) (observe that in the notation of loc. cit., all b_{i} are 1 in our case) and shows that the characters χ_{x,w} = ⨂_{i=1}^{r} ⨂_{j=1}^{d_{i}} χ(π_{x,i,w_{i}(j)}) of Z(M) for all sets w of permutations w_{i} of {1, …, d_{i}} occur in r_{P}^{G}(π̃_{x}). The result follows. □

Let Z(M)^{++} ⊂ Z(M) be the subgroup generated by elements with the property that the p-adic valuations are non-decreasing on the diagonal of each block GL_{e_{i}}^{d_{i}}. Clearly, Z(M)^{+} ⊂ Z(M)^{++}.

4.9. Corollary. The conclusion of Lemma 4.8 holds for all t ∈ Z(M)^{++}.

Proof. There is a permutation of {1, …, r} which induces a permutation on the factors GL_{e_{i}}^{d_{i}} of M such that the image of t under that permutation has non-decreasing p-adic valuations. Let M′ be the Levi subgroup of G with the permuted blocks as factors. Abstractly, M′ ≃ M and by Proposition 6.4 of [Zel80], we know that the induction i_{P}^{G} π_{M} is independent of the ordering of the τ_{i}. We conclude by applying Lemma 4.8 to the Levi M′ instead of M. □

As discussed in Sections 3.3 and 3.6, we have a semisimple Bushnell–Kutzko type (J, λ) such that σ_{sm} is a direct summand of Ind_{J}^{K}(λ), and the natural map s_{P} : H(G, λ) → H(σ_{sm}) induces an isomorphism Z(H(G, λ)) ⟶^{∼} H(σ_{sm}). In particular, this means that π̃_{x}|_{J} contains λ. Then in the notation of Section 3.3, π_{x,M} contains the type (J ∩ M, λ_{M}). Let χ_{π_{x,M}} be the character by which H(M, λ_{M}) acts on Hom_{J∩M}(λ_{M}, π_{M}).

4.10. Corollary. Let t ∈ Z(M) and let ν_{t} ∈ H(M, λ_{M}) be an intertwiner supported on t(J ∩ M). Then there exists a constant C_{t} such that for all points x of X^{rig} we have |χ_{π_{x,M}} (ν_{t})|_{p} ≤ C_{t}.

Proof. Assume ν_{t} ≠ 0. Note that since t ∈ Z(M), ν_{t}(t) commutes with the action of J ∩M on λ_{M} and since λ_{M} is irreducible we deduce that ν_{t}(t) is a nonzero scalar. Rescaling, we may assume that ν_{t}(t) := id_{λ_{M}}. Let s ∈ Z(M)^{++} be such that s = ^{w}t for some w ∈ W_{[M,π_{x,M}]}. It follows from the definitions that χ_{π_{x,M}} (ν_{t}) = χ_{x,w}(s), and the corollary then follows from Corollary 4.9. □

4.11. Corollary. Let ν ∈ H(M, λ_{M}). Then there exists a constant C_{ν} such that |χ_{π_{x,M}} (ν)|_{p} ≤ C_{ν} for all points x of X^{rig}.

Proof. Since ν has compact support and we only need some bounded constant C_{ν}, it suffices to prove the claim in the case that ν is an element of the basis of H(M, λ_{M}) given by Lemma 3.4. Let ν′ be the e-fold convolution of ν with itself, where e is as in the statement of Lemma 3.4; then from Corollary 4.10 applied to ν′, we see that there is some constant C_{ν} such that |χ_{π_{x,M}} (ν′)|_{p} ≤ C_{ν}^{e} for all x. This implies that |χ_{π_{x,M}} (ν)|_{p} ≤ C_{ν}, as required. □

4.12. Proposition. For any ν ∈ H(σ), there is a constant C_{ν} such that

|χ_{π_{x}}(ι_{σ}^{−1}(ν))|_{p} ≤ C_{ν}

for all points x of X^{rig}.

Proof. Recall from Section 3.6 that we have an isomorphism

(s_{P} ∘ t_{P}) : H(M, λ_{M})^{W_{[M,π_{x,M}]}} → H(σ_{sm}).

Set ν_{t} := (s_{P} ∘ t_{P})^{−1}(ι_{σ}^{−1}(ν)). Corollary 4.11 implies that there is a constant C_{ν} such that |χ_{π_{x,M}} (ν_{t})|_{p} ≤ C_{ν} for all x.

Now, in order to conclude, we just need to relate χ_{π_{x,M}} (ν_{M}) to χ_{π_{x}}((s_{P} ∘t_{P})(ν_{M})) for any ν_{M} ∈ H(M, λ_{M})^{W_{[M,π_{x,M}]}}. As recalled in Section 3.3, i_{P}^{G} corresponds on the level of Hecke modules to pushforward along the map t_{P}. More precisely, if we let M := Hom_{J∩M}(λ_{M}, π_{x,M}) and N := Hom_{J}(λ, π_{x}), then N ≃ Hom_{H(M,λ_{M})}(H(G, λ), M). Here we view H(G, λ) as a left H(M, λ_{M})-module via t_{P} and the action of H(G, λ) on the space of homomorphisms is via right translation. For z ∈ Z(H(G, λ)), we note that the right action is also a left action, so the eigenvalue of z on N is the same as the eigenvalue of (t_{P})^{−1}(z) on M. On the other hand, Hom_{K}(σ, π_{x}) ≃ e_{K}N for the idempotent e_{K} in H(G, λ) which defines σ. Therefore, any eigenvalue of e_{K} ∗ z on e_{K}N is an eigenvalue of z on N. We deduce that χ_{π_{x,M}} (ν_{M})=χ_{π_{x}}((s_{P} ∘ t_{P})(ν_{M})). □

