PATCHING AND THE p-ADIC LOCAL LANGLANDSCORRESPONDENCEANA 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 GLn (F ),F a finite extension of Qp . We use our construction to prove many new casesof the Breuil–Schneider conjecture.

Contents
1. Introduction
1
2. The patching argument7
3. Hecke algebras and types
19
4. Local-global compatibility
30
5. The Breuil–Schneider conjecture42
6. Relationship with a hypothetical p-adic local Langlands correspondence 45
References46

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 GLn (F ), where F is an arbitrary finite extension of Qp , andp - 2n. At present, the existence of such a correspondence is only known for GL1 (F )(where it is given by local class field theory), and for GL2 (Qp ) (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 sayenough about our construction to prove many new cases of the Breuil–Schneider
conjecture, and to reduce the general case of the Breuil–Schneider conjecture (undersome mild technical hypotheses) to standard conjectures related to automorphy
lifting theorems.
The idea that global methods could be used to construct the correspondence isa 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
A.C. was partially supported by the NSF Postdoctoral Fellowship DMS-1204465. M.E. waspartially supported by NSF grants DMS-1003339, DMS-1249548, and DMS-1303450. T.G. waspartially 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

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

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 firsthalf of this idea (although, in contrast to the constructions of [HT01], the directionof the correspondence we construct is from representations of the local Galois group
GF to representations of GLn (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 conjecturalextensions of the existing p-adic Langlands correspondence for GL2 (Qp ).)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 GLn (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 caseof GL2 (Qp ), where it is an immediate consequence of known properties of the p-adic
local Langlands correspondence.) In one direction, Hu [Hu09] showed that if sucha norm exists, the locally algebraic representation necessarily comes from a regularde Rham representation.
The converse direction is largely open. We recall the conjecture in more detailin Section 5 below, to which the reader should refer for any unfamiliar notationor terminology. Given a de Rham representation r : GF → GLn (Qp ) of regularweight, in [BS07] there is associated to r a locally algebraic Qp -representation
BS(r) of GLn (F ). The following is [BS07, Conjecture 4.3] (in the open direction).
A. Conjecture. If r : GF → GLn (Qp ) is de Rham and has regular weight, thenBS(r) admits a nonzero unitary Banach completion.
In fact, the known properties of the p-adic local Langlands correspondence forGL2 (Qp ) suggest that there should even be a nonzero admissible completion. Conjecture A was proved in the case that πsm (r) is supercuspidal in [BS07, Theorem5.2], 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 notpossible 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 GL2 (Qp ), the only previous results in the general principal seriescase that we are aware of are those of [AKdS13, Ies12, KdS12, Vig08], which provethe conjecture for certain principal series cases for GL2 (F ), under additional restrictions on the Hodge filtration of r. The methods of these papers do not seem toshed 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PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

3

any continuous representation r : GF → GLn (Qp ). 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 toshow 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 associatedto 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 : GF →GLn (Qp ) is potentially crystalline of regular weight, and that r is generic. Supposethat r lies on an automorphic component of the corresponding potentially crystalline deformation ring. Then BS(r) admits a nonzero unitary admissible Banachcompletion.
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 : GF → GLn (Qp ) is de Rham of regularweight, and that r is generic. Suppose further that either(1) n = 2, and r is potentially Barsotti–Tate, or(2) F/Qp is unramified and r is crystalline with Hodge–Tate weights in theextended Fontaine–Laffaille range, and n 6= p.
Then BS(r) admits a nonzero unitary admissible Banach completion.
Actually, we prove a more general result (Corollary 5.4) which establishes theconjecture for potentially diagonalizable representations; conjecturally, every potentially crystalline representation is potentially diagonalizable. We remark thatwhile we expect these results to extend to potentially semistable (rather than justpotentially crystalline) representations, and to non-generic representations, we haverestricted 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 GLn (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 atpresent no analogue of the theory of [BZ77, Zel80] in the setting of admissible padic Banach space representations, and even in the supercuspidal case, there areuncountably many p-adic Galois representations up to twist, but only countably
many automorphic representations.

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

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 fromthe 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 notknown 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 eachcomponent 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–Tateweights, 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 GLn (OF ); by keeping trackof the action of the Hecke operators at p, we are able to promote this to an actionof GLn (F ). Taking the fiber of this patched module at the point corresponding toa 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 suspectthat 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 GL2 (Qp ); we will report on this in a future paper.
We refer to Section 6 below for a slightly more detailed discussion of how theconstruction 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 GLn . We then prove some refinements of some of these results, and explain
how to descend them from algebraically closed coefficient fields to finite extensionsof Qp . In Section 4 we begin by proving a result of independent interest, showing

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

5

that the classical local Langlands correspondence interpolates over a (potentiallysemistable) 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 Qp of Qp . Throughout
the paper we work with a finite extension E/Qp in Qp , which will be our coefficientfield. We write O = OE for the ring of integers in E, $ = $E for a uniformiser, andF := O/$ 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 courseof our arguments. Furthermore, we will often assume without further commentthat E and F are sufficiently large, and in particular that if we are working withrepresentations of the absolute Galois group of a p-adic field F , then the images ofall embeddings F ,→ E are contained in E.
If F is a field, we let GF denote its absolute Galois group. Let ε denote thep-adic cyclotomic character, and ε the mod p cyclotomic character. If F is a finite
extension of Qp for some p, we write IF for the inertia subgroup of GF , and $F
for a uniformiser of the ring of integers OF of F . If Fe is a number field and v is a
finite place of Fe then we let Frobv denote a geometric Frobenius element of GFev .
If F is a p-adic field, W is a de Rham representation of GF over E, and κ :F ,→ E, then we will write HTκ (W ) for the multiset of Hodge–Tate weights of W
with respect to κ. By definition, the multiset HTκ (W ) contains i with multiplicityb (i))GF . Thus for example HT (ε) = {−1}.
dim (W ⊗
F
E

κ,F

κ

We say that W has regular Hodge–Tate weights if for each κ, the elements ofHTκ (W ) are pairwise distinct. Let Zn+ denote the set of tuples (ξ1 , . . . , ξn ) ofintegers with ξ1 ≥ ξ2 ≥ · · · ≥ ξn . Then if W has regular Hodge–Tate weights, there
is a ξ = (ξκ,i ) ∈ (Zn+ )HomQp (F,E) such that for each κ : F ,→ E,
HTκ (W ) = {ξκ,1 + n − 1, ξκ,2 + n − 2, . . . , ξκ,n },and we say that W is regular of weight ξ. For any ξ ∈ Zn+ , view ξ as a dominant
weight (with respect to the upper triangular Borel subgroup) of the algebraic group
GLn in the usual way, and let Mξ0 be the algebraic OF -representation of GLn givenbynMξ0 := IndGL
Bn (w0 ξ)/OF
where Bn is the Borel subgroup of upper-triangular matrices of GLn , and w0 is thelongest element of the Weyl group (see [Jan03] for more details of these notions,6

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

and note that Mξ0 has highest weight ξ). Write Mξ for the OF -representation ofGLn (OF ) obtained by evaluating Mξ0 on OF . For any ξ ∈ (Zn+ )HomQp (F,E) we writeLξ for the O-representation of GLn (OF ) defined byLξ := ⊗κ:F ,→E Mξκ ⊗OF ,κ O.
If F is a p-adic field, then an inertial type is a representation τ : IF → GLn (Qp )with open kernel which extends to the Weil group WF .
Then we say that a de Rham representation ρ : GF → GLn (E) has inertial type
τ if the restriction to IF of the Weil–Deligne representation WD(ρ) associated toρ is equivalent to τ . Given an inertial type τ , there is a finite-dimensional smooth
irreducible Qp -representation σ(τ ) of GLn (OF ) associated to τ by the “inertial local
Langlands correspondence”; see Theorem 3.7 below. (Note that by the results ofSection 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 Qp , and let rec denote the local Langlands correspondence from isomorphism classes of irreducible smooth representations ofGLn (F ) over C to isomorphism classes of n-dimensional Frobenius semisimple Weil–
Deligne representations of WF as in the introduction to [HT01]. Fix once and for∼
all an isomorphism ı : Qp −→ C. We define the local Langlands correspondence√
recp over Qp by ı ◦ recp = rec ◦ ı. This depends only on ı−1 ( p), and if we definerp (π) := recp (π ⊗ | det |(1−n)/2 ), then rp is independent of the choice of ı. Furthermore, if V is a Frobenius semisimple Weil–Deligne representation of WF overE, then rp−1 (V ) is also defined over E by [Clo90, Prop 3.2] and the fact that rp
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∨
Homcont
O (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 7→ Ad := Homcont
O (A, O)
induces an anti-equivalence of categories between the category of compact, Otorsion-free linear-topological O-modules A and the category of $-adically complete and separated O-torsion-free O-modules. A quasi-inverse is given by B 7→
B d := HomO (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 Ad is the unit ball in the E-Banach
space HomO (A, E).
If r : GF → GLn (E) is de Rham of regular weight a, then we write πalg (r) :=d
La ⊗O E, and πsm (r) := rp−1 (WD(r)F −ss ), both of which are E-representationsof GLn (F ). (The GLn (OF )-action on Lda extends linearly to a GLn (F )-action on
πalg (r).) As the names suggest, πalg (r) is an algebraic representation, and πsm (r)is a smooth representation. Note that πalg (r) = Lξ ⊗O E for ξκ,i := −aκ,n+1−i .
∼We let ArtF : F × −→ WFab 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 Gndenote the group scheme over Z defined to be the semidirect product of GLn × GL1
by the group {1, j}, which acts on GLn × GL1 byj(g, µ)j −1 = (µ · t g −1 , µ).

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

7

We have a homomorphism ν : Gn → GL1 , sending (g, µ) to µ and j to −1.
Further notation is introduced in the course of our arguments; we mention justsome of it here, for the reader’s convenience.
From Subsection 2.6 on, we will have fixed a particular finite extension F of Qp ,with ring of integers OF and uniformiser $F . To ease notation we will typicallywrite G := GLn (F ), K := GLn (OF ), and Z := Z(G). For each m ≥ 0, we
write Γm = GLn (OF /$Fm ) and Km := ker GLn (OF ) → GLn (OF /$Fm ) , so that∼
K/Km −→ Γm .
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 ourmajor object of study in the paper, is a patched G-representation that we denoteby M∞ . Beginning in Section 4, we will also writed◦ d
M∞ (σ ◦ ) := Homcont
(M
,
(σ
)
)
,
∞
O[[K]]
when σ ◦ is a K-invariant OE -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 iG
P to denote normalised parabolic induction.
If σ is a representation of K, then we will write H(σ) := End(c-IndG
K σ) todenote the Hecke algebra of G with respect to σ. Sometimes, when it is helpful toemphasise 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 unitarygroups. 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 GLn (F )-representation overthat ring; we refer to the discussion at the beginning of Subsection 2.6 below for amore 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 thatit actually produces a GLn (F )-representation is verified in Proposition 2.8. In
Subsection 2.10 we explain how our patched GLn (F )-representation gives rise toadmissible unitary Banach representations attached to local Galois representations.
2.1. Globalisation. Let F/Qp be a finite extension, and fix a continuous representation r̄ : GF → GLn (F). Our goal in this subsection is to give a criterion for r̄ tobe 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8

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

• r̄ admits a potentially crystalline lift of regular weight, which is potentiallydiagonalizable in the sense of [BLGGT13].
Conjecturally, the second hypothesis is always satisfied; this is Conjecture A.3of [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 seethat by lifting these characters to crystalline characters, we can find a potentiallycrystalline lift which has regular weight, and is a sum of inductions of characters.
Such a lift is obviously potentially diagonalizable (indeed, after restriction to somefinite extension, it is a sum of crystalline characters).

Having assumed these hypotheses, Corollary A.7 of [EG13] (with K our F , andF our Fe) provides us with an imaginary CM field Fe with maximal totally realsubfield Fe+ , and a continuous irreducible representation ρ : GFe+ → Gn (F) suchthat ρ is a suitable globalisation of r̄ in the sense of Section 5.1 of [EG13]. In
particular, ρ is automorphic, it is unramified outside of p, and it globalises r̄ in thesense that:• each place v | p of Fe+ splits in Fe, and has Fev+ ∼= F ; we fix a choice of suchisomorphisms.
• For each place v | p of Fe+ , there is a place ve of Fe lying over v with ρ|GFeve
isomorphic to r̄.
In fact, in order to arrange that our patched modules have a certain multiplicity
one property, we will also demand that:• ρ(GFe ) = GLn (F0 ) for some subfield F0 ⊆ F with #F0 > 3n, andker ad ρ|G e
F

does not contain Fe(ζp ).
• Fe
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 theproof of Proposition A.2 of [EG13] allows us to arrange that ρ(GFe ) = GLn (Fpm )for any sufficiently large m.
2.3. Unitary groups. We now use the globalisation ρ of our local Galois representation r̄ to carry out the Taylor–Wiles–Kisin patching argument as in Section 5of [EG13]. The definitions of Hecke algebras, the choices of auxiliary primes andso 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 thesedefinitions, indicating only the differences in our construction.
e Fe+
As in Sections 5.2 and 5.3 of [EG13], we fix a certain definite unitary group G/ee
together with a model (which we will also denote by G) over OFe+ . (The group G
is denoted G in [EG13], but we will later use G to denote GLn (F ).) This modelhas the property that for each place v of Fe+ which splits as wwc in Fe, there is∼e e+ ) −an isomorphism ιw : G(O→ GLn (OFew ); we fix a choice of such isomorphisms.
Fv
We also choose a finite place v1 of Fe+ which is prime to p, with the properties that• v1 splits in Fe, say as v1 = ve1 ve1c ,• v1 does not split completely in Fe(ζp ), andPATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

9• ρ(FrobFeve ) has distinct F-rational eigenvalues, no two of which have ratio
1(Nv1 )±1 .