Proof of Theorem 4.1. By [Kis08, Thm 3.3.8] we know that R□ ; 𝔭̃(σ)[1/p] is a regular ring. Proposition 7.3.6 of [dJ95] (which is applicable because R□ ; 𝔭̃(σ)[1/p] is in particular normal) then implies that the ring of rigid analytic functions on X^{rig} whose absolute value is bounded by 1 coincides precisely with the normalisation of R□ ; 𝔭̃(σ) in R□ ; 𝔭̃(σ)[1/p], and so the ring of bounded rigid analytic functions on X^{rig} is equal to R□ ; 𝔭̃(σ)[1/p].

The result then follows immediately from Proposition 4.12 and the defining property of η. □

We now return to the global setting. Fix a K-stable O-lattice σ^{∘} in σ. Set

M_{∞}(σ^{∘}) := (Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘})^{d}))^{d} .

Here we are considering continuous homomorphisms for the profinite topology on M_{∞} and the p-adic topology on (σ^{∘})^{d}. We equip Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘})^{d}) with the padic topology. Note that M_{∞}(σ^{∘}) is an O-torsion free, profinite, linear-topological O-module.

4.13. Lemma. There is a natural isomorphism of topological O-modules

M_{∞}(σ^{∘}) ≅ \underleftarrow{lim}_{n} (Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘}/ϖ^{n})^{∨}))^{∨} .

Proof. Let H := Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘})^{d}), so that M_{∞}(σ^{∘}) = H^{d}. Then H^{d} ≅ \underleftarrow{lim}_{n} Hom_{O}(H, O/ϖ^{n}) ≅ \underleftarrow{lim}_{n} Hom_{O}(H/ϖ^{n}, O/ϖ^{n}) ≅ \underleftarrow{lim}_{n}(H/ϖ^{n})^{∨}. Since M_{∞} is a projective O[[K]]-module, the short exact sequence

0 → (σ^{∘})^{d} →^{ϖ^{n}·} (σ^{∘})^{d} → (σ^{∘})^{d}/ϖ^{n} → 0

yields an isomorphism H/ϖ^{n} ≅ Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘})^{d}/ϖ^{n}). Finally, (σ^{∘})^{d}/ϖ^{n} ≅ Hom_{O}(σ^{∘}/ϖ^{n}, O/ϖ^{n}) ≅ (σ/ϖ^{n})^{∨}. □

4.14. Remark. One may modify the proof of Lemma 4.13 to show that M_{∞}(σ^{∘}) is

naturally isomorphic to (Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘})^{∨}))^{∨} .

4.15. Remark. M_{∞}(σ^{∘}) is essentially the patched module constructed in Section 5.5 of [EG13], although as the conventions and constructions of the current paper differ slightly from those of [EG13] (see e.g. the difference in the choices of v_{1}, noted in the discussion of Subsection 2.3, as well as Remark 2.7) we will not make this precise.

Let H(σ^{∘}) := End_{G}(c-Ind_{K}^{G}σ^{∘}); this is an O_{E}-subalgebra of H(σ). Note that since σ^{∘} is a free O-module of finite rank, it follows from the proof of Theorem 1.2 of [ST02] that Schikhof duality induces an isomorphism

Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘})^{d}) ≅ Hom_{K}(σ^{∘}, (M_{∞})^{d}).

Frobenius reciprocity gives Hom_{K}(σ^{∘}, (M_{∞})^{d}) ≅ Hom_{G}(c-Ind_{K}^{G}σ^{∘}, (M_{∞})^{d}). Thus M_{∞}(σ^{∘}) is equipped with a tautological Hecke action of H(σ^{∘}), which commutes with the action of R_{∞}.

Let R_{∞}(σ) be the quotient of R_{∞} which acts faithfully on M_{∞}(σ^{∘}). (It follows from Lemma 2.16 of [Paš12] that this is independent of the choice of lattice σ^{∘} ⊂ σ.)

Set R_{∞}(σ)′ := R_{∞} ⊗_{R□ ; 𝔭̃} R□ ; 𝔭̃(σ).

4.16. Lemma. (1) R_{∞}(σ) is a reduced O-torsion free quotient of R_{∞}(σ)′.

(2) If h ∈ H(σ^{∘}) is such that η(h) ∈ R□ ; 𝔭̃(σ), then the action of h on M_{∞}(σ^{∘}) agrees with the action of η(h) via the natural map R□ ; 𝔭̃(σ) → R_{∞}(σ)′.

Proof. (1) That R_{∞}(σ) is O-torsion free follows immediately from the fact that by definition it acts faithfully on the O-torsion free module M_{∞}(σ^{∘}). The fact that it is actually a quotient of R_{∞}(σ)′ is then essentially an immediate consequence of classical local-global compatibility at 𝔭̃, but to see this will require a little unraveling of the definitions. Note that if N is sufficiently large, then K_{N} acts trivially on (σ^{∘})^{d}/ϖ^{N}. Using Lemma 4.13, we see that

M_{∞}(σ^{∘})^{∨} = \underleftarrow{lim}_{N} Hom_{O[Γ_{2N}]} ((M_{∞}/𝔟_{N})_{K_{2N}} , (σ^{∘})^{d}/ϖ^{N})^{∨}

= \underleftarrow{lim}_{N} Hom_{O[Γ_{2N}]} ((M_{1,Q_{N′(N)}}^{□}/𝔟_{N})_{K_{2N}} , (σ^{∘})^{d}/ϖ^{N})^{∨} ,

so it suffices to show that if N ≫ 0 then the action of R_{∞} on

Hom_{O[[K]]}^{cont}(M_{1,Q_{N′(N)}}^{□}, (σ^{∘})^{d}/ϖ^{N})

factors through R_{∞}(σ)′. Now, by definition we have

M_{1,Q_{N′(N)}}^{□} =

pr^{∨}(S_{ξ,τ} (U_{1}(Q_{N′(N)})_{2N′(N)}, O/ϖ^{N′(N)})_{𝔪_{Q_{N′(N)}}}^{∨} ) ⊗_{R_{S_{Q_{N′(N)}}}^{univ}} R_{S_{Q_{N′(N)}}}^{□_{T}} ,

so it suffices to prove the same result for

Hom_{O[[K]]}^{cont}(S_{ξ,τ} (U_{1}(Q_{N′(N)})_{2N′(N)}, O/ϖ^{N′(N)})_{𝔪_{Q_{N′(N)}}}^{∨} ⊗_{R_{S_{Q_{N′(N)}}}^{univ}} R_{S_{Q_{N′(N)}}}^{□_{T}} ,

(σ^{∘})^{d}/ϖ^{N}),

which is equal to

Hom_{O[[K]]}^{cont}(S_{ξ,τ} (U_{1}(Q_{N′(N)})_{2N′(N)}, O/ϖ^{N′(N)})_{𝔪_{Q_{N′(N)}}}^{∨} ,

(σ^{∘})^{d}/ϖ^{N}) ⊗_{R_{S_{Q_{N′(N)}}}^{univ}} (R_{S_{Q_{N′(N)}}}^{□_{T}} )^{∨},

which in turn equals

S_{ξ,τ} (U_{1}(Q_{N′(N)})_{0}, (σ^{∘})^{d}/ϖ^{N})_{𝔪_{Q_{N′(N)}}} ⊗_{R_{S_{Q_{N′(N)}}}^{univ}} (R_{S_{Q_{N′(N)}}}^{□_{T}} )^{∨}.

Therefore it would suffice to prove the same result for

S_{ξ,τ} (U_{1}(Q_{N′(N)})_{0}, (σ^{∘})^{d})_{𝔪_{Q_{N′(N)}}} ⊗_{R_{S_{Q_{N′(N)}}}^{univ}} (R_{S_{Q_{N′(N)}}}^{□_{T}} )^{∨}.

If 𝕋 denotes the image of 𝕋^{S_{p}∪Q_{N′(N)},univ} in the endomorphism ring

End_{O}(S_{ξ,τ} (U_{1}(Q_{N′(N)})_{0}, (σ^{∘})^{d})_{𝔪_{Q_{N′(N)}}} ),

then the action of R_{S_{Q_{N′(N)}}}^{univ} on S_{ξ,τ} (U_{1}(Q_{N′(N)})_{0}, (σ^{∘})^{d})_{𝔪_{Q_{N′(N)}}} is given by an

O-algebra homomorphism R_{S_{Q_{N′(N)}}}^{univ} → 𝕋. Since the space of automorphic forms

S_{ξ,τ} (U_{1}(Q_{N′(N)})_{0}, (σ^{∘})^{d})_{𝔪_{Q_{N′(N)}}} is O-torsion free (by the choice of U_{m,v_{1}} in Sec-

tion 2.3) and the algebra 𝕋 is reduced (by the usual comparison between algebraic modular forms and classical automorphic forms, and the semisimplicity of the space of cuspidal automorphic forms; cf. [CHT08, Corollary 3.3.3 and §3.4]), then by the definition of R□ ; 𝔭̃(σ), we need to show that if 𝕋 → \overline{ℚ}_{p} is a closed point, then the restriction to G_{F𝔭̃} of the corresponding Galois representation G_{F̃^{+},T ∪Q_{N′(N)}} →

G_{n}(\overline{ℚ}_{p}) is potentially crystalline of type σ; but this is immediate from classical local-global compatibility (see e.g. Theorem 1.1 of [Car12]).

Finally, to see that R_{∞}(σ) is reduced, we note that by part (2) of Lemma 4.17 below, the ring R_{∞}(σ)[1/p] is a direct factor of the regular (by [Kis08, Thm 3.3.8]) ring R_{∞}(σ)′[1/p]. Thus R_{∞}(σ)[1/p] is regular, and in particular reduced, and hence so is its subring R_{∞}(σ). (The reader can easily check that this reducedness is not used in the proof of Lemma 4.17, and hence no circularity is involved in this argument.)

(2) Again, this is essentially an immediate consequence of classical local-global compatibility at 𝔭̃, but a little explanation is needed in order to make this plain.