(It is possible to find such a place v1 by our assumptions that ρ(GFe ) = GLn (F0 )ker ad ρ|GeFwith #F0 > 3n, and that Fe
does not contain Fe(ζp ). Note that this differs
slightly from the choice of place v1 in the first paragraph of Section 5.3 of [EG13],
but that it is still the case that any deformation of ρ|GFe is unramified. We havev
e1

made this choice in order to be able to arrange that our patched modules satisfy
multiplicity one.)
Let Sp denote the set of primes of Fe+ dividing p. We now fix a place p | Q
p of Fe+ ,and for each integer m ≥ 0 we consider the compact open subgroup Um = v Um,ve ∞ ), whereof G(Ae+F
e e+ ) for all v which split in Fe other than v1 and p;• Um,v = G(OFv• Um,v1 is the preimage of the upper triangular matrices under∼e e+ ) → G(ke v )−G(O→ GLn (kv1 )
1
Fv
1

ιve1


e O e+ ) → G(Oe e+ /$m+ ;• Um,p is the kernel of the map G
e
Fp
Fp
F
p

e Fev+ ) if v is inert• Um,v is a hyperspecial maximal compact subgroup of G(
in Fe.
By the choice of v1 and Um,v1 we see that Um is sufficiently small. Write U := U0 .
In order to make the patching argument, we will need to consider certain compact
open subgroups of the Um corresponding to choices of sets of auxiliary primes
QN that will be introduced in Section 2.5. Specifically, for each integer N ≥ 1,
we will have a finite set of primes QN of Fe+ disjoint from Sp ∪ {v1 } as well as
e e+ ) for each v ∈ QN and i = 0, 1. We
open compact subgroups Ui (QN )v of G(OFvQthen define subgroups Ui (QN )m = v Ui (QN )m,v ⊂ Um , for i = 0, 1 by setting
U0 (QN )m,v = U1 (QN )m,v = Um,v for v 6∈ QN , and Ui (QN )m,v = Ui (QN )v forv ∈ QN .
By assumption, r̄ has a potentially diagonalizable lift of regular weight, sayrpot.diag : GF → GLn (O). Suppose that rpot.diag has weight ξ and inertial typeτ (in the sense of Section 1.5). Extending E if necessary, we may assume that
the GLn (OF )-representation σ(τ ) is defined over E. Then we have two representations Lξ and Lτ of GLn (OF ) on finite free O-modules in the following way: therepresentation Lξ is the one defined in the notation section, and Lτ is a choice ofGLn (OF )-stable lattice in σ(τ ). Set Lξ,τ := Lτ ⊗O Lξ , a finite free O-module withan action of GLn (OF ).
ReturningQ
to our global situation, let Wξ,τ denote the finite free O-module withan action of v∈Sp \{p} Um,v given by Wξ,τ = ⊗v∈Sp \{p},O Lξ,τ where Um,v acts on∼∼
e e+ ) −the factor corresponding to v via Um,v = G(O→ GLn (O e ) −→ GLn (OF ). In
Fv

ιve

Fve

order to avoid duplication of definition, we allow QN = ∅ in the definitions we now
make. For any finite O-module V with a continuous action of Um,p , we have spacesof algebraic modular forms Sξ,τ (Ui (QN )m , V ); these are just the functions
e Fe+ )\G(Ae ∞+ ) → Wξ,τ ⊗O V
f : G(eF

10 A. CARAIANI, M. EMERTON, T. GEE, D. GERAGHTY, V. PAŠKŪNAS, AND S. W. SHINe ∞ ) and u ∈ Ui (QN )m then f (gu) = u−1 f (g),with the property that if g ∈ G(Ae+F
Q
where Ui (QN )m acts on Wξ,τ ⊗O V via projection to v∈Sp Um,v . (For example:
when V = O is the trivial representation, then after extending scalars from O to C∼
via O ⊂ Qp −→ C, this space corresponds to classical automorphic forms of fixedıtype σ(τ ) at the places in Sp \ {p}, full level pm at p, and whose weight (via our
fixed isomorphism ı : Qp → C) is 0 at places above p, and given by ξ at each of theplaces in Sp \ {p}.)We let TSp ∪QN ,univ be the commutative O-polynomial algebra generated by for(j)mal variables Tw for all 1 ≤ j ≤ n, w a place of Fe lying over a place v of Fe+ which
splits in Fe and is not contained in Sp ∪ QN ∪ {v1 }, together with formal variables(j)
Tve1 for 1 ≤ j ≤ n. The algebra TSp ∪QN ,univ acts on Sξ,τ (Ui (QN )m , V ) via the
Hecke operators




0
(j)
−1 $w 1j
Tw := Um,w ιw
Um,w
0
1n−j
where $w is a fixed uniformiser in OFew .
Choose an ordering δ1 , . . . , δn of the (distinct) eigenvalues of ρ(Frobve1 ). Since ρ isa suitable globalisation of r̄, it is in particular automorphic in the sense of Definition
5.3.1 of [EG13], and we let mQN be the maximal ideal of TSp ∪QN ,univ corresponding(j)to ρ, and containing Tve1 − (Nv1 )j(1−j)/2 (δ1 · · · δj ) for each 1 ≤ j ≤ n.
We will write m for m∅ .
2.4. Galois deformations. Let S be a set of places of Fe+ which split in Fe, withSp ⊆ S. As in [CHT08], we will write Fe(S) for the maximal extension of Fe
unramified outside S, and from now on we will write GFe+ ,S for Gal(Fe(S)/Fe+ ).
We will freely make use of the terminology (of liftings, framed liftings etc.) ofSection 2 of [CHT08].
Let T = Sp ∪{v1 }. For each v ∈ Sp , we let ve be a choice of a place of Fe lying overv, with the property that ρ|GFe ∼= r̄. (Such a choice is possible by our assumptionvethat ρ is a suitable globalisation of r̄.) We let Te denote the set of places ve, v ∈ T .
For each v ∈ T , we let Rve denote the maximal reduced and p-torsion free quotientof the universal O-lifting ring of ρ|GFe . For each v ∈ Sp \{p}, we write Rve,ξ,τ for thevereduced and p-torsion free quotient of Rve corresponding to potentially crystalline
lifts of weight ξ and inertial type τ .
Consider (in the terminology of [CHT08]) the deformation problem
S := Fe/Fe+ , T, Te, O, ρ, ε1−n δFne/Fe+ , {Rve1 } ∪ {Rep } ∪ {Rve,ξ,τ }v∈Sp \{p} .

There is a corresponding universal deformation ρuniv: GFe+ ,T → Gn (RSuniv ) of ρ. In
S
addition, there is a universal T -framed deformation ring RST in the sense of Definition 1.2.1 of [CHT08], which parameterises deformations of ρ of type S together
with particular local liftings for each ve ∈ Te.
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 ≥ [Fe+ :e N with the following propQ]n(n − 1)/2 and for each N ≥ 1 sets of primes QN , Q
erties:PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

11• QN is a finite set of finite places of Fe+ of cardinality q which is disjoint
from T and consists of places which split in Fe;
e N consists of a single place ve of Fe above each place v of QN ;• Q
• Nv ≡ 1 mod pN for v ∈ QN ;• for each v ∈ QN , ρ|GFe ∼= s̄ve ⊕ ψ ve where ψ ve is an eigenspace of Frobeniusveon which Frobenius acts semisimply.
e e+ ) denote the parahoric
For each N , v ∈ QN and i = 0, 1, we let Ui (QN )v ⊂ G(OFv

open compact subgroups defined in Section 5.5 of [EG13], following [Tho12].
ψFor each v ∈ QN , a quotient Rve ve of Rve is defined in section 5.5 of [EG13](following [Tho12]). We let SQN denote the deformation problem
e N , O, ρ, ε1−n δ n + ,SQN := Fe/Fe+ , T ∪ QN , Te ∪ Q
e /F
e
F

ψ
{Rve1 } ∪ {Rep } ∪ {Rve,ξ,τ }v∈Sp \{p} ∪ {Rve ve }v∈QN .
denote the corresponding universal deformation ring, and we let RSQT
We let RSuniv
QN
N
denote the corresponding universal T -framed deformation ring. We define
b ⊗b v∈Sp \{p} Rve,ξ,τ ⊗R
b ve
Rloc := Rep ⊗1

where all completed tensor products are taken over O. By the choice of the sets ofprimes QN , we also know that• the ring RSQT can be topologically generated over Rloc byN
q − [Fe+ : Q]n(n − 1)/2 elements.
For each v ∈ QN we choose a uniformiser $ve ∈ OFeve , so that we have the projection operator pr$ve ∈ EndO (Sξ,τ (Ui (QN )m , O/$r )mQN ) defined as in Proposition
5.9 of [Tho12]. We define pr to be the composite of the projections pr$ve . (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 theunderlying set QN is.) Then, as in Section 5.5 of [EG13]:(1) The map
pr : Sξ,τ (Um , O/$r )m → pr Sξ,τ (U0 (QN )m , O/$r )mQN
is an isomorphism. Moreover, since pr is defined using Hecke operators
e Fep+ ) on the spaces
at places in QN , it commutes with the action of G(
e Fep+ ) satisfies
of algebraic automorphic forms. More precisely, if g ∈ G(
−10
g Um0 ,p g ⊆ Um,p for some positive integers m ≤ m , then we have acommutative diagram
prSξ,τ (Um , O/$r )m −−−−→ pr Sξ,τ (U0 (QN )m , O/$r )mQN


g
g
y
y


pr
Sξ,τ (Um0 , O/$r )m −−−−→ pr Sξ,τ (U0 (QN )m0 , O/$r )mQN
e Fep+ ) (namely:
where both vertical arrows are induced by the action of g ∈ G(
00
(gf )(g ) = f (g g)).

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

Q
(2) Let Γm = GLn (OF /$Fm ) ∼= U/Um and ∆QN = v∈QN U0 (QN )v /U1 (QN )v .
Then U0 (QN )0 acts on Sξ,τ (U1 (QN )m , O/$r ) via (gf )(g 0 ) = gp f (g 0 g),
and this action factors through ∆QN × Γm . With respect to this action,

pr Sξ,τ (U1 (QN )m , O/$r )mQN is a projective (O/$r )[∆QN ][Γm ]-module,
and there is a natural Γm -equivariant isomorphism∆QN∼
pr Sξ,τ (U1 (QN )m , O/$r )mQN−
→ Sξ,τ (Um , O/$r )m .

(The projectivity follows from the proof of Lemma 3.3.1 of [CHT08]. The
fact that there is a Γm -equivariant isomorphism follows immediately frompoint (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.)Sp ∪QN(3) Let Tξ,τ(Ui (QN )m , O/$r ) be the image of TSp ∪QN ,univ in the ring
EndO (pr (Sξ,τ (Ui (QN )m , O/$r ))). Then there exists a deformationSp ∪QN(U1 (QN )m , O/$r )mQN
GFe+ ,T ∪QN → Gn Tξ,τof ρ which is of type SQN . In particular, pr Sξ,τ (Ui (QN )m , O/$r )mQN
-module.
is a finite RSunivQ
N)× obAs in Section 5.5 of [EG13], there is a homomorphism ∆QN → (RSuniv
QN
tained by identifying ∆QN with the productQ of the inertia subgroups in the maximal abelian p-power order quotient of v∈QN GFeve , and thus a homomorphism


r
univ
-modulestructure
on
pr
S
(U
(Q
)
,
O/$
)
.
The
R
O[∆QN ] → RSuniv
ξ,τ
1
N
m
m
Q
S
QN
QN
N


thus induces an action of O[∆QN ] on pr Sξ,τ (U1 (QN )m , O/$r )mQN , 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 GLn (OF )-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 GLn (OF )-representation over the local deformation ring (again, perhaps with patching variables added). More precisely, interms of such a GLn (OF )-representation M∞ , the patching functor can be defined∨ ∨as Homcont
O[[GLn (OF )]] (M∞ , V ) , if V is a continuous representation of GLn (OF ) on a
finitely generated O-module. The exactness of the patching functor can be encodedin the requirement that M∞ be a projective O[[GLn (OF )]]-module.
In this paper our approach is to construct the representing object M∞ directly,and (most importantly) to promote it from being merely a GLn (OF )-representation
to being a representation of the full p-adic group GLn (F ). (In terms of patching functors, one can somewhat loosely think of this as extending the patching functor from the category of GLn (OF )-representations to a category that we

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

13

might call the Hecke category, whose objects are the same, but in which the morphisms between any two GLn (OF )-representations U and V are defined to beGL (F )GL (F )
HomGLn (F ) c-IndGLnn (OF ) U, c-IndGLnn (OF ) V .)From now on, to ease notation we write K = GLn (OFee ) = GLn (OF ), G =p
GLn (Feep ) = GLn (F ), and Z = Z(G). For each integer N ≥ 0, we set KN :=∼
ker GLn (OF ) → GLn (OF /$FN ) , so that K/KN −→ ΓN . We have the Cartan
decomposition G = KAK, where A is the set of diagonal matrices whose diagonal
entries are powers of the uniformiser $F , and we let AN be the subset of A consisting of matrices with the property that the ratio of any two diagonal entries is of
the form $Fr with |r| ≤ N , and set GN = KAN K. Note that GN is not a subgroupof G unless N = 0, but that each K\GN /KZ is finite, and G = ∪N ≥0 GN .
N
If (σ, W ) is a representation of KZ, then we write IndG
KZ σ for the space offunctions f : GN → W with f (kg) = σ(k)f (g) for all g ∈ GN , k ∈ KZ; this isnaturally a KZ-representation via (kf )(g) := f (gk). We define IndG
KZ σ in the00same way; then IndGσisarepresentation
of
G
via
(gf
)(g
)
:=
f
(g
g).
KZ
For each N , we set
∨
Mi,QN := pr Sξ,τ (Ui (QN )2N , O/$N )mQN .
Note that Mi,QN depends on the integer N as well as on the set of primes QN(it could happen that QM = QN for M 6= N ), but we will only include QN in
the notation for the sake of simplicity. Note also that we could have equivalently
defined
Mi,QN := pr∨ Sξ,τ (Ui (QN )2N , O/$N )∨,mQ
Nsince pr is an endomorphism of Sξ,τ (Ui (QN )2N , O/$N )mQN and Pontrjagin duality
is an exact contravariant functor.
Let ∆QN be as above; it is of p-power order by the definitions of the Ui (QN )2N .
It follows from point (2) in the previous section that M1,QN is a finite projective(O/$N )[∆QN ][Γ2N ]-module. Since Z centralises U1 (QN )2N , there is also a naturalaction of Z on M1,QN .
2.7. Remark. The reason for including a Pontrjagin dual in the definition of Mi,QN
is that Sξ,τ (Ui (QN )2N , O/$N ) is a space of automorphic forms, and so is mostnaturally 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 Sξ,τ (Ui (QN )2N , O/$N ) is a space of automorphic forms on thee it is a space of functions on a finite set, and so has a natdefinite unitary group G,
ural self-duality. Thus, by exploiting this self-duality to convert its contravariant
functoriality into a covariant functoriality, we could omit the Pontrjagin dual in
the preceding definition, and indeed it is traditionally omitted (see e.g. [CHT08],[BLGG11], [Tho12], and [EG13]). However, we have found it conceptually clearer
to include this duality in our definitions and constructions.
We now define a KZ-equivariant map
N
αN : M1,QN → IndG
KZ ((M1,QN )KN )(where (M1,QN )KN denotes the KN -coinvariants in M1,QN ) in the following way.
Note firstly that there is a natural identification(M1,QN )KN = pr∨ Sξ,τ (U1 (QN )N , O/$N )∨,mQN

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

so it suffices to define a KZ-equivariant map
N ∨N
αN : Sξ,τ (U1 (QN )2N , O/$N )∨ → IndG
KZ Sξ,τ (U1 (QN )N , O/$ ) .