Note first that the natural action of H(σ^{∘}) on M_{∞}(σ^{∘}) is induced via Frobenius reciprocity from the G-action on M_{∞}, which is patched from the partial G-actions defined by

\overline{α}_{N′(N)} : (M_{∞}/𝔟_{N})_{K_{2N}} → c-Ind_{KZ}^{G_{N}}((M_{∞}/𝔟_{N})_{K_{N}} )

and these in turn, after taking homomorphisms into (σ^{∘})^{d}/ϖ^{N}, induce partial H(σ^{∘})-actions on spaces of algebraic modular forms of weight (σ^{∘})^{d}. (More precisely, the G-action on ϖ-adically completed cohomology

S̃_{ξ,τ} (U_{1}^{𝔭}(Q_{N′(N)}), O) := \underleftarrow{lim}_{s}(\underrightarrow{lim}_{m} S_{ξ,τ} (U_{1}(Q_{N′(N)})_{m}, O/ϖ^{s}))

gives rise, via Frobenius reciprocity and the identification

S_{ξ,τ} (U_{1}^{𝔭}(Q_{N′(N)})_{0}, (σ^{∘})^{d}) ≃ Hom_{K}(σ^{∘}, S̃_{ξ,τ} (U_{1}^{𝔭}(Q_{N′(N)}), O)),

to a natural action of H(σ^{∘}) on S_{ξ,τ} (U_{1}(Q_{N′(N)})_{0}, (σ^{∘})^{d}).) We see therefore, as in part (1), that it is enough to consider the natural action of each h ∈ H(σ^{∘}) on the spaces

S_{ξ,τ} (U_{1}(Q_{N′(N)})_{0}, (σ^{∘})^{d})_{𝔪_{Q_{N′(N)}}} ⊗_{R_{S_{Q_{N′(N)}}}^{univ}} (R_{S_{Q_{N′(N)}}}^{□_{T}} )^{∨}

for N ≫ 0. In addition to the natural action of H(σ^{∘}), this is equipped with an action of R□ ; 𝔭̃(σ) via the composite R□ ; 𝔭̃ → R^{loc} → R_{S_{Q_{N′(N)}}}^{□_{T}}, which factors through

R□ ; 𝔭̃(σ) by part (1). By classical local-global compatibility and the defining property of the morphism η of Theorem 4.1, we see that, after inverting p, the action of h on this space agrees with the action of η(h). The desired result now follows from the fact that S_{ξ,τ} (U_{1}(Q_{N′(N)})_{0}, (σ^{∘})^{d})_{𝔪_{Q_{N′(N)}}} is O-torsion free. □

We now use the usual commutative algebra arguments underlying the Taylor– Wiles–Kisin method to study the support of M_{∞}(σ^{∘}).

4.17. Lemma. (1) The module M_{∞}(σ^{∘}) is finitely generated over R_{∞}(σ), and moreover M_{∞}(σ^{∘})[1/p] is locally free of rank one over R_{∞}(σ)[1/p]. The topology on M_{∞}(σ^{∘}) coincides with its 𝔪-adic topology, where 𝔪 denotes the maximal ideal of R_{∞}(σ).

(2) The support of M_{∞}(σ^{∘}) in Spec R_{∞}(σ)′ is a union of irreducible components of R_{∞}(σ)′.

(3) Let \overline{R}_{∞}(σ) be the normalisation of R_{∞}(σ) inside R_{∞}(σ)[1/p]. Then the action of H(σ^{∘}) on M_{∞}(σ^{∘}) induces an O-algebra map α : H(σ^{∘}) → \overline{R}_{∞}(σ).

Proof. Since M_{∞} is a finite projective S_{∞}[[K]]-module, the module M_{∞}(σ^{∘}) is finite and projective (equivalently, free) over S_{∞}. Indeed, we may write M_{∞} as a direct summand of S_{∞}[[K]]^{r} for some r ≥ 0, and so Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘})^{d})^{d} is a direct

summand of Hom_{O[[K]]}^{cont}(S_{∞}[[K]]^{r}, (σ^{∘})^{d})^{d} ≅ (Hom_{O[[K]]}^{cont}(S_{∞}[[K]], (σ^{∘})^{d})^{d})^{r}. Thus

it suffices to note that since S_{∞}[[K]] ≅ S_{∞} ˆ⊗_{O}O[[K]] as an O[[K]]-module, there is a natural isomorphism

Hom_{O[[K]]}^{cont}(S_{∞}[[K]], (σ^{∘})^{d})^{d} ≅ Hom_{O[[K]]}^{cont}(O[[K]], Hom_{O}^{cont}(S_{∞}, (σ^{∘})^{d}))^{d}
≅ Hom_{O}^{cont}(S_{∞} ⊗_{O} σ^{∘}, O)^{d} ≅ S_{∞} ⊗_{O} σ^{∘}.

Since the S_{∞}-action on M_{∞}(σ^{∘}) factors through the action of R_{∞}, which in turn factors through R_{∞}(σ) by definition, we find that in particular M_{∞}(σ^{∘}) is finitely generated over R_{∞}(σ).