Now, given g ∈ GN , we have g −1 K2N g ⊆ KN , so that there is a natural map
g ∗ : Sξ,τ (U1 (QN )N , O/$N ) → Sξ,τ (U1 (QN )2N , O/$N )given by (g ∗ · f )(x) := f (xg), and a map
∨g∗ := (g −1 )∗ : Sξ,τ (U1 (QN )2N , O/$N )∨ → Sξ,τ (U1 (QN )N , O/$N )∨ .
(The latter is well defined since GN is stable under taking inverses. We note that g ∗(resp. g∗ ) may be interpreted as the natural pullback (resp. pushforward) map one ∞)cohomology (resp. homology) under the natural right (resp. left) action of G(Ae We have (gh)∗ = g ∗ ◦ h∗ and henceon the tower of arithmetic quotients of G.)(gh)∗ = g∗ ◦ h∗ , whenever all are defined. Then we define αN by
αN (x) (g) := g∗ (x).
In order to check that this is KZ-equivariant, we must check that for all k ∈ KZ
we have αN (kx) (g) = (kαN (x))(g); this is equivalent to checking that g∗ (kx) =(gk)∗ (x), which is immediate from the definition.

Now set M1,Q:= M1,QN ⊗RSuniv RSQT . We have an induced KZ-equivariant
N
QN

N



N
map αN : M1,Q
→ IndGKZ (M1,QN )KN . DefineN

R∞ := Rloc [[x1 , . . . , xq−[F̃ + :Q]n(n−1)/2 ]],S∞ := O[[z1 , . . . , zn2 #T , y1 , . . . , yq ]],for formal variables x1 , . . . , xq−[F̃ + :Q]n(n−1)/2 , y1 , . . . , yq and z1 , . . . , zn2 #T . For
each N , we fix a surjection R∞  RSQT of Rloc -algebras (which, as recalled inN
Section 2.5, is possible by the choice of the sets QN ). These choices allow us toas an R∞ -module. Also, we fix choices of lifts representing
regard each M1,Q
N
such that our chosen lift
the universal deformations over RSuniv and each RSuniv
QN
univ
univ
over each RSQ reduces to our chosen lift over RS . These choices give rise toN∼
b O O[[z1 , . . . , zn2 #T ]] compatible with a fixed isoisomorphisms RST −→ RSuniv ⊗
QN

QN

∼
b O O[[z1 , . . . , zn2 #T ]]; they also allow us to regard eachmorphism RST −→ RSuniv ⊗
M1,QN as an O[[z1 , . . . , zn2 #T ]]-module. Finally, for each N , choose a surjectionO[[y1 , . . . , yq ]]  O[∆QN ]. This gives each M1,Q
the structure of an S∞ -moduleNand hence the structure of an S∞ [[K]]-module (where the action of K factors
through Γ2N ).
We now apply the Taylor–Wiles method in the usual way to pass to a subse

quence, and patch the modules M1,Q
together with the maps αN : M1,Q
→
N
N
GN

IndKZ (M1,QN )KN . More precisely, for each N ≥ 1, let bN denote the ideal of S∞
N

generated by $N , ziN and (1 + yi )p − 1. Let a denote the ideal of S∞ generated
by the zi and the yi . Fix a sequence (dN )N ≥1 of open ideals of RSuniv such that
• $N RSuniv ⊂ dN ⊂ Ann RSuniv (Sξ,τ (U2N , O/$N )∨m );• dN ⊃ dN +1 for all N ;• ∩N ≥1 dN = (0).

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

15N ∨)m ).)(For example, we may take dN = mNuniv ∩ Ann Runiv (Sξ,τ (U2N , O/$RSSAt level N , we consider tuples consisting of• a surjective homomorphism of Rloc -algebras φ : R∞  RSuniv /dN ;• a finite projective (S∞ /bN )[Γ2N ]-module M  which carries a commuting
action of R∞ ⊗O O[Z] such that the action of S∞ can be factored throughthat of R∞ ;∼→ Sξ,τ (U2N , O/$N )∨• a Γ2N and Z-equivariant isomorphism ψ : (M  /aM  ) −mcompatible with φ;N• a KZ-equivariant and R∞ ⊗O S∞ -linear map α : M  → IndG
KZ [(M )KN ],whose reduction modulo a is compatible with the map
GNN ∨Sξ,τ (U2N , O/$N )∨m → IndKZ [Sξ,τ (UN , O/$ )m ]induced by the action of GN on the spaces of algebraic modular forms.
0

We consider two such tuples (φ, M  , ψ, α) and (φ0 , M  , ψ 0 , α0 ) to be equivalent if
φ = φ0 and if there is an isomorphism of (S∞ /bN )[Γ2N ] ⊗O R∞ ⊗O O[Z]-modules
0∼M −→ M  which identifies ψ with ψ 0 and α with α0 . Note that there are at
most finitely many equivalence classes of such tuples. (For a given M  there areN
only finitely many KZ-equivariant homomorphisms α : M  → IndG
KZ [(M )KN ],
because M and K\GN /KZ are finite.) Note also that a tuple (φ, M , ψ, α) of
0level N gives rise to a tuple (φ0 , M  , ψ 0 , α0 ) of level (N − 1) by setting φ0 := φ0
mod dN −1 , M  := (M  /bN −1 )K2(N −1) and ψ 0 := ψ mod $N −1 . The map α0 isdefined by the formula α0 (m)(g) = α(m)(g). Here m denotes the image of m ∈ M 
00in M  and α(m)(g) denotes the image of α(m)(g) in (M  )KN −1 . Note that forany m ∈ M  , γ ∈ K2(N −1) and g ∈ GN −1 , we haveα (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 ∈ KN −1 it is straightforward
to see that α0 is well-defined.
For each pair of integers N 0 ≥ N ≥ 1, we define a tuple (φ, M  , ψ, α) of level
N as follows: we set φ equal to R∞  RSQT 0  RSuniv /dN and we set M  =
N


(M1,Q
⊗S∞ S∞ /bN )K2N . The map ψ comes from points (1) and (2) in the previous
N0
section. The map α comes from αN 0 defined above (in the same way that α0 isdefined 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 0 is allowed to be arbitrarily large, we can apply a diagonal argument to finda subsequence of pairs (N 0 (N ), N )N ≥1 indexed by N such that for each N ≥ 2, thetuple 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∞ := lim(M1,Q⊗S∞ S∞ /bN )K2N ,N 0 (N )←−N

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

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 thediscrete topology). Moreover, the action of S∞ on M∞ can be factored through amap 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∞ /bN )K2N −→ (M1,Q/bN )K2N .
N 0 (N )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∞ → IndG
KZ M∞
by taking the projective limit of the maps


GN


(M1,Q
⊗
S
/b
)
→
Ind
(M
⊗
S
/b
)
.
S
∞
N
K
S
∞
N
K
1,Q
∞
∞
2N
N
0
0
KZ
N (N )N (N )

GN 0 (N )induced by αN 0 (N ) : M1,QN 0 (N ) → IndKZ(M1,QN 0 (N ) )KN 0 (N ) . We denote thisinduced map by αN 0 (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 M1,QN is a projective(O/$N )[∆QN ][Γ2N ]-module, we see that
(M∞ /bN )K2N ∼/bN )K2N= (M1,QN 0 (N )is a projective S∞ /bN [Γ2N ]-module, so we have a cofinal system of ideals (namely,
the ideals generated by bN + ker(O[[K]] → O[Γ2N ])) defining the topology of
S∞ [[K]] modulo which the surjection splits. The sets of possible splittings at thesefinite levels then give us a projective system of non-empty finite sets, and an elementof the projective limit of this projective system gives the required splitting.
We now check that α∞ is injective. Note that by definition, for each αN :N
M1,QN → IndG
(M
)
we
have
α
(m)
(1) = 1∗ (m) = m, where m de1,QN KN
N
KZ
notes the image of m in (M1,QN )KN . From this (with N replaced by N 0 (N )) we/bN )K2N
deduce that αN 0 (N ) )(m) (1) = m where m is the image of m ∈ (M1,Q
N 0 (N )in (M1,QN 0 (N ) /bN )KN . This then implies that α∞ (m) (1) = m for each m ∈ M∞ ,and α∞ is certainly injective.

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

17

In order to show that the image of α∞ is G-stable, we will show that for allg ∈ 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 thatg, h, gh ∈ GN . This certainly means that g, h, gh ∈ G2N as well. Since N can bearbitrarily large, it is enough to show that both sides of the equation above become
equal in (M∞ /bN )KN and we do this by explicit computation.
We let πN : M∞ → (M∞ /bN )K2N and σN : M∞ → (M∞ /bN )KN denote theprojection maps. Then by definition, we have
σN α∞ (m)(hg) = αN 0 (N ) πN (m) (hg)and

σN α∞ (α∞ (m)(g))(h) = αN 0 (N ) πN (α∞ (m)(g)) (h)= αN 0 (N ) σ2N (α∞ (m)(g)) mod bN (h)= αN 0 (N ) αN 0 (2N ) (π2N (m))(g) mod bN (h).
e , N 00 ≥ 1 with Ne ≤ N 0 (N ), we let U1 (QN 0 (N ) , Ne )N 00 be theNow, for integers N, N
open compact subgroup intermediate to U1 (QN 0 (N ) )N 00 and U0 (QN 0 (N ) )N 00 suchee )N 00 ∼that U0 (QN 0 (N ) )N 00 /U1 (QN 0 (N ) , N= (Z/pN Z)q . Then we have a commutative
diagram∨g∗−−−−→ Sξ,τ U1 (QN 0 (2N ) , 2N )2N , O/$2N


ynat
∨

∨
Sξ,τ U1 (QN 0 (2N ) , N )2N , O/$N
Sξ,τ U1 (QN 0 (2N ) , 2N )4N , O/$2N



h
ynat
y ∗

∨
(hg)∗
∨
−−−−→ Sξ,τ U1 (QN 0 (2N ) , N )N , O/$N .
Sξ,τ U1 (QN 0 (2N ) , N )2N , O/$N
Sξ,τ U1 (QN 0 (2N ) , 2N )4N , O/$2N

∨

Localising at mQN 0 (2N ) , applying the projector pr∨ , tensoring over RSuniv
Q 0

with

N (N )RSQT 0and reducing modulo b2N or bN as appropriate, we obtain a commutative

N (N )

diagram:g∗(M∞ /b2N )K4N −−−−→ (M∞ /b2N )K2N

ynat
(M∞ /b2N )K4N


ynat

(M∞ /bN )K2N

h
y ∗
(hg)∗

(M∞ /bN )K2N −−−−→ (M∞ /bN )KN .
Now, on the one hand, the element π2N (m) lies in the space in the upper left
corner of this diagram. By definition, we have
αN 0 (N ) αN 0 (2N ) (π2N (m))(g) mod bN (h) = h∗ ◦ nat ◦ g∗ π2N (m)18 A. CARAIANI, M. EMERTON, T. GEE, D. GERAGHTY, V. PAŠKŪNAS, AND S. W. SHIN

On the other hand, πN (m) lies in the space in the lower left corner of the diagram,
and we haveαN 0 (N ) πN (m) (hg) = (hg)∗ πN (m) = (hg)∗ ◦ nat π2N (m) .
The desired equality now follows from the commutativity of the above diagrams. 
Note that for each positive integer m, the compact open subgroups Um have the
e ∞,p ) denote that common level. Define thesame level away from p. Let U p ⊂ G(Ae+F$-adically completed cohomology spaceS̃ξ,τ (U p , O)m := lim lim Sξ,τ (Um , O/$s )m .
←− −→
s

m

The space S̃ξ,τ (U p , O)m is equipped with a natural G-action, induced from theaction of G on algebraic automorphic forms. Moreover, the space S̃ξ,τ (U p , O)m is
equipped with a natural action of the Hecke algebra
S

S

p
p
Tξ,τ
(U p , O)m := lim Tξ,τ(Um , O)m .
←
−
m

S

p
By taking the inverse limit of the Tξ,τ(Um , O)m -valued deformations of ρ̄ of type SSpwe obtain a map RSuniv → Tξ,τ(U p , O)m . Therefore, S̃ξ,τ (U p , O)m is also equippedwith an action of the local deformation ring Rep via the composition Rep → RSuniv →

S

p
Tξ,τ
(U p , O)m .

2.9. Corollary. There is a G-equivariant isomorphism∼(M∞ /aM∞ ) −→ S̃ξ,τ (U p , O)dm ,which is compatible with the Rep -action on both sides.
∼

Proof. We have a KZ-equivariant isomorphism ψ∞ : (M∞ /aM∞ ) −→ S̃ξ,τ (U p , O)dm
obtained by taking the projective limit of the Γ2N - and Z-equivariant isomorphisms
ψ at level N .
We first show that ψ∞ is compatible with the Rep -action on both sides. The
Rep -action on the left hand side is via the map Rep → R∞ and each ψ at level N
is compatible with φ : R∞  RSuniv /dN . On the other hand, the action of Rep on
the right hand side is via the map Rep → RSuniv and ∩N ≥1 dN = (0). The desired
compatibility follows.
Moreover, ψ∞ fits into a commutative diagram
M∞ /aM∞

g
y

ψ∞



IndG
KZ M∞ /aM∞

−−−−→



ψ∞

S̃ξ,τ (U p , O)dm

g
y

p
d
−−−−→ IndG
KZ S̃ξ,τ (U , O)m

where both vertical maps are induced by the action of g ∈ G. The G-equivarianceof ψ∞ now follows from Proposition 2.8.


PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

19

2.10. Admissible unitary Banach representations. Fix a lifting r : GF →GLn (O) of r̄. We now explain how M∞ allows us to associate an admissible unitary
Banach representation V (r) of GLn (F ) to r.
As above, we identify F with Feep . By definition, r comes from a homomorphismof O-algebras x : Rep → O. We extend this to a homomorphism of O-algebrasb ⊗b v∈Sp \{p} Rve,ξ,τ → O
x0 : Rep ⊗by using the homomorphisms Rve,ξ,τ → O corresponding to our given potentiallydiagonalizable representation rpot.diag , and then extend x0 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 GLn (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) 6= 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 GLn (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,
KN
is finite-dimensional. Writing y for
the F-vector space (M∞ ⊗R∞ ,y O)d ⊗ F
the composite y : R∞ → O → F, we must check that (M∞ ⊗R∞ ,y F)KN is finitedimensional. Since M∞ is a finite S∞ [[K]]-module, and ker y induces an open idealof 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 notneeded for our main results, which concern representations r : GF → GLn (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 r0 which factors throughGLn (O) and whose reduction r̄0 is semisimple, so that r̄0 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 andthe 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 orderto 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 generalcontext of representations with arbitrary monodromy, so we will make it clear when
we impose this restriction. We begin by collecting and explaining various resultsfrom the literature that we will need.

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

Let F/Qp be a finite extension. Recall that we write K = GLn (OF ), G =GLn (F ), and that WF is the Weil group of F . Although several of our references
work over C, we work consistently over Qp (except in Subsection 3.12, where we
fix a single Bernstein component, and work over a finite extension of Qp ). The
various results over C are transferred to our context over Qp via the fixed choice∼of ı : Qp → C everywhere so this is harmless. (The transfer depends on ı since anautomorphism of C does not fix isomorphism classes of representations in general.)
As recalled in Section 1.5, the local Langlands correspondence recp gives a bijection between the isomorphism classes of irreducible smooth representations of G
over Qp , and the n-dimensional Frobenius semisimple Weil–Deligne representationsof WF .
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 = n1 + · · · + nr , let P = M N 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 ∼=GLn1 (F ) × · · · × GLnr (F ). For any smooth representations πi of GLni (F ), thetensor product π1 ⊗ · · · ⊗ πr is naturally a representation of M and thus of P , andwe denote by π1 × · · · × πr the normalised induction iG
P π1 ⊗ · · · ⊗ πr . If π1 , . . . , πr
are irreducible supercuspidal representations then π1 × · · · × πr has finite length
by [Rod82, Proposition 4].
Any smooth irreducible representation π of G is necessarily a subrepresentationof some π1 × · · · × πr for some P and some irreducible supercuspidal representations
π1 , . . . , πr . By [Rod82, Proposition 5], π determines the multiset {π1 , . . . , πr } (andthus the corresponding partition of n, up to reordering) uniquely, and we refer to it
as the cuspidal support of π. For any representation πi of GLni (F ), and any integers, we write πi (s) := πi ⊗ | det |s . Then by [Rod82, Théorème 1] (and the remark
which follows it), π1 × · · · × πr is reducible if and only if there are some i, j withni = nj and πj ∼= πi (1).
We define a segment to be a set of isomorphism classes of irreducible cuspidal
representations GLni (F ) of the form ∆ = {πi , πi (1), . . . , πi (r − 1)} for some r ≥ 1,
and we write ∆ = [πi , πi (r − 1)]. We say that two segments ∆1 , ∆2 are linked ifneither contains the other, and ∆1 ∪ ∆2 is also a segment. If ∆1 = [πi , πi0 ] and∆2 = [πi00 , πi000 ] are linked, we say that ∆1 precedes ∆2 if πi00 = πi (r) for some r ≥ 0.
If ∆ = [πi , πi (r−1)] is a segment, then πi ×· · ·×πi (r−1) has a unique irreducible
subrepresentation Z(∆) and a unique irreducible quotient L(∆). By [Rod82, Théorème 2], if ∆1 , . . . , ∆r are segments such that if i < j, then ∆i does not precede ∆j ,then the representation Z(∆1 )×· · ·×Z(∆r ) has a unique irreducible subrepresentation, which we denote Z(∆1 ×· · ·×∆r ). Every irreducible smooth representation of
G is of the form Z(∆1 ×· · ·×∆r ) for some segments ∆1 , . . . , ∆r , which are uniquelydetermined up to reordering. By [Rod82, Théorème 3], the analogous statement
also holds for the L(∆i ). By [Rod82, Théorème 5], Z(∆1 , . . . , ∆r ) occurs withmultiplicity one as a subquotient of Z(∆1 ) × · · · × Z(∆r ).
The link with the local Langlands correspondence is as follows (see [Rod82,§4.4]). The representation recp (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

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

21

monodromy operator has rank r−1. Then if ∆ = [πi , πi (r−1)] is a segment, we haverecp L(∆) = recp (πi ) ⊗ Sp(r), and more generally we have recp L(∆1 , . . . , ∆r ) =⊕ri=1 recp L(∆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 GLn ) ina fashion adapted to our needs. The Bernstein spectrum is the set of G-orbits ofpairs (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 (GLn1 (F ) × · · · × GLnr (F ), π1 ⊗ · · · ⊗ πr ), as in
the preceding section. As we will explain below, the Bernstein spectrum naturally
has the structure of an algebraic variety over Qp (with infinitely many connectedcomponents), the Bernstein variety. Given an irreducible smooth G-representation
π, we obtain a point of the Bernstein spectrum by passing to the cuspidal supportof 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 iG
P ω (and indeed
these are precisely the π for which (M, ω) arises as the cuspidal support; here P isany parabolic subgroup of G admitting M as a Levi quotient — the collection of
Jordan–Hölder factors of iG
P 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 0 , ω 0 ) are inertially equivalent
if they are in the same Bernstein component, and write [M, ω] for the equivalenceclass. 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 categoryof 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 ZΩ of Ω is the centre of the corresponding Bernstein component (that is,
the endomorphism ring of the identity functor), so that ZΩ acts naturally on each
irreducible smooth representation π ∈ Ω. Since π is irreducible, each element ofZΩ will act on π through a scalar. In fact this scalar depends only on the cuspidal
support of π, and in this way ZΩ 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 GLni (F ),and we will make use of this extension below without further comment (in order tocompare 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 0 of G lie in the same Bernstein component
if and only if recp (π)|IF ∼= recp (π 0 )|IF (where we ignore the monodromy operators).

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

3.3. Bushnell–Kutzko theory. Fix a Bernstein component Ω. In [BK99], thereis 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 Qp -representation of J.
The pair (J, λ) has the property that if π is an irreducible smooth representationof G, then HomJ (λ, π) 6= 0 if and only if π ∈ Ω, and in fact the functor HomJ (λ, ∗)induces an equivalence of categories between Ω and the category of left modulesof the Hecke algebra H(G, λ) := H(G, J; λ) := EndG (c-IndG
J λ), with the inverse
functor being given by c-IndG
λ⊗(∗).
H(G,λ)J
Let π be an irreducible element of Ω, and let (M, ω) be a representative for theinertial support of π. We can and do suppose that M is a standard Levi subgroup
Qr Qdi⊗dir, where πi and πi0 are not inertially
i=1
j=1 GLei (F ), and that ω = ⊗i=1 πiequivalent (that is, do not differ by a twist by an unramified quasicharacter) if
i 6= i0 . Then by the construction of (J, λ) in [BK99], there is a pair (J ∩ M, λM )which is a semisimple Bushnell–Kutzko type for the Bernstein component ΩM of
M determined by ω, in the sense that if πM is an irreducible smooth representationof M , then HomJ∩M (λM , πM ) 6= 0 if and only if πM ∈ ΩM , and the functor
HomJ (λM , ∗) induces an equivalence of categories between ΩM and the category ofleft modules of H(M, λM ) in the same way as above.
Let P be a parabolic subgroup of G with Levi factor M . Then the normalised
induction iG
P restricts to a functor from ΩM to Ω, and there is a unique injective
algebra homomorphism tP : H(M, λM ) → H(G, λ) (which is denoted jQ on page
55 of [BK99]) such that under the equivalences of categories explained above, iGPcorresponds to the pushforward along tP .
It will be useful to us to have a somewhat more explicit description of the pair
Qr Qdi(J ∩ M, λM ). Recall that we may write M = i=1 j=1
GLei (F ), and that ω =⊗ri=1 πi⊗di . Then as is explained in [BK99, §1.4-5] (see also the paragraph afterQr QdiNr
Lemma 7.6.3 of [BK93]), we may write J ∩ M = i=1 j=1
Ji , λM = i=1 λi⊗di ,where (Ji , λi ) is a maximal simple type occurring in πi in the sense of [BK98]. There⊗di, and eachis a corresponding
isomorphism H(M, λM ) ∼= ⊗ri=1 H GLei (F ), λi
H GLei (F ), λi is commutative, so that H(M, λM ) is commutative. The following
somewhat technical lemma will be useful to us in Section 4. We remind the reader
that the algebra H(M, λM ) is naturally isomorphic to the convolution algebra ofcompactly supported functions f : M → EndQp (λM ) such that f (jmj 0 ) = j ◦f (m)◦j 0 for all m ∈ M, j, j 0 ∈ J ∩ M (see [BL94, §2.2]).
3.4. Lemma. There is an integer e ≥ 1 and a Qp -basis for H(M, λM ) with theproperty that if ν is an element of this basis, then the e-fold convolution of ν withitself is supported on tν (J ∩ M ) for some tν ∈ Z(M ).
Proof. By the above remarks, we need only prove the corresponding
result for theHecke algebra of a maximal simple type H GLei (F ), λi . 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 tJi 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 suchthat if t ∈ D(B) then the e-fold composition of a Hecke operator ψt supported on
tJi with itself is supported on sJi , where s is a scalar matrix. But D(B) is a cyclic
group, generated by a uniformiser $E of some finite extension of fields E/F inside
GLei (F ), so we can just take e to be the ramification degree e(E : F ). In that

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

23

case, the e-fold composition of a Hecke operator supported on $E Ji with itself is
supported on $F Ji , as remarked on the bottom of page 201 of [BK93].

3.5. Remark. If λi is a maximal simple type for GLni (F ), then the ramification
degree e(E : F ) is equal to ni /fi , where fi is the number of unramified charactersof F × which preserve a supercuspidal GLni (F )-representation containing λi . This
follows from Lemma 6.2.5 of [BK93]. Therefore, det($E ) ∈ F × has valuation fi .
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 constructedin the paper [SZ99]. Note that there is not quite a bijection between irreduciblesmooth representations of G and characters of the Bernstein centre; as explainedin Section 3.2, any two Jordan–Hölder constituents of a parabolic induction froman 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, twoirreducible representations lie in the same Bernstein component if and only if thecorresponding 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 recp (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 inthe terminology of [BS07]. A partial ordering is defined on these functions, andthere is a unique maximal element for this ordering, which we will denote by P
from now on. In [SZ99, §6] an irreducible direct summand σP (λ) of IndKJ λ is constructed, with the property that if π is an irreducible smooth G-representation,then HomK (σP (λ), π) 6= 0 if and only if π ∈ Irr(Ω)0 , in which case σP (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 proofof Proposition 6.5.3 of [BC09].)
If τ is the inertial type corresponding to Ω, then we write σ(τ ) for σP (λ). As
remarked on page 201 of [SZ99], since P is maximal, σ(τ ) coincides with the representation ρs = eK IndKJ λ constructed in Theorem 4.1 of [Dat99]. Here eK is acertain idempotent in H(G, λ) which is constructed in [Dat99, §4.2].
Let ZΩ be the centre of Ω. By Theorem 4.1 of [Dat99], the action of ZΩ on
c-IndGK σ(τ ) induces an isomorphismZΩ ∼= EndG c-IndG
K σ(τ ) =: H G, σ(τ ) ,
so in particular H G, σ(τ ) is commutative.
Let W[M,ω] := NG ([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 tP : H(M,
 λM ) → H(G, λ) induces∼
an isomorphism tP : H(M, λM )W[M,ω] −→ Z H(G, λ) . It follows from the discussion in [Dat99, §4.2] that there is an algebra homomorphism sP : H(G, λ) →

24 A. CARAIANI, M. EMERTON, T. GEE, D. GERAGHTY, V. PAŠKŪNAS, AND S. W. SHIN
H G, σ(τ ) , which makes H G, σ(τ ) a direct summand of H(G, λ), and the com∼
posite sP ◦ tP induces an isomorphism H(M, λM )W[M,ω] −→ H G, σ(τ ) . (The map
sP is just given by h 7→ eK ∗ h ∗ eK , where eK 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 Qp -representation σ(τ ) of K such that if π is any irreducible smooth
Qp -representation of G, then the restriction of π to K contains (an isomorphic
copy of) σ(τ ) as a subrepresentation if and only if recp (π)|IF ∼ τ and N = 0 on
recp (π). Furthermore, in this case the restriction of π to K contains a unique copy
of σ(τ ). The ring H G, σ(τ ) := EndG c-IndG
K σ(τ ) is commutative.
3.8. Proposition. Let πi be an irreducible supercuspidal representation of GLni (F )for 1 ≤ i ≤ r, and n = n1 + . . . + nr . If πj ∼6 πi (1) (the condition being empty if=
ni 6= nj ) for every i < j then the G-socle and the G-cosocle of π1 × . . . × πr are
irreducible. Moreover, the socle occurs as a subquotient with multiplicity one andis the only generic subquotient of π1 × · · · × πr .
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 ∆1 = {π1 }, . . . , ∆r = {πr }, then by assumption the ∆i are segments suchthat ∆i does not precede ∆j for i < j. Since the segments are of length one, wehave L(∆i ) ∼= Z(∆i ) ∼= πi by definition, so that as recalled above π1 × · · · × πr
has a unique irreducible subrepresentation Z(∆1 , . . . , ∆r ) and a unique irreducible
quotient L(∆1 , . . . , ∆r ). In addition, Z(∆1 , . . . , ∆r ) occurs as a subquotient withmultiplicity one, so we only need to show that it is the unique generic subquotient.
Let Un be the subgroup of unipotent upper-triangular matrices in GLn (F ), andPn−1
××
let θn : Un → Qp be the character (uij ) 7→ ψ( i=1 ui,i+1 ), where ψ : F → Qp isa fixed smooth non-trivial character. If π is a representation of G, we let πθn be thelargest quotient of π on which Un acts by θn . If π is an irreducible representationof G then the dimension of πθn is at most one, and is equal to one if and only if
π is generic. Since Un is equal to the union of its compact open subgroups, thefunctor π 7→ πθn is exact. Thus it is enough to show that (π1 × · · · × πr )θn is one
dimensional and Z(∆1 , . . . , ∆r )θn is non-zero.
In [BZ77, Zel80] the authors define a family of exact functors π 7→ π (i) for 0 ≤ i ≤n from the category of smooth representations of GLn (F ) to the category of smooth
representations of GLn−i (F ). The representation π (i) is called the i-th derivativeof π. We have π (0) = π and π (n) = πθn . If π is irreducible and supercuspidalthen 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 (π1 × · · · × πr )(n) ∼= (π1 )(n1 ) ⊗ . . . ⊗ (πr )(nr ) ,which is one dimensional, since the πi are supercuspidal representations of GLni (F ).
It follows from [Zel80, Theorem 6.2] that Z(∆1 , . . . , ∆r )(n) 6= 0, as required.

If π in Ω is irreducible, then the action of ZΩ on π defines a maximal idealχπ : ZΩ → EndG (π) ∼= Qp . The following is a strengthening of Theorem 4.1 of[Dat99].
3.9. Proposition. Let π be an irreducible smooth Qp -representation of G, such thatrecp (π)|IF ∼ τ and N = 0 on recp (π). Then
c-IndG σ(τ ) ⊗Z ,χ Qp ∼= π1 × · · · × πr ,KΩπPATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

25

where πi is a supercuspidal representation of GLni (F ), such that if i < j thenπj ∼6 πi (1) (the condition being empty if ni 6= nj ), and recp (π) ∼= ⊕ri=1 recp (πi ).
=G
Moreover, π is the unique irreducible quotient of c-IndK σ(τ ) ⊗ZΩ ,χπ Qp .
Proof. Since N = 0 on recp (π), we may write it as a direct sum of irreducible
representations of the Weil group WF , and there is a partition n = n1 + · · · + nr
and supercuspidal representations πi of GLni (F ) such that recp (π) ∼= ⊕ri=1 recp (πi ).
∼After reordering we may assume that if i < j then πj =6 πi (1). It follows from
Proposition 3.8 that π1 × · · · × πr has a unique irreducible quotient π 0 .
Then recp (π 0 ) ∼= ⊕ri=1 recp (πi ) as representations of WF , and N = 0 on recp (π 0 )since all the segments have length one, as in the proof of Proposition 3.8. Thus
recp (π 0 ) ∼= π 0 . Since the socle of π1 × · · · × πr is= recp (π), which implies that π ∼irreducible and occurs as a subquotient with multiplicity one, the action of ZΩ on
π1 × · · · × πr factors through a maximal ideal χ : ZΩ → Qp . Since the cosocle isisomorphic to π, we deduce that χ = χπ .
G
Theorem 3.7 implies that π is a quotient of c-IndG
K σ(τ ). Since c-IndK σ(τ ) isG
projective there exists a G-equivariant map ϕ : c-IndK σ(τ ) → π1 × · · · × πr suchthat the composition with π1 × · · · × πr  π is surjective. The G-cosocle of thecokernel of ϕ is zero. Since π1 × · · · × πr is of finite length, so is the cokernel ofϕ, and we deduce that ϕ is surjective. Since ZΩ acts naturally on everything, ϕinduces a surjection ϕ̄ : c-IndG
K σ(τ ) ⊗ZΩ ,χπ Qp  π1 × · · · × πr .
Then [Dat99, Theorem 4.1] implies that the semisimplification of c-IndG
K σ(τ )⊗ZΩ
χπ is isomorphic to the semisimplification of π10 ×· · ·×πs0 , where πi0 are supercuspidal
representations of GLn0i (F ) for some integers n0i , such that n = n01 +. . .+n0s . Since π
is an irreducible subquotient of both π10 ×. . .×πs0 and π1 ×· · ·×πr , [BZ77, Theorem
2.9] implies that π10 × . . . × πs0 and π1 × · · · × πr have the same semisimplification.
Thus c-IndG
K σ(τ ) ⊗ZΩ ,χπ Qp and π1 × · · · × πr 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 ZΩ be the centre of Ω. Let χ : ZΩ → Qp be a maximal ideal of ZΩ . Thenthe G-socle of c-IndG
K σ(τ ) ⊗ZΩ ,χ Qp 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-IndG
K σ(τ ) ⊗ZΩ ,χπ Qp .
Proof. The first part follows from Propositions 3.8 and 3.9. The converse may beseen as follows. There exist supercuspidal representations πi of GLni (F ) such thatn = n1 + . . . + nr and π is a subquotient of π1 × · · · × πr . If w is a permutation of{1, . . . , r} then π1 ×· · ·×πr and πw(1) ×· · ·×πw(r) have the same semisimplification
by [BZ77, Theorem 2.9], so we may assume that the πi satisfy the conditions of
Proposition 3.8. Since the socle of π1 × · · · × πr is irreducible and occurs as asubquotient with multiplicity one, the action of ZΩ on π1 × · · · × πr factors througha maximal ideal, which is equal to χπ , as π occurs as a subquotient. If we let π 0
be the G-cosocle of π1 × · · · × πr then π 0 satisfies the conditions of Proposition 3.9,
and we have χπ0 = χπ . The assertion follows from Propositions 3.8 and 3.9.

3.11. Corollary. Let π be an irreducible smooth generic Qp -representation of G,
such that recp (π)|IF ∼ τ and N = 0 on recp (π). Then we have a natural isomor∼
phism c-IndG
K σ(τ ) ⊗ZΩ ,χπ Qp = π.

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

Proof. By Corollary 3.10, we see that the G-socle of c-IndG
K σ(τ )⊗ZΩ ,χπ Qp is isomorphic to π and occurs with multiplicity one as a subquotient. Theorem 3.7 implies
that π is a quotient of c-IndGK σ(τ ) ⊗ZΩ ,χπ Qp . 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 Qp as coefficientfield, 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 Qp , and we then deduce the resultsfrom those over Qp 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 → Qp , 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 ρT of T , such that IndM
T ρT is isomorphic to a
finite direct sum of copies of ω. Thus χ ∈ X (M ) lies in X (M )(ω) if and only if therestriction 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 Qp , which are trivial on M 0 .
Let D be the Bernstein component (for M ) containing ω, let ZD be the centreof D and let Π(D) := c-IndM
M 0 ρ. It is shown in [Ren10, VI.4.1] that Π(D) is a
projective generator
for
D.
Thus
we may identify ZD with the centre of the ring
EndM Π(D) . Since ρ is irreducible, Π(D) is a finitely generated Qp [M ]-module.
Let ZΩ be the Bernstein centre of Ω and let Π(Ω) := iG
P 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
 Qp [G]-module.
Thus we may identify ZΩ with the centre of the ring EndG Π(Ω) .
M0∼
It follows from Bushnell–Kutzko theory that ω ∼= c-IndM
J Λ, and ρ = c-IndJ 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 Qp . By compactly
inducing these realisations, we deduce that both ω and ρ can be realised over E.
We denote these representations by ωE and ρE , respectively. It is shown in [Ren10,
Lemme V.2.7] that X (M )(ω) is a finite group. Let W (D) be the subgroup ofNG (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 char√acters ξ are E-valued. By further enlarging E we may assume that q, where q
is the number of elements in the residue field of F , is contained in E. Then themodulus character of P is defined over E.
MLet Π(D)E := c-IndM
M 0 ρE and let ZD,E be the centre of EndM (c-IndM 0 ρE ). Since
ρE ⊗E Qp ∼= ρ, we have Π(D)E ⊗E Qp ∼= Π(D). We may express the generators of
Π(D) (as a Qp [M ]-module) as a finite Qp -linear combination of elements of Π(D)E .

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

27

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 Qp . In particular, Π(D)E is a
finitely generated E[M ]-module.
Let Π(Ω)E := iG
P Π(D)E and let ZΩ,E be the centre of EndG (Π(Ω)E ). The
smooth parabolic induction commutes with ⊗E Qp , as the set P \G/H is finite forevery open subgroup H of G and tensor products commute with inductive limits,so Π(Ω)E ⊗E Qp ∼= Π(Ω). Since Π(Ω) is a finitely generated Qp [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 0 are representations of some group G on E-vector spaces, such that π isa finitely generated E[G]-module, then
(3.14)
HomE[G] (π, π 0 ) ⊗E Qp ∼(π ⊗E Qp , π 0 ⊗E Qp ).
= Hom
Qp [G]

It follows from (3.14) that
(3.15)


EndM (Π(D)E ) ⊗E Qp ∼= EndM Π(D) ,(3.16)
EndG (Π(Ω)E ) ⊗E Qp ∼= EndG Π(Ω) .

Let DE be the full subcategory of smooth representation ω 0 of M on E-vector
spaces, such that ω 0 ⊗E Qp is in D. It follows from (3.14) that HomM (Π(D)E , ω 0 )⊗E
Qp ∼= HomM (Π(D), ω 0 ⊗E Qp ). Since Π(D) is a projective generator of D, wededuce that Π(D)E is a projective generator of DE . (This follows from the factthat Qp is faithfully flat over E; we will repeatedly use this fact below without
further comment.) In particular, ZD,E is naturally isomorphic to the centre of thecategory DE .
Similarly we let ΩE be the full subcategory of smooth representations π 0 of G
on E-vector spaces, such that π 0 ⊗E Qp is in Ω. The same argument as above givesthat Π(Ω)E is a projective generator of ΩE and ZΩ,E is naturally isomorphic to thecentre of ΩE .
3.17. Lemma. Let A be an E-algebra and let Z be an E-subalgebra of A. If Z ⊗E Qp
is the centre of A ⊗E Qp then Z is the centre of A.
Proof. For each z ∈ Z, we define an E-linear map φz : A → A, a 7→ az − za. Sincez ⊗1 is central in A⊗E Qp by assumption, we deduce that (Im φz )⊗E Qp = 0, which
implies that Im φz = 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 Qp and thus in Z ⊗E Qp
by assumption. Hence (Z(A)/Z) ⊗E Qp = 0, which implies that Z = Z(A).

3.18. Lemma. The isomorphism (3.15) induces an isomorphism ZD,E ⊗E Qp ∼= ZD .
T
Proof. Since Π(D) ∼= IndMT (c-IndM 0 ρ) and induction is a functor, we have an inT
clusion EndT (c-IndM 0 ρ) ⊂ EndM Π(D) . It follows from [Ren10, Thm.VI.4.4]
that this inclusion identifies EndT (c-IndTM 0 ρ) with the centre of EndM Π(D) .
T
The assertion follows
 from Lemma 3.17 applied to Z = EndT (c-IndM 0 ρE ) and
A = EndM 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

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

X (M )/X (M )(ω) → Irr(D), χ 7→ ω ⊗ χ. Composing this bijection with (3.13) we
obtain a bijection∼=

Irr(D) → X (T ).

(3.19)

Now X (T ) is naturally isomorphic to the set of homomorphisms of Qp -algebras fromQp [T /M 0 ] to Qp . It is explained in [Ren10, VI.4.4] that we have identifications
ZD ∼= Qp [T /M 0 ];= EndT (IndTM 0 ρ) ∼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 ZD .
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 ξw be the restriction of ξ
to T . Then ξw depends only on w and not on the choice of ξ. If χ ∈ X (M ) then(ω ⊗ χ)w ∼= ω ⊗ χw ξ. Thus the action of W (D) on X (T ) via (3.19) is given byw
w.χ = χ ξw . It is immediate that this action is induced by the action of W (D) on−1
Qp [T /M 0 ] given by w.(tM 0 ) = ξw(t)tw M 0 .
3.20. Lemma. The action of W (D) on ZD preserves ZD,E .
Proof. Since ω and ρ can both be defined over E, so can the representation ρH defined at the beginning of [Ren10, VI.4.4], and in particular so can ρT , its restriction
to T . Hence, if we identify ZD with Qp [T /M 0 ] as in [Ren10, Prop.VI.4.4] then ZD,E
is identified with E[T /M 0 ]. Since the characters ξw are E-valued by the choice ofE, we get the assertion.
W (D)3.21. Lemma. ZΩ,E = ZD,E .
Proof. Since Π(Ω) = iG
P Π(D)
 and parabolic induction is a functor, we have an
inclusion ZD ⊂ EndG Π(Ω) . It follows from the discussion immediately precedingW (D)the proof of Theorem VI.10.4
with of [Ren10] that this inclusion identifies ZD
the centre of EndG Π(Ω) . The assertion follows from Lemma 3.17 applied to

W (D)

Z = ZD,E and A = EndG Π(Ω)E .
3.22. Proposition. The isomorphism (3.16) induces an isomorphism ZΩ,E ⊗E Qp ∼=ZΩ .
Proof. Using Lemmas 3.18 and 3.21 we obtain
W (D) ∼∼ ZW (D) ⊗E Qp =∼ (ZD,E ⊗E Qp )W (D) ∼ZΩ,E ⊗E Qp ==Z
= ZΩ ,
D,E

D

the last isomorphism following from [Ren10, VI.10.4] as in the proof of Lemma 3.21.

3.23. Lemma. ZΩ,E coincides with the ring E[B] constructed in [Che09, Prop. 3.11].
Proof. Let ∆ be the subgroup of X (M ) o W (D) consisting of pairs (ξ, w), suchthat ω w ∼= ω ⊗ ξ. This subgroup acts naturally on E[M/M 0 ]. The map ξ 7→ (ξ, 1)
identifies X (M )(ω) with a normal subgroup of ∆ and the quotient is isomorphic to
W (D). We have
Qp [M/M 0 ]∆ ∼= (Qp [M/M 0 ]X (M )(ω) )W (D) ∼= Qp [T /M 0 ]W (D) ∼
= ZD

W (D)

,

see [Ren10, Rem.VI.4.4] for the second isomorphism. Chenevier defines E[B] to be
E[M/M 0 ]∆ . This subring gets identified with E[T /M 0 ]W (D) inside Qp [T /M 0 ]W (D) ,PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE
W (D)

W (D)

and with ZD,E inside ZD
from Lemma 3.21.

29, see the proof of Lemma 3.20. The assertion follows
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 Qp ∼= σ(τ ). Then c-IndG
K σ(τ )E is anobject in ΩE . Since ZΩ,E is the centre of ΩE it acts on c-IndGσ(τ),
E thus inducing
KGa homomorphism ZΩ,E → EndG c-IndK σ(τ )E .