Since the identification of M_{∞} as a direct summand of S_{∞}[[K]] is compatible with the natural topologies on each of M_{∞} and S_{∞}[[K]], one easily verifies that the topology on M_{∞}(σ^{∘}) coincides with its 𝔫-adic topology, where 𝔫 denotes the maximal ideal of S_{∞}. Furthermore, since by definition R_{∞}(σ) embeds into End_{S_{∞}}(M_{∞}(σ^{∘})), we find that R_{∞}(σ) is finite as an S_{∞}-algebra, and so in particular the 𝔫-adic topology and 𝔪-adic topology on M_{∞}(σ^{∘}) coincide (where, as in the statement of the lemma, 𝔪 denotes the maximal ideal of R_{∞}(σ)). Thus the topology on M_{∞}(σ^{∘}) coincides with its 𝔪-adic topology.

By Lemma 3.3 of [BLGHT11], Lemma 2.4.19 of [CHT08], and Theorem 3.3.8 of [Kis08], we see that the ring R_{∞}(σ)′ is equidimensional of the same Krull dimension as S_{∞}. Since M_{∞}(σ^{∘}) is free of finite rank over S_{∞}, and the image of R_{∞}(σ)′ in End(M_{∞}(σ^{∘})) is an S_{∞}-algebra, we see that the depth of M_{∞}(σ^{∘}) as an R_{∞}(σ)′-module is at least the Krull dimension of S_{∞}. Since this is equal to the Krull dimension of R_{∞}(σ)′, it follows immediately from Lemma 2.3 of [Tay08] that the support of M_{∞}(σ^{∘}) is a union of irreducible components of R_{∞}(σ)′. (Of course, conjecturally R_{∞}(σ) is actually equal to R_{∞}(σ)′.)

That M_{∞}(σ^{∘})[1/p] is locally free over R_{∞}(σ)[1/p] follows by an argument of Diamond (cf. [Dia97]). More precisely, R□ ; 𝔭̃(σ)[1/p] and each of the rings Rσ,ξ,τ ; ṽ [1/p] for places v | p, v ≠ 𝔭 are regular by Theorem 3.3.8 of [Kis08], and R□ ; ṽ_{1} is smooth by the choice of v_{1}, so R_{∞}(σ)′[1/p] is regular. Therefore R_{∞}(σ)[1/p] is also regular, so M_{∞}(σ^{∘})[1/p] is locally free over R_{∞}(σ)[1/p] by Lemma 3.3.4 of [Kis09b] (or rather by its proof, which goes over unchanged to our setting, where we do not assume that R_{∞}(σ)[1/p] is a domain).

That it is actually locally free of rank one can be checked at finite level, where it follows from the multiplicity one assertion in Theorem 3.7, and the choice of v_{1} (and the fact that we have fixed the action mod p of the Hecke operators at ṽ_{1}).

This completes the proof of parts (1) and (2), and so we turn to proving (3). To this end, let A be the R_{∞}-subalgebra of the endomorphism algebra of M_{∞}(σ^{∘}) generated by H(σ^{∘}). Since M_{∞}(σ^{∘}) is a finite type R_{∞}(σ)-module, we see that A is a finite R_{∞}(σ)-algebra. Since M_{∞}(σ^{∘})[1/p] is in fact locally free of rank one over R_{∞}(σ)[1/p] we have End_{R_{∞}(σ)[1/p]}(M_{∞}(σ^{∘})[1/p]) = R_{∞}(σ)[1/p], so that A[1/p] = R_{∞}(σ)[1/p]. So the natural map H(σ^{∘}) → A lands inside \overline{R}_{∞}(σ). □

The morphism α of Lemma 4.17 induces an E-algebra morphism α : H(σ) → R_{∞}(σ)[1/p].

4.18. Theorem. The map α coincides with the composition

H(σ) →^{η} R□ ; 𝔭̃(σ)[1/p] → R_{∞}(σ)[1/p].

Proof. Note firstly that if h ∈ H(σ^{∘}) is such that η(h) ∈ R□ ; 𝔭̃(σ), then the two maps agree on h by Lemma 4.16 (2). Since R_{∞}(σ) is p-torsion free by Lemma 4.16 (1), it is therefore enough to show that H(σ) is spanned over E by such elements.

Now, H(σ^{∘}) certainly spans H(σ) over E, so it is enough to show that for any element h′ ∈ H(σ^{∘}), we have η(p^{C}h′) ∈ R□ ; 𝔭̃(σ) for some C ≥ 0; but this is obvious.

□

4.19. Remark. It follows from Lemma 4.17 (2) that the locus of closed points of Spec R□ ; 𝔭̃(σ)[1/p] which come from closed points of Spec R_{∞}(σ)[1/p] is a union of irreducible components, which we call the set of automorphic components of Spec R□ ; 𝔭̃(σ)[1/p]. (Note that we do not know a priori that this notion is independent of the choice of global setting, although of course we expect that in fact every component of Spec R□ ; 𝔭̃(σ)[1/p] is an automorphic component.)

4.20. Remark. We expect that η(H(σ^{∘})) is contained in the normalisation of R□ ; 𝔭̃(σ) in R□ ; 𝔭̃(σ)[1/p]; it may well be possible to prove this via our methods, but as we do not need this result, we have not pursued it. It is easy to see that the analogous result holds for the quotient of R□ ; 𝔭̃(σ) corresponding to the automorphic components in the sense of Remark 4.19.

5. The Breuil–Schneider conjecture

Continue to assume that p ∤ 2n, and that F is a finite extension of ℚ_{p}. If r : G_{F} → GL_{n}(E) is a de Rham representation of regular weight then we say that r is generic if π_{sm}(r) is generic. In this case, we set

BS(r) := π_{alg}(r) ⊗ π_{sm}(r).

(In fact, our BS(r) differs from the definition made in [BS07] in that π_{alg}(r) and π_{sm}(r) are their analogues in [BS07] times the characters det^{n−1} and | det |^{n−1}, respectively. Since (det | det |)^{n−1} is a unitary character, this makes no difference to the following conjecture. See also Section 2.4 of [Sor12] for a discussion of the difference between these conventions.) The following is [BS07, Conjecture 4.3] (in the open direction, in the generic case).

5.1. Conjecture. If r : G_{F} → GL_{n}(E) is de Rham and has regular weight, then BS(r) admits a nonzero unitary Banach completion.

5.2. Remark. In fact, it seems reasonable (particularly in the light of Corollary 5.4 below) to conjecture that there is even a nonzero admissible completion.

Fix a representation r : G_{F} → GL_{n}(E), and assume from now on that r is potentially crystalline of regular weight, and that r is generic. By Remark 2.13 (and possibly replacing E with a finite extension if necessary), we may replace r with a conjugate representation so that r : G_{F} → GL_{n}(O), and r̄ satisfies the hypotheses of Section 2. We can therefore carry out the construction of Section 2, obtaining the patched module M_{∞}. Recall that r is induced from an O-algebra homomorphism x : R□ ; 𝔭̃ → O, which we extended to an O-algebra homomorphism y : R_{∞} → O. Then V (r) is obtained from the fiber of (M_{∞})^{d}[1/p] above the closed point of R_{∞}[1/p] determined by y.

Write σ_{sm}(r) for σ(τ), σ_{alg}(r) for π_{alg}(r)|_{K}, and let σ := σ_{alg}(r)⊗σ_{sm}(r), keeping in mind the convention at the end of Section 3.12. (Also enlarge E to another finite extension if necessary as explained in that section.) As above, we write H(σ) for End_{G}(c-Ind_{K}^{G}σ), which is isomorphic to End_{G}(c-Ind_{K}^{G}σ(τ)) via ι_{σ}, so that π_{sm}(r) determines a character χ_{π_{sm}(r)} : H(σ) → E. Since r is generic, we see from Corollary 3.11 that π_{sm}(r) ≅ (c-Ind_{K}^{G}σ_{sm}(r)) ⊗_{H(σ),χ_{πsm(r)}} E. Tensoring with π_{alg}(r), we have

BS(r) ≅ (c-Ind_{K}^{G}σ) ⊗_{H(σ),χ_{πsm(r)}} E.

Since our representations V (r) are unitary Banach representations, and since (because we have assumed that r is generic) BS(r) is irreducible, in order to prove Conjecture 5.1 it would be enough to check that Hom_{G}(BS(r), V (r)) ≠ 0. While we cannot at present do this in general, we are able to reinterpret the problem in terms of automorphy lifting theorems, and deduce new cases of Conjecture 5.1. In particular, Corollary 5.5 below gives the first general results in the principal series case.

5.3. Theorem. Suppose that p ∤ 2n, and that r : G_{F} → GL_{n}(E) is a generic potentially crystalline representation of regular weight. If r corresponds to a closed point on an automorphic component of R□ ; 𝔭̃(σ)[1/p] (in the sense of Remark 4.19), then BS(r) admits a non-zero unitary admissible Banach completion.

Proof. As remarked above, it suffices to show that Hom_{G}(BS(r), V (r)) ≠ 0. Proposition 2.20 of [Paš12] implies that

dim_{E} Hom_{K}(σ, V (r)) = dim_{E} M_{∞}(σ^{∘}) ⊗_{R_{∞},y} E.

(More specifically, in the notation of that paper we take R = R_{∞}, Θ = σ^{∘}, V = σ, N = M_{∞}, and m^{∘} = O, regarded as an R_{∞}-module via y. Note that [Paš12] assumes that N is finitely generated as an R[[K]]-module, which is satisfied in our case: M_{∞} is a finitely generated R_{∞}[[K]]-module, since it is finite over S_{∞}[[K]] and the S_{∞}-action on M_{∞} factors through a map S_{∞} → R_{∞}.)

Together with Frobenius reciprocity, this shows that Hom_{G}(c-Ind_{K}^{G}σ, V (r)) = Hom_{K}(σ, V (r)) ≠ 0, as y is in the support of M_{∞}(σ^{∘})[1/p] by assumption. Since BS(r) ≅ (c-Ind_{K}^{G}σ) ⊗_{H(σ),χ_{πsm(r)}} E, we need only show that the action of H(σ^{∘}) on M_{∞}(σ^{∘}) ⊗_{R_{∞},y} O factors through the character χ_{π_{sm}(r)}; but this is immediate from Theorem 4.18. □

5.4. Corollary. Suppose that p ∤ 2n and that r : G_{F} → GL_{n}(E) is de Rham of regular weight, and is potentially diagonalizable in the sense of [BLGGT13]. Suppose also that r is generic. Then BS(r) admits a nonzero unitary admissible Banach completion.