3.24. Lemma. The map ZΩ,E → EndG c-IndG
K 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 Qp . This implies the assertion. Let R := EndG Π(Ω) . Since Π(Ω) is a projective generator the functors M 7→
M ⊗R Π(Ω) and π 7→ HomG (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 ZΩ
on π factors through χπ : ZΩ → Qp . It follows from [Ren10, Lem.VI.10.4] that R
is a finitely generated ZΩ -module , which implies that the module corresponding to
π is a finite dimensional Qp -vector space. Since ZΩ,E is a finitely generated algebra
over E, E(χπ ) := χπ (ZΩ,E ) is a finite extension of E.
In the above E was only required to be sufficiently large. Thus if E 0 is a subfield of Qp containing E, then we let ΩE 0 , Π(Ω)E 0 be the corresponding objects
defined over E 0 instead of E. Then Π(Ω)E 0 is a projective generator of ΩE 0 andthe functors M 7→ M ⊗RE0 Π(Ω)E 0 and π 7→ HomG (Π(Ω)E 0 , π) induce an equivalence of categories between the category of right RE 0 -modules and ΩE 0 , where
RE 0 := EndG (Π(Ω)E 0 ).
3.25. Lemma. Every irreducible generic π ∈ Ω can be realised over E(χπ ).
0
Proof. To ease the notation we let E 0 := E(χπ ) and let π 0 := c-IndG
K σ(τ )E ⊗ZΩ,E E .
Then∼
π 0 ⊗E 0 Qp ∼= c-IndG
K σ(τ ) ⊗ZΩ ,χπ Qp = π1 × . . . × πr ,where the last isomorphism is given by Proposition 3.9. Hence, π 0 ⊗E 0 Qp is of finite
length, which implies that HomG (Π(Ω), π 0 ⊗E 0 Qp ) is a finite dimensional Qp -vector
space, which implies that M 0 := HomG (Π(Ω)E , π 0 ) is a finite dimensional E 0 -vector
space.
If π 00 is an irreducible E 0 -subrepresentation of π 0 , and if we define M 00 :=HomG (Π(Ω)E 0 , π 00 ), then M 00 is an irreducible RE 0 -module which is finite dimensional over E 0 . It follows from [Bou12, Cor.12.7.1a)] that M 00 ⊗E 0 Qp is a semi-simple
R-module. Hence, π 00 ⊗E 0 Qp is a semi-simple G-representation. Proposition 3.8
implies that the G-socle of π 00 ⊗E 0 Qp is irreducible and is isomorphic to π. Thus
π 00 ⊗E 0 Qp ∼= π.
Henceforth for each Bernstein component Ω we will fix a sufficiently large E asabove 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 ZΩ , σ(τ ), and so on,in place of ZΩ,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.

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

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 bypatching 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 willthen 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 inproving 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 = GLn (F ) defined over E. Then bydefinition σ is an absolutely irreducible representation of K = GLn (OF ) over Eof the form σsm ⊗ σalg , where σsm is a smooth type for K (i.e. σsm = σ(τ ) forsome inertial type τ ) and σalg is the restriction to K of an irreducible algebraic
representation of ResF/Qp G; we will sometimes also write σalg 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) :=EndG (c-IndG
K σ).
We say that a continuous representation r : GF → GLn (E) is potentially crystalline of type σ if it is potentially crystalline with inertial type τ and Hodge–Tateweights prescribed by σalg . We will also say a global representation has type σ if it
restricts to such an r. Let Rep (σ) be the local framed universal deformation ring oftype σ at ep (i.e. the unique reduced and p-torsion free quotient of Rep corresponding
to potentially crystalline lifts of type σ).
Let X = Spf Rep (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 Uj 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 unionof closed polydiscs. By a standard abuse of notation, we will write OX rig for the
ring of rigid-analytic functions on X rig . Then OX rig = limj Γ(Uj , OUj ) and we equip←−
it with the inverse limit topology. We note that by [dJ95, Lemma 7.1.9], there isa bijection between the points of X rig and the closed points of Spec Rep (σ)[1/p].
The universal lift over Rep (σ) gives rise to a continuous family of representations
rig
ρrig : GF → GLn (OX rig ). (The continuity of ρ is equivalent to that of each of
the representations GF → GLn Γ(Uj , OUj ) obtained by restricting elements ofGLn (OX rig ) to Uj .) If x is a point of X rig with residue field Ex , we denote byρx : GF → GLn (Ex ) the specialisation of ρrig at x. We define the locally algebraic
G-representation πl.alg,x := πsm (ρx )⊗E σalg = πsm (ρx )⊗Ex πalg (ρx ). (Recall the notation πsm (ρx ) and πalg (ρx ) from §1.5; in particular, πsm (ρx ) = rp−1 (WD(ρx )F −ss ).)Note that H(σ) acts via a character on the space HomK (σ, πl.alg,x ), the latter being

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

31

one-dimensional (by Theorem 3.7 together with the argument of [ST06, Lemma1.4]).
The following theorem, which may be of independent interest, gives our interpolation of the local Langlands correspondence. Its proof will occupy much of thissection.
4.1. Theorem. There is an E-algebra homomorphism η : H(σ) → Rep (σ)[1/p]which interpolates the local Langlands correspondence rp . More precisely, for any
closed point x of Spec Rep (σ)[1/p], the action of H(σ) on HomK (σ, πl.alg,x ) factors
as η composed with the evaluation map Rep (σ)[1/p] → Ex .
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) → OX rig whichinterpolates the local Langlands correspondence rp . More precisely, for any pointx ∈ X rig , the action of H(σ) on HomK (σ, πl.alg,x ) factors as η composed with theevaluation map OX rig → Ex .
Let ρrig
WD : WF → GLn (OX rig ) be the family of Weil–Deligne representations
associated to the family of representations ρrig by Corollaire 3.19 of [Che09]. Moreprecisely, this result of Chenevier generalises Théorème C of [BC08] to the case ofan arbitrary reduced quasi-compact and quasi-separated rigid space. Applying it
to each Uj , we obtain a compatible family of Weil–Deligne representations ρrig
WD :
WF → GLn (Γ(Uj , OUj )) and thus a Weil–Deligne representation ρrig
:
W
F →
WD
rig
GLn (OX rig ). Note that ρWD has N = 0.
For a point x of X rig , we denote by ρWD,x the specialisation of ρrig
WD at x. Then
ρWD,x |IF ' τ for all points x of X rig . Recall that ZΩ is the Bernstein centre for theBernstein component Ω corresponding to σ(τ ).
4.3. Proposition. There exists a unique E-algebra map I : ZΩ → OX rig suchthat for any point x of X rig with residue field Ex , the smooth G-representation πx
corresponding to ρWD,x via the local Langlands correspondence recp determines via
specialisation the map x ◦ I : ZΩ → Ex .
Proof. Consider the following map, obtained by specialisation:YγG : ZΩ →Ex0 ,x∈X rig

where γG is defined on the factor corresponding to x by evaluating ZΩ at the closed
point in the Bernstein component Ω determined via local Langlands by x, andEx0 /Ex is a sufficiently large finite extension.
Consider as well the following map, also obtained by specialisation:
Y
γWD : OX rig →Ex0 .
x∈X rig
rig

This is an injection since X
is reduced and since each Γ(Uj , OUj ) 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
h iat closed points, but these arethe same as the completed local rings of Rp̃ (σ) p1 by [dJ95, Lemma 7.1.9]. The
latter is reduced (being a localization of Rp̃ (σ), which is reduced by definition)

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

and excellent, since Rp̃ (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 ZΩ under γG
is contained in the image of γWD . Let T : WF → ZΩ be the pseudo-representation
constructed in Proposition 3.11 of [Che09]. (Note that Chenevier’s E[B] is our ZΩ
by Lemma 3.23.) By the construction of T , we have γG ◦ T = γWD ◦ tr(ρrig
WD ).
Therefore the proof of the proposition is reduced to Lemma 4.5 below.

Write v : WF  Z for the valuation map assigning 1 to any lift of the geometric
Frobenius. Let φ ∈ WF be an element of valuation 1. For w ∈ WF and any
IF -representation r0 , let r0w be the twist r0w (γ) := r0 (w−1 γw).
4.4. Lemma. Let r be an irreducible continuous representation of WF over Qp .
(1) The restriction r|IF decomposes as a direct sum of non-isomorphic irrei
ducible IF -representations ⊕fi=1 r1φ for some integer f ≥ 1. If t ∈ Z
ithen r(φt ) respects the decomposition (i.e. r(φt ) sends r1φ into itself for1 ≤ i ≤ f ) exactly when f | t.
(2) We have tr r(w) 6= 0 for some w ∈ WF of valuation t if and only if f | t.
(3) The unramified characters χ of WF satisfying r ⊗ χ ' r are exactly thecharacters of order dividing f .
Proof. (1) The representation r|IF factors through a finite quotient IF /H, so it
decomposes as a direct sum of irreducible IF -representations ⊕fi=1 ri , for some integer f ≥ 1. The fact that r is irreducible as a WF -representation implies thatr(φ) acts transitively on (the representation spaces of) the ri . Up to reordering theri , we may assume that it sends ri to ri+1 , where rf +1 := r1 . Moreover, we alsofdeduce that ri+1 ' riφ and that r1 ' r1φ . Finally, all the representations ri are
non-isomorphic, since if there was an isomorphism between them, we could define aproper WF -subrepresentation of r and thus contradict the irreducibility of r. (Moreprecisely, if we had an isomorphism r1 ' r1+s for some 1 ≤ s < f , then we could
assume that f = sf 0 for some integer f 0 and get IF -isomorphisms∼→ r1+sk ⊕ · · · ⊕ rs(1+k)αsk : r1 ⊕ · · · ⊕ rs −for each 1 ≤ k < f 0 . In that case, we could take the IF -subrepresentation of r
generated by v + αs (v) + · · · + αs(f 0 −1) (v) with v ∈ r1 ⊕ · · · ⊕ rs ; it is easy to checkthat this space is also stable under φ if we choose the αsk appropriately.) The factthat r(φ) induces a cyclic permutation of the f irreducible constituents implies thestatement about r(φt ).
(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(φt ) has thesame block decomposition as r|IF . Note that the group algebra of IF /H surjects
onto ⊕fi=1 EndQp (ri ), since the ri are non-isomorphic irreducible representations oftheP finite group IF /H. Therefore, there is some linear combination of matrices
non-zero trace against the non-zero matrix r(φt ). Thish∈IF ch · r(h) which hast
implies that tr r(h · φ ) 6= 0 for some h ∈ IF .
(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 allw ∈ WF such that f |v(w), or equivalently that χf = 1. Hence part 3 is verified. 

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

33

4.5. Lemma. The image of T generates ZΩ as an E-algebra.
Proof. It suffices (by the faithful flatness of the extension Qp /E) to prove the result after replacing E with Qp . Since the inertial type τ factors through a finite
quotient IF /H, it decomposes as a direct sum ⊕ri=1 (τi )di , where the τi are nonisomorphic inertial types such that σ(τi ) is cuspidal. As in the proof of Proposition 3.11 of [Che09], the Bernstein component Ω decomposes as Ω1 × · · · × Ωr andZΩ = ⊗ri=1 ZΩi , where each Ωi corresponds to the simple type σ((τi )di ). If we let
Ti : WF → ZΩi be the pseudo-representation
associated to Ωi by Proposition 3.11
Prof [Che09], then by definition T (g) := i=1 Ti (g). It suffices to show that the image of T generates each ZΩi for i = 1, . . . , r, where ZΩi is regarded as a subalgebraof ZΩ .
Let ri be an irreducible WF -representation such that ri |IF ' τi , and let fi be the⊗diinteger associated to ri by Lemma 4.4. By choosing rec−1
as a base point,p (ri )
each closed point of Spec ZΩi may be represented by an unramified character χi =i
(χi,1 , . . . , χi,di ) (or more precisely by ⊗dj=1(rec−1p (ri ) ⊗ χi,j ) up to a permutation offactors), where the χi,j are unramified characters of F × . Then each Ti (g) is defined
by
di
X

Ti (g)(χi ) := tr(ri )(g)
χi,j ArtF (g) .
j=1
ti

Consider elements g of the form h · φ , with h ∈ IF and ti a multiple of fi . By
Lemma 4.4(1), the matrix ri (φti ) is non-zero and consists of fi blocks which matchthe block decomposition of τi . Because the constituents of τi are non-isomorphic
for different i’s, we may choose the ch such thatXch · tr ri (h) · ri (φti ) 6= 0h∈IF /HP
and h∈IF /H ch · ri0 (h) = 0 for i0 6= i. In particular, this means that h∈IF /H ch ·T (hφti ) ∈ ZΩi .
Now notice that ArtF (φti ) ∈ F × has valuation ti , which by Lemma 4.4(3) andRemark 3.5 coincides with the valuation of det(πEi )ti /fi , where Ei /F is the exten
Pdision in Lemma 3.4 for the cuspidal type σ(τi ). Therefore, j=1
χi,j ArtF (g) asPdi
t /fa regular function on the Bernstein component ZΩi is exactly j=1
Xiji i , whereXi1 is a generator of the Bernstein centre for σ(τi ) and the Xij are obtained fromXi1 from the Weyl group action. This implies that for our choice of coefficients ch ,PPditi /fi
ti∈ ZΩi , up to a non-zero scalar.
h∈IF /H ch · T (hφ ) is equal to
j=1 Xij
Note that we can ensure that ti /fi is any integer. Therefore, we can generate
all elements in ZΩi which are sums of k-th powers of the di generators of theBernstein centre for a single supercuspidal, for any integer k. However, since ZΩi
is obtained by taking Weyl-group invariants, it is generated as a Qp -algebra by theelementary symmetric polynomials in those di variables together with the productof the inverses of the variables. Over Qp , which is a field of characteristic 0, we may
take the sums of powers of di variables as generators for the elementary symmetricpolynomials in those variables. We may also generate the product of the inversesof the variables from sums of powers with negative exponents.
P

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

in families, and hence that the family of G-representations parameterised by ZΩ— and thus the parameter ring ZΩ itself — should be completely determined by
the corresponding family of Weil group representations, which are encoded by theZΩ -valued pseudo-representation T .
If we let AΩ denote the E-subalgebra of ZΩ generated by the image of T , thenthis is a finite type E-algebra, and we have a morphism Spec ZΩ → Spec AΩ . It
is not hard to see (e.g. by applying local Langlands over the fraction field of AΩ )that this is a birational map, which is in fact a bijection on points (as one sees byapplying local Langlands at the closed points). Unfortunately, we were unable tofind a completely conceptual proof in general that this morphism is an isomorphismof varieties over E.
In the case when ZΩ 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 ZΩ . Let πx be the supercuspidal
G-representation corresponding to x, and let Tx : WF → Ex the specialisation of
T to the image of x in Spec AΩ . Let Rx be the universal formal deformation ringof Tx , so that we have morphismsc
Spf ZcΩx → Spf AΩ → Spf Rx ,the second being induced by T . Let rx : WF → GLn (Ex0 ) denote the (absolutely)
irreducible representation attached to πx via local Langlands, where Ex0 /Ex is a
finite extension, and let Tx0 denote the composite Tx : WF → Ex → Ex0 . Then Tx0
is the pseudo-representation attached to rx . Since rx is irreducible, the universal
formal deformation rings of rx and Tx0 coincide ([Nys96, Théorème 3], [Rou96,
Corollaire 6.2]), and are thus both given by Rx ⊗Ex Ex0 . A direct analysis, using thatthe source and target are both obtained simply by forming unramified twists, andthat local Langlands gives a bijection on isomorphism classes that is compatible
with twisting, shows that the composite of the base change to Ex0 of the abovemorphisms is an isomorphism. Since the first of them is dominant, it is also an
isomorphism. Thus the morphism Spec ZΩ → Spec AΩ 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 ZΩ parameterises a family of supercuspidal representations, where the preceding argument
then applies. In this way, one could give a slightly more conceptual proof of theLemma.
Proof of Proposition 4.2. We adopt the notation of §3.12. In particular M is theLevi 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
1−n∼group automorphism X (M ) −→ X (M ) given by χM 7→ χM | det | 2 gives rise to an∼
E-isomorphism Spec ZD −→ Spec ZD . 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 ZΩ −→ Spec ZΩ in view of Lemma 3.21. Let tw : ZΩ → ZΩdenote the induced isomorphism.
∼
We have a natural isomorphism ισ : H(σsm ) → H(σ); viewing Hecke algebras
as endomorphism-valued functions on G, this is given by ψ 7→ ψ · σalg . (This is a

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

35

priori only an injection, but in fact is an isomorphism by the proof of Lemma 1.4of [ST06].) Now we construct η as the following composite map
ι−1
σ

tw

I

H(σ) ' H(σsm ) ' ZΩ → ZΩ → OX rig ,where the second map comes from Lemma 3.24. Note that η is already definedover E. To verify the desired interpolation property of η, we let x : OX rig → Ex
be an E-algebra map. Then x ◦ I ◦ tw : ZΩ → Ex gives the supercuspidal supportof πsm (ρx ) = rp−1 (WD(ρx )F −ss ) by Proposition 4.3; indeed, since I interpolates
recp by that proposition, I ◦ tw interpolates rp . In order to complete the proof,
we can and do base change to Qp . Then Proposition 3.9 shows us that ZΩ acts on
HomK σsm , πsm (ρx ) through x ◦ I ◦ tw.
To conclude, it is enough to observe that the action of H(σsm ) on the space
HomK σsm , πsm (ρx ) is compatible with the ZΩ -action on the same space via theisomorphism H(σsm ) ' ZΩ , and also with the H(σ)-action on HomK (σ, πl.alg,x ) via
the canonical isomorphisms between the algebras (via ισ ) and the modules. These
are readily checked.
In order to deduce Theorem 4.1 from Proposition 4.2, we will now use the resultsof [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 = σalg ⊗ σsm , where σsm = σ(τ ) for some inertial type τ : IF →GLn (E). Let x : Rp̃ (σ)[1/p] → Ex be a closed point, so that Ex is a finite extension
field of E. Then x defines a local Galois representation ρx : GF → GLn (Ex ) whichis potentially crystalline, and has Hodge–Tate weights determined by the highest
weight ξ of σalg . Set πx := πsm (ρx ). Recall that πl.alg,x is the locally algebraic
representation defined over Ex corresponding to ρx (see §1.5), whose smooth partis πx and which determines the character χπx ◦ ι−1
σ : H(σ) → Ex (via the action ofH(σ) on HomK (σ, πl.alg,x ).
Let P = M N be a parabolic subgroup of G, with Levi M and unipotent radical N . Let Z(M ) be the centre of M , let N0 ⊂ N be a compact open subgroupand define Z(M )+ := {t ∈ Z(M )|tN0 t−1 ⊂ N0 }. 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
JP 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 = M N as above, with modulus
character δP , every χ : Z(M ) → E × such that HomZ(M ) χ, JP (V ) 6= 0, and everyt ∈ Z(M )+ , we have |χ(t)δP (t)−1 |p ≤ 1.
As above, we write τ = ⊕ri=1 (τi )di , where the τi are pairwise non-isomorphic
IF -representations corresponding via the inertial
Qr local Langlands correspondence
to cuspidal types σ(τi ) of GLei (OF ). Let M = i=1 GLei (F )di be a standard Leviof G, with corresponding standard parabolic P = M N .
From now on until the end of this section, we will replace πx (as well as χπx ,
σsm , and so on) by its base extension to Qp . Then as recalled in Section 3.1,
πx is the unique irreducible quotient of a normalised parabolic induction π̃x :=diriG
P πx,M , where πx,M = ⊗i=1 (⊗j=1 πx,i,j ) such that each πx,i,j is a supercuspidal

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

representation of GLei (F ) containing the type σ(τi ) and where for each i, we have
πx,i,j 6' πx,i,j 0 (1) for 1 ≤ j 0 < j ≤ di .
By Proposition 3.9 we have a G-equivariant homomorphism ϕ : c-IndG
K σsm → π̃x ,Q.
We
identify
Wwhich identifies π̃x with c-IndG
σ⊗[M,πx,M ] withH(σsm ),χπx
p
K sm
Qr
i=1 Sdi in the obvious way, where Sdi is the symmetric group on {1, . . . , di }.
Note that W[M,πx,M ] and the identification are independent of x.
4.8. Lemma. Let χ(πx,i,j ) denote the central character of πx,i,j . For an element
×w = {wi }ri=1 ∈ W[M,πx,M ] , define characters χx,w : Z(M ) → Qp by χx,w =Nr Ndii=1
j=1 χ(πx,i,wi (j) ).
For every t ∈ Z(M )+ , there exists a constant Ct such that |χx,w (t)|p ≤ Ct for
all points x of X rig .
Proof. We know that σalg ⊗ π̃x , after twisting by a unitary character (det | det |)1−n(this twist is discussed at the beginning of §5 below), corresponds to the potentiallycrystalline Galois representation ρx with Hodge–Tate weights determined by σalg ,in the sense that π̃x | det |1−n ↔ WD(ρx )F −ss via the modified local Langlands
correspondence as in Section 4 of [BS07]. Moreover, note that by Lemma 4.2 ofSection 4 of [BS07], σalg ⊗ π̃x actually has a model over a sufficiently large finite
extension of Qp , so the characters χx,w then take values in some sufficiently large
finite extension Ex0 /Qp .
By the equivalence between parts (ii) and (iv) of Theorem 1.2 of [Hu09] (where
our coefficient field is taken to be Ex0 ), we know that σalg (det)1−n ⊗ π̃x | det |1−n has
a unitary central character and satisfies Condition 4.7 [Eme06a]. Therefore so does
σalg ⊗ π̃x .
∼N
Note that by Proposition 4.3.6 of [Eme06a] we have JP (σalg ⊗ π̃x ) → σalg⊗
1/2

rPG (π̃x )δP , where rPG 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 ≤N(t) · δP (t)−1/2 |−1 for every χ occurring in rPG (π̃x ).
|σalg
Now, Proposition 3.2(2) of [Hu09] computes rPG (π̃x ) (observe that in the notation of loc. cit., all bi are 1 in our case) and shows that the characters χx,w =Nr Ndii=1
j=1 χ(πx,i,wi (j) ) of Z(M ) for all sets w of permutations wi of {1, . . . , di }occur in rPG (π̃x ). 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 GLdeii .
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 GLdeii of M such that the image of t under that permutation has non-decreasing
p-adic valuations. Let M 0 be the Levi subgroup of G with the permuted blocks asfactors. Abstractly, M 0 ' M and by Proposition 6.4 of [Zel80], we know that theinduction iG
P πM is independent of the ordering of the τi . We conclude by applying
Lemma 4.8 to the Levi M 0 instead of M .

As discussed in Sections 3.3 and 3.6, we have a semisimple Bushnell–Kutzko type(J, λ) such that σsm is a direct summand of IndKJ (λ),and the natural map sP :∼H(G, λ) → H(σsm ) induces an isomorphism Z H(G, λ) −→ H(σsm ). In particular,

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

37

this means that π̃x |J contains λ. Then in the notation of Section 3.3, πx,M contains
the type (J ∩ M, λM ). Let χπx,M be the character by which H(M, λM ) acts on
HomJ∩M (λM , πM ).
4.10. Corollary. Let t ∈ Z(M ) and let νt ∈ H(M, λM ) be an intertwiner supportedon t(J ∩ M ). Then there exists a constant Ct such that for all points x of X rig wehave |χπx,M (νt )|p ≤ Ct .
Proof. Assume νt 6= 0. Note that since t ∈ Z(M ), νt (t) commutes with the actionof J ∩M on λM and since λM is irreducible we deduce that νt (t) is a nonzero scalar.
Rescaling, we may assume that νt (t) := idλM . Let s ∈ Z(M )++ be such that s = w t
for some w ∈ W[M,πx,M ] . It follows from the definitions that χπx,M (νt ) = χx,w (s),and the corollary then follows from Corollary 4.9.

4.11. Corollary. Let ν ∈ H(M, λM ). Then there exists a constant Cν such that|χπx,M (ν)|p ≤ Cν for all points x of X rig .
Proof. Since ν has compact support and we only need some bounded constant Cν , itsuffices to prove the claim in the case that ν is an element of the basis of H(M, λM )given by Lemma 3.4. Let ν 0 be the e-fold convolution of ν with itself, where e is asin the statement of Lemma 3.4; then from Corollary 4.10 applied to ν 0 , we see that
there is some constant Cν such that |χπx,M (ν 0 )|p ≤ Cνe for all x. This implies that|χπx,M (ν)|p ≤ Cν , as required.
4.12. Proposition. For any ν ∈ H(σ), there is a constant Cν such that|χπx ι−1σ (ν) |p ≤ Cνfor all points x of X rig .
Proof. Recall from Section 3.6 that we have an isomorphism
W(sP ◦ tP ) : H(M, λM ) [M,πx,M ] → H(σsm ).
Set νt := (sP ◦ tP )−1 ι−1σ (ν) . Corollary 4.11 implies that there is a constant Cνsuch that |χπx,M (νt )|p ≤ Cν for all x.
Now, in order to conclude, we just need to relate χπx,M (νM ) to χπx (sP ◦tP )(νM )W
for any νM ∈ H(M, λM ) [M,πx,M ] . As recalled in Section 3.3, iG
P correspondson the level of Hecke modules to pushforward along the map tP . More precisely, if we let M := HomJ∩M (λM , πx,M ) and N := HomJ (λ, πx ), then N 'HomH(M,λM ) (H(G, λ), M). Here we view H(G, λ) as a left H(M, λM )-module via
tP 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 (tP )−1 (z) on M. On theother hand, HomK (σ, πx ) ' eK N for the idempotent eK in H(G, λ) which defines
σ. Therefore, any eigenvalue of eK ∗ z on eK N is an eigenvalue of z on N . We
deduce that χπx,M (νM )=χπx (sP ◦ tP )(νM ) .
Proof of Theorem 4.1. By [Kis08, Thm 3.3.8] we know that Rep (σ)[1/p] is a regularring. Proposition 7.3.6 of [dJ95] (which is applicable because Rep (σ)[1/p] is inparticular 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

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

Rep (σ) in Rep (σ)[1/p], and so the ring of bounded rigid analytic functions on X rig
is equal to Rep (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d◦ d
M∞ (σ ◦ ) := Homcont
(M
,
(σ
)
)
.
∞
O[[K]]
Here we are considering continuous homomorphisms for the profinite topology on◦ d
M∞ and the p-adic topology on (σ ◦ )d . We equip Homcont
O[[K]] (M∞ , (σ ) ) with the p◦
adic topology. Note that M∞ (σ ) is an O-torsion free, profinite, linear-topologicalO-module.
4.13. Lemma. There is a natural isomorphism of topological O-modules∨
M∞ (σ ◦ ) ∼Homcont(M∞ , (σ ◦ /$n )∨ ) .
= lim
O[[K]]
←−
n

◦ d
◦
d
d ∼
Proof. Let H := Homcont=
O[[K]] (M∞ , (σ ) ), so that M∞ (σ ) = H . Then H
n
n ∼
n ∨
limn HomO (H, O/$n ) ∼
lim
Hom
(H/$
,
O/$
)
lim
(H/$
)
.
Since
M
= ←−n
= ←−n
O
∞ is
←−

a projective O[[K]]-module, the short exact sequence$ n·

0 → (σ ◦ )d → (σ ◦ )d → (σ ◦ )d /$n → 0
◦ d
n
◦ d
n ∼
yields an isomorphism H/$n ∼= HomcontO[[K]] (M∞ , (σ ) /$ ). Finally, (σ ) /$ =
n ∨
◦
n
n ∼
HomO (σ /$ , O/$ ) = (σ/$ ) .

4.14. Remark. One may modify the proof of Lemma 4.13 to show that M∞ (σ ◦ ) is∨
◦ ∨naturally isomorphic to Homcont.
O[[K]] (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 v1 , notedin the discussion of Subsection 2.3, as well as Remark 2.7) we will not make this
precise.
◦Let H(σ ◦ ) := EndG (c-IndG
K σ ); this is an OE -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
Homcont (M∞ , (σ ◦ )d ) ∼= HomK (σ ◦ , (M∞ )d ).
O[[K]]◦d
Frobenius reciprocity gives HomK (σ ◦ , (M∞ )d ) ∼= HomG (c-IndG
K σ , (M∞ ) ). Thus◦◦
M∞ (σ ) is equipped with a tautological Hecke action of H(σ ), which commuteswith 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)0 := R∞ ⊗R Rep (σ).
ep4.16. Lemma.
(1) R∞ (σ) is a reduced O-torsion free quotient of R∞ (σ)0 .
◦(2) If h ∈ H(σ ) is such that η(h) ∈ Rp̃ (σ), then the action of h on M∞ (σ ◦ )agrees with the action of η(h) via the natural map Rp̃ (σ) → R∞ (σ)0 .

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

39

Proof. (1) That R∞ (σ) is O-torsion free follows immediately from the fact that bydefinition it acts faithfully on the O-torsion free module M∞ (σ ◦ ). The fact that itis actually a quotient of R∞ (σ)0 is then essentially an immediate consequence ofclassical local-global compatibility at ep, but to see this will require a little unravelingof the definitions. Note that if N is sufficiently large, then KN acts trivially on(σ ◦ )d /$N . Using Lemma 4.13, we see that∨M∞ (σ ◦ )∨ = lim HomO[Γ2N ] (M∞ /bN )K2N , (σ ◦ )d /$N
←−
N
∨


◦ d
N
,
= lim HomO[Γ2N ] (M1,Q
/b
)
,
(σ
)
/$
N
K
2N
0
N (N )←−
N

so it suffices to show that if N  0 then the action of R∞ on


◦ d
N
Homcont
O[[K]] M1,QN 0 (N ) , (σ ) /$factors through R∞ (σ)0 . Now, by definition we have=M1,Q
N 0 (N )0pr∨ Sξ,τ (U1 (QN 0 (N ) )2N 0 (N ) , O/$N (N ) )∨mQ 0

N (N )

so it suffices to prove the same result for
N 0 (N ) ∨Homcont
O[[K]] Sξ,τ U1 (QN 0 (N ) )2N 0 (N ) , O/$m

Q

N 0 (N )⊗RSuniv
Q

N 0 (N )⊗RSuniv
Q

N 0 (N )RSQT 0,N (N )RSQT 0N (N ),(σ ◦ )d /$N ,which is equal to
N 0 (N ) ∨Homcont
O[[K]] Sξ,τ U1 (QN 0 (N ) )2N 0 (N ) , O/$m

Q,
N 0 (N )(σ ◦ )d /$N ⊗RSunivQ

N 0 (N )RSQT 0∨,N (N )

which in turn equalsSξ,τ U1 (QN 0 (N ) )0 , (σ ◦ )d /$N m

Q

N 0 (N )⊗RSuniv
Q

N 0 (N )Therefore it would suffice to prove the same result for
Sξ,τ U1 (QN 0 (N ) )0 , (σ ◦ )d m
⊗RSuniv
Q

N 0 (N )Q

N 0 (N )∨RSQT 0.

N (N )RSQT 0∨.

N (N )

If T denotes the image of TSp ∪QN 0 (N ) ,univ in the endomorphism ringEndO Sξ,τ U1 (QN 0 (N ) )0 , (σ ◦ )d m,
Q 0
N (N )
then the action of RSunivon Sξ,τ U1 (QN 0 (N ) )0 , (σ ◦ )d m
is given by an
Q 0
N (N )Q

N 0 (N )O-algebra homomorphism RSuniv→ T. Since the space of automorphic forms
Q 0
N (N )
◦dSξ,τ U1 (QN 0 (N ) )0 , (σ ) m
is O-torsion free (by the choice of Um,v1 in SecQN 0 (N )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 spaceof cuspidal automorphic forms; cf. [CHT08, Corollary 3.3.3 and §3.4]), then by thedefinition of Rep (σ), we need to show that if T → Qp is a closed point, then

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

the restriction to GFpe of the corresponding Galois representation GFe+ ,T ∪Q 0

N (N )→Gn (Qp ) is potentially crystalline of type σ; but this is immediate from classicallocal-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)0 [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 thisargument.)(2) Again, this is essentially an immediate consequence of classical local-globalp, but a little explanation is needed in order to make this plain.
compatibility at e
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 byN
αN 0 (N ) : (M∞ /bN )K2N → c-IndG
KZ (M∞ /bN )KN
and these in turn, after taking homomorphisms into (σ ◦ )d /$N , induce partialH(σ ◦ )-actions on spaces of algebraic modular forms of weight (σ ◦ )d . (More precisely, the G-action on $-adically completed cohomologyS̃ξ,τ U1p (QN 0 (N ) ), O := lim lim Sξ,τ U1 (QN 0 (N ) )m , O/$s
←− −→
s

m

gives rise, via Frobenius reciprocity and the identificationSξ,τ U1p (QN 0 (N ) )0 , (σ ◦ )d ' HomK σ ◦ , S̃ξ,τ U1p (QN 0 (N ) ), O ,to a natural action of H(σ ◦ ) on Sξ,τ U1 (QN 0 (N ) )0 , (σ ◦ )d .) We see therefore, as inpart (1), that it is enough to consider the natural action of each h ∈ H(σ ◦ ) on thespaces∨⊗RSuniv
RSQT 0Sξ,τ U1 (QN 0 (N ) )0 , (σ ◦ )d m
Q

N 0 (N )Q

N 0 (N )N (N )

for N  0. In addition to the natural action of H(σ ◦ ), this is equipped with anaction of Rep (σ) via the composite Rep → Rloc → RSQT 0 , which factors through
N (N )

Rep (σ) by part (1). By classical local-global compatibility and the defining propertyof the morphism η of Theorem 4.1, we see that, after inverting p, the action of hon this space agrees with the action of η(h). The desired result now follows fromthe fact that Sξ,τ U1 (QN 0 (N ) )0 , (σ ◦ )d m
is O-torsion free.
Q

N 0 (N )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), andmoreover 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)0 is a union of irreducible components of R∞ (σ)0 .
(3) Let 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 ◦ ) → R∞ (σ).

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

41

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◦ d dsummand of S∞ [[K]]r for some r ≥ 0, and so Homcont
is a direct
O[[K]] M∞ , (σ )

 r

cont
cont
◦ d d
r
◦ d d ∼
. Thus
S∞ [[K]], (σ )summand of Hom
S∞ [[K]] , (σ )= HomO[[K]]

O[[K]]b O O[[K]] as an O[[K]]-module, there isit suffices to note that since S∞ [[K]] ∼= S∞ ⊗
a natural isomorphism


cont
◦ d d
◦ d d ∼
Homcont= HomcontO[[K]] O[[K]], HomO (S∞ , (σ ) )O[[K]] S∞ [[K]], (σ )∼= S∞ ⊗O σ ◦ .
= Homcont (S∞ ⊗O σ ◦ , O)d ∼O◦

Since the S∞ -action on M∞ (σ ) factors through the action of R∞ , which in turnfactors 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
EndS∞ 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 3.3.8of [Kis08], we see that the ring R∞ (σ)0 is equidimensional of the same Krull dimension as S∞ . Since M∞ (σ ◦ ) is free of finite rank over S∞ , and the image ofR∞ (σ)0 in End(M∞ (σ ◦ )) is an S∞ -algebra, we see that the depth of M∞ (σ ◦ ) asan R∞ (σ)0 -module is at least the Krull dimension of S∞ . Since this is equal to
the Krull dimension of R∞ (σ)0 , it follows immediately from Lemma 2.3 of [Tay08]
that the support of M∞ (σ ◦ ) is a union of irreducible components of R∞ (σ)0 . (Ofcourse, conjecturally R∞ (σ) is actually equal to R∞ (σ)0 .)That M∞ (σ ◦ )[1/p] is locally free over R∞ (σ)[1/p] follows by an argument of
Diamond (cf. [Dia97]). More precisely, Rep (σ)[1/p] and each of the rings Rveσ,ξ,τ [1/p]for places v | p, v 6= p are regular by Theorem 3.3.8 of [Kis08], and Rve1 is smooth by
the choice of v1 , so R∞ (σ)0 [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 v1(and the fact that we have fixed the action mod p of the Hecke operators at ve1 ).
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 EndR∞ (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 R∞ (σ).
The morphism α of Lemma 4.17 induces an E-algebra morphism α : H(σ) →R∞ (σ)[1/p].

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

4.18. Theorem. The map α coincides with the compositionηH(σ) → Rep (σ)[1/p] → R∞ (σ)[1/p].
Proof. Note firstly that if h ∈ H(σ ◦ ) is such that η(h) ∈ Rp̃ (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 h0 ∈ H(σ ◦ ), we have η(pC h0 ) ∈ Rp̃ (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 ofSpec Rep (σ)[1/p] which come from closed points of Spec R∞ (σ)[1/p] is a unionof irreducible components, which we call the set of automorphic components ofSpec Rep (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 Rep (σ)[1/p] is an automorphic component.)
4.20. Remark. We expect that η(H(σ ◦ )) is contained in the normalisation of Rep (σ)
in Rep (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 Rep (σ) corresponding to the automorphic components
in the sense of Remark 4.19.
5. The Breuil–Schneider conjectureContinue to assume that p - 2n, and that F is a finite extension of Qp . If
r : GF → GLn (E) is a de Rham representation of regular weight then we say thatr is generic if πsm (r) is generic. In this case, we set
BS(r) := πalg (r) ⊗ πsm (r).
(In fact, our BS(r) differs from the definition made in [BS07] in that πalg (r) andπsm (r) are their analogues in [BS07] times the characters detn−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 thedifference between these conventions.) The following is [BS07, Conjecture 4.3] (inthe open direction, in the generic case).
5.1. Conjecture. If r : GF → GLn (E) is de Rham and has regular weight, thenBS(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 : GF → GLn (E), and assume from now on that r ispotentially 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 : GF → GLn (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 : Rep → O, which we extended to an O-algebra homomorphism

PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

43

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 σsm (r) for σ(τ ), σalg (r) for πalg (r)|K , and let σ := σalg (r)⊗σsm (r), keeping
in mind the convention at the end of Section 3.12. (Also enlarge E to anotherfinite extension if necessary as explained in that section.) As above, we writeGH(σ) for EndG (c-IndG
σ),
which
is
isomorphic
to
End
c-Ind
σ(τ
)
via ισ , soG
KK
that πsm (r) determines a character χπsm (r) : H(σ) → E. Since r is generic, we seefrom Corollary 3.11 that πsm (r) ∼= c-IndG
K σsm (r) ⊗H(σ),χπsm (r) E. Tensoring with
πalg (r), we haveBS(r) ∼= (c-IndG
K σ) ⊗H(σ),χπsm (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 HomG (BS(r), V (r)) 6= 0. While
we cannot at present do this in general, we are able to reinterpret the problem interms 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 seriescase.
5.3. Theorem. Suppose that p - 2n, and that r : GF → GLn (E) is a generic
potentially crystalline representation of regular weight. If r corresponds to a closed
point on an automorphic component of Rep (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 HomG BS(r), V (r) 6= 0. Proposition 2.20 of [Paš12] implies thatdimE HomK σ, V (r) = dimE 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 ourcase: 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
HomG c-IndG
K σ, V (r) = Frobenius reciprocity, this shows that
HomK σ, V (r) 6= 0, as y is in the support of M∞ (σ ◦ )[1/p] by assumption. Since◦
BS(r) ∼= (c-IndG
K σ) ⊗H(σ),χπsm (r) E, we need only show that the action of H(σ ) on◦
M∞ (σ ) ⊗R∞ ,y O factors through the character χπsm (r) ; but this is immediate from
Theorem 4.18.

5.4. Corollary. Suppose that p - 2n and that r : GF → GLn (E) is de Rhamof 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 Rep (σ)[1/p]. Recalling that y was chosen to correspondto the potentially diagonalizable representation rpot.diag at the places v | p, v 6= 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 ofRep (σ)[1/p] as r (cf. the proof of Corollary 4.4.3 of [GK12]).

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

Indeed, the existence of a global automorphic Galois representation corresponding to a point on Rep (σ)[1/p] shows that Sξ,τ (U0 , (σ ◦ )d )m [1/p] is supported at thispoint. We have Rep (σ)[1/p]-equivariant isomorphismsSξ,τ (U0 , (σ ◦ )d )m [1/p] ' HomK σ ◦ , S̃ξ,τ (U p , O)m [1/p]◦ d' HomcontO[[K]] M∞ /aM∞ , (σ ) [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 isodmorphisms imply that M∞ (σ ◦ )/aM∞ (σ ◦ ) [1/p] is supported at the same pointof Rep (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 Rep (σ)[1/p] corresponding to r is
automorphic in the sense of Remark 4.19.

5.5. Corollary. Suppose that p > 2, that r : GF → GLn (E) is de Rham of regularweight, and that r is generic. Suppose further that either(1) n = 2, and r is potentially Barsotti–Tate, or(2) F/Qp is unramified, r is crystalline, n 6= p and r has Hodge–Tate weights
in the extended Fontaine–Laffaille range; that is, for each κ : F ,→ E, any
two elements of HTκ (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 incase (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 thisversion, in the following way: given an (admissible) unitary Banach completion ofBS(r) ⊗E E 0 , where E 0 /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 therequired 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 HomG 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 thesocle 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 otherconstituent 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-IndG→ BS(r); rather, there should be a surjection from theK σ) ⊗H(σ),χπsm (r) E −PATCHING AND THE p-ADIC LOCAL LANGLANDS CORRESPONDENCE

45

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 thiskernel will admit non-trivial unitary completions.
5.8. Remark. Our techniques also apply to the converse question of describing therepresentations 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 :GF → GLn (E) is a representation with the property that V (r) has non-zero locallyalgebraic 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 : GF → GLn (F) with trivial endomorphisms1 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 : GF → GLn (E) arising from an OE valued point x of Spec Rr̄ , we may associate a unitary Banach space representation
B(r) := (L∞ ⊗Rr̄ ,x OE )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 : GF → GLn (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 andthe Hecke algebra that acts on it introduced in the discussion preceding CorolSplary 2.9, there is an isomorphism of Tξ,τ(U p , O)m [G]-modules
SpSeξ,τ (U p , O)dm ∼(U p , O)m ⊗Rr̄ L∞= Tξ,τ
Sp(here Tξ,τ(U p , O)m is regarded as an Rr̄ -algebra via the natural maps Rr̄ →SpRSuniv → Tξ,τ(U p , O)m , where the first morphism corresponds to restricting global
Galois representations to the decomposition group at p, and the second morphism

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 withthe framed deformation ring Rr̄ in everything that follows, and the representation L∞ would beendowed with a further equivariant structure for the group GLn acting by “change of framing”.

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

is induced by the universal automorphic deformation of ρ), as well as analogous isomorphisms when we add auxiliary QN -level structure, as in the patching process.
6.2. The relationship between L∞ and M∞ . If Condition 6.1.2 held, thenwe 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 (M1 , QN 0 (N ) ⊗S∞ S∞ /bN )K2N that arise in the patchingprocess. 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-backto 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 ρ corresponding to a point of RSuniv , then it would contribute to the locally algebraic vectors ofcompleted 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. (Thisis an abstraction of the strategy used to deduce the Fontaine–Mazur conjecture forodd two-dimensional representations of GQ in [Eme].)6.4. Concluding remarks. The preceding discussion shows that the question ofwhether 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 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 apurely local construction over Spec Rr̄ .
Furthermore, it seems quite possible that our hypothetical formulation of thep-adic local Langlands correspondence is too naive; even if a local correspondenceof 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