Proof. By Theorem 5.3, we need only prove that r corresponds to a point on an automorphic component of R□ ; 𝔭̃(σ)[1/p]. Recalling that y was chosen to correspond to the potentially diagonalizable representation r_{pot.diag} at the places v | p, v ≠ 𝔭, this follows from Theorem A.4.1 of [BLGG13], which constructs a global automorphic Galois representation corresponding to a point on the same component of R□ ; 𝔭̃(σ)[1/p] as r (cf. the proof of Corollary 4.4.3 of [GK12]).

Indeed, the existence of a global automorphic Galois representation corresponding to a point on R□ ; 𝔭̃(σ)[1/p] shows that S_{ξ,τ} (U_{0}, (σ^{∘})^{d})_{𝔪}[1/p] is supported at this point. We have R□ ; 𝔭̃(σ)[1/p]-equivariant isomorphisms

S_{ξ,τ} (U_{0}, (σ^{∘})^{d})_{𝔪}[1/p] ≃ Hom_{K}(σ^{∘}, S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪})[1/p]
≃ Hom_{O[[K]]}^{cont}(M_{∞}/𝔞M_{∞}, (σ^{∘})^{d})[1/p],

where the former comes by identifying locally algebraic vectors in completed cohomology and the latter follows from Schikhof duality and Corollary 2.9. These isomorphisms imply that (M_{∞}(σ^{∘})/𝔞M_{∞}(σ^{∘}))^{d}[1/p] is supported at the same point of R□ ; 𝔭̃(σ)[1/p] coming from a global automorphic Galois representation. Therefore, M_{∞}(σ^{∘})^{d}[1/p] is supported at a point of R_{∞}(σ)[1/p] on the same component as r. Finally, we conclude that the component of R□ ; 𝔭̃(σ)[1/p] corresponding to r is automorphic in the sense of Remark 4.19. □

5.5. Corollary. Suppose that p > 2, that r : G_{F} → GL_{n}(E) is de Rham of regular weight, and that r is generic. Suppose further that either

(1) n = 2, and r is potentially Barsotti–Tate, or

(2) F/ℚ_{p} is unramified, r is crystalline, n ≠ p and r has Hodge–Tate weights in the extended Fontaine–Laffaille range; that is, for each κ : F ↪ E, any two elements of HT_{κ}(r) differ by at most p − 1.

Then BS(r) admits a nonzero unitary admissible Banach completion.

Proof. By Corollary 5.4, it is enough to check that our hypotheses imply that r is potentially diagonalizable; in case (1), this is Lemma 4.4.1 of [GK12], and in case (2), it is the main result of [GL12]. □

5.6. Remark. The attentive reader will have noticed that since throughout the paper we assumed that E is sufficiently large (and allowed it to be enlarged in the course of making our argument), we have not proved cases of Conjecture 5.1 as it is written, but rather an apparently weaker version, which allows a finite extension of scalars. However, Conjecture 5.1 is an immediate consequence of this version, in the following way: given an (admissible) unitary Banach completion of BS(r) ⊗_{E} E′, where E′/E is a finite extension, we may regard this completion as being a representation over E, and then the closure of BS(r) inside it gives the required representation.

5.7. Remark. It is natural to wonder whether our methods also prove the conjecture when r is non-generic, in which case the representation BS(r) defined in [BS07] is reducible. Our methods should still give a nonzero element of Hom_{G}(BS(r), V (r)) in these cases, and it will then suffice to prove that this homomorphism is injective. We anticipate that in most cases none of the constituents of BS(r) other than the socle will admit unitary completions, from which the injectivity would follow. For example, when n = 2, the representation BS(r) has length at most 2, and the other constituent is a finite-dimensional algebraic representation, which can only admit a unitary completion if it is one-dimensional.

Similar issues arise if we study potentially semistable representations r which are not potentially crystalline. The definition of σ(τ) can be modified to extend to this case, but even if r is generic, we no longer expect to have an isomorphism (c-Ind_{K}^{G}σ) ⊗_{H(σ),χ_{πsm(r)}} E ⟶^{∼} BS(r); rather, there should be a surjection from the left hand side (which will no longer be irreducible) to the right-hand side, with a non-trivial kernel. Again, we expect that typically none of the constituents of this kernel will admit non-trivial unitary completions.

5.8. Remark. Our techniques also apply to the converse question of describing the representations r such that V (r) has non-zero locally algebraic vectors. Although we have not discussed the general potentially semistable case in this paper, it is not hard to modify the proofs of Lemma 4.16(1) and Theorem 5.3 to show that if r : G_{F} → GL_{n}(E) is a representation with the property that V (r) has non-zero locally algebraic vectors, then r is a regular de Rham representation. It is presumably also possible to develop the theory of Section 3 to encompass representations with non-zero monodromy, and then deduce further that the locally algebraic vectors in V (r) are exactly BS(r) (at least if r is generic).

6. Relationship with a hypothetical p-adic local Langlands correspondence

In this section we describe the relationship between our construction and a hypothetical p-adic local Langlands correspondence.

6.1. A hypothetical formulation of the p-adic local Langlands correspondence. Perhaps the strongest hypothesis one might make regarding a p-adic local Langlands correspondence is the following: that given r̄: G_{F} → GL_{n}(𝔽) with trivial endomorphisms^{1} and with associated deformation ring Rr̄, there exists a finitely generated Rr̄[[K]]-module L_{∞} which is O-torsion free, and is equipped with an Rr̄-linear G-action which extends the K-action arising from its Rr̄[[K]]-module structure.

Given such an object L_{∞}, then for any r : G_{F} → GL_{n}(E) arising from an O_{E}- valued point x of Spec Rr̄, we may associate a unitary Banach space representation B(r) := (L_{∞} ⊗_{Rr̄,x} O_{E})^{d}[1/p] of G, which should be “the” representation of G associated to r via the p-adic local Langlands correspondence.

One might conjecture that such a structure should exist and satisfy the following properties:

6.1.1. Relationship with classical local Langlands. For any potentially semistable lift r : G_{F} → GL_{n}(E) of r̄ with regular Hodge–Tate weights, the locally algebraic vectors of B(r) are isomorphic to BS(r).

6.1.2. Local-global compatibility. Using the notation for completed cohomology and the Hecke algebra that acts on it introduced in the discussion preceding Corollary 2.9, there is an isomorphism of 𝕋_{ξ,τ}^{S_{p}} (U^{𝔭}, O)_{𝔪}[G]-modules

S̃_{ξ,τ} (U^{𝔭}, O)_{𝔪}^{d} ≅ 𝕋_{ξ,τ}^{S_{p}} (U^{𝔭}, O)_{𝔪} ⊗_{Rr̄} L_{∞}

(here 𝕋_{ξ,τ}^{S_{p}} (U^{𝔭}, O)_{𝔪} is regarded as an Rr̄-algebra via the natural maps Rr̄ → R_{S}^{univ} → 𝕋_{ξ,τ}^{S_{p}} (U^{𝔭}, O)_{𝔪}, where the first morphism corresponds to restricting global Galois representations to the decomposition group at 𝔭, and the second morphism is induced by the universal automorphic deformation of \overline{ρ}), as well as analogous isomorphisms when we add auxiliary Q_{N}-level structure, as in the patching process.

1We make this assumption in what follows for simplicity, since the discussion is purely hypothetical in any case. If r̄ admits non-trivial endomorphisms, then we would instead work with the framed deformation ring R□ ; r̄ in everything that follows, and the representation L_{∞} would be endowed with a further equivariant structure for the group GL_{n} acting by “change of framing”.

6.2. The relationship between L_{∞} and M_{∞}. If Condition 6.1.2 held, then we would find that our patched R_{∞}[G]-module M_{∞} is obtained by restricting R_{∞} ⊗_{Rr̄} L_{∞} to the smallest closed subscheme of Spec R_{∞} which contains the support of all the modules (M_{1}, Q_{N′(N)}^{□} ⊗_{S_{∞}} S_{∞}/𝔟_{N})_{K_{2N}} that arise in the patching process. Optimistic conjectures (of “big R equals big 𝕋”-type) might suggest that this support is all of R_{∞}, and thus that M_{∞} is obtained from L_{∞} simply by pullingback it back along the natural map Spec R_{∞} → Spec Rr̄. For this reason, we are hopeful that our patched representation M_{∞} is a good candidate for (the pull-back to R_{∞} of) p-adic local Langlands.

6.3. The relationship with the Fontaine–Mazur conjecture. Note that if both Conditions 6.1.1 and 6.1.2 held, together with an appropriate “big R equals big 𝕋” result, then we would find that if ρ is de Rham deformation of \overline{ρ} corresponding to a point of R_{S}^{univ}, then it would contribute to the locally algebraic vectors of completed cohomology. However, locally algebraic vectors in the completed cohomology arise precisely from algebraic automorphic forms (see [Eme06b]), and hence we would conclude that ρ would be an automorphic Galois representation. (This is an abstraction of the strategy used to deduce the Fontaine–Mazur conjecture for odd two-dimensional representations of G_{ℚ} in [Eme].)

6.4. Concluding remarks. The preceding discussion shows that the question of whether one can in fact relate M_{∞} to a purely local correspondence which satisfies the above two conditions is closely related to the Fontaine–Mazur conjecture for deformations of \overline{ρ}. Given this, we expect it to be a difficult question to precisely determine the support of M_{∞} and the related modules M_{∞}(σ^{∘}) in general, and we likewise expect it to be difficult to analyze the extent to which M_{∞} arises from a purely local construction over Spec Rr̄.

Furthermore, it seems quite possible that our hypothetical formulation of the p-adic local Langlands correspondence is too naive; even if a local correspondence of some kind exists, it may be of a more subtle nature. In this case, we would still expect it to have a strong relationship to our patched modules M_{∞}, but perhaps not as direct as the one considered in the above discussion.

In any case, whatever the eventual truth might be, the preceding discussion suggests that the further investigation of the patched representation M_{∞} is a problem of substantial interest, which we hope to return to in future work.

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E-mail address: caraiani@princeton.edu

Department of Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, NJ 08544, USA

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

Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA

E-mail address: toby.gee@imperial.ac.uk

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

E-mail address: david.geraghty@bc.edu

Department of Mathematics, 301 Carney Hall, Boston College, Chestnut Hill, MA 02467, USA

E-mail address: paskunas@uni-due.de

Universität Duisburg-Essen, Fakultät für Mathematik, 45117 Essen, Germany

E-mail address: swshin@math.mit.edu

Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA / Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea