THE LOCAL LANGLANDS CORRESPONDENCE FOR GLn IN
FAMILIES
MATTHEW EMERTON AND DAVID HELM

Contents
1. Introduction
2. Representation theory — general background
3. Representation theory — the case of GLn
4. The local Langlands correspondence in characteristic zero
5. The local Langlands correspondence in characteristic p
6. The local Langlands correspondence in families
References

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1. Introduction
The goal of this paper is to extend, for any non-archimedean local field E ofresidue characteristic `, the local Langlands correspondence between n-dimensional
Weil-Deligne representations of the Weil group WE and admissible smooth representations of GLn (E) to a correspondence defined on p-adic families of representationsof the absolute Galois group GE of E (for primes p distinct from `).
Fix an algebraically closed field K containing Qp . Then on one side of thelocal Langlands correspondence are the n-dimensional, Frobenius-semisimple WeilDeligne representations of WE ; that is, pairs (ρ, N ), where ρ is an n-dimensional,
semisimple representation of WE over K, and N is a nilpotent endomorphism of
the representation space of ρ such that ρ(w)N ρ(w)−1 = q |w| N for all w ∈ WE .
On the other side are the admissible smooth K-representations of GLn (E); thatis, representations π of GLn (E) such that every element of π is fixed by somecompact open subset U of GLn (E), and such that, for all such U , the space π U of
U -invariant vectors in π is finite-dimensional.
The most naive thing one could hope for would be the following: given a familyof representations of GE over a complete, Noetherian, local, p-torsion free ring A ofcharacteristic zero and residue characteristic p (that is, a representation ρ : GE →GLn (A) for such an A), one could hope to find an admissible smooth A[GLn (E)]module π such that, for every characteristic zero prime ideal p of A, with residue
field κ(p), the representation π ⊗A κ(p) of GLn (E) corresponds, via local Langlands,
with the Frobenius-semisimple Weil-Deligne representation attached to ρ ⊗A κ(p).
A moment’s consideration will show that this is far too much to hope for. Indeed, if ρ is a direct sum of two unramified characters that specializes at a single
The first author was supported in part by NSF grants DMS-0401545, DMS-0701315, andDMS-1002339.
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MATTHEW EMERTON AND DAVID HELM

characteristic zero point p0 to the trivial character plus the cyclotomic character,then the local Langlands correspondence would tell us that π ⊗A κ(p0 ) would be
one-dimensional. On the other hand at general points p one would need π ⊗A κ(p)to be infinite-dimensional.
Indeed, as this example suggests, even formulating a precise statement of a “local
Langlands correspondence in families” is nontrivial. Our approach is motivated by
global considerations, suggested by work of the first author [6].
1.1. Global motivation. In the setting of [6], the ring A is typically a p-adicallycompleted Hecke algebra, and ρ is the two dimensional representation of GQ over A
arising from the theory of p-adic modular forms. For a certain finite set of primes
Σ containing p, the Hecke algebra A acts on a suitable localization of p-adicallycompleted cohomology H 1 (XΣ ) of the tower of modular curves of levels divisible
by primes in Σ. This cohomology is also equipped with commuting actions of GQ ,of GL2 (Qp ), and of GL2 (Q` ) for ` a prime of Σ not equal to p.
What is shown in [6] is that the space H 1 (XΣ ) has a natural tensor factorization, and that the tensor factor corresponding to ` interpolates the local Langlands
correspondence in a natural way. More precisely, one has a factorization:O
H 1 (XΣ ) ∼π`= ρ ⊗ πp ⊗`∈Σ\{p}where πp is a certain representation of GL2 (Qp ) attached to ρ|GQp via considerations
arising from the p-adic local Langlands correspondence, and the representations π`
depend only on the restriction of ρ to GQ` , and interpolate the local Langlands
correspondence in a natural way. (We refer the reader to Conjecture 6.1.6 of [6] fora precise statement; Proposition 6.2.13 of [6] establishes this conjecture under mild
hypotheses.)
The representations π` have a number of nice properties, which suggest the“shape” that the desired “local Langlands correspondence in families” should take:• π` is the smooth W (k)-dual of a (unique) admissible smooth A[GL2 (Q` )]module π`0 , where k is the residue field of A. (Modules with this property
are called “coadmissible” and studied in detail in Appendix C of [6].)• π` is “A-cotorsion free” in the sense of Definition C.37 of [6].
• For a Zariski dense set of characteristic zero primes p of A, the space π` [p]of p-torsion vectors in π` is the representation attached to ρ ⊗A κ(p) bya “generic” version of the local Langlands correspondence due to Breuiland Schneider. (This correspondence differs from the usual local Langlands correspondence in several ways. Rather than being a bijection it isa map from isomorphism classes of n-dimensional representations of GE
to indecomposable (but not necessarily irreducible) generic representationsof GL2 (E). We refer the reader to Section 4 for the definition and basic
properties of the Breuil-Schneider correspondence.)• The space (π` /pπ` )[m] (where m is the maximal ideal of A) has an absolutely
irreducible generic socle, and no other generic subquotients. (Here recall
that the socle of a k[GLn (E)]-module is its maximal semisimple submodule;
that is, the sum of all of its irreducible submodules. A generic representation of GLn (E) is one that whose restriction to the unipotent radical of a
Borel contains a generic character; for GL2 (Q` ) an irreducible admissible
representation is generic if, and only if, it is infinite dimensional.) In theLOCAL LANGLANDS IN FAMILIES

3

setting above, the genericity of the socle of (π` /pπ` )[m] is a consequence of
Ihara’s lemma.
Given an arbitrary A, and an arbitrary family ρ : GE → GLn (A), it is natural inlight of the above properties to ask if one can attach an A[GLn (E)]-module π to ρ,with properties similar to those above, and to ask if such properties characterize asuch a module uniquely. In fact, in order to avoid the complication of dealing withthe theory of coadmissible modules, we will work in a setting dual to the above
picture.
1.2. The local Langlands correspondence for GLn in p-adic families. We
are now in a position to state our main result. Before we do so, we introduce further
notation: Let K be a field of characteristic zero. Then given ρ : GE → GLn (K)as above, we write π(ρ) for the representation attached to ρ by the generic local
Langlands correspondence of Breuil-Schneider, and we write πe(ρ) to denote thesmooth contragredient of π(ρ).
1.2.1. Theorem. Let A be a reduced complete p-torsion free Noetherian local ringwith maximal ideal m and finite residue field k of characteristic p. If ρ : GE →GLn (A) is continuous (when the target is given its m-adic topology), then there
exists at most one admissible smooth GLn (E)-representation V over A, up to isomorphism, satisfying the following conditions:(1) V is A-torsion free.
(2) If a is a minimal prime of A, with residue field κ(a), then there is a κ(a)linear GLn (E)-equivariant isomorphism ∼πe κ(a) ⊗A ρ −→ κ(a) ⊗A V.
(3) If we write V := k ⊗A V , then the GLn (E) cosocle cosoc(V ) of V is absolutely irreducible and generic, while the kernel of the surjection V →cosoc(V ) contains no generic Jordan–Hölder factors.
Furthermore, if such a V exists, then:1(4) There exists an open dense subset U of Spec A[ ], such that for each primep
p in U , there is a GLn (E)-equivariant, nonzero surjectionπe κ(p) ⊗A ρ → κ(p) ⊗A V,
where κ(p) is the residue field of p. Moreover, there is an open dense subset
U 0 of U such that if p lies in U 0 then this surjection is an isomorphism.
The admissibility condition on V , together with properties (1) through (3) above,are (roughly) dual to the properties satisfied by the tensor factors π` . Indeed,
we prove a “recognition theorem” (Theorem 6.2.15 below) that states that if thesmooth W (k)-dual of V satisfies the properties listed for π` above (for a given ρ),
then V is admissible and satisfies properties (1) through (3).
If a representation V satisfying the conditions of this theorem with respect toa given Galois representation ρ : GE → GLn (A) exists, then we write V := πe(ρ).
(Note that the theorem ensures that V is unique up to isomorphism, so that πe(ρ)
is then uniquely determined by ρ, up to isomorphism, if it exists.) We justify thisnotation by thinking of representations of GE over fields of characteristic zero as“families over a single point”, so that the map ρ 7→ πe(ρ), where it is defined, extends

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MATTHEW EMERTON AND DAVID HELM

the dual of the Breuil-Schneider correspondence from one-point families to families
over local rings A as in the theorem.
Part (4) of the theorem describes the precise sense in which V interpolates the
local Langlands correspondences attached to the Galois representations κ(p) ⊗A ρ
1
as p ranges over the points of Spec A[ ]. Conjecturally, we can take U equal to allp
1of Spec A[ ] in statement (4), although our results fall short of establishing this.
p(We refer the reader to Theorems 6.2.5 and 6.2.6 for the precise results.) On theother hand, we will give examples in Section 6 showing that the subset U 0 is, in
1general, not equal to all of Spec A[ ].
p1.2.2. Remark. Our convention for the generic local Langlands correspondence isthat the GLn (K)-representation π(ρ) attached to a continuous Galois representation ρ : GE → GLn (K) should have generic socle. It is this convention that seems
to fit best with global applications of the type considered in [6] and [7], for example.
On the other hand, when working with families, it turns out to be easier to interpolate representations whose cosocle is generic. This is because one may work withadmissible A[GLn (E)]-modules, rather than the less-familiar coadmissible modulesof [6]. This explains the appearance of the various contragredient representationsin Theorem 1.2.1, and in our notation for the representations that it describes.
1.3. Global applications. Although we have used the results of [6] to motivate
our formulation of the local Langlands correspondence in families, in fact Theorem 1.2.1, together with the “recognition theorem” (Theorem 6.2.15) describedabove, are essential ingredients in the proofs of the main results of [6].
In the language of Theorem 1.2.1, the results of [6] state that (again, under
mild hypotheses) when A is a suitable p-adically completed Hecke algebra, andρ : GQ → GLn (A) is the representation arising from Eichler-Shimura theory, thenthe representations πe(ρ|GQ` ) exist for all ` not equal to p, and one has a tensor
factorization:O
H 1 (XΣ ) ∼πe(ρ|G )∨ .
= ρ ⊗ πp ⊗Q``∈Σ\{p}

The resulting structure theory for completed cohomology of the modular tower
has striking arithmetic applications; in particular [6] establishes many cases of
the Fontaine-Mazur conjecture as a corollary of the tensor factorization describedabove (see in particular [6], corollary 1.2.2.) It thus is natural to attempt to seek aframework in which one can describe completed cohomology of Shimura towers inas broad a context as possible; a primary goal of this paper is to develop a language
which should apply to the factors of completed cohomology at primes ` 6= p.
It is important to note that the arguments of [6] do not require one to know,a priori, that representations of the form πe(ρ) exist. Instead, they rely on Theorem 6.2.15 below to deduce from considerations at a dense set of points (together
with a genericity condition that, in the setting of [6], essentially reduces to Ihara’s
lemma) that a module with an action of GLn (Q`i ) for each `i in a finite collectionof primes admits a tensor factorization as a product of families of the form πe(ρ).
Whereas the proof of Theorem 1.2.1 is relatively elementary (essentially relying on
an integral version of the Bernstein-Zelevinski theory of the derivative), the proof

LOCAL LANGLANDS IN FAMILIES

5

of Theorem 6.2.15 requires detailed information about the behavior of the generic
local Langlands correspondence under specialization, which we develop in section 4.
1.4. Existence of the correspondence: results and conjectures. Just as in
the traditional setting, it seems to be easier to characterize the local Langlands
correspondence in families than to prove its existence. However, we make thefollowing conjecture.
1.4.1. Conjecture. If A is a reduced p-torsion free complete Noetherian local ringwith maximal ideal m and finite residue field k of characteristic p, and if ρ : GE →GLn (A) is continuous, then πe(ρ) exists.
One instance in which we can verify the conjecture is the case when A is thering of integers in a finite extension of Qp . (In this case, it is a consequence ofTheorem 3.3.2.) Forthcoming work of the second author [11], [10] will establish
many additional cases of this conjecture, using the second author’s theory of theintegral Bernstein center [9]. In particular the conjecture holds for n = 2 and podd, and also for p a banal prime; that is, a prime for which the integers 1, q, . . . , q n
are distinct modulo p, where q is the order of the residue field of E. (This meansthat for any fixed n, there are only finitely many p at which the conjecture canfail.)
1.5. A mod p local Langlands correspondence for GLn . In [20], Vigneras
defines a mod p local Langlands correspondence for GLn . However, this correspondence seems ill-suited for arithmetic applications (and in particular for describing
the mod p cohomology of the modular tower.) In particular, it lacks the good
behavior of the Breuil-Schneider correspondence under specialization.
In Section 5 we define a “modified” mod p local Langlands correspondence, thatis more suited to our needs. It differs from the correspondence of Vigneras in thefollowing ways:(a) The input is a Galois representation (not a Weil–Deligne representation).
(b) The output is an admissible smooth GLn (E)-representation that is possibly
reducible, but always generic.
(c) The correspondence is compatible with reduction modulo p in the direct
sense given by parts (2) and (3) of Theorem 1.5.1 below. (The Zelevinski
involution does not intervene.)
This being said, we rely on the results of [20] for the construction of our correspondence. The key point, whose proof relies on [20], is that for any deformation ρe(ρ) reduces modulo m to a representation whose cosocle isof ρ, the representation πabsolutely irreducible and generic, and is independent, up to isomorphism, of the
choice of ρ. (See the discussion following Corollary 5.1.2 below.)
The key properties of the correspondence of Section 5 are summarized in thefollowing theorem:
1.5.1. Theorem. There is a map ρ 7→ π(ρ) from the set of isomorphism classesof continuous representations GE → GLn (k) (where k is a finite field of characteristic p) to the set of isomorphism classes of finite length admissible smoothGLn (E)-representations on k-vector spaces, uniquely determined by the following
conditions:

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MATTHEW EMERTON AND DAVID HELM

(1) For any ρ, the G-socle soc π(ρ) of the associated GLn (E)-representationπ(ρ) is absolutely irreducible and generic, and the quotient π(ρ)/ soc π(ρ)
contains no generic Jordan–Hölder factors.
(2) Given ρ : GE → GLn (k), together with a deformation ρ : GE → GLn (O)of ρ, where O is a characteristic zero discrete valuation ring with uniformizer $ and residue field k 0 containing k, there is GLn (E)-equivariante(ρ) ⊗k k 0 → πe(ρ)/$eπ (ρ). (Note that πe(ρ) must exist as O is asurjection π
finite extension of Qp .)(3) The representation π(ρ) is minimal with respect to satisfying conditions (1)and (2), i.e. given any continuous representation ρ : GE → GLn (k) andany representation π of GLn (E) satisfying these two conditions with respectto ρ, there is a GLn (E)-equivariant embedding π(ρ) ,→ π.
As with the local Langlands correspondence in families, our motivation for introducing this modified mod p local Langlands correspondence is that it arises in
global contexts. Indeed, consider the limit H 1 (XΣ , k) of the cohomology of thetower XΣ of modular curves, where the levels of curves in XΣ are divisible precisely by the primes in Σ. This has an action of a completed Hecke algebra A,
and if one considers the maximal ideal of A corresponding to a suitable irreducible
representation ρ : GQ → GL2 (k), then one has an isomorphism:O
H 1 (XΣ , k)[m] ∼π(ρ|GQ` ).
= ρ ⊗ πp ⊗`∈Σ\{p}Thus the modified mod p local Langlands correspondence gives a framework inwhich one may describe the mod p cohomology of towers of modular curves, andone expects that this should apply to more general towers of Shimura varieties as
well.
The modified mod p local Langlands correspondence admits a completely concrete description for n = 2 and p odd; we briefly describe some cases of this in
section 5, and refer the reader to [12] for the complete picture.
1.5.2. Remark. One can consider the following stronger form of condition (2) of
Theorem 1.5.1:(20 ) Given ρ : GE → GLn (k), together with a deformation ρ : GE → GLn (A)of ρ, where A is a reduced complete Noetherian local W (k)-algebra, flatover W (k), with maximal ideal m and residue field k, and πe(ρ) exists, theree(ρ) → π
is a GLn (E)-equivariant surjection πe(ρ)/$eπ (ρ).
In some circumstances, we are able to verify that π(ρ) is in fact minimal withrespect to conditions (1) and (20 ). (In other words, π(ρ) contains as a submodule
the dual of any module that arises by specializing a family πe(ρ) attached to a
deformation of ρ.) This is essentially a characteristic p analog of Theorems 6.2.5
and 6.2.6. We conjecture that this stronger minimality property holds in general.
1.5.3. Remark. In general, π(ρ) is not irreducible, and if this is the case, thenit is not possible to strengthen “surjection” to “isomorphism” in the statement ofpart (2) of Theorem 1.5.1.
1.6. The organization of the paper. We begin by establishing some basic facts
about admissible smooth representations of certain topological groups over a Noetherian local ring A; we apply this machinery in section 2.2 to the study of invariant

LOCAL LANGLANDS IN FAMILIES

7

lattices in representations of topological groups over the field of fractions of a complete discrete valuation ring. The key result we establish is Lemma 2.2.6, which incertain circumstances allows us to construct an invariant lattice in such a representation whose reduction has a prescribed socle.
Section 3 establishes results specific to the representation theory of GLn (E) overcertain local rings A. In 3.1 we construct a theory of Kirillov models over a large
class of base rings. Our approach essentially follows that of [1], but as we are
not working over algebraically closed fields issues of descent arise. In spite of thisone recovers almost all of the theory of the Kirillov functors developed in [1, §4].
Particularly useful for us is the notion of a “generic” irreducible representation overan arbitrary field.
Section 3.2 introduces an essential concept for our results: that of an essentially absolutely irreducible and generic representation (“essentially AIG” in short.)
These are representations whose socles are absolutely irreducible and generic, andthat satisfy a certain finiteness property. The importance of these representations
stems from the fact that the “modified Langlands correspondences” we consider
send Galois representations to essentially AIG representations. In section 3.3 weapply the results of section 2.2 to establish some basic facts about the reduction
theory of essentially AIG representations.
In section 4 we study the behavior of the local Langlands correspondence (overfields of characteristic zero) under specialization. As we have previously discussed,
the usual local Langlands correspondence is not suitable for our purposes, and we
instead consider a modification of this correspondence due to Breuil and Schneider.
Our first main result (Corollary 4.3.3) establishes that the admissible representations of GLn (E) produced by the Breuil-Schneider correspondence are essentially
AIG. Once we have this, we apply the reduction theory of section 3.3, together withideas from the Zelevinski classification, to establish Theorem 4.5.7, which relates
the behaviour of a Galois representation under specialization to a characteristic zero
residue field to the behaviour (under the same specialization) of the corresponding
admissible representation constructed by Breuil-Schneider.
Section 5 constructs a “modified local Langlands correspondence” in characteristic p, by analogy with the Breuil-Schneider correspondence; in particular we define
this correspondence to be the “minimal” correspondence that satisfies a mod p
analogue of Theorem 4.5.7. We refer the reader to Theorem 5.1.5 for the precise
definition.
We finally turn to the study of the local Langlands correspondence for families ofadmissible representations in section 6. Section 6.2 discusses the main results of ourtheory; to avoid obscuring this discussion with technicalities we postpone the proofs
to section 6.3. Surprisingly little beyond the theory of Kirillov models is necessary toprove the basic uniqueness result of Theorem 6.2.1. On the other hand, establishing
more precise results about the structure of the family of admissible representations
attached to a given family of Galois representations (for instance, the interpolation
theorems 6.2.5 and 6.2.6) requires the full strength of the specialization results in
section 4.
1.7. Acknowledgments. Matthew Emerton would like to thank Kevin Buzzard,Frank Calegari, Gaetan Chenevier, Mark Kisin, and Eric Urban for helpful conversations on the subject of this note. He first began to investigate some of thequestions considered in this paper at the conference “Open questions and recent

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MATTHEW EMERTON AND DAVID HELM

developments in Iwasawa theory”, held at Boston University in June 2005, in honour of Ralph Greenberg’s 60th birthday, and was given the opportunity to further
explore them in a series of lectures given during the special semester on eigenvarieties at Harvard in April 2006. He would like to take the opportunity to thank both
institutions, as well as Robert Pollack and Barry Mazur, the principal organizersof the two events, for providing such stimulating mathematical environments.
David Helm would like to thank Kevin Buzzard for his ideas and perspective on
the questions addressed by this note, and Richard Taylor for his continued interest,
advice, and encouragement.
Both authors benefited from a mutual exchange of ideas occasioned by the conference “Modular forms and arithmetic”, held at MSRI in June/July 2008, andsponsored by MSRI and the Clay Math Institute. They would like to thank these
institutions, as well as the organizers of the conference, for creating this fruitful
opportunity.
2. Representation theory — general background2.1. Admissible smooth representations. Let A be a Noetherian local ringwith maximal ideal m and residue field k. In this subsection we recall some basic
facts about admissible smooth representations over A.
2.1.1. Definition. An A-linear representation of a topological group H on an Amodule V is called smooth if any element of V is fixed by an open subgroup of H.
Clearly any H-invariant sub- or quotient A-module of a smooth H-representation
over A is again a smooth H-representation over A.
2.1.2. Definition. A smooth representation of a topological group H on an Amodule V is called admissible if for any open subgroup H0 ⊂ H, the A-module of
fixed points V H0 is finitely generated.
Clearly any H-invariant A-submodule of an admissible smooth H-representation
over A is again an admissible smooth H-representation over A. (For the case of
H-invariant quotients, see Lemma 2.1.6 below.)
Consider the following condition on H:
2.1.3. Condition. H contains a profinite open subgroup, admitting a countablebasis of neighbourhoods of the identity, whose pro-order is invertible in A.
Suppose that H satisfies Condition 2.1.3, and let {Hi }i≥0 denote a decreasing
sequence of open subgroups of H, each of whose pro-order is invertible in A, andwhich forms a neighbourhood basis of the identity in H. If V is a smooth Hrepresentation over A, Rthen for each n ≥ 1, we may define the idempotent projector
πi : V → V Hi via v 7→ Hi hvdµi , where µi denotes Haar measure on Hi , normalizedTso that Hi has total measure 1. If we define Vi := ker πi V Hi+1 , then the inclusions
Vi ⊂ V induces an isomorphism of A-modules
M
∼
(2.1)
Vi −→ V.
i

The formation of Vi is evidently functorial on the category of smooth representationsof H over A, and thus so is the direct sum decomposition (2.1). In fact one can say
something more precise:LOCAL LANGLANDS IN FAMILIES

9

2.1.4. Lemma. Suppose that H satisfies Condition 2.1.3. If W is an H-invariant
A-submodule
T of the smooth
T H-representation A over V , then the natural maps
Wi ,→ Vi W and Vi /(Vi W ) → (V /W )i are isomorphisms.
Proof. This is evident.



2.1.5. Lemma. Suppose that H satisfies Condition 2.1.3. A smooth H-representation V over A is admissible if and only if each of the A-modules Vi is finitely
generated.
L∼
Hi
Proof. This follows from the isomorphisms
for each i, and thej≤i Vj −→ Vfact that the sequence {Hi } is cofinal in the collection of all of open subgroupsof H.

2.1.6. Lemma. Suppose that H satisfies Condition 2.1.3. If V is an admissible
smooth H-representation over A, and if W is a G-invariant A-submodule of V ,then V /W is again an admissible smooth H-representation over A.
Proof. This follows from the preceding lemma, and the fact that Vi → (V /W )i is
surjective.

If V is an A-module equipped with an admissible smooth H-representation, thentypically V itself will not be finitely generated as an A-module. Nevertheless, theexistence of the decomposition (2.1) allows us to extend many results about finitely
generated A-modules to the situation of admissible smooth G-representations.
2.1.7. Lemma. If H satisfies Condition 2.1.3, and if V is an admissible smooth
H-representation for which V /mV = 0, then V = 0.
L∼
Proof. The decomposition (2.1) yields the isomorphism i Vi /mVi −→ V /m. Thus
V /mV = 0 implies that Vi /mVi = 0 for each value of i. Since each Vi is finitely
generated over A, this in turn implies that Vi = 0 for each i, by Nakayama’s lemma.
Thus V = 0, as claimed.

2.1.8. Lemma. If H satisfies Condition 2.1.3, and if V is an admissible smooth
representation such that V /mV is finitely generated over k[H], then V is finitely
generated over A[H].
Proof. Let S ⊂ A be a finite subset whose image in V /mV generates this quotient
over k[H], and let W be the A[H]-submodule of V generated by S. Lemma 2.1.6 implies that (V /W ) is admissible, and by construction we see that (V /W )/m(V /W ) =0. Thus Lemma 2.1.7 shows that W = V , and so V is also finitely generated.
It will be technically useful to consider a related notion of admissible representation.
2.1.9. Definition. If V is an A-module equipped with an A-linear representationof H, we say that V is an admissible continuous H-representation if:(1) V is m-adically complete and separated.
(2) The H-action on V is continuous, when V is equipped with its m-adic
topology (i.e. the action map H × V → V is jointly continuous).
(3) The induced H-representation on V /mV (which is automatically smooth,by (1)) is admissible smooth.

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2.1.10. Lemma. Suppose that H satisfies Condition 2.1.3. If V is a continuous admissible H-representation over A, then for each n > 0, the induced Hrepresentation on V /mn V is admissible smooth.
Proof. Condition (2) of Definition 2.1.9 implies that the H-action on V /mn V is
continuous, when the latter is equipped with its discrete topology. In other words,
V /mn V is a smooth representation of H. Since the formation of Hi -invariants is∼exact, for any i ≥ 0, we find that (V /mn V )Hi /m(V /mn V )Hi −→ (V /mV )Hi isfinite dimensional over A/m, by Condition (3) of Definition 2.1.9. Lemma 2.1.16
below (applied to the module (V /mn V )Hi of the Artinian local ring A/mn ) thenshows that (V /mn V )Hi is finitely generated over A/mn . Consequently V /mn V isadmissible.

2.1.11. Definition. If V is an A-module, we let Vb denote the m-adic completionof V .
2.1.12. Definition. If V is an A-module equipped with an H-representation, we
let Vsm denote the subset of V consisting of vectors which are smooth, i.e. which
are fixed by some open subgroup of H. One immediately verifies that Vsm is
an A-submodule of V , closed under the action of H. Thus Vsm is a smooth Hrepresentation.
2.1.13. Proposition. Suppose that H satisfies Condition 2.1.3, and let V be anadmissible smooth H-representation over A.
(1) Vb is a continuous admissible H-representation over A.
(2) If A is m-adically complete, then the natural map V → Vbsm is an isomorphism.
Proof. The H-action on V is smooth, and thus so is the H-action on V /mn V , foreach n ≥ 0. Passing to the projective limit over n, we find that the H-action on
Vb is m-adically continuous. Since Vb /mVb = V /mV, it follows from Lemma 2.1.6
that the H-action on Vb /mVb is admissible. Thus Vb satisfies both the conditions ofDefinition 2.1.9. This proves (1).
We now turn to proving (2), and so in particular, assume that A is m-adicallycomplete. For each i ≥ 0, we find that∼∼∼Vb Hi −→ lim(V /mn V )Hi −→ lim V Hi /mn V Hi −→ V Hi
←−
n

←−
n

(the second isomorphism following from the exactness of the formation of Hi invariants, and the third following from the fact that V Hi is finitely generated
over A, by assumption, and hence m-adically complete, since A is m-adically complete). Consequently, the map V Hi → Vb Hi is an isomorphism for each i ≥ 0, and∼thus, passing to the inductive limit over i, we find that V −→ Vbsm , as claimed. 
2.1.14. Proposition. Suppose that H satisfies Condition 2.1.3 and that A is madically complete, and let V be an admissible continuous H-representation over A.
(1) Vsm is an admissible smooth H-representation.
d(2) The natural map V
sm → V is an isomorphism.
Proof. Since V is m-adically complete and separated, we see that V Hi is m-adicallycomplete and separated for each i ≥ 0. Since the formation of Hi -invariants is exact,∼
we see that V Hi /mV Hi −→ (V /mV )Hi , which by assumption is finite dimensional

LOCAL LANGLANDS IN FAMILIES

11

over A/m. Lemma 2.1.16 below then implies that V Hi is finitely generated over A.
Since (Vsm )Hi = V Hi by the very definition of Vsm , we see that Vsm is admissible,
proving (1).
If i ≥ 0 and n > 0, then∼(Vsm )Hi /mn (Vsm )Hi = V Hi /mn V Hi −→ (V /mn V )Hi ,the equality holding (as was already noted above) by the very definition of Vsm ,and the isomorphism following from the exactness of the formation of Hi -invariants.
Passing to the inductive limit over i, and taking into account the fact that Vsm and∼V /mn V are both smooth H-representations, we find that Vsm /mn Vsm −→ V /mn V.
∼dPassing to the projective limit over n, we find that Vsm −→ V, proving (2).
2.1.15. Remark. It follows from the preceding propositions that if A is m-adicallycomplete, then the functors V 7→ Vb and V 7→ Vsm are mutually quasi-inverse, andinduce an equivalence of categories between the category of admissible smooth Hrepresentations over A, and the category of admissible continuous H-representations
over A.
We close this subsection by recalling a version of Nakayama’s lemma in thesetting of m-adically separated modules over complete local rings.
2.1.16. Lemma. Suppose that A is m-adically complete. If M is an m-adicallyseparated A-module such that M/mM is finite dimensional over A/m, then M is
finitely generated over A.
Proof. Choose S = {s1 , . . . , sm } ⊂ M to be finite, and such that the image of S in
M/mM spans M over A/m, and let N denote the A-submodule of M generated by
S. We then have that M = N + mM, and so arguing inductively, for each v ∈ M
we may find, for each i = 1, . . . , s, a sequence of elements ai,n of A with ai,n ∈ mn
for each n, such that for each n we havev∈mX
(ai,0 + ai,1 + · · · + ai,n )si + mn+1 M.
i=1Writing ai = ai,0 + ai,1 + · · · (a well-defined element of A, since A is m-adicallycompleteby assumption), we then find (since M is m-adically separated) that v =Psas∈ N, and thus that M = N , proving the lemma.

i
i
i=1
2.2. Invariant lattices. Let O be a complete discrete valuation ring, with field of
fractions K and residue field K of characteristic different from `. Let $ be a choiceof uniformizer of O. If V is a K-vector space, then by a lattice in V we mean an
O-submodule V ◦ which spans V over K.
2.2.1. Definition. We say that a representation V of a group H over K is a good
integral representation if V contains a $-adically separated H-invariant lattice V ◦◦with the property that V := V ◦ /$V ◦ has finite length as a K[H]-module.
We now prove some basic results pertaining to this definition.
2.2.2. Lemma. Any subrepresentation of a good integral representation V of H
over K is again a good integral representation of H.

12

MATTHEW EMERTON AND DAVID HELM◦

Proof. If V ◦ is a $-adically separated H-invariant lattice in V for which V is offinite length over K[H], then W ◦ := V ◦ ∩ W is a $-adically separated H-invariant◦◦◦
lattice in W , and since the natural map W → V is injective, we see that W also
has finite length over K[H].

2.2.3. Lemma. Let V be a good integral representation of a group H, and fix a◦$-adically separated H-invariant lattice V ◦ ⊂ V such that V has finite length
over K[H]. If M is an H-invariant O-submodule of V /V ◦ , then either M is ofbounded exponent as an O-module, or else M contains a non-zero H-invariant
divisible O-submodule.
Proof. If M is not of bounded exponent, then the map M [$n ] → M [$m ] inducedby multiplication by $n−m has non-zero image for each n ≥ m ≥ 1, and hence,since each M [$n ] has finite length, we find that lim
M [$n ] 6= 0 (the transition←−nmaps being given by multiplication by $). If (mn )n≥0 is a non-zero element of
this projective limit, then the O-submodule of M generated by the elements mn isevidently non-zero and divisible. Thus the maximal divisible submodule of M is
non-zero; it is also clearly H-invariant.

2.2.4. Lemma. If H is a topological group satisfying Condition 2.1.3, and if V isa good integral admissible smooth representation of H over K, then:(1) Any two $-adically separated H-invariant lattices in V are commensurable.
(2) If V ◦ is a $-adically separated H-invariant lattice in V , then V ◦ is finitely◦
generated over O[H], the K[H]-module V := V ◦ /$V ◦ is of finite length,◦◦
and the isomorphism class of (V )ss (the semisimplification of V as a◦k[H]-module) is independent of the choice of V .
Proof. Since V is good integral by assumption, we may and do choose an $-adically◦◦separated H-invariant lattice V ◦ ⊂ V such that V is of finite length. Then V◦
is certainly finitely generated over K[H], and so Lemma 2.1.8 implies that V isfinitely generated over O[H].
Let V  be another $-adically separated H-invariant lattice in V . We will provethat V  is commensurable with V ◦ . This will prove (1). An easy (and standard)◦argument then proves that V is of finite length over K[H], and that (V )ss and ss(V ) are isomorphic. Also Lemma 2.1.8 will imply that V  is finitely generated
over O[H]. Thus (2) will also follow.
Since V ◦ is finitely generated over O[H], we may find m ≥ 0 such that V ◦ ⊂−m $ V . In proving the commensurability of V ◦ and V  , it is clearly no loss ofgenerality to replace V  by $−m V  , and so we may and do assume for the remainderof the proof that V ◦ ⊂ V  .
Consider now the quotient V  /V ◦ ⊂ V /V ◦ . If V  /V ◦ is not of bounded exponent, then Lemma 2.2.3 shows that it contains a non-zero H-invariant divisible
submodule D. The Tate module Tp D := lim
D[$n ] (the transition maps being←−ngiven by multiplication by $) is then a non-zero $-adically complete and separated O-module, equipped with an action of H, and an injection∼(2.2)
Tp D ,→ Tp (V /V ◦ ) −→ Vc◦ .
Now∼

Tp D/$Tp D −→ D[$] ⊂1 ◦ ◦V /V ,$

LOCAL LANGLANDS IN FAMILIES

13

and hence Tp D/$Tp D is an admissible smooth H-representation. Thus Tp D isan admissible continuous H-representation over O, and so by Remark 2.1.15, theinjection (2.2) is obtained from the induced embedding (Tp D)sm ,→ V ◦ by passing
to $-adic completions. In particular (Tp D)sm 6= 0. Also, the image of the composite
K ⊗O (Tp D)sm → V → V /V ◦is precisely D, and so (since D ⊂ V  /V ◦ ), we conclude that K ⊗O (Tp D)sm ⊂ V  .
But K ⊗O (Tp D)sm is a non-zero K-vector space, and hence is not $-adically separated. This contradicts our assumption on V  , and hence we conclude that V  /V ◦is indeed of bounded exponent, and thus that V ◦ and V  are commensurable, as
required.

2.2.5. Definition. Let H be a topological group satisfying Condition 2.1.3, andlet V be a good integral admissible smooth representation of H over K. If V ◦ is◦a $-adically separated H-invariant lattice in V such that V := V ◦ /$V ◦ is ofssfinite length as a K[H]-module (which exists, by assumption), then we write V◦
to denote the semisimplification of V as a K[H]-module. (The preceding lemmassshows that, up to isomorphism, V is independent of the choice of V ◦ .)The following lemma will allow us to choose lattices in good integral admissible
smooth representations whose reductions modulo $ have certain specified H-socles.
2.2.6. Lemma. Let H be a topological group satisfying Condition 2.1.3, and let V
be a good integral admissible smooth representation of H over K. Let S denotess
the set of isomorphism classes of Jordan–Hölder factors of V (as a K[H]-module;
the discussion of Definition 2.2.5 shows that this set is well-defined), let T be asubset of S, and suppose that V contains no non-zero subrepresentation W (necessarily also good integral, by Lemma 2.2.2 ) such that every Jordan–Hölder factor
ssof W belongs to T . Then there exists a $-adically separated H-invariant latticeV  contained in V ◦ with the property that V := V  /$V  contains no subobject
isomorphic to an element of T .
Proof. Choose (as we may, by assumption) a $-adically separated H-invariant◦
lattice V ◦ with the property that V := V ◦ /$V ◦ is of finite length as a K[H]◦
module. Let M ⊂ V /V be the maximal O[H]-submodule all of whose Jordan–Hölder factors are isomorphic to an element of T .
If we form the projective limit lim
M [$n ] 6= 0 (the transition maps being given←−n

1 ◦ ◦ ∼ b◦
V /V −→ V , with saturatedn$nn
and $-adically complete image. If we write W = K ⊗O (limM [$n ])sm , then Re←−by multiplication by $), then lim
M [$n ] ,→ lim
←−
←−

n

mark 2.1.15 implies that W vanishes if and only if lim
M [$n ] does. On the other←−nother hand, by construction W is a subrepresentation of V with the property thatss
all the Jordan–Hölder factors of W belong to T , and so by assumption W mustnvanish. Thus lim
M [$ ] = 0, and so Lemma 2.2.3 implies that M is of bounded←−nexponent, say M = M [$n ].
Let V  denote the preimage of M in V . Since V ◦ ⊂ V  ⊂ $−n V ◦ , we see that ∼
V is $-adically separated. Since V −→ $−1 V  /V  ,→ V /M, our choice of Mensures that V contains no subobject isomorphic to an element of T .
14

MATTHEW EMERTON AND DAVID HELM

3. Representation theory — the case of GLn
3.1. Kirillov models. Let k be a perfect field. Let ` be a prime distinct from thecharacteristic of k, and let e
k be a Galois extension of k containing all `-power rootsof unity.
In this section we set G = GLn (E), where E is a non-archimedean local fieldof residue characteristic `. We will define a notion of Kirillov models for smooth
representations of G over a W (k)-algebra A.
We begin by recalling the basic properties of Kirillov models associated to smooth
W (ek)[G]-modules.
In the case of smooth C[G]-modules, these results are found in [1, §4]; over moregeneral algebraically closed fields they can be found in [17, Ch. III.1]. The extensionof these results to coefficients in W (ek) is more or less immediate. We summarize
the key facts:
Define subgroups Pn and Nn of GLn (E) by setting:a b
Pn = {| a ∈ GLn−1 (E), b ∈ E n−1 },0 1
Idn−1 b
Nn = {| b ∈ E n−1 },01
We consider GLn−1 (E) as a subgroup of Pn in the obvious way, and identify Nn
with E n−1 . Note that Pn = GLn−1 (E)Nn . Any character ψ : E n−1 → W (ek)×
induces a character of Nn via
Idn−1 b7→ ψ(b),01which we again denote by ψ.
We fix, for the remainder of this section, a character ψ : E → W (ek)× whosekernel is equal to the subgroup OE of E. We consider ψ as a character of E n−1
by setting ψ(e1 , . . . , en−1 ) = ψ(en−1 ), and also as a character of Nn via the isomorphism of E n−1 with Nn . The subgroup GLn−1 (E) of Pn normalizes Nn , andtherefore acts on the set of characters of Nn by conjugation. The stabilizer of ψunder this action is the subgroup Pn−1 of GLn−1 (E).
3.1.1. Definition. For a W (ek)-algebra A, let RepA (G) denote the category ofsmooth A[G]-modules. Define functors Ψ− , Ψ+ , Φ− , Φ+ , Φ̂+ by:• Ψ− : RepW (ek) (Pn ) → RepW (ek) (GLn−1 (E)) is given by Ψ− (V ) = VNn , themodule of Nn -coinvariants of V .
• Ψ+ : RepW (ek) (GLn−1 (E)) → RepW (ek) (Pn ) is the functor that takes a
GLn−1 -module V and extends the action of GLn−1 to Pn by letting Nn
act trivially.
• Φ− : RepW (ek) (Pn ) → RepW (ek) (Pn−1 ) is given by Φ− (V ) = Vψ , where Vψ
is the largest quotient of V on which Nn acts by ψ. (As Pn−1 normalizes
ψ, Vψ is naturally a Pn−1 -module.)• Φ+ : RepW (ek) (Pn−1 ) → RepW (ek) (Pn ) is given by0nΦ+ (V ) = c − IndP
Pn−1 Nn V ,where V 0 is the Pn−1 Nn -module obtained from V by letting Nn act via ψ,and c − Ind denotes smooth induction with compact support.

LOCAL LANGLANDS IN FAMILIES

15• Φ̂+ : RepW (ek) (Pn−1 ) → RepW (ek) (Pn ) is given by0nΦ̂+ (V ) = IndP
Pn−1 Nn V .

The natural surjections of V onto Ψ− V and of V onto Φ− V are GLn−1 (E) andPn−1 -equivariant, respectively.
3.1.2. Remark. Note that these functors differ from the ones defined in [1] in thatthey are not “normalized.” More precisely, the functors defined in [1] are twists of
the above functors by the square roots of certain modulus characters. This makes
them unsuitable for most of our purposes, as the descent arguments we make atthe end of this section do not apply to the twisted functors defined in [1]. We willthus use the “non-normalized” functors defined above throughout the bulk of thepaper.
An unfortunate exception to this is in the proof of Proposition 4.3.2. While itwould in principle be possible to give a proof of Proposition 4.3.2 using the nonnormalized functors that we use elsewhere, the normalization of [1] simplifies the
combinatorics immensely. We have thus chosen to adopt this normalization for the
purposes of that proof only.
The arguments of [1, §3.2] carry over to this setting to show:
3.1.3. Proposition.
(1) The functors Ψ− , Ψ+ , Φ− , Φ+ , Φ̂+ are exact.
+(2) Φ is left adjoint to Φ− , Ψ− is left adjoint to Ψ+ , and Φ− is left adjoint
to Φ̂+ .
(3) Ψ− Φ+ = Φ− Ψ+ = 0.
(4) The composite functors Ψ− Ψ+ , Φ− Φ̂+ , and Φ− Φ+ are naturally isomorphic
to identity functors.
(5) One has an exact sequence of functors:
0 → Φ+ Φ− → Id → Ψ+ Ψ− → 0.
For our purposes, it will be necessary to have versions of these functors for representations over W (k), rather than W (ek). The key difficulty is that the characterψ is not defined over W (k). Nonetheless, one has:
3.1.4. Proposition. The functors Ψ− , Ψ+ , Φ− , Φ+ , Φ̂+ descend to functors definedon representations over W (k). That is, one has a functor:Ψ− : RepW (k) (Pn ) → RepW (k) (GLn−1 (E))
such that for any W (k)[Pn ]-module V , one has
Ψ− (V ⊗W (k) W (ek)) = Ψ− (V ) ⊗W (k) W (ek),
and similarly for the remaining functors. Moreover, the statements of Proposition 3.1.3 apply to these functors.
Proof. For the functors Ψ− and Ψ+ this is clear, as the character ψ does notintervene in their definition. We thus begin with the functor Φ− .
Note that Gal(ek/k) acts naturally on W (ek), and fixes W (k). Moreover, W (e
k) is
σ
e
faithfully flat over W (k). For σ ∈ Gal(k/k), let ψ be the character σ ◦ ψ of E × .
×
There exists a unique eσ ∈ OE
such that ψ σ (e) = ψ(eσ e). The map σ 7→ eσ is a×ehomomorphism from Gal(k/k) to OE.

16

MATTHEW EMERTON AND DAVID HELM

Consider each eσ as an element of Pn−1 via the inclusions:×
OE⊂ E × ⊂ GLn−1 (E) ⊂ Pn−1 .

If we consider ψ as a character of Nn , we have ψ σ (u) = ψ(eσ ue−1σ ) for all u ∈ Nn .
Now let V be a smooth Pn -representation over W (k), and let Ve be the representation V ⊗W (k) W (ek). Then we have a W (ek)-semilinear action of Gal(ek/k) on Ve ,−ethat fixes V . By definition, Φ V is the quotient of Ve by the W (ek)[Pn−1 ]-submodule
generated by all vectors of the form uv − ψ(u)v, for u ∈ Nn and v ∈ Ve .
The Galois action on Ve does not descend to Φ− Ve , but a twist of it does. For
an element [v] ∈ Φ− Ve , represented by an element v of Ve , define σ[v] = [eσ σv].
This is well-defined, since if v = uw − ψ(u)w for some w ∈ Ve and u ∈ Nn , we can
0 0
00
set w0 = eσ σw, u0 = eσ ue−1
σ , and then eσ σv = u w − ψ(u )w . We thus obtain aW (ek)-semilinear action of Gal(ek/k) on Φ− Ve ; we define Φ− V to be the invariants
under this action. This is clearly functorial with the desired properties. (Note,however, that the surjection Ve → Φ− Ve does not descend to a natural surjection of
V onto Φ− V .)Now let V be a smooth Pn−1 -representation over W (k). Then Φ+ Ve and Φ̂+ Ve
can both be realized as spaces of functions: f : Pn → Ve , such that for any h ∈ Pn−1
and any u in Nn , we have f (ghu) = ψ(u)hf (g). Define an action of Gal(ek/k) onthe space of such functions by setting (σf )(g) = e−1
σ σf (g). This preserves theidentity f (ghu) = ψ(u)hf (g), and so defines a W (ek)-semilinear action of Gal(e
k/k)
+e
+e
+
+
on Φ V and Φ̂ V . We set Φ V and Φ̂ V to be the invariants of this action in
Φ+ Ve and Φ̂+ Ve , respectively. Note that if V is a smooth Pn -representation, then
the natural maps: Φ+ Φ− Ve → Ve and Ve → Φ̂+ Φ− Ve are Gal(ek/k)-equivariant, andhence descend to V .
Using this, one easily verifies the adjointness property (2) of Proposition 3.1.3
for the functors over W (k). Properties (1), (3), (4), and (5) then follow by basechange and the fact that W (ek) is faithfully flat over W (k).
Note that if A is a Noetherian W (k)-algebra, and V is a smooth representationof P n−1 over A, then the modules Ψ− V , Φ− V obtained by treating V as a representation of P n−1 over W (k) and applying the appropriate functors inherit an
A-module structure. We can thus define the functors Ψ− , Ψ+ , Φ− , etc. on suitable
categories of smooth representations over A. It is then clear that if B is a Noetherian A-algebra, one has Ψ− (V ⊗A B) = Ψ− (V )⊗A B, and similarly for the remaining
functors.
Finally, observe that the functors Ψ− , Ψ+ , Φ− , Φ+ , Φ̂+ commute with tensor
products; that is, if M is an A-module, then Ψ− (V ⊗A M ) ∼= Ψ− (V ) ⊗A M , andsimilarly for the other functors.
We now define the “derivatives” of a smooth Pn -representation V . For 0 ≤ r ≤ n,
we set V (r) = Ψ− (Φ− )r−1 V ; V (r) is a representation of GLn−r (E). If A is a W (ek)algebra, then one has a GLn−r (E)-equivariant surjection V → V (r) (but this is nottrue if A is only a W (k)-algebra.)
Note that V (n) is simply an A-module. The adjointness properties of Proposition
3.1.3 give, for any V , maps:
V → (Φ̂+ )(n−1) Ψ+ (V (n) ).

LOCAL LANGLANDS IN FAMILIES

17(Φ+ )n−1 Ψ+ (V (n) ) → V.
(Note that in the setting of the second of these maps, the Ψ+ that appears, andthe Ψ− that arises in the definition of V (n) , are both the identity functor, and thus
the fact that Ψ+ is usually a right adjoint to Ψ− rather than a left adjoint is not
an issue.) The image of V in (Φ̂+ )(n−1) Ψ+ (V (n) ) is called the Kirillov model of V .
The exact sequence of Proposition 3.1.3 implies that the map(Φ+ )n−1 Ψ+ (V (n) ) → V
is injective; we denote its image by J(V ). The space J(V ) is often referred to as
the space of Schwartz functions in V .
e = A ⊗W (k) W (e3.1.5. Lemma. Let V be a smooth Pn -module over A. Set Ak), andlet Ve = V ⊗W (k) W (ek). The modules V and J(V ) each contain an A-submodule
e maps isomorphically to Ve (n)W such that W is isomorphic to V (n) , and W ⊗A A
(n)under the surjection Ve → Ve .
Proof. It suffices to show that J(V ) contains such a submodule, as J(V ) embeds in
V . We haveJ(V ) = (Φ+ )n−1 Ψ+ V (n) = [(Φ+ )n−1 Ψ+ W (k)] ⊗W (k) V (n) .
Moreover, the map J(Ve ) → Ve (n) arises from the surjection:(Φ+ )n−1 Ψ+ W (ek) → W (ek)
by tensoring over W (ek) with Ve (n) . It thus suffices to construct a submodule W of
+ n−1 +[(Φ )
Ψ W (k)] that is free of rank one over W (k) and such that W ⊗W (k) W (ek)
maps isomorphically onto W (ek) under the surjection(Φ+ )n−1 Ψ+ W (ek) → W (ek).
This amounts to simply choosing any element of (Φ+ )(n−1) Ψ+ W (k) that maps toan element of W (ek)\$W (ek), where $ is the uniformizer of W (ek). Such an element
clearly exists, as otherwise the image of the composition:∼[(Φ+ )n−1 Ψ+ W (k)] ⊗W (k) W (ek) −→ (Φ+ )n−1 Ψ+ W (ek) → W (ek)
would be contained in $W (ek).

Over a field, the top derivatives are multiplicative with respect to parabolic
induction:
3.1.6. Proposition. Let V and W be admissible k-representations of GLn (E) andGLm (E), respectively. Let P ⊂ GLn+m (E) be the parabolic subgroup of G given by:


a b
P = {
| a ∈ GLn (E), d ∈ GLm (E) },0 dand consider V ⊗ W as a representation of P by letting the unipotent radical of P
act trivially. ThenGL

(IndP n+m

(E)

V ⊗ W )(n+m) = V (n) ⊗ W (m) .

Proof. This is a special case of [17, Lem. 1.10]. Note that the derivative functors
used in [17] are normalized differently than ours; this normalization does not affectthe top derivative V (n) of a representation V of GLn (E).


18

MATTHEW EMERTON AND DAVID HELM

We also have:
3.1.7. Theorem. Let V be an absolutely irreducible admissible representation ofGLn (E) over a field K that is a W (k)-algebra. Then V (n) is either zero or a
one-dimensional K-vector space, and is one-dimensional if V is cuspidal.
Proof. This is [17, III.5.10], over Fp or Qp ; the same argument works over anarbitrary field containing the `-power roots of unity. The general case follows byextending scalars.

3.1.8. Definition. We say that an absolutely irreducible admissible representation
V of GLn (E) over a field K that is a W (k)-algebra is generic if V (n) is onedimensional over K.
We now turn to finiteness properties of the derivative.
3.1.9. Lemma. Let A be a Noetherian local ring, with maximal ideal m, and supposethat M is a submodule of a direct sum of finitely generated A-modules. If M/mM
is finite dimensional, then M is finitely generated.
Proof. Our assumption on M/mM allows us to choose a finitely generated submodule N of M such that N + mM = M, or equivalently such that m(M/N ) = M/N .
Nakayama’s Lemma then shows that any finitely generated quotient of M/N mustvanish. Since by assumption M embeds into a direct sum of finitely generated Amodules, we may find a finitely generated A-module X, and an A-module L
Y which isa direct sum of finitely generated A-modules,
andanembeddingM⊂X
Y , suchTthat N ⊂ X. On the one hand M/(M X) is a quotient of M/N , and hence has
Lno
non-vanishing finitely generated quotients.
Ontheother,
theprojection
of
X
Y
T
T
onto Y induces an embedding of M/(M X) into Y . Thus M/(M X) = 0, andso M ⊂ X is finitely generated.

We note for future reference the following corollary of the preceding lemma.
3.1.10. Corollary. Let A be a Noetherian local ring, and suppose that M is asubmodule of a direct sum of finitely generated A-modules. If M/mM vanishes,
then M itself vanishes.
Proof. The preceding result implies that M is finitely generated. The corollary
thus follows from Nakayama’s Lemma.

3.1.11. Theorem. Suppose that A is a Noetherian local W (k)-algebra with maximal
ideal m. Let V be an admissible representation of GLn over A, and suppose that(V /mV )(n) is finite dimensional over A/m. Then V (n) is a finitely generated Amodule.
Proof. As derivatives commute with tensor products, we have
∼ (V /mV )(n) .
V (n) /mV (n) =On the other hand, we have already observed that J(V ) (and hence V ) contains
an A-submodule isomorphic to V (n) . As V is a direct sum of finitely generated
A-modules, the result follows from Lemma 3.1.9.

3.1.12. Remark. In the setting of Theorem 3.1.11, if V /mV has finite length, then(V /mV )(n) is finite dimensional over A/m by Theorem 3.1.7. Thus Theorem 3.1.11
applies to all admissible representations of GLn over A such that V /mV has finite
length.

LOCAL LANGLANDS IN FAMILIES

19

Theorem 3.1.11 allows us to establish the following extension of Proposition 3.1.6:
3.1.13. Corollary. Let A be a reduced Noetherian local W (k)-algebra with maximal
ideal m. Let V and W be admissible smooth A-representations of GLn (E) andGLm (E), and let P be the parabolic subgroup of GLn+m (E) defined in the statementof Proposition 3.1.6. Then, if V (n) and W (m) are free of rank one over A, so is the
A-module:
GL(E)(IndP n+m
V ⊗ W )(n+m) .
Proof. For each minimal prime a of A, let κ(a) be its residue field. Set Va =V ⊗A κ(a) and Wa = W ⊗A κ(a). We have isomorphisms:
GL(IndP n+m

(E)

∼

GL

V ⊗ W )(n+m) ⊗A κ(a) −→ (IndP n+m(E)

Va ⊗ Wa )(n+m) ,and the latter is one-dimensional by Proposition 3.1.6. Thus in particular theGL(E)annihilator of (IndP n+m
V ⊗ W )(n+m) as an A-module is the zero ideal of A.
On the other hand, let V = V /mV and W = W/mW . Then we have isomorphisms:
GL(IndP n+m

(E)

∼

GL

V ⊗ W )(n+m) ⊗A A/m −→ (IndP n+m(E)

V ⊗ W )(n+m) ,and the latter is again one-dimensional by Proposition 3.1.6. Theorem 3.1.11 showsGL(E)
that (IndP n+m
V ⊗ W )(n+m) is furthermore finitely generated over A, and thusGL(E)
it follows by Nakayama’s lemma that (IndP n+m
V ⊗ W )(n+m) is a cyclic Amodule, and hence (taking into account that it is faithful, as we proved above) is
free of rank one.

We will also need a generalization of this machinery to a product of GLn (Ei )for various local fields Ei of residue characteristics `i , all prime to the residue
characteristic of k. Fix a finite collection of such Ei , indexed by a set S, and let
G be the product of the groups GLn (Ei ) for all i. Let Pn be the product of thesubgroups Pn (Ei ) of GLn (Ei ).
Now if we fix for each i a character ψi : Nn (Ei ) → W (ek)× , we can definefunctors Ψ−,i , Φ−,i , Ψ+,i , Φ+,i , Φ̂+,i as follows: if H is any topological group, andV is a Pn (Ei ) × H-module, then Ψ−,i (V ) is the GLn−1 (Ei ) × H-module defined byapplying Ψ− to V (considered as a Pn (Ei )-module,) and then taking the naturalaction of H on Ψ− (V ). The other functors are defined similarly. Note that if V
is a G-module over A (or even a module over the product of the Pn (Ei ),) thenΨ−,i Ψ−,j V = Ψ−,j Ψ−,i V (here the equality denotes a natural isomorphism), andthe other functors have similar commutativity properties. If S 0 is a subset of S,
the composition of functors Ψ−,i for all i ∈ S 0 is a thus a functor that takes an
A-module V over the product of the groups Pn (Ei ) to an A-module with an actionof Pn−1 (Ei ) for each i ∈ S 0 and of Pn (Ei ) for each i not in S 0 . This composition
is independent (up to natural isomorphism) of the order in which we compose the000functors; we denote it by Ψ−,S . Similarly define Φ−,S , Ψ+,S , etc. Finally, if
000
V is an A[G]-module, define V (n),S to be the representation Ψ−,S (Φ−,S )n−1 V .
00Note that the functors Φ−,S , Ψ+,S , etc. satisfy analogues of properties (1)-(4) of
Proposition 3.1.3.
When it is clear from the context what S and Ei we are working with, we will
denote Φ+,S , Ψ+,S , etc. by Φ+ , Ψ+ , and so forth.

20

MATTHEW EMERTON AND DAVID HELM

Similarly, if V is an A[G]-module, we define Ji (V ) to be space of Schwartz
functions in V for the action of GLn (Ei ) on V ; this has an action of Pn (Ei ) and ofGLn (Ej ) for j not equal to i. Note that Ji Jj (V ) = Jj Ji (V ), so that we can define,for any S 0 ⊂ S, the functor JS 0 to be the composition (in any order) of the functors
Ji for i in S 0 . Then JS 0 (V ) is the smallest A-submodule of V , stable under Pn (Ei )00for i in S 0 and GLn (Ei ) for i not in S 0 , such that the map JS (V )(n),S → V (n),Sis an isomorphism. By construction, the functor JS 0 is left adjoint to the functor0
V 7→ V (n),S (when the latter is considered as a functor from A[Pn ]-modules to
A-modules).
Now, precisely as in the proof of Theorem 3.1.11, we have:
3.1.14. Theorem. Let V be an admissible representation of G over a Noetherian
local W (k)-algebra A, and suppose that (V /mV )(n) is finite dimensional over A/mA.
Then V (n) is a finitely generated A-module.
We also have an analogue of Theorem 3.1.7
3.1.15. Theorem. Let V be an absolutely irreducible admissible representation of
G over k. Then V (n) is either zero or a one-dimensional k-vector space.
Proof. The representation V splits as a tensor product of absolutely irreducible
representations Vi of GLi (Ei ) for all i ∈ I. It follows that V (n) is the tensor product(n)of the Vi . Hence this result is an immediate consequence of Theorem 3.1.7. 
3.1.16. Proposition. Let V be an A[G]-module, and suppose that V (n) is free ofrank one over A. Then the map A → EndPn (J(V )) is an isomorphism.
Proof. By the adjointness properties of the functors Ψ+ and Φ+ we have natural
isomorphisms:∼∼

EndPn (J(V )) −→ HomA (V (n) , Ψ− (Φ− )n−1 J(V )) −→ EndA (V (n) ).
The result follows immediately.



3.2. Essentially AIG representations. Let K be a field of characteristic different from `.
3.2.1. Definition. We say that a smooth representation V of G := GLn (E) isessentially absolutely irreducible and generic (“essentially AIG” for short) if:(1) The G-socle soc(V ) is absolutely irreducible and generic.
(2) The quotient V / soc(V ) contains no generic Jordan–Hölder factors; equiv(n)alently, V / soc(V )= 0.
(3) The representation V is the sum (or equivalently, the union) of its finite
length submodules.
3.2.2. Lemma.
(1) If V is an essentially AIG smooth representation of G,
and if χ : E × → k × is a continuous character, then (χ ◦ det) ⊗ V is again
essentially AIG.
(2) If V is an essentially AIG smooth G-representation, and if U ⊂ V is a
non-zero smooth G-subrepresentation, then U is also essentially AIG, andfurthermore soc(U ) = soc(V ).
(3) If U and V are essentially AIG admissible smooth G-representations, thenrestricting to socles induces an embedding
HomG (U, V ) ,→ HomG soc(U ), soc(V ) .

LOCAL LANGLANDS IN FAMILIES

21(4) Any non-zero G-equivariant homomorphism between essentially AIG admissible smooth G-representations is an embedding.
Proof. Claim (1) is clear.
If U ⊂ V is as in (2), then 0 6= soc(U ) ⊂ soc(V ). Since the latter is absolutely
irreducible, we find that soc(U ) = soc(V ), and so in particular soc(U ) is absolutely
irreducible and generic. Furthermore, we see that U/ soc(U ) ,→ V / soc(V ). Since
the latter representation contains no generic Jordan–Hölder factors, neither does
the former. Finally, every element of U is contained in a finite length submoduleof V ; the intersection of this with U is also finite length. Thus U is the union of its
finite length submodules, and is therefore essentially AIG, proving (2).
Now suppose that φ : U → V is a map of essentially AIG representations, asin (3). If φ soc(U ) = 0, then φ factors to induce a map U/ soc(U ) → V. But
the source of this map has no generic Jordan–Hölder factors, while its target has
generic socle. Thus this map vanishes, and hence φ vanishes. This proves (3).
To prove (4), suppose given φ : U → V as above. If ker φ 6= 0, then it has a
non-zero socle. As soc(U ) is irreducible, we conclude that soc(U ) ⊂ ker φ. Part (3)then implies that φ = 0.
3.2.3. Lemma. If V is an essentially AIG smooth representation of G over K, and
if U is a non-zero submodule of V, then HomG (U, V ) is one-dimensional over K.
In particular, EndG (V ) = K.
Proof. Part (2) of the preceding lemma shows that U is again essentially AIG andthat soc(U ) = soc(V ). Part (3) of the same lemma shows that restriction to socles
induces an embedding
HomG (U, V ) ,→ HomG soc(U ), soc(V ) = EndG soc(V ) = K(where the first equality follows from the already noted equality of socles, andthe second equality following from the absolute irreducibility of soc(V )). Since
HomG (U, V ) is non-zero (as U embeds into V by assumption), it must therefore be
one-dimensional, as claimed.

3.2.4. Lemma. If V is an essentially AIG smooth representation of G over K, thenV admits a central character.
Proof. The preceding lemma shows that AutG (V ) = K × . Since the centre Z of G
acts as automorphisms of V , the lemma follows.

3.2.5. Lemma. Let V and W be essentially AIG smooth representations of G overK, and let K 0 be a finite separable extension of K. For any map f : V ⊗K K 0 →W ⊗K K 0 , there exists a scalar c ∈ (K 0 )× such that cf descends uniquely to a map
V → W.
Proof. We may assume K 0 is Galois over K, and that f is nonzero (and thus
injective). For σ ∈ Gal(K 0 /K), define f σ by f σ (x) = σf (σ −1 x). Then f σ = cσ f
for a scalar cσ ∈ (k 0 )× . The cσ give a cocycle in H 1 (Gal(K 0 /K), (K 0 )× ) and arecfor alltherefore a coboundary; that is, there exists a c ∈ (K 0 )× such that cσ = σcσ. Then cf is Galois-equivariant, and thus descends to K.

3.2.6. Definition. If V is an essentially AIG admissible smooth G-representation,then we say that a smooth representation W is an essentially AIG envelope of V if:(1) W is itself essentially AIG.

22

MATTHEW EMERTON AND DAVID HELM

(2) There is an G-equivariant embedding V ,→ W (which Lemma 3.2.3 showsis then unique up to multiplication by a non-zero scalar).
(3) W is maximal with respect to properties (1) and (2), i.e. if V ,→ Y is any
G-equivariant embedding with Y essentially AIG admissible smooth, thenthere is a G-equivariant embedding Y ,→ W.
3.2.7. Proposition. If V is an essentially AIG admissible smooth G-representation,then V admits an essentially AIG envelope, which is unique up to isomorphism.
Proof. Let V ,→ I be an injective envelope of V in the category of smooth representations. Let U denote the subrepresentation of I/V obtained by taking the sumof all the non-generic subrepresentations of I/V (so U is the maximal subrepresentation of I/V for which U (n) = 0), and define X to be the preimage of U in I. Let
W be the sum of all of the finite length submodules of X. By construction, the(n)socle of W is generic, W/ soc(W )= 0, and W is the sum of its finite length
submodules, so W is essentially AIG.
If V ,→ Y is an embedding as in (3), then (since I is injective) we may extend
the embedding of V into I to an embedding of Y into I. Since every Jordan–Hölder
constituent of Y /V is nongeneric, the image of Y lies in X. Moreover, Y is the sumof its finite length submodules, so the image of Y lies in W .

If V is an essentially AIG smooth G-representation, then we write env(V ) todenote the essentially AIG envelope of V (which by the preceding proposition exists,
and is unique up to isomorphism).
3.2.8. Lemma. Let V be an essentially AIG admissible smooth G-representation.
If χ : E × → K × is a continuous character, then there is an isomorphism ∼env (χ ◦ det) ⊗ V −→ (χ ◦ det) ⊗ env(V );i.e. the formation of essentially AIG envelopes is compatible with twisting.
Proof. This is immediate from Lemma 3.2.2.



In fact, essentially AIG representations actually have finite length. This will be
proven in forthcoming work of the second author [10]. In this paper, we will content
ourselves with somewhat weaker results whose proofs are more elementary. In thecase when K is of characteristic zero, this finiteness follows from Theorem 4.3.9
below, while in the case when n = 2, it is easy to establish for arbitrary K. (See
Proposition 3.2.18 below.) In the remainder of this section, we establish a weaker
finiteness result for essentially AIG representations that will suffice for our purposes.
The key tool will be the notions of cuspidal and supercuspidal support; we recall
the definitions below.
Let {π1 , . . . , πr } be a multiset of irreducible cuspidal
P representations of thegroups GLn1 (E), . . . , GLnr (E), for n1 , . . . , nr such thatni = n.
3.2.9. Definition. Let π be an irreducible representation of G over K. The multiset{π1 , . . . , πr } is a cuspidal support of π if there exists a parabolic subgroup P = M U
of G, with M isomorphic to the product of the GLni , such that π is isomorphic toa quotient of the normalized parabolic induction
IndG
P π1 ⊗ · · · ⊗ πr ,for some choice of ordering of π1 , . . . , πr .

LOCAL LANGLANDS IN FAMILIES23

It is an easy consequence of the “geometric lemma” of Bernstein-Zelevinski,
and adjointness of parabolic induction and restriction, that every irreducible π
has a cuspidal support, and this cuspidal support is uniquely determined (as amultiset of isomorphism classes of irreducible representations.) We refer the reader
to [17], II.2.4, for the details. By contrast, the notion of supercuspidal support lies
considerably deeper:
3.2.10. Definition. Let π be an irreducible representation of G over K. The multiset {π1 , . . . , πr } is a supercuspidal support of π if each πi is supercuspidal, andthere exists a parabolic subgroup P = M U of G, with M isomorphic to the product of the GLni , such that π is isomorphic to a Jordan–Hölder constituent of thenormalized parabolic induction
IndG
P π1 ⊗ · · · ⊗ πr .
In characteristic zero the notions of cuspidal and supercuspidal support coincide.
This is false, however, in characteristic p, where there are cuspidal representations
that are not supercuspidal. In this setting it is a deep result of Vigneras ([18], V.4)
that every irreducible π has a supercuspidal support that is uniquely determinedby π.
Let scs(π) (resp. cs(π)) denote the supercuspidal support (resp. cuspidal support) of π.
3.2.11. Proposition.
Let P = M U be a parabolic subgroup of G, with M isomorQ
phic to i GLni . Then:(1) Let πi be an irreducible admissible representation of GLni for all i. If π isan irreducible submodule or quotient ofIndG
P π1 ⊗ · · · ⊗ πr ,then cs(π) is the multiset sum of cs(πi ) for all i.
(2) Let πi be an irreducible admissible representation of GLni for all i. If π isa Jordan–Hölder constituent ofIndG
P π1 ⊗ · · · ⊗ πr ,then scs(π) is the multiset sum of scs(πi ) for all i.
(3) Let π be an irreducible admissible representation of G over k, and let π 0 =π1 ⊗ · · · ⊗ πr be a Jordan–Hölder constituent of ResP
G π. Then scs(π) is themultiset sum of scs(πi ) for all i.
Proof. Statement (1) and (2) are easy consequences of the definitions, together withthe transitivity of parabolic induction. The details of the proof of (1) can be foundin [17], II.2.20, for instance. Statement (3) follows from the “geometric lemma” ofBernstein-Zelevinski (which holds in characteristic p as well as characteristic zeroby [17], II.2.19.)

3.2.12. Proposition. Let {π1 , . . . , πr } be a multiset of supercuspidal representationsof GLni over K. There exists, up to isomorphism, exactly one irreducible genericrepresentation π of G over K with supercuspidal support equal to {π1 , . . . , πr }.
Proof. A representation π with supercuspidal support {π1 , . . . , πr } is isomorphic toa generic Jordan–Hölder constituent of IndG
P π1 ⊗ · · · ⊗ πr . By Theorem 3.1.7 and
Proposition 3.1.6, the top derivative of (IndG
P π1 ⊗ · · · ⊗ πr ) is one-dimensional, so
it has exactly one generic Jordan–Hölder constituent.
24

MATTHEW EMERTON AND DAVID HELM

We are now in a position to show:
3.2.13. Theorem. Let π, π 0 be irreducible admissible representations of G over K,or more generally of a Levi subgroup M of G. If Exti (π, π 0 ) is nonzero for some i,then π and π 0 have the same supercuspidal support.
Proof. By [18, IV.6.2], the category of smooth representations of M factors as a
product of blocks; two irreducible representations of M are in the same block if,
and only if, their supercuspidal supports are inertially equivalent, that is, if andonly if they coincide up to twisting by unramified characters. We thus reduce to thecase where the supercuspidal supports of π and π 0 differ by at most an unramified
twist.
First consider the case in which π and π 0 are both cuspidal representations of
G. In this case, Vigneras’ classification of the supercuspidal supports of cuspidal
representations (see for instance the proof of [18], V.4) shows that π 0 is an unramified twist of π. In this case, there exists a maximal distinguished cuspidal type(K, κ ⊗ σ) (in the sense of [18], section IV) contained in both π and π 0 .
We recall some basic facts from [18] about the pair (K, κ ⊗ σ). In particular,
K is a compact open subgroup of G, with a normal pro-` subgroup K1 . Moreover,there exists a finite extension E 0 of E, an embedding of GLr (OE 0 ) into K, and an
isomorphism of K/K1 with G := GLr (OE 0 /$E 0 ) such that the composition∼G
GLr (OE 0 ) → K → K/K1 =is equal to the reduction map GLr (OE 0 ) → G. Moreover, κ is a representation of
K whose restriction to K1 is irreducible, and σ is a cuspidal representation of G
viewed as a representation of K trivial on K1 .
The composition
×OE
0 → GLr (OE 0 ) → K
0 ×
extends to a map (E ) → G such that the G-intertwining of (K, κ ⊗ σ) is precisely(E 0 )× K. It follows that there are extensions τ and τ 0 of κ ⊗ σ to (E 0 )× K such thatG00π = c − IndG
(E 0 )× K τ and π = c − Ind(E 0 )× K τ .
The choice of extension τ defines an action of (E 0 )× K/K1 on HomK1 (κ|K1 , V )for any representation V of (E 0 )× K; this makes the functor HomK1 (κ|K1 , −) into an
equivalence between the category of representations of (E 0 )× K whose restriction toK1 is a direct sum of copies of κ and the category of representations of (E 0 )× K/K1 .
Frobenius reciprocity and the Mackey formula yield an isomorphism of ExtiG (π, π 0 )with Exti(E 0 )× K/K1 (τ, τ 0 ). Applying the equivalence of the previous paragraph, andusing the fact that τ 0 is an unramified twist of τ , we find that the latter is isomorphic to Exti(E 0 )× K/K1 (σ, σ ⊗ χ), for some unramified character χ of (E 0 )× K/K1 .
On the other hand, we have an isomorphism of (E 0 )× K/K1 with ($E 0 )Z × G. It isthus clear that Exti(E 0 )× K/K1 (σ, σ ⊗ χ) can only be nonzero if $E 0 acts on σ andσ ⊗ χ via the same scalar; if this happens then σ and σ ⊗ χ are isomorphic and soτ is isomorphic to τ 0 . Then π and π 0 are isomorphic as claimed.
The case in which π and π 0 are cuspidal representations of M now follows, as any
such representation is a tensor product of cuspidal representations of the factors
GLni of M .
We handle the remaining cases by induction on i; the case i = 0 is clear. Now
assume exactly one of π and π 0 is cuspidal; without loss of generality (taking duals ifnecessary) we may assume π is cuspidal. We then have a proper parabolic subgroup

LOCAL LANGLANDS IN FAMILIES

25

P 0 = M 0 U 0 of M , and a cuspidal representation σ of M 0 , such that π 0 arises as a
submodule of IndM
P 0 σ. We thus have an exact sequence:
0 → π 0 → IndM
P0 σ → C → 0
where C is the cokernel of the inclusion of π 0 in IndM
P 0 σ. By Frobenius reciprocityMjP0jExt (π, IndP 0 σ) = Ext (ResM π, σ) for all j, and the latter vanishes because π
is cuspidal. Thus Exti (π, π 0 ) is isomorphic to Exti−1 (π, C). By the inductive
hypothesis, the latter vanishes unless π has the same supercuspidal support assome subquotient of C; since every subquotient of C has the same supercuspidal
support as π 0 the claim follows in this case.
Finally, assume that neither π nor π 0 is cuspidal. We again fix a proper parabolic
subgroup P 0 = M 0 U 0 of M , and a cuspidal representation σ of M 0 , such that π 0arises as a submodule of IndM
P 0 σ. Now the exact sequence:
0 → π 0 → IndM
P0 σ → C → 0
yields an exact sequence:Exti−1 (π, C) → Exti (π, π 0 ) → Exti (π, IndM
P 0 σ).
By the inductive hypothesis Exti−1 (π, C) is nonzero only if π has the same supercuspidal support as π 0 . On the other hand, we again have an isomorphism ofi
P0
Exti (π, IndM
P 0 σ) with Ext (ResM π, σ), and (as σ is cuspidal), the previous caseshows that this too vanishes unless π and π 0 have the same supercuspidal support.

3.2.14. Corollary. If V is an essentially AIG representation of G over K, then all
the Jordan–Hölder constituents of V have the same supercuspidal support.
Proof. Suppose otherwise. As V is the sum of its finite length submodules, thereis then a finite length submodule W of V that is minimal among submodules of V
that have a Jordan–Hölder constituent with supercuspidal support different fromthat of soc(V ). Let W 0 be the kernel of the map W → cosoc(W ). The minimality ofW implies that cosoc(W ) is irreducible and that every Jordan–Hölder constituentof W 0 has the same supercuspidal support as soc(V ). Thus Exti (W 0 , cosoc(W ))vanishes for all i, by the preceding theorem; in particular cosoc(W ) is a direct
summand of W . This is impossible, since Lemma 3.2.3 implies that any essentially
AIG representation is indecomposable.

3.2.15. Corollary. Let V be an essentially AIG representation of G over K. If V
is admissible, then V has finite length.
Proof. By Corollary 3.2.14 there are only finitely many isomorphism classes of
Jordan–Hölder constituents of V , and one can bound the number of times any given
Jordan–Hölder constituent appears in terms of the dimension of the U -invariantsof V for a sufficiently small compact open subgroup U of G.

Corollary 3.2.14 has additional finiteness implications for essentially AIG representations. More precisely, for a smooth representation V of G, define socc (V )inductively by setting soc1 (V ) = soc(V ), and defining socc (V ) to be the preimageof soc(V / socc−1 (V )) under the surjection
V → V / socc−1 (V ).
We then have:

26

MATTHEW EMERTON AND DAVID HELM

3.2.16. Theorem. Let V be an essentially AIG representation of G over K. Thensocc (V ) has finite length for all c.
Proof. By induction it suffices to show that socc (V )/ socc−1 (V ) has finite length
for all c ≥ 2. The space socc (V )/ socc−1 (V ) is semisimple, and every irreduciblesummand of socc (V )/ socc−1 (V ) is an irreducible non-generic representation of G
with the same supercuspidal support as soc(V ). There are finitely many isomorphism classes of such representations. It thus suffices to show, for every irreducible
non-generic representation π of G with the same supercuspidal support as soc(V ),that Hom(π, socc (V )/ socc−1 (V )) is finite dimensional. We have an exact sequence:
0 → socc−1 (V ) → socc (V ) → socc (V )/ socc−1 (V ) → 0.
As the socle of socc (V ) is generic, we have Hom(π, socc (V )) = 0. We thus obtainan injection:
Hom(π, socc (V )/ socc−1 (V )) → Ext1 (π, socc−1 (V )).
But socc−1 (V ) has finite length by the induction hypothesis, and so by [19], Corollary 2.12, Ext1 (π, socc−1 (V )) is finite dimensional.

3.2.17. Corollary. Let V be an essentially AIG representation of G, let c be apositive integer, and let Vi be an arbitrary collection of submodules of V of length
less than or equal to c. Then the sum of the Vi has finite length.
Proof. Each Vi is contained in socc (V ), so their sum is as well. The result thus
follows immediately from the theorem above.

We close this subsection with the following result treating essentially AIG representations in the case n = 2.
3.2.18. Proposition. Any essentially AIG representation over GL2 (E) is of finite
length.
Proof. Let V be an essentially AIG representation over GL2 (E); we must show that
V has finite length. Clearly we may check this after making an extension of scalars,
and so without loss of generality we may and do assume that K = K.
If V / soc(V ) is trivial then V = soc(V ) is irreducible, and we are done. Thus weassume from now on that V / soc(V ) is non-trivial. The quotient V / soc(V ) contains
no generic constituent, hence its Jordan–Hölder factors are all one-dimensional, andso each is of the form χ ◦ det for some character χ. Moreover, if there exist Jordan–Hölder factors of V / soc(V ) isomorphic to χ ◦ det and χ0 ◦ det, then Corollary 3.2.14
implies χ◦det and χ0 ◦det have the same supercuspidal support. From this it is easy
to see that χ2 = (χ0 )2 . Replacing V by an appropriate twist, we may thus assume
that the center E × of G acts trivially on each Jordan–Hölder factor of V / soc(V ).
Since V / soc(V ) is the sum of its finite length subrepresentations (as V is; thisis one of the conditions of being essentially AIG), and since each of its Jordan–Hölder factors is one-dimensional, V is a sum of finite-dimensional representationsof GL2 (E). The action of GL2 (E) on any finite dimensional representation factors
through det, so the action of GL2 (E) on V / soc(V ) must also factor through det. In
this way we regard V / soc(V ) as a representation of E × . Since V admits a central
character, by Lemma 3.2.4, and since the centre acts trivially on each Jordan–Hölder
factor of V / soc(V ), we see that the centre must act trivially on V . (This is where
we use the non-triviality of V / soc(V ).) Thus V / soc(V ) is in fact a representation

LOCAL LANGLANDS IN FAMILIES

27

of the group E × /(E × )2 . Theorem 3.2.16 shows that the socle of V / soc(V ) is finite
length, and it follows from the fact that E × /(E × )2 is finite that V / soc(V ) itself isof finite length, as required.

3.3. Invariant lattices. We now prove some results about the reduction of finite
length essentially AIG representations of GLn (E). Let O be a complete discrete
valuation ring, with field of fractions K and residue field K of characteristic different
from `. Fix a uniformizer $ of O.
If V is an integral admissible smooth representation of GLn (E) of finite length,
then by the “Brauer-Nesbitt Theorem” of ([17, Ch. II.5.11]), V is a good integral
representation in the sense of Definition 2.2.1, and hence Lemma 2.2.4 shows that
if V ◦ is a GLn (E)-invariant O-lattice in V , then (V ◦ /$V ◦ )ss is independent of V ◦ ;ss
we denote it V .
3.3.1. Proposition. If V is an essentially AIG admissible smooth representationssof GLn (E) over K which is integral and of finite length, then V contains a unique
irreducible generic summand.
Proof. Fix an invariant O-lattice V ◦ in V . Then (V ◦ )(n) is a finitely generated
O-submodule of V (n) , and the latter is a one-dimensional K-vector space. Thus
(V ◦ )(n) is free of rank one over O. As the derivative commutes with tensor products,
ss
it follows that (V )(n) is a one-dimensional K-vector space; the result follows. 
3.3.2. Theorem. If V is an essentially AIG admissible smooth representationof GLn (E) over K which is integral and of finite length, then V admits a $adically separated GLn (E)-invariant lattice V ◦ which is admissible as a GLn (E)◦representation, and such that V := V ◦ /$V ◦ is essentially AIG. Moreover, V ◦ isunique up to homothety.
Proof. We apply Lemma 2.2.6 to V , taking T to consist of all the nongeneric◦
Jordan–Hölder factors. This yields an O-lattice V ◦ , such that V contains no nonss
generic subrepresentations. As V has only one irreducible generic submodule, thisssssss (n)submodule is the socle of V , and V / soc(V )= 0. If H is any open sub◦ H
group of G, then (V ) is $-adically separated, and its E-span coincides with V H ,which is finite dimensional, since V is admissible. It follows that (V ◦ )H is finitely
generated over O, and so V ◦ is an admissible smooth representation of GLn (E).
◦Thus V is admissible smooth, and therefore essentially AIG.
Suppose now that V  is a second lattice in V satisfying the conditions of thecorollary. Scaling it appropriately, we may assume that V  ⊂ V ◦ , but that the◦
induced map V → V is non-zero. Since both source and target are essentially
AIG, this map is necessarily injective by Lemma 3.2.2, and hence (since source andtarget are of the same length) an isomorphism.

4. The local Langlands correspondence in characteristic zero
Let F be an algebraically closed field of characteristic zero. The local Langlands
correspondence for GLn (E) [8] establishes a certain bijection between the set of
isomorphism classes of irreducible admissible smooth representations of GLn (E)on F -vector spaces, and the set of isomorphism classes of n-dimensional Frobenius
semisimple Weil–Deligne representations over F (as defined in [5, §8] or [16, §4]).

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MATTHEW EMERTON AND DAVID HELM

In fact there are various choices of correspondence, depending on the desired
normalization. The so-called unitary correspondence is uniquely determined by therequirement that the local L- and ε-factors attached to a pair of corresponding isomorphism classes should coincide. On the other hand, this correspondence depends
on the choice of a square root of ` in F , and (because of this) is not compatible
in general with change of coefficients (although a suitably chosen twist will be; werefer the reader to 4.2 for details.)
However, even if we normalize the local Langlands correspondence to be compatible with change of coefficients, the correspondence as usually defined is notsuitable for the applications we have in mind. In particular, the usual local Langlands correspondence fails to be compatible with specialization. More precisely, let
O be a complete discrete valuation ring containing Qp , with field of fractions K andresidue field K, and let ρ : GE → GLn (O) be a continuous Galois representation.
Then the local Langlands correspondence associates admissible representations π
and π to the Weil–Deligne representations induced by ρ⊗O K and ρ⊗O K, but there
need not be a close relationship between π and the reduction of π. (For example,
π could be a character even if π is infinite-dimensional.)
We therefore work with a modification of the usual local Langlands correspondence, which we describe fully in 4.2. We denote this correspondence by ρ 7→ π(ρ),
where ρ is a continuous n-dimensional representation of GE over an extension Kof Qp . The correspondence ρ 7→ π(ρ) is essentially the generic local Langlands
correspondence introduced by Breuil and Schneider in [2]. Unlike more standard
formulations of local Langlands, the representation π(ρ) of GLn (E) will in general be reducible (in fact, it will be an essentially AIG representation of GLn (E)).
The map ρ 7→ π(ρ) will not be a bijection in any meaningful sense but simply amap from isomorphism classes of n-dimensional representations of GE over K to
indecomposable admissible representations of GLn (E) over K. The advantage ofthis choice is that the map ρ 7→ π(ρ) will be compatible with change of coefficients(in the sense that π(ρ ⊗K K 0 ) will be isomorphic to π(ρ) ⊗K K 0 for an extension
K 0 of K,) and also compatible with specialization (in the sense of Theorem 4.5.7
below.)
4.1. Galois representations and Weil–Deligne representations. In order togive a precise description of the map ρ 7→ π(ρ), we first recall some basic facts about
Frobenius-semisimple Weil–Deligne representations. Recall that a Weil–Deligne
representation over a field K containing Qp is a pair (ρ0 , N ), where ρ0 : WE →GLn (K) is a smooth representation of WE with coefficients in K and N is a nilpotent endomorphism of K n satisfying ρ0 (w)N ρ0 (w)−1 = |w|N . The representation(ρ0 , N ) is called Frobenius-semisimple if ρ0 is absolutely semisimple.
We first consider absolutely irreducible representations ρ0 : WE → GLn (K).
Let IE be the inertia subgroup of E. Then ρ0 (IE ) is a finite group, and so all of
its irreducible representations are defined over a finite extension K0 of Qp . Afterreplacing K with an algebraic extension we may assume K contains a subfield
isomorphic to K0 . Then the restriction of ρ0 to IE splits as a direct sum of absolutely
irreducible representations τi of IE over K0 .
Let Φ be a Frobenius element of WE , and let V be an IE -stable subspace of
K n isomorphic to τ1 ⊗K0 K as an IE -representation. Then IE acts on ΦV by the
conjugate τ1Φ of τ1 . In fact, we have:LOCAL LANGLANDS IN FAMILIES

29

4.1.1. Lemma. Let r be the order of the orbit of τ1 under the action of Φ on theset of isomorphism classes of absolutely irreducible representations of IE over K0 .
Then we have a direct sum decomposition:
ρ0 |IE =r−1
Miτ1Φ ⊗K0 K

i=0and the action of Φ on this decomposition permutes the summands.
Proof. As IE is normal in WE , this is a standard result in Clifford theory.

In particular, the vector space HomK[IE ] (τ1 ⊗K0 K, ρ0 |IE ) is one-dimensional. Ifr∼we fix an isomorphism τ1 −→ τ1Φ , then we get an endomorphism Ψ of this vector
space via:
Φr

r

∼

HomK[IE ] (τ1 ⊗K0 K, ρ0 ) → HomK[IE ] (τ1Φ ⊗K0 K, ρ0 ) −→ HomK[IE ] (τ1 ⊗K0 K, ρ0 ).
The action of Ψ is given by a scalar λ in K × .
4.1.2. Lemma. For any λ ∈ K × there is a unique absolutely irreducible representation ρ0 over K (up to isomorphism) such that ρ0 |IE contains τ1 ⊗K0 K and Ψ actson HomK[IE ] (τ1 ⊗K0 K, ρ0 ) via λ.
Proof. If r = 1, then the restriction of ρ0 to IE is given by τ1 ⊗K0 K, and soto determine ρ0 it suffices to give an action of Φ on the representation space ofτ1 , compatible with the action of IE . This amounts to giving an isomorphism∼∼
τ1 ⊗K0 K −→ (τ1 ⊗K0 K)Φ . As we have already fixed an isomorphism τ1 −→ τ1Φ ,such an isomorphism is determined by λ.
If r > 1, let E 0 be the unramified extension of E of degree r. The restriction of0ρ to WE 0 breaks up as a sum of irreducible representations ρ00 , . . . , ρ0r−1 such thatithe restriction of ρ0i to IE is isomorphic to τ1Φ . Thus ρ0i is determined by λ and τ1 ,W
and ρ0 is isomorphic to IndWE
ρ00 by Frobenius reciprocity.
E0
4.1.3. Lemma. Let K be a field containing Qp , and let ρ0 be an absolutely irreducible
representation of WE over K. Then there exists an unramified character χ : WE →×
K such that the twist ρ0 ⊗ χ is defined over Qp .
Proof. The representation τ1 is defined over a finite extension of Qp , so it suffices
to show that after a twist we can take the scalar λ to be in Qp . Twisting by anunramified χ changes λ to χ(Φ)r λ, so this is clear.

4.1.4. Definition. Let ρ0 be an absolutely irreducible smooth representation WE →GLn (K), and let d be a positive integer. The special representation Spρ0 ,d is thepairSpρ0 ,d = (V0 ⊕ · · · ⊕ Vd−1 , N ),where WE acts on Vi by |0 ≤ i ≤ d − 2.

|i ρ0 and N maps Vi isomorphically onto Vi+1 for

The representation Spρ0 ,d is well-defined up to isomorphism, and is an absolutely
indecomposable Weil–Deligne representation. If K is algebraically closed, thenevery indecomposable Frobenius-semisimple Weil–Deligne representation has theform Spρ0 ,d for a unique absolutely irreducible representation ρ0 of WE over K.
Combining this with the previous lemma, we find:

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MATTHEW EMERTON AND DAVID HELM

4.1.5. Lemma. Let K be a field containing Qp , and let (ρ0 , N ) be an indecomposable
Frobenius-semisimple Weil–Deligne representation over K. Then there exists a×character χ : WE → K such that the twist (ρ0 ⊗ χ, N ) is defined over Qp .
In those situations in which we will apply the local Langlands correspondence,
we will be beginning not with Weil–Deligne representations, but with Galois representations. Thus we recall the recipe of Deligne for associating a Weil–Deligne
representation to a continuous Galois representation, in a slightly broader context
than that in which it is usually considered.
Let A be a complete Noetherian local domain of residue characteristic p different
from `, maximal ideal m, and field of fractions K of characteristic zero. Let R be
any subring of K containing A and p1 . (In most applications, R will equal either K,or else a complete discrete valuation ring O containing A and contained in K.)
For any n ≥ 0, we say that a representation ρ : GE → GLn (R) is continuous
if we can find a finitely generated A-submodule M of Rn that is invariant underρ(GE ), spans Rn over R, and such that the induced GE -action on M is m-adicallycontinuous. (Note that if R = K, and K is a finite extension of Qp , then thiscoincides with the usual notion of continuity of a GE -representation.)
As in [16, (4.2)], we fix a non-zero homomorphism tp : IE → Qp . (When comparing the present discussion with that of [16], note that the roles of ` and p arereversed.) This homomorphism is uniquely determined up to scaling by an element of Q×
p . The following result then extends a theorem of Deligne [5, §8], [16,Thm. (4.2.1)] (which treats the case when the coefficient field is a finite extensionof Qp ).
4.1.6. Proposition. A continuous representation ρ : GE → GLn (R) uniquely determines the following data:(1) a representation ρ0 : WE → GLn (R) that is continuous when the target is
equipped with its discrete topology;(2) a nilpotent matrix N ∈ Mn (R);subject to the following condition:(3) ρ(Φr σ) = ρ0 (Φr σ) exp(tp (σ)N ) for all σ ∈ IE and r ∈ Z.
Furthermore, as a Weil–Deligne representation, the pair (ρ0 , N ) is independent,
up to isomorphism, of the choice of tp and Φ.
Proof. Let (ρ01 , N1 ) and (ρ02 , N2 ) be two Weil–Deligne representations satisfying thecondition of the proposition. Then there is an open subgroup of IE on which both
ρ01 and ρ02 are trivial; we can thus find an element σ of IE for which ρ01 (σ) and ρ02 (σ)
are the identity but tp (σ) is nonzero. Then N1 = tp 1(σ) log ρ(σ) = N2 . The identity
ρ(Φr σ) = ρ0i (Φr σ) exp(tp (σ)Ni )then forces ρ01 = ρ02 .
It thus suffices to construct a (ρ0 , N ) as above. Choose a finitely-generated Asubmodule M of Rn that is preserved by ρ and spans Rn over R. Then GE acts
via ρ on M/mi+1 M for all i, and these A-modules are discrete with respect to them-adic topology. In particular for each i the subgroup Hi of IE that acts triviallyon M/mi M is a compact open subgroup of IE .
The group of automorphisms of M/mi+1 M that reduce to the identity in M/mi M
is an abelian p-group for all i ≥ 2. Thus the action of H2 on M factors throughLOCAL LANGLANDS IN FAMILIES

31

the map tp : IE → Qp . Let σ be an element of H2 ; the action of σ on M yields
an element of α of End(M ) that is congruent to the identity modulo m2 . The
power series log(α) thus converges in the m-adic topology on End(M ); set N =1
tp (σ) log(α). Then any τ ∈ H2 acts on M via exp(tp (τ )N ). It follows that for allτ ∈ GE , ρ(τ )N ρ(τ )−1 = |τ |N . In particular N must be nilpotent.
We can then set ρ0 (Φr σ) = ρ(Φr σ) exp(tp (σ)N )−1 for all σ ∈ IE and r ∈ Z; thisgives a well-defined ρ0 that is trivial on the compact open subgroup H2 of WE . 
Let O be a discrete valuation ring containing A and contained in K, with residue
field K of characteristic zero and uniformizer $. We will be interested in the
reduction mod $ of both Galois representations and Weil–Deligne representations
over O. One has:
4.1.7. Lemma. Let ρ0 : WE → GLn (O) be a representation of WE over O such that
ρ0 ⊗O K is absolutely irreducible. Then ρ0 := ρ0 ⊗O K is also absolutely irreducible.
Proof. By Lemma 4.1.1, over a finite extension of K, the restriction of ρ0 to IE splits
as a direct sum of absolutely irreducible representations ρ0i of IE , each of which
factors through a finite quotient if IE and is defined over Qp . The representations
ρ0i are distinct and cyclically permuted by conjugation by Φ. As K has characteristic
zero, the ρ0i remain irreducible and distinct after “reduction mod $”.
Thus, over a finite extension of K, ρ0 splits as a direct sum of absolutely irreducible representations ρ0i which are distinct and cyclically permuted under conjugation by Φ. It is thus clear that ρ0 is absolutely irreducible.
It follows that if ρ0 : WE → GLn (K) is absolutely irreducible and contains an
O-lattice L, then the mod $ reduction of L is independent, up to isomorphism, ofthe lattice L. We denote this reduction by ρ0 .
The passage from Galois representations to Weil–Deligne representations commutes with reduction modulo $:
4.1.8. Lemma. Let ρ : GE → GLn (O) be a continuous Galois representation, with
mod $ reduction ρ, and let (ρ0 , N ) and (ρ0 , N ) be the Weil–Deligne representations
attached to ρ and ρ, respectively. Then (ρ0 , N ) is isomorphic to (ρ0 ⊗O K, N ⊗O K).
Proof. This follows immediately from the identities
ρ(Φr σ) = ρ0 (Φr σ) exp(tp (σ)N )ρ(Φr σ) = ρ0 (Φr σ) exp(tp (σ)N )and the fact that the latter identity characterizes (ρ0 , N ) up to isomorphism.

Given a Weil–Deligne representation (ρ0 , N ) over K, one can associate a natural
Frobenius-semisimple representation (ρ0 , N )F−ss , (the Frobenius-semisimplificationof (ρ0 , N ).) We recall the definition; see [5, 8.5] for details.
The matrix ρ0 (Φ) factors uniquely as a product su, with s and u elements ofGLn (K) that are semisimple and unipotent, respectively, and commute with eachother. Moreover, if ρ0 (Φ) lies in GLn (O), then so do s and u. The element u thencommutes with N , and one defines (ρ0 )F−ss to be the representation of WF thatsatisfies (ρ0 )F−ss (Φr σ) = u−r ρ0 (Φr σ). Then (ρ0 )F−ss is a semisimple representationof WF over K, and the pair ((ρ0 )F−ss , N ) is a Frobenius-semisimple Weil–Deligne
representation which we write (ρ0 , N )F−ss .

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MATTHEW EMERTON AND DAVID HELM

It will be necessary for us to understand how Frobenius-semisimplification commutes with reduction modulo $. Note that even if (ρ0 , N ) is a Frobenius-semisimple
Weil–Deligne representation over O, its reduction modulo $ need not be, as themod $ reduction of a semisimple element of GLn (O) need not be semisimple.
4.1.9. Lemma. Let (ρ0 , N ) be a Weil–Deligne representation over O, and let (ρ0 , N )be its reduction mod $. Then (ρ0 , N )F−ss is defined over O. Moreover, the reduction
mod $ of (ρ0 , N )F−ss has Frobenius-semisimplification (ρ, N )F−ss .
Proof. If ρ0 (Φ) decomposes as su, with s and u as above, then s and u lie in GLn (O),so (ρ0 , N )F−ss is defined over O. Thus ρ0 (Φ) = su, where s and u are the mod $reductions of s and u.
The element s decomposes uniquely as s0 u0 , where s0 is semisimple and u0 is
unipotent and commutes with s0 . As u commutes with s, s also decomposes as a
product of us0 u−1 with uu0 u−1 ; the uniqueness of this decomposition shows thatthese two decompositions coincide. That is, u commutes with s0 and u0 .
We have ρ0 (Φ) = s0 u0 u, and the unipotent element u0 u commutes with s0 . Thus
the Frobenius-semisimplification of (ρ0 , N ) sends Φ to s0 . On the other hand, thereduction of (ρ0 , N )F−ss takes Φ to s, which equals s0 u0 . Hence the Frobeniussemisimplification of the reduction of (ρ0 , N )F−ss takes Φ to s, and therefore coin0
cides with (ρ0 , N )F−ss .
4.2. The generic local Langlands correspondence of Breuil and Schneider.
We are now in a position to describe the “generic local Langlands correspondence”
of Breuil and Schneider [2, pp. 162–164]. This is a map (ρ0 , N ) 7→ π(ρ0 , N ) from
Frobenius-semisimple Weil–Deligne representations over a finite extension K of Qp
1
to indecomposable admissible representations of GLn (E) over K. Fix a choice of ` 2
in Qp (and thus a choice of square root of the character | | ◦ det of GLn (E), as well
as a unitary local Langlands correspondence for representations over Qp ). With
this choice, the properties of this correspondence can be summarized as follows(c.f. [2, 4.2]):
00(1) For any character χ : WF → Q×
p , one has π(ρ ⊗ χ, N ) = π(ρ , N ) ⊗ χ.
0000(2) If K is a finite extension of K, then π(ρ ⊗K K , N ) = π(ρ , N ) ⊗K K 0 .
(3) If (ρ0 , N ) is a direct sum of representations of the form Spρ0i ,ni over Qp ,then π(ρ0 , N ) is defined by the parabolic induction:n−1

GL (E)π(ρ0 , N ) = (| | ◦ det)− 2 IndQ nStπ1 ,n1 ⊗ · · · ⊗ Stπr ,nr ,where πi corresponds to ρi under the unitary local Langlands correspondence, Stπi ,ni is the generalized Steinberg representation (and thus corresponds to Spρi ,ni under unitary local Langlands,) and Q is the upper
triangular parabolic subgroup of GLn (E) whose Levi subgroup is block diGL (E)agonal with block sizes (n1 dim ρ01 , . . . , nr dim ρ0r ). The symbol IndQ ndenotes normalized parabolic induction. The representations Stρi ,ni areordered so that the condition of [14, Def. 1.2.4] holds. (As long as thiscondition is satisfied, the resulting parabolic induction is independent, up
to isomorphism, of the precise choice of ordering, as well as of the choice1
the square root of ` needed to define (| | ◦ det) 2 .)

LOCAL LANGLANDS IN FAMILIES

33

These properties uniquely characterize the generic local Langlands correspondence. We will need a slight extension of this correspondence to the case of coefficients in an arbitrary field extension K of Qp . Let (ρ0 , N ) be a Frobenius-semisimple
Weil–Deligne representation over K, and suppose that it decomposes over K as adirect sum of representations of the form Spρ0i ,ni . Then by Lemma 4.1.5, there×

exist characters χi : WE → K such that ρ0i ⊗ χi is defined over Qp . For such
representations the unitary local Langlands correspondence is defined, and we cantake πi to be the representation over K such that πi ⊗ χi corresponds to ρ0i ⊗ χi
via the unitary local Langlands correspondence over Qp .
Stπi ,ni = Stπi0 ⊗χi ,ni ⊗ (χ−1i ◦ det).
n−1

GL (E)π(ρ0 , N ) = (| | ◦ det)− 2 IndQ nStπ1 ,n1 ⊗ · · · ⊗ Stπr ,nr ,where the Stπi ,ni are ordered as before. A priori, this is a representation of GLn (E)over K, but the argument of [2, Lem. 4.2] shows that π(ρ0 , N ) is defined over K
itself. Moreover, π(ρ0 , N ) is independent of the choices of χi .
As was the case over finite extension of Qp , the map (ρ0 , N ) 7→ π(ρ0 , N ) is
compatible with twists, and also with arbitrary field extensions.
We extend this definition to a map from representations of GE to admissible
representations of GLn (E) as follows:
4.2.1. Definition. Let ρ be a continuous n-dimensional representation of GE overK, and let (ρ0 , N ) be the corresponding Weil–Deligne representation. We defineπ(ρ) to be π((ρ0 , N )F−ss ).
4.3. Segments and the Zelevinski classification. Our next goal is to establish
key properties of the generic local Langlands correspondence (in particular, we willshow that π(ρ0 , N ) is essentially AIG).
Following [21], we define a segment to be a set of supercuspidal representationsof the form: [π, (| | ◦ det)π, . . . , (| | ◦ det)r−1 π], where π is an irreducible supercuspidal representation of GLn (E) over K. We think of the segment ∆ givenby [π, (| | ◦ det)π, . . . , (| | ◦ det)r−1 π] as corresponding to the generalized Steinberg representation Stπ,r ; this gives a bijection between segments and generalized
Steinberg representations. If Stπ,r corresponds to a segment ∆, we will often writeSt∆ for Stπ,r . Similarly, we will write Sp∆ for the indecomposable Weil–Deligne
representation Spρ,r , where ρ is the irreducible Weil–Deligne representation corresponding to π under the unitary local Langlands correspondence.
Two segments ∆, ∆0 are said to be linked if neither contains the other, and if∆ ∪ ∆0 is a segment. The segment ∆ precedes ∆0 if ∆ and ∆0 are linked and ∆0
has the form [(| | ◦ det)π, . . . , (| | ◦ det)r−1 π] for some π in ∆.
We consider the following condition on a sequence S~ of segments ∆i :4.3.1. Condition. For all i < j, the segment ∆i does not precede the segment ∆j .
It is clear that any multiset of segments can be given an ordering that satisfies Condition 4.3.1. If S is a multiset of segments (henceforth referred to as amultisegment), then we let π(S) denote the normalized parabolic induction
GL (E)

IndQ nSt∆1 ⊗ · · · ⊗ St∆n ,where ∆1 , . . . , ∆n are the segments in S, taken with multiplicities and ordered sothat Condition 4.3.1 holds, and Q is a suitable, block upper triangular parabolic

34

MATTHEW EMERTON AND DAVID HELM

subgroup of GLn (E). By [21, Prop. 6.4], the representation π(S) does not depend,
up to isomorphism, on the order of the collection of segments in S (as long as Condition 4.3.1 holds). Note that if (ρ0 , N ) is an n-dimensional Frobenius-semisimple
Weil–Deligne representation that decomposes as the direct sum of Sp∆i for ∆i ∈ S,
n−1
then π(ρ0 , N ) is isomorphic to (| | ◦ det)− 2 π(S). By [14, 1.2.5], π(S) admits aunique irreducible quotient Q(S), and Q(S) is the irreducible representation corresponding to (ρ0 , N ) under the unitary local Langlands correspondence.
4.3.2. Proposition. If S is an multisegment, then every irreducible submodule ofπ(S) is generic.
Proof. We will prove a stronger statement — namely, that every irreducible Pn submodule of the restriction π(S)|Pn is generic. (In other words, π(S) embeds in its
Kirillov model.) Over the complex numbers this is a result of Jacquet-Shalika [13].
Their argument does not seem to adapt easily to other fields of characteristic zero.
One could reduce this proposition to their result by choosing an isomorphism of
K with C; we instead give an algebraic argument over K that is an adaptation of
the argument of [1, 4.15]. Their argument necessarily uses Ψ± and Φ± functorsthat are normalized differently from ours, to avoid unpleasant combinatorial issues.
Therefore, for the purposes of this proof only we take the functors Ψ± and Φ± to
be normalized as in [1], rather than as in section 3.1.
Let S be the multisegment (∆1 , . . . , ∆n ), where ∆i does not precede ∆j forany j > i. We can assume without loss of generality that the ∆i are ordered sothat if ∆i = [πi , (| | ◦ det)πi , . . . (| | ◦ det)ri −1 πi ], where πi is a supercuspidal
representation of GLni (E), then (| | ◦ det)ri πi is not contained in any segment ∆jwith j > i; clearly for such an ordering ∆i never precedes a ∆j with j > i. We
proceed by induction on the sum of the lengths of the segments ∆i . Note that theresult is clear for a single segment, as St∆i is absolutely irreducible and generic. Let
S 0 be the multisegment (∆2 , . . . , ∆n ); by the induction hypothesis every irreducible
submodule of π(S 0 ) is generic.
Suppose we have an irreducible, non-generic submodule ω of π(S)|Pn . We haveπ(S) = St∆1 × π(S 0 ), where “×” is the product defined in [1, 4.12]. By [1, 4.13a],
we have an exact sequence:
0 → (St∆1 )|Pr1 n1 × π(S 0 ) → π(S)|Pn → St∆1 × π(S 0 )|Pn−r1 n1 → 0.
In particular, ω is a submodule of one of (St∆1 )|Pr1 n1 ×π(S 0 ) or St∆1 ×π(S 0 )|Pn−r1 n1 .
(k)Suppose first that ω is contained in (St∆1 )|Pr1 n1 × π(S 0 ). By [21, 9.6], St∆i is
(kn )

(k)

zero if k is not divisible by ni , whereas St∆i i is St∆(k) , where ∆iis the segment

i

k

r−1

[(| | ◦ det) πi , . . . , (| | ◦ det)
− i

− kn1

(Φ ) ((Φ )πi ]. It follows by [1, 4.13c], that, for i < n1 ,(St∆1 |Pr1 n1 ) × π(S 0 )) = (Φ− )kn1 +i (St∆1 |Pr1 n1 ) × π(S 0 ),so that for such i, ((Φ− )kn1 (St∆1 |Pr1 n1 ) × π(S 0 ))(i) = 0. For i = n1 , [1, 4.13c]shows that the representation (Φ− )(k+1)n1 (St∆1 |Pr1 n1 ) × π(S 0 ) is instead a proper
submodule of (Φ− )n1 ((Φ− )kn1 (St∆1 |Pr1 n1 ) × π(S 0 )); the quotient of the latter by
the former is isomorphic to St∆(k+1) × π(S 0 )|Pn−r1 n1 .
1Since ω is contained in (St∆1 )|Pr1 n1 × π(S 0 ), we have ω (i) = 0 for i < n1 . As ω
has at least one nonzero derivative it follows that (Φ− )n1 −1 ω is nonzero. On theLOCAL LANGLANDS IN FAMILIES

35

other hand, we have (Φ− )n1 −1 ω ⊂ (Φ− )n1 −1 ((St∆1 )|Pr1 n1 × π(S 0 )); by [1, 4.13d] it
follows that (Φ− )n1 ω is nonzero.
Then (Φ− )n1 ω is a non-generic submodule of (Φ− )n1 ((St∆1 )|Pr1 n1 × π(S 0 )), andis therefore a non-generic submodule of either (Φ− )n1 ((St∆1 )|Pr1 n1 ) × π(S 0 ), orSt∆(1) × π(S 0 )|Pn−r1 n1 . It is easy to rule out the latter case: by the inductive
1
hypothesis π(S 0 )|Pn−r1 n1 has no non-generic submodules; by [21, 5.3] neither doesSt∆(1) × π(S 0 )|Pn−r1 n1 .
1Thus (Φ− )n1 ω is a non-generic submodule of (Φ− )n1 ((St∆1 )|Pr1 n1 ) × π(S 0 ). In
particular ((Φ− )n1 ω)(i) = 0 for i < n1 ; it follows as above that (Φ− )2n1 −1 ω isnonzero, and by [1, 4.13d], that (Φ− )2n1 ω is nonzero. Then (Φ− )2n1 ω is a nonzero
non-generic submodule of (Φ− )n1 ((Φ− )n1 ((St∆1 )|Pr1 n1 ) × π(S 0 )), and hence (withanother use of the inductive hypothesis and [21, 5.3]), is a nonzero non-genericsubmodule of (Φ− )2n1 ((St∆1 )|Pr1 n1 ) × π(S 0 ).
Proceeding in this fashion we find that (Φ− )kn1 ω is a nonzero non-generic submodule of (Φ− )kn1 ((St∆1 )|Pr1 n1 ) × π(S 0 ) for all k, which is impossible since thelatter vanishes for large k.
We have thus ruled out the possibility that ω is contained in (St∆1 )|Pr1 n1 ×π(S 0 ).
The other alternative is that ω is contained in St∆1 × π(S 0 )|Pn−r1 n1 . Suppose thiswere the case, and let k be the largest integer such that ω (k) is nonzero. Thenω (k) is nonzero and embeds in the k-the derivative of St∆1 × π(S 0 )|Pn−r1 n1 , whichis St∆1 × π(S 0 )(k) . It follows that the supercuspidal support of ω (k) contains thatof St∆1 ; in particular it contains (| | ◦ det)r−1 π1 . By [1, 4.7b], it follows that(| | ◦ det)r π1 is contained in the supercuspidal support of π(S); this is impossible
by our choice of ordering on the ∆i .

4.3.3. Corollary. If π is an admissible representation of GLn over a field K ofcharacteristic zero, such that π ⊗K K is isomorphic to π(S) for some S satisfying Condition 4.3.1, then π is essentially AIG. In particular, every representation
π(ρ0 , N ) over a field K of characteristic zero is essentially AIG.
Proof. It suffices to show that π ⊗K K is essentially AIG, as then the socle of π
must be absolutely irreducible and generic, and π must contain no other irreducible
generic subquotients. But π ⊗K K has the form π(S) for some S, so the previous
proposition shows that the socle of π(S) is a direct sum of irreducible genericrepresentations. It thus suffices to show that π(S)(n) is one-dimensional; this follows
from the fact that St∆i is irreducible and generic, together with Theorem 3.1.7 and
Proposition 3.1.6.

If S and S 0 are two multisegments, we say S 0 arises from S by an elementary
operation if S 0 is obtained from S by replacing a pair of linked segments ∆, ∆0 in
S with the pair ∆ ∪ ∆0 , ∆ ∩ ∆0 . We say that S 0  S if S 0 can be obtained from
S by a sequence of elementary operations. This partial order contains information
about the Jordan–Hölder constituents of a given π(S). In particular, one has thefollowing:
4.3.4. Theorem ([15], Theorem 5.3). If S is a multisegment, then every Jordan–Hölder constituent of π(S) is isomorphic to Q(S 0 ) for some S 0  S, and each suchQ(S 0 ) appears with positive multiplicity in π(S).

36

MATTHEW EMERTON AND DAVID HELM

In fact, the relationship between π(S) and π(S 0 ) is considerably stronger than
the theorem above suggests. We will construct maps of π(S 0 ) into π(S) for all
S 0  S, and show that any nonzero such map is an embedding, and unique up toscaling. Before we do so, however, we need a preliminary result about the partial
order . Let S be a multisegment, and suppose that S 0 is obtained from S by asingle elementary operation. We say this elementary operation is primitive if there
is no multisegment S 00 with S 0  S 00  S other than S 00 = S and S 00 = S 0 .
4.3.5. Lemma. Let S be a multisegment, let ∆ and ∆0 be two linked segments in
S, such that ∆ precedes ∆0 . Suppose that the elementary operation that replaces ∆and ∆0 with ∆ ∩ ∆0 , ∆ ∪ ∆0 is primitive. Then there exists an ordering on S thatsatisfies Condition 4.3.1, and in which ∆0 and ∆ appear consecutively.
Proof. Choose an ordering on S that satisfies Condition 4.3.1, and that minimizes
the number of segments that appear between ∆0 and ∆. Suppose there is a segment∆00 between ∆ and ∆0 .
By our assumption on the chosen ordering, the ordering on S obtained by moving∆00 after ∆ fails to satisfy Condition 4.3.1. There must thus be a segment ∆000 thatappears between ∆00 and ∆ in the chosen ordering, for which ∆000 precedes ∆00 .
Similarly, ∆00 must precede a segment that appears between ∆0 and ∆00 in thechosen ordering.
Applying these considerations repeatedly we obtain a chain:∆0 = ∆0 , ∆1 , . . . , ∆r = ∆such that each ∆i precedes ∆i−1 , and appears after ∆i−1 in the chosen order on
S. Moreover, since ∆ precedes ∆0 , it follows that ∆ precedes ∆1 . The elementary
operation on S that replaces ∆ and ∆0 with ∆ ∩ ∆0 and ∆ ∪ ∆0 then factors as:(1) Replace ∆ and ∆1 with ∆ ∪ ∆1 and ∆ ∩ ∆1 .
(2) Replace ∆0 and ∆ ∪ ∆1 with ∆ ∪ ∆1 ∪ ∆0 and ∆0 ∩ (∆ ∪ ∆1 ). (Note that∆ ∪ ∆1 ∪ ∆0 is equal to ∆ ∪ ∆0 .)(3) Replace ∆0 ∩ (∆ ∪ ∆1 ) and ∆ ∩ ∆1 with [∆0 ∩ (∆ ∪ ∆1 )] ∪ [∆ ∩ ∆1 ] (whichis equal to ∆1 ), and [∆0 ∩ (∆ ∪ ∆1 )] ∩ [∆ ∩ ∆1 ] (which is equal to ∆ ∩ ∆0 .)In particular the elementary operation that replaces ∆ and ∆0 with ∆ ∩ ∆0 and∆ ∪ ∆0 is not primitive, as required.

4.3.6. Proposition. Suppose that S and S 0 are multisegments, and that S 0  S.
Then HomK[GLn (E)] (π(S 0 ), π(S)) is one-dimensional over K, and every nonzero
map π(S 0 ) → π(S) is an embedding.
Proof. As π(S 0 ) and π(S) are essentially AIG, it suffices to show that there exists
a nonzero map π(S 0 ) → π(S). Moreover, we may reduce to the case where S 0
and S differ by a single, primitive, elementary operation. Let S 0 differ from S byreplacing ∆, ∆0 with ∆ ∪ ∆0 , ∆ ∩ ∆0 , where ∆0 precedes ∆. By the above lemma
we may choose an ordering on S that satisfies Condition 4.3.1 in which ∆ and ∆0
are adjacent. We obtain from this ordering on S an ordering on S 0 in which ∆ ∩ ∆0
replaces ∆ and ∆ ∪ ∆0 replaces ∆0 ; this ordering also satisfies Condition 4.3.1.
Let S0 be the multisegment consisting of the segments that appear in S before ∆and ∆0 in this chosen ordering, and let S1 be the multisegment consisting of thesegments that appear in S after ∆ and ∆0 .

LOCAL LANGLANDS IN FAMILIES

37

By [15, Prop. 4.3], we find that π(∆ ∪ ∆0 , ∆ ∩ ∆0 ) embeds in π(∆, ∆0 ). But π(S 0 )is isomorphic ton
IndGLπ(S0 ) ⊗ π(∆ ∪ ∆0 , ∆ ∩ ∆0 ) ⊗ π(S1 ),P

for P a suitable block upper triangular parabolic subroup of GLn , and π(S) is
isomorphic ton
IndGLπ(S0 ) ⊗ π(∆, ∆0 ) ⊗ π(S1 ).
P
The embedding of π(∆ ∪ ∆0 , ∆ ∩ ∆0 ) in π(∆, ∆0 ) thus gives rise to a nonzero mapof π(S 0 ) into π(S), as required.
Moreover, the embeddings of π(S 0 ) into π(S) constructed above descend to fields
of definition:
4.3.7. Proposition. Let π and π 0 be admissible representations over K, and suppose
there are multisegments S and S 0 , with S 0  S, such that π ⊗K K is isomorphic
to π(S) and π 0 ⊗K K is isomorphic to π(S 0 ). Then HomK[GLn (E)] (π 0 , π) is onedimensional over K, and every nonzero map π 0 → π is an embedding.
Proof. As π and π 0 are essentially AIG, HomK[GLn (E)] (π 0 , π) is either zero or onedimensional over K, and every nonzero map π 0 → π is an embedding. It thus
suffices to construct a nonzero map from π 0 to π. Let φ : π 0 ⊗K K → π ⊗K K
be an embedding. By Lemma 3.2.5, a scalar multiple of φ descends to the desired
embedding of π 0 in π.

We immediately deduce:
4.3.8. Corollary. Let ρ be a continuous n-dimensional representation of GE over K,
and let π be an admissible representation of GLn (E) over K, such that π ⊗K K is
isomorphic to π(ρ ⊗K K). Then π is isomorphic to π(ρ).
The above results allow us to establish some useful facts about essentially AIG
envelopes in characteristic zero, that will be useful in the proof of Proposition 6.2.8.
4.3.9. Theorem. Let π be an irreducible generic representation of GLn (E) over analgebraically closed field K of characteristic zero, and let π1 , . . . , πr be the supercuspidal support of π, ordered so that Condition 4.3.1 holds (when the πi are treatedas one-element segments.) Then the parabolic induction
GL (E)

IndP nπ1 ⊗ · · · ⊗ πr

is an essentially AIG envelope of π. (Here P = M U is a suitable block upper triangular parabolic subgroup of GLn (E), with Levi subgroup M and unipotent radical
U .)Proof. By Corollary 4.3.3, the representation
GL (E)

IndP nπ1 ⊗ · · · ⊗ πr

is essentially AIG. Its socle is thus an irreducible generic representation with thesame supercuspidal support as π, and is therefore isomorphic to π, by Proposition 3.2.12. It thus remains to show that any essentially AIG representation W
whose socle is isomorphic to π embeds in
GL (E)

IndP nπ1 ⊗ · · · ⊗ πr .

38

MATTHEW EMERTON AND DAVID HELM

Note that as any map of essentially AIG representations is injective, it suffices toconstruct a map:
GL (E)
W → IndP n
π1 ⊗ · · · ⊗ πr .
By Frobenius Reciprocity, this is equivalent to constructing a map:
ResP
GLn (E) W → π1 ⊗ · · · ⊗ πr .
As every Jordan–Hölder constituent of W has supercuspidal support {π1 , . . . , πr }by Corollary 3.2.14, it follows that every Jordan–Hölder constituent of ResP
GLn (E) W
is a supercuspidal representation of M , and at least one of these Jordan–Hölder
constituents is isomorphic to π1 ⊗ · · · ⊗ πr . By Theorem 3.2.13, π1 ⊗ · · · ⊗ πr only
admits nontrivial extensions (as an M -representation) with irreducible representations isomorphic to π1 ⊗ · · · ⊗ πr . Thus ResP
GLn (E) W admits a quotient isomorphic
to π1 ⊗ · · · ⊗ πr and the result follows.

4.3.10. Corollary. Let W be an essentially AIG representation of GLn (E) over a
field K of characteristic zero. Then W has finite length.
Proof. Let W 0 be the essentially AIG envelope of soc(W ). By the preceding lemma,
W 0 ⊗K K has finite length, so W 0 , and hence W , has finite length.
It will be useful later, in the proof of Proposition 6.2.8, to have control over themultiplicities of Jordan–Hölder constituents of essentially AIG representations. We
observe:
4.3.11. Corollary. Let W be an essentially AIG representation of GL2 (E) or
GL3 (E) over a field K of characteristic zero. Then no Jordan–Hölder constituentof W appears with multiplicity greater than one.
Proof. Theorem 4.3.9 above shows that W embeds in some parabolic induction
GL (E)

IndP nπ1 ⊗ · · · ⊗ πrwith each πi cuspidal. Zelevinski’s computations of multiplicities of the Jordan–Hölder constituents of such inductions ([21, §11]) shows that when r ≤ 3, each
Jordan–Hölder constituent of such an induction occurs with multiplicity one. The
result follows immediately.

4.3.12. Remark. In contrast to the preceding proposition, if n = 4, and if wechoose a Levi subgroup of the form (E × )4 of GL4 (E),
GL (E)

IndP n| |2 ⊗ | | ⊗ | | ⊗1

has a Jordan–Hölder constituent that appears with multiplicity two.
4.4. Reduction of π(S). We now turn to integrality considerations. We continue
to suppose that O is a discrete valuation ring, with residue field K of characteristic
zero, uniformizer $, and field of fractions K. We say an admissible representation
π over K is O-integral if it contains a $-adically separated O-lattice.
4.4.1. Lemma. Let π be an absolutely irreducible supercuspidal representation ofGLn (E) over K. Then π is O-integral if and only if its central character takes
values in O× . In this case there is a $-adically separated GLn (E)-stable O-latticeπ ◦ in π, unique up to homothety, such that the reduction π ◦ /$π ◦ is absolutely
irreducible and supercuspidal.

LOCAL LANGLANDS IN FAMILIES

39

Proof. Clearly if π is O-integral, then its central character takes values in O× . Let
K0 be a finite Galois extension of K, such that there exists a character χ : E × → K×
whose nth power is the central character of π. If the central character of π takes
values in O× , then χ takes values in (O0 )× , where O0 is the integral closure of O
in K0 .
The central character of π ⊗ χ−1 ◦ det is trivial. By [17, II.4.9], π ⊗ χ−1 ◦ det
is defined over a finite extension F of Qp , contained in K0 . That is, there exists anadmissible representation π0 over F such that π0 ⊗F K0 is isomorphic to π⊗χ−1 ◦det.
As O0 has residue characteristic zero, F is contained in O0 . Thus π ◦ := (π0 ⊗F O0 )⊗(χ ◦ det) is a $0 -adically separated O0 -lattice in π ⊗K K0 , where $0 is a uniformizerof O0 .
The reduction modulo $0 of π ◦ is (π0 ⊗F K 0 ) ⊗ (χ ◦ det), where K 0 is the residue
field of O0 and χ is the reduction of χ modulo $0 . In particular π ◦ /$π ◦ is absolutely
irreducible and supercuspidal (and therefore π ◦ is unique up to homothety.) It
follows that π ◦ is stable under the action of Gal(K0 /K), and hence descends to aGLn (E)-stable lattice in π (which must also be unique up to homothety.)Given an O-integral absolutely irreducible supercuspidal representation π ofGLn (E) over K, we can thus define π to be the reduction mod $ of any $-adicallyseparated GLn (E)-stable O-lattice in π. For a segment ∆ = [π, (| | ◦ det)π, . . . , (| |◦ det)r−1 π], let ∆ be the segment [π, (| | ◦ det)π, . . . , (| | ◦ det)r−1 π]. If S is amultiset of integral segments, define S to be the multisegment consisting of thesegments ∆i for ∆i ∈ S.
4.4.2. Lemma. Let π be an O-integral, absolutely irreducible supercuspidal representation of GLn (E) over K, and let ∆ be the segment [π, (| | ◦ det)π, . . . , (| |◦ det)r−1 π]. There is a $-adically separated, GLn (E)-stable O-lattice St◦∆ in St∆ ,unique up to homothety, and St◦∆ /$St◦∆ is isomorphic to St∆ .
Proof. This follows by precisely the same argument as in Lemma 4.4.1.



If we want to consider the reduction mod $ of representations of the form π(S),then the situation is more complicated, as π(S) typically contains more than one
homothety class of lattices. However, Theorem 3.3.2 allows us to single out apreferred such homothety class.
4.4.3. Proposition. If S is a multisegment over K that are O-integral, then there
is an O-lattice π(S)◦ in π(S), unique up to homothety, such that π(S)◦ /$π(S)◦ isessentially AIG. Moreover, π(S)◦ /$π(S)◦ is isomorphic to π(S).
Proof. Theorem 3.3.2 shows that π(S)◦ exists and is unique up to homothety. Let∆1 , . . . , ∆r be the segments in S, and fix for each i a $-adically separated O-lattice
Li in St∆i . Then Li /$Li is isomorphic to St∆i . Recall that
GL (E)

π(S) = IndP n

St∆1 ⊗ · · · ⊗ St∆r ,GL (E)and hence contains the integral induction IndP n
L1 ⊗ · · · ⊗ Lr as a lattice.
The mod $ reduction of this lattice is clearly isomorphic to π(S), which is esGL (E)sentially AIG. Thus IndP n
L1 ⊗ · · · ⊗ Lr is homothetic to π(S)◦ , and henceπ(S)◦ /$π(S)◦ is indeed isomorphic to π(S), as claimed.

4.4.4. Corollary. Let K0 be a finite Galois extension of K, and let O0 be the integral
closure of O in K0 . Let π be an admissible representation of GLn (E) over K, and40

MATTHEW EMERTON AND DAVID HELM

let S be a multisegment consisting of O0 -integral segments over K0 . Suppose thatπ ⊗K K0 is isomorphic to π(S). Then π is O-integral, and there is a $-adicallyseparated O-lattice π ◦ in π, unique up to homothety, such that π ◦ /$π ◦ is essentially
AIG. Moreover, π ◦ /$π ◦ ⊗K K 0 is isomorphic to π(S).
Proof. It suffices to show that the lattice π(S)◦ constructed in the previous proposition is stable under the action of Gal(K0 /K). This is clear since π(S)◦ is unique
up to homothety.

4.5. Compatibility with specialization. We now use the results of the previous
sections to understand the relationship between π(ρ ⊗O K) and π(ρ ⊗O K), whereρ : GE → GLn (O) is a continuous Galois representation. The key idea is a geometricinterpretation of the partial order  on multisegments, due to Zelevinski [22].
Let V = ⊕π Vπ be a finite-dimensional vector space over a field F , “graded”
by the set of isomorphism classes of irreducible supercuspidal representations ofGLm (E) over K, for all m. We denote the automorphisms of V as a graded F -vector
space by Aut(V ), and let End+ (V ) denote the space of F -linear endomorphisms of
V that take Vπ to V(| |◦ det)π for all π. Let NV be an element of End+ (V ); it is a
nilpotent endomorphism of V .
We construct a bijection between the set of isomorphism classes of pairs (V, NV )and the set of multisegments S over K, as follows: For any segment ∆ = [π, (| |◦ det)π, . . . , (| | ◦ det)r−1 π], let V∆,F be the vector space defined by (V∆,F )π0 = F
if π 0 is in ∆, and zero otherwise. We define an endomorphism N∆,F of V∆,F that is
an isomorphism (V∆,F )(| |◦ det)i π → (V∆,F )(| |◦ det)i+1 π for 0 ≤ i < r − 1, and zerootherwise.
For a multisegment S, we define:M(VS,F , NS,F ) =(V∆,F , N∆,F ).
∆∈S

It is easy to see (for instance, by the structure theory of graded F [N ]/N r -modules)
that the association S 7→ (VS,F , NS,F ) yields a bijection between multisegments
and isomorphism classes of pairs (V, NV ).
4.5.1. Theorem ([22, §2]). Let S 0 and S be multisegments over K. Then S 0  S if
and only if VS,F is isomorphic to VS 0 ,F as a graded F -vector space, and NS,F is in
the closure of the orbit of NS 0 ,F under the action of Aut(VS 0 ,F ) on End+ (VS 0 ,F ).
i
As a result, if S 0  S, then, for all i, the rank of NS,F
is less than or equal toithat of NS 0 ,F . These ranks are equal for all i if, and only if, S 0 = S.
If (ρ0 , N ) is a Frobenius-semisimple Weil–Deligne representation over K, andn−1
if S is the multisegment such that π(S) = (| | ◦ det) 2 π(ρ0 , N ), then the pair
(VS,K , NS,K ) can be described easily in terms of (ρ0 , N ). Indeed, one has:

4.5.2. Lemma. For any supercuspidal representation π of GLm (E) over K, there
is a natural isomorphism:∼(VS,K )π −→ HomK[WE ] (ρ, ρ0 ),where ρ is the absolutely irreducible representation of WE that corresponds to π
under the unitary local Langlands correspondence. Moreover, under these isomorphisms, the map
NS,K : (VS,K )π → (VS,K )(| |◦ det)πLOCAL LANGLANDS IN FAMILIES

41

is identified with the map
HomK[WE ] (ρ, ρ0 ) → HomK[WE ] (| | ⊗ρ, ρ0 )induced by N .
Proof. This is true by construction if (ρ0 , N ) is indecomposable, and extends to thegeneral case by taking direct sums.

Zelevinski’s result strongly suggests a connection between the Zelevinski partial
order and reduction of Weil–Deligne representations. In order to make this connection precise we need a compatibility between the reduction mod $ and local
Langlands:
4.5.3. Lemma. If ρ0 is absolutely irreducible, and π ⊗K K corresponds to ρ0 ⊗K
K under the unitary local Langlands correspondence, then π ⊗K K and ρ0 ⊗K K
correspond under the unitary local Langlands correspondence.
Proof. We first translate this into a statement in terms of the generic local Langlands correspondence. From this point of view the representation π(ρ0 ) is isomorphic
n−1
n−1
to (| | ◦ det)− 2 π, and we must show that π(ρ0 ) is isomorphic to (| | ◦ det)− 2 π.
0
0 ×There is a finite extension K of K, and a character χ : WE → (K ) , such that
ρ0 ⊗ χ is defined over Qp ; as ρ0 is integral χ takes values in (O0 )× , where O0 is
the integral closure of O in K 0 . In particular, there is a finite extension K0 of Qp ,contained in K0 , and a representation ρ0 : WE → GLn (K0 ), such that ρ0 ⊗K0 K0 is
isomorphic to ρ0 ⊗χ. If we let K 0 be the residue field of O0 , and let χ be the reduction
mod $0 of the character χ, then K0 is contained in K 0 and ρ0 ⊗K0 K 0 is isomorphicn−1
to ρ0 ⊗ χ. It follows that π(ρ0 ) ⊗ χ is isomorphic to (| | ◦ det)− 2 π(ρ0 ) ⊗ χ.
n−1
Let π0 = (| | ◦ det) 2 π(ρ0 ). As the generic local Langlands correspondence iscompatible with twists and base change, the representation π ⊗ χ is isomorphic ton−1
π0 ⊗K0 K0 . Thus π ⊗χ is isomorphic to π0 ⊗K0 K 0 , and hence to (| | ◦ det) 2 π(ρ0 )⊗χ. The result follows.

If S is obtained from a multisegment S by reduction mod $, the pairs (VS,F , NS,F )and (VS,F , NS,F ) are related by (VS,F )π = ⊕π0 (VS,F )π0 , where the sum is over π 0
with π 0 = π.
Now let (ρ0 , N ) be a Weil–Deligne representation over O such that the restrictionof ρ0 ⊗O K to IE is a direct sum of absolutely irreducible representations of IE overK, and such that ρ0 ⊗O K is a direct sum of absolutely irreducible representations
ρ0i of WE . (We can always arrange this by replacing K with a finite extension.) Let
S be the segment associated to (ρ0 , N ) ⊗O K; we have
(VS,K )πi = HomK[WE ] (ρi , ρ0 ⊗O K),
where πi corresponds to ρi under unitary local Langlands. We also consider the
K-vector-space VS,K .
Let (ρ0 , N ) be the Weil–Deligne representation (ρ0 , N ) ⊗O K, and let S 0 be themultisegment associated to (ρ0 , N )F−ss . Our goal is to compare S 0 to S; we will dothis by comparing VS,K to VS 0 ,K . The key difficulty is to construct an O-lattice in
VS,K , stable under NS,K , whose reduction mod $ is isomorphic to VS 0 ,K .
By Lemma 4.1.1 there is a finite extension K0 of Qp and representations τ1 , . . . , τnof IE over K0 , each in its own orbit under conjugation by Φ, such that the restriction

42

MATTHEW EMERTON AND DAVID HELM

ρ0 ⊗O K is a direct sum of Φ-conjugates of the τi , each with multiplicity one. For
each i, let Li be the O-module HomO[IE ] (τi ⊗K0 O, ρ0 ).
For each j such that the restriction of ρ0j to IE contains a copy of τi ⊗K0 K, therestriction map
HomK[WE ] (ρ0j , ρ0 ) → HomK[IE ] (τi ⊗K0 K, ρ0 )is an injection. We thus obtain an isomorphism:∼∼

Li ⊗O K −→ HomK[IE ] (τi ⊗K0 K, ρ0 ) −→M

HomK[WE ] (ρ0j , ρ0 ),jwhere the latter sum is over those j such that the restriction of ρ0j to IE contains a
copy of τi ⊗K0 K. This condition is satisfied if, and only if, the restriction of ρ0j to
IE contains a copy of τi ⊗K0 K. We can thus view Li as a sublattice of ⊕j (VS,K )πj ,where the sum is over those j such that the restriction of ρ0j to IE contains a copyof τi ⊗K0 K. For any such j, let Lπj be the intersection Li ∩ (VS,K )πj .
In fact, this gives a direct sum decomposition of Li . To see this, first observe:
4.5.4. Lemma. Let M be a free O-module of finite rank, and Ψ is an O-linear
endomorphism of M that acts semisimply on M ⊗O K. Suppose that all of theeigenvalues of Ψ ⊗O K lie in K, and for each eigenvalue λ̃ of Ψ ⊗O K, let Mλ̃ be
the intersection of M with the λ̃-eigenspace of Ψ ⊗O K. Then M decomposes as
M = ⊕λ M λ ,where λ runs over the eigenvalues of Ψ : M/$M → M/$M and Mλ is the sum ofMλ̃ for those λ̃ congruent to λ modulo $. The endomorphism Φ acts on Mλ /$Mλas the product of λ with a unipotent endomorphism of Mλ /$Mλ .
Proof. Let P (t) be the minimal polynomial of Ψ, and consider M as an O[t]/P (t)module on which t acts by Ψ. The connected components of Spec O[t]/P (t) are in
bijection with the roots λ of the mod $ reduction Ψ of Ψ; these are the eigenvaluesof Ψ. Thus, considered as a sheaf on Spec O[t]/P (t), M decomposes as a direct sumof sheaves Mλ supported on each connected component. On each Mλ , the minimal
polynomial of Ψ is a power of t − λ, so λ−1 Ψ is unipotent on Mλ .

4.5.5. Lemma. We have a direct sum decomposition: Li = ⊕j Lπj , where the sum
is over those j such that the restriction of ρ0j to IE contains a copy of τi ⊗K0 K.
Proof. Let r be the size of the orbit of τi under the conjugation action of Φ, andr∼
fix an isomorphism τiΦ −→ τi . This isomorphism induces an endomorphism Ψ of0
HomO[IE ] (τi ⊗K0 O, ρ ) via
Φr

∼

r

HomO[IE ] (τi ⊗K0 O, ρ0 ) → HomO[IE ] (τiΦ ⊗K0 O, ρ0 ) −→ HomO[IE ] (τi ⊗K0 O, ρ0 ).
Let ρ0j be an absolutely irreducible summand of ρ0 ⊗O K whose restriction to IE
contains a copy of τi ⊗K0 K. This copy is unique, and yields a restriction map:
HomK[WE ] (ρ0j , ρ0 ⊗O K) → HomK[IE ] (τi , ρ0 ⊗O K).
This restriction map is injective, and its image can be characterized in terms of Ψ.
In particular, the endomorphism:
Φr

r

∼

HomK[IE ] (τi ⊗K0 K, ρ0j ) → HomK[IE ] (τiΦ ⊗K0 K, ρ0j ) −→ → HomK[IE ] (τi ⊗K0 K, ρ0j ).

LOCAL LANGLANDS IN FAMILIES

43

is an endomorphism of one-dimensional K vector spaces and is thus given by ascalar λ̃; it follows by Lemma 4.1.2 that ρ0j is determined by λ̃ and τi , and that theimage of the map:
HomK[WE ] (ρ0j , ρ0 ⊗O K) → HomK[IE ] (τi , ρ0 ⊗O K)
is the λ̃-eigenspace of Ψ.
Now let ρj be an absolutely irreducible summand of ρ0 ⊗O K whose restriction
to IE contains a copy of τi ⊗K0 K, and let π j be the corresponding admissible
representation. Then the endomorphism Ψ of HomK[IE ] (τi ⊗K0 K, ρi ) is a scalar λ,and, by the same reasoning as above, (VS,K )πj is the sum of the λ̃-eigenspaces ofΨ for those λ̃ congruent to λ modulo $. Thus, by the preceding lemma, Lπj is adirect summand of Li .
Let L be the lattice in VS,K defined by:
L = ⊕π L π .
Note that as NS,K preserves each Li , it also preserves L.
∼

4.5.6. Lemma. There is a natural isomorphism L/$L −→ VS 0 ,K . Moreover, theendomorphism NS,K of L reduces to NS 0 ,K under this isomorphism.
Proof. Recall that S 0 is the multisegment associated to (ρ0 , N )F−ss . Let ρi be
any absolutely irreducible Jordan–Hölder constituent ρi of (ρ0 )ss , corresponding toan admissible representation π i under unitary local Langlands. Then (VS 0 ,K )πi isequal to HomK[WE ] (ρi , (ρ0 )ss ). It thus suffices to construct, for each i, a natural
isomorphism of (L/$L)πi with (VS 0 ,K )πi .
Let τ be an absolutely irreducible representation of IE over K0 such that ρicontains τ ⊗K0 K; let r be the order of the orbit of τ under conjugation by Φ, andr
fix an isomorphism of τ with τ Φ . Let λ ∈ K × be the scalar giving the action ofrΦ on Homk[IE ] (τ ⊗K0 K, ρi |IE ) under this identification.
We also have an action of Φr on HomK[IE ] (τ ⊗K0 K, (ρ0 )ss |IE ); this yields a linearss
endomorphism Ψ of HomK[IE ] (τ ⊗K0 K, (ρ0 )ss |IE ). The natural map(VS 0 ,K )πi → HomK[IE ] (τ ⊗K0 K, (ρ0 )ss |IE )ssidentifies (VS 0 ,K )πi with the λ-eigenspace of Ψ .
On the other hand, the previous lemma shows that Lπi is the sum of the λ̃eigenspaces of Ψ on Hom O[IE ](τ ⊗K0 O, ρ0 |IE ); it follows that Lπi /$Lπi is the
λ-generalized eigenspace of Ψ/$Ψ on Hom K[IE ](τ ⊗K0 K, ρ0 |IE ).
ss
Finally, observe that Ψ is the semisimplification of Ψ/$Ψ, so that the λss
generalized eigenspace of Ψ/$Ψ is equal to the λ-eigenspace of Ψ , and hence
to (VS 0 ,K ). One verifies easily that these identifications are all compatible with themonodromy operators.

By Theorem 4.5.1 it follows that for S and S 0 as in Lemma 4.5.6, we must havei
S  S 0 . Moreover, we have equality if, and only if, the ranks of the operators NS,Kiand NS 0 ,K agree for all i. We are thus finally in a position to prove:
4.5.7. Theorem. Let ρ : GE → GLn (O) be a continuous Galois representation, and(ρ0 , N ) the Frobenius-semisimplification of the corresponding Weil–Deligne representation. Then there is a $-adically separated O-lattice π(ρ)◦ in π(ρ⊗O K), unique

44

MATTHEW EMERTON AND DAVID HELM

up to homothety, such that π(ρ)◦ /$π(ρ)◦ is essentially AIG, and an embeddingπ(ρ)◦ /$π(ρ)◦ → π(ρ),
where ρ = ρ ⊗O K. This embedding is an isomorphism if, and only if, the K-rankiof N equals the K-rank of (N ⊗O K)i for all i.
Proof. Let (ρ0 , N ) be the reduction mod $ of (ρ0 , N ). Then, by Lemmas 4.1.8
and 4.1.9, (ρ0 , N )F−ss is the Frobenius-semisimplification of the Weil–Deligne representation attached to ρ.
Over a finite extension K0 of K, we may assume that ρ0 splits as a direct sum ofabsolutely irreducible representations of WE , and similarly for its restriction to IE .
The corresponding statements then hold for the semisimplification of ρ0 .
Let O0 be the integral closure of O in K0 , and let K 0 be its residue field. Let
S and S 0 be the segments associated to (ρ0 , N ) ⊗O O0 and (ρ0 , N )F−ss ⊗K K 0 . We
have shown that S  S 0 .
n
On the other hand, we have π(ρ ⊗O K0 ) is isomorphic to (| | ◦ det)− 2 π(S); by
Corollary 4.4.4 there is, up to homothety, a unique lattice π(ρ)◦ in π(ρ ⊗O K) suchthat π(ρ)◦ /$π(ρ)◦ is essentially AIG; moreover one has an isomorphism∼[π(ρ)◦ /$π(ρ)◦ ] ⊗K K 0 −→ π(S).
nAs π(ρ ⊗K K 0 ) is isomorphic to (| | ◦ det)− 2 π(S 0 ), and S  S 0 , we have anembedding of [π(ρ)◦ /$π(ρ)◦ ] ⊗K K 0 in π(ρ ⊗K K 0 ). This embedding descends to
K by Proposition 4.3.7.
Finally, this embedding is an isomorphism if, and only if, S is equal to S 0 . Thisi
is true if, and only if, the ranks of NS,K
and NSi 0 ,K agree for all i; it is easy toisee this is equivalent to requiring that the ranks of (N ⊗O K)i and N agree forall i.

4.5.8. Remark. An alternative approach to some of the above questions is given
in [3], particularly Proposition 3.11. Chenevier constructs, for each Bernstein component B of the category of smooth representations of GLn (E), a pseudocharacterof WE valued in the algebra of functions on B that “is compatible with the local
Langlands correspondence”, in the sense that if one specializes this pseudocharacter at any irreducible representation of GLn (E) that lies in B, one obtains thepseudocharacter of the semisimplification of the corresponding representation of
WE . From our perspective, this result allows us to deduce that the supercuspidal
support of π(ρ) is the reduction modulo $ of the supercuspidal support of π(ρ),
but it does not contain any information about the monodromy operator.
In cases where the embedding arising in the previous proposition is an isomorphism, we say that ρ is a minimal lift of ρ. (Such lifts are lifts of ρ in which theramification arising from the monodromy operator is as small as possible.) We willneed this language in a broader context than that of representations over discrete
valuation rings:
4.5.9. Definition. Let A be a reduced complete Noetherian local ring with finite
residue field k of characteristic p, that is flat over W (k), and let ρ be a continuous
representation of GE into GLn (A). Let (ρ0 , N ) be the associated Weil–Deligne
1
representation over GLn (A[ ]). If p is a characteristic zero prime of A, and a is ap

LOCAL LANGLANDS IN FAMILIES

45

prime of A whose closure contains p, we say ρa is a minimal lift of ρp if, for all i,
the rank of (N ⊗A κ(a))i is equal to the rank of (N ⊗A κ(p))i .
Note that, for any given a, the locus of p such that ρa is a minimal lift of ρp is
1
Zariski open in the closure of a in Spec A[ ].
p
5. The local Langlands correspondence in characteristic p
5.1. Definition and basic properties. We now construct an analogue of theBreuil-Schneider local Langlands correspondence for representations of GE overfinite fields of characteristic p. Such a correspondence should satisfy an analog
of Theorem 4.5.7 for representations over discrete valuation rings of characteristic
zero and residue characteristic p. Throughout this section we fix a finite field k ofcharacteristic p, and let O denote a complete discrete valuation ring of characteristic
zero with field of fractions K and finite residue field k 0 containing k.
Our starting point is the semisimple mod p local Langlands correspondence ofVigneras [20]. This is a map ρ 7→ π ss (ρ) that associates to each n-dimensional
irreducible representation ρ : WE → GLn (Fp ) an irreducible supercuspidal representation π ss (ρ) over Fp . If q denotes the order of the residue field of E, and if
k 0 is a finite field of characteristic p containing a square root of q, then this correspondence is defined over k 0 ; that is, if ρ is defined over k 0 , then π(ρ) descends
uniquely to a representation over k 0 . Moreover, the correspondence is compatible
with “reduction mod p” in the following sense:
5.1.1. Theorem ([20, Thm. 1.6]). Suppose that k 0 contains a square root of q. Let(ρ, N ) be an n-dimensional Frobenius-semisimple Weil–Deligne representation of
WE over O, and let π be the irreducible representation of GLn (E) over K attachedto (ρ, N ) ⊗O K by the unitary local Langlands correspondence. Let ρ = ρ ⊗O Fp ,and letρss = ρ1 ⊕ · · · ⊕ ρr
be a decomposition of ρss into irreducible representations of WE over Fp . Then π
is O-integral, and for any GLn (E)-stable O-lattice L in π, and any Jordan–Hölder
constituent π of L ⊗O Fp , one has:scs(π) = {π ss (ρ1 ) . . . π ss (ρr )}.
5.1.2. Corollary. Suppose that k 0 contains a square root of q. Let ρ : GE → GLn (k)be a Galois representation, and let ρ : GE → GLn (O) be a lift of ρ ⊗k k 0 . Thenπ(ρ ⊗O K) is O-integral, and for any O-lattice L in π(ρ ⊗O K), the supercuspidal
support of any Jordan–Hölder constituent π of L ⊗O Fp depends only on ρ.
Proof. Let (ρ0 , N ) be the Frobenius-semisimple Weil–Deligne representation overK attached to ρ. Then ρ0 is O-integral and the semisimplification of its reduction
mod p depends only on ρ. By the definition of the Breuil-Schneider local Langlands
correspondence, π(ρ ⊗O K) is (up to a twist by an integral character) a parabolic
induction of representations that correspond (via unitary local Langlands) to irreducible summands of ρ0 ⊗O K. These summands are integral, so π(ρ ⊗O K) isn−1
as well, and so is π(ρ ⊗O K). Moreover, (up to a twist by (| | ◦ det) 2 ), every
Jordan–Hölder constituent of π(ρ ⊗O K) corresponds via unitary local Langlands
to a Weil–Deligne representation of the form (ρ0 ⊗O K, N 0 ) for some choice of monodromy operator N 0 .

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MATTHEW EMERTON AND DAVID HELM

Now if L is a lattice in π(ρ ⊗O K), and π is a Jordan–Hölder constituent of
L ⊗O Fp , then there exists a Jordan–Hölder constituent of π(ρ ⊗O K), and a lattice
L0 in this constituent, such that π is a Jordan–Hölder constituent of the mod p
reduction of L0 . The result thus follows from Theorem 5.1.1.

Let L be a lattice in π(ρ⊗O K), where ρ : GE → GLn (O) is a lift of ρ⊗k k 0 for some
ρ : GE → GLn (k). As L⊗O Fp has a unique generic Jordan–Hölder constituent, andup to isomorphism there is only one irreducible generic representation of G withgiven supercuspidal support, the generic Jordan–Hölder constituent of L ⊗O Fp
likewise depends only on ρ.
We will also need to control the length of L/$L, for lattices L of the sortappearing in the Corollary above. We first show:
5.1.3. Proposition. Let P = M U be a parabolic subgroup of GLn (E), and let
π = π1 ⊗ · · · ⊗ πr be an irreducible representation of M . There exists an integer c,
GL (E)
depending only on n, such that the length of IndP n
π1 ⊗ · · · ⊗ πr is bounded
above by c.
Proof. This follows easily from [17], Proposition III.1.12.



5.1.4. Proposition. Let ρ : GE → GLn (k) be a Galois representation, let ρ : GE →GLn (O) be a lift of ρ ⊗k k 0 , and let L be a GLn (E)-stable lattice in π(ρ ⊗O K).
There exists an integer c, depending only on n, such that the length of L/$L isbounded above by c.
Proof. The length of L/$L is independent of L. As π(ρ ⊗O K) is a parabolic
induction of a tensor product of integral Steinberg representations, we can write
GL (E)π(ρ ⊗O K) = IndP n

Stπ1 ,n1 ⊗ · · · ⊗ Stπi ,ni ,where the πi are integral cuspidal representations of GLn . For each i, Stπi ,ni arises
as the normalized parabolic induction of a tensor product of the form:(| | ◦ det)−

ni −1
2

πi ⊗ · · · ⊗ (| | ◦ det)

ni −1
2

πi .

Thus there is a parabolic induction of a tensor product of irreducible, integral,
cuspidal representations πj0 (all of which are twists of the πi ) that maps surjectively
onto π(ρ ⊗O K); if we choose a lattice Lj inside each of the πj0 , the parabolic
induction of the tensor product of the Lj maps into a lattice L in π(ρ⊗O K). We thenhave a surjection of the parabolic induction of the tensor product of Lj /$Lj onto
L/$L. As each πj0 is cuspidal, so is Lj /$Lj ; as (Lj /$Lj )(n) is one-dimensional wemust have Lj /$Lj irreducible for all j. Thus the length of L/$L is bounded aboveby the maximum length of a parabolic induction of an irreducible representation ofa Levi subgroup of GLn (E), and the desired result follows by Proposition 5.1.3. 
We can now prove the main result of this subsection.
5.1.5. Theorem. There is a map ρ 7→ π(ρ) from the set of isomorphism classesof continuous representations GE → GLn (k) to the set of isomorphism classes offinite length admissible smooth GLn (E)-representations on k-vector spaces, uniquelydetermined by the following three conditions:(1) For any ρ, the associated GLn (E)-representation π(ρ) is essentially AIG.

LOCAL LANGLANDS IN FAMILIES

47(2) If K is a finite extension of Qp , with ring of integers O, uniformizer $, andresidue field k 0 containing k, ρ : GE → GLn (O) is a continuous representation lifting ρ ⊗k k 0 , and L is a GLn (E)-invariant O-lattice in π(ρ) suchthat L/$L is essentially AIG, then there is a GLn (E)-equivariant embedding L/$L ,→ π(ρ) ⊗k k 0 . (Note that Theorem 3.3.2 shows that such an L
always exists, and that it is unique up to homethety.)(3) The representation π(ρ) is minimal with respect to satisfying conditions (1)and (2), i.e. given any continuous representation ρ : GE → GLn (k), andany representation π of GLn (E) satisfying these two conditions with respectto ρ, there is a GLn (E)-equivariant embedding π(ρ) ,→ π.
Furthermore:(4) The formation of π(ρ) is compatible with extending scalars, i.e. given ρ :GE → GLn (k), and a finite extension k 0 of k, one hasπ(ρ ⊗k k 0 ) ∼= π(ρ) ⊗k k 0 .
(5) The formation of π(ρ) is compatible with twists, i.e. given ρ : GE →GLn (k), and a continuous character χ : GE → k × , one hasπ(ρ ⊗ χ) = π(ρ) ⊗ (χ ◦ det).
(6) EndGLn (E) (π(ρ)) = k.
n(n−1)(7) The representation π(ρ) has central character equal to | | 2 (det ρ).
(8) Suppose (ρ⊗k k)ss is the direct sum of irreducible representations ρ1 , . . . , ρr .
Then every Jordan–Hölder constituent of π(ρ) has supercuspidal support
n−1
equal to | | 2 {π ss (ρ1 ), . . . , π ss (ρr )}Proof. We first establish uniqueness: If π and π 0 are two finite length admissible
smooth representations of GLn (E) that satisfy properties (1), (2), and (3) withrespect to ρ, then by property (3) we have embeddings of π in π 0 and vice versa.
As both π and π 0 have finite length these embeddings are isomorphisms.
We now turn to the construction of π(ρ). Let ρ : GE → GLn (O) be a lift ofρ ⊗k k 0 , for some O, k 0 as in property 2. (Choi ([4], Theorem 3.0.13) has shownthat the characteristic zero fiber of the universal framed deformation ring of ρ isgenerically smooth of dimension n2 , so such a lift always exists.) Suppose L isan O-lattice in π(ρ) such that L/$L is essentially AIG. The socle V of L/$L isabsolutely irreducible and generic, and its supercuspidal support depends only on ρ
and not the specific lift ρ chosen. As there is a unique generic representation withgiven supercuspidal support, V depends only on ρ up to isomorphism. In particular
V is defined over k, as we can take O to have residue field k.
Let env(Vk ) be the essentially AIG envelope of V ⊗k k. For each lift ρ of ρ, andeach lattice L in π(ρ) such that L/$L is essentially AIG, the socle of (L/$L) ⊗k k
is isomorphic to V ⊗k k. Hence (L/$L) ⊗k0 k embeds uniquely (up to the action of
×k ) in env(Vk ). Let π(ρ)k be the sum, in env(Vk ), of the images of (L/$L) ⊗k k
in env(Vk ) as ρ ranges over all lifts of ρ.
By construction, Gal(k/k) acts on env(Vk ). This action preserves π(ρ)k , as it
permutes the images of L/$L for various O and ρ. Thus π(ρ)k descends uniquely toa submodule π(ρ) of env(V ). Clearly, π(ρ) satisfies properties (1) and (2). On theother hand, if π is any other representation satisfying properties (1) and (2), thenthe socle of π is isomorphic to V and hence env(V ) contains a unique submodule

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MATTHEW EMERTON AND DAVID HELM

isomorphic to π. As π satisfies property (2), π ⊗k k contains the images of L/$L
in env(Vk ) for all lifts ρ of ρ, and thus contains π(ρ) ⊗k k. It follows that π contains
π(ρ), so π(ρ) satisfies property (3). Finally, π(ρ) is finite length by Proposition 5.1.4
and Corollary 3.2.17.
Now let k 0 be a finite extension of k. Then π(ρ) ⊗k k 0 clearly satisfies properties(1) and (2) with respect to ρ⊗k k 0 , and thus admits an embedding of π(ρ⊗k k 0 ) thatis unique up to rescaling. The above construction shows that π(ρ) ⊗k k and π(ρ ⊗kk 0 ) ⊗k k coincide as submodules of env(Vk ), so this embedding is an isomorphism.
Similarly, if χ is a character of E × with values in k × , we can choose a lift χ of
χ to a character with values in W (k)× . Then if ρ ⊗ χ is a lift of ρ ⊗ (χ ◦ det) to arepresentation over O, and L ⊗ χ is a lattice in π(ρ ⊗ χ) with (L ⊗ (χ ◦ det))/$(L ⊗(χ ◦ det)) essentially AIG, then L is a lattice in π(ρ) with L/$L essentially AIG.
Thus L/$L embeds in π(ρ) ⊗k k 0 , so (L/$L) ⊗ (χ ◦ det) embeds in π(ρ) ⊗ (χ ◦ det).
Thus π(ρ) ⊗ (χ ◦ det) has property (2) and hence contains π(ρ ⊗ χ). Conversely,
replacing ρ with ρ ⊗ χ, we find that π(ρ ⊗ χ) ⊗ (χ−1 ◦ det) contains π(ρ). Thus
π(ρ) and π(ρ ⊗ χ) have the same length, and so π(ρ ⊗ χ) and π(ρ) ⊗ (χ ◦ det) are
isomorphic.
The endomorphisms of π(ρ) are all scalar because π(ρ) is essentially AIG. In
particular the center of GLn (E) acts on π(ρ) (and hence on all of its submodules)
via a character. To compute this character, let ρ be any lift of ρ, and let L bea lattice in π(ρ) such that L/$L is essentially AIG. The center of GLn (E) actsn(n−1)det ρ, and hence on L/$L via the characteron π(ρ) via the character | | 2
n(n−1)| | 2 det ρ. As L/$L embeds in π(ρ), this character is also the central characterof π(ρ).
As π(ρ) is essentially AIG, every Jordan–Hölder constituent of π(ρ) has the same
supercuspidal support. To determine this supercuspidal support, let ρ be any liftof ρ, and let L be a lattice in π(ρ) such that L/$L is essentially AIG. The repren−1
sentations | |− 2 π(ρ) and ρ then correspond under unitary local Langlands, andso, by Theorem 5.1.1, the supercuspidal support of any Jordan–Hölder constituentn−1
of | |− 2 L/$L is equal to {π ss (ρ1 ), . . . π ss (ρr )}.

5.2. The local Langlands correspondence for GL2 in characteristic p. For
GL2 (E), at least in odd characteristic, the correspondence ρ 7→ π(ρ) can be madefairly concrete. We give a complete picture (for p odd) in [12]; the approach described there runs into difficulties when p = 2 because the essentially AIG envelopeof a cuspidal (but not supercuspidal) representation is rather complicated in thiscase. The first thing to observe is:5.2.1. Proposition. Let ρ : GE → GL2 (Fp ) be a representation, and suppose that
ρss is not a twist of 1⊕ | |. Then π(ρ) is the unique representation of GL2 (E)whose supercuspidal support is given by part (8) of Theorem 5.1.5.
Proof. Part (8) of Theorem 5.1.5, together with our hypothesis on ρ implies thatthe supercuspidal support of any Jordan–Hölder constituent of π(ρ) is either asingle supercuspidal representation of GL2 (E), or a pair of characters of GL1 (E)that do not differ by a factor of | |. In either case, there is, up to isomorphism, aunique irreducible representation π of GL2 that has that particular supercuspidal
support; in particular π is generic. Thus every Jordan–Hölder constituent of env(π)
is isomorphic to π; as π is generic and env(π) is essentially AIG there is only one

LOCAL LANGLANDS IN FAMILIES

49

such Jordan–Hölder constituent. In particular π = env(π). and so As π(ρ) is
contained in env(π), we must have π(ρ) = π.

When ρss is a twist of 1⊕ | |, the situation is more complicated, as π(ρ) will
typically not be irreducible. As the correspondence ρ 7→ π(ρ) is compatible with
twists, it suffices to describe π(ρ) when ρss = 1⊕ | |. In this case π(ρ) has
supercuspidal support {1, | |}. The details of this will be carried out in [12]; here
we content ourselves with summarizing the results.
First, assume that the order q of the residue field of E is not congruent to ±1
modulo p. (This is the so-called banal situation.) Here there are two irreducible
representations of G with supercuspidal support {1, | |}: the character | |◦det andthe twisted Steinberg representation St ⊗ (| | ◦ det). The latter representation is
generic, and its envelope is the unique nonsplit extension of | |◦det by St⊗(| |◦det).
On the Galois side there is, up to isomorphism, a unique nonsplit ρ whose
semisimplification is 1⊕ | |. Then π(ρ) is equal to St ⊗ (| | ◦ det) if ρ is nonsplit, and to the unique nonsplit extension of | | ◦ det by St ⊗ (| | ◦ det) if ρ issplit.
Next, assume that p is odd and q is congruent to −1 modulo p. In this casethere are three irreducible representations of G with supercuspidal support {1, | |}:the trivial character, the character | | ◦ det, and a cuspidal generic representationthat Vigneras denotes by π(1) (see [17, II.2.5] for a discussion of this). Up toisomorphism, there is a unique nonsplit extension of the trivial character by π(1)and similarly a unique nonsplit extension of | |◦det by π(1). The envelope env(π(1))
is the unique extension of 1 ⊕ (| | ◦ det) by π(1) that contains both of these nonsplit
extensions as submodules.
In this case π(ρ) = env(π(1)) if ρ is split. If ρ is not split, it is either an extension
of | | by 1 or an extension of 1 by | |. In the first case, π(ρ) is the nonsplit extensionof (| | ◦ det) by π(1); in the second case π(ρ) is the nonsplit extension of the trivial
character by π(1).
Finally, assume that p is odd and q is congruent to 1 modulo p. In this case | |is the trivial character. The only irreducible representations of G with supercuspidal support {1, 1} in this case are the Steinberg representation St and the trivial
representation. Moreover, Ext1 (1, St) is two-dimensional, and naturally isomorphic
to H 1 (GE , 1). The envelope env(St) is isomorphic to the universal extension of 1
by St, and thus has length three.
In this case π(ρ) is equal to env(St) if ρ is split. On the other hand, the nonsplit
ρ with trivial semisimplification are in bijection with the one-dimensional subspacesof H 1 (GE , 1), and hence in natural bijection with the one-dimensional subspacesof Ext1 (1, St). These in turn correspond to the nonsplit extensions of 1 by St. For
any nonsplit ρ, π(ρ) is the corresponding extension of 1 by St.
6. The local Langlands correspondence in families
6.1. The set-up. Throughout this section we will be considering representations
over rings A satisfying the following condition:
6.1.1. Condition. A is a complete reduced Noetherian local ring, with finite residue
field k of characteristic p, which is flat over the ring of Witt vectors W (k).

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MATTHEW EMERTON AND DAVID HELM

We will typically write m for the maximal ideal of A. Note that the conditionof being flat over W (k) is equivalent to A being p-torsion free, or again (since A
is reduced), to each minimal prime of A being of residue characteristic 0. We will
write κ(p) to denote the residue field of a prime ideal p of A; thus Yκ(p) is the fraction
field of the complete local domain A/p. We write K(A) :=κ(a) (where,a minimal

as indicated, the product is taken over the finitely many minimal primes of A) for
the total quotient ring of A. Since A is reduced, the natural map A → K(A) is anembedding.
6.2. Statement of the correspondence and related results. Now let E be anumber field. Let v be a non-archimedean place of E, and let ρ : GEv → GLn (A) be
a continuous representation (when the target is equipped with its m-adic topology).
For each prime ideal p of A, let ρp : GEv → GLn κ(p) denote the representation
obtained from ρ by extending scalars from A to κ(p). In the particular case of the
maximal ideal, we also write ρ := ρm . If p is a prime of A with residue characteristic
zero, we write πe(ρp ) for the smooth κ(p)-dual of the representation π(ρp ) definedin Definition 4.2.1.
In the situations that we will consider below, we will have a finite S of nonarchimedean places of E, all prime to p, and for each v ∈ S we will have a continuous
representation ρv : GEv → GLn (A).
We are now ready to describe the local Langlands correspondence for local Galois
representations over A.
6.2.1. Theorem. Let S denote a finite set of non-archimedean places of E, noneof which lie over p, and suppose for eachQ v ∈ S that we are given a representation
ρv : GEv → GLn (A). If we write G := v∈S GLn (Ev ), then there is (up to isomorphism) at most one admissible smooth representation V of G over A satisfying thefollowing conditions:(1) V is A-torsion free (i.e. all associated primes of V are minimal primesof A, or equivalently, the natural map V → K(A) ⊗A V is an embedding).
(2) For each minimal prime a of A, there is a G-equivariant isomorphismO∼πe(ρv,a ) −→ κ(a) ⊗A V.
v∈S(3) The G-cosocle cosoc(V /mV ) of V /mV is absolutely irreducible and generic,while the kernel of the natural surjection V /mV → cosoc(V /mV ) contains
no generic subrepresentations. (In other words, the smooth dual of V /mV
is essentially AIG.)
Any such V satisfies the following additional conditions:(4) V is cyclic as an A[G]-module.
(5) EndA[G] (V ) = A.
We postpone the proof of the theorem to the following subsection.
6.2.2. Definition. If in the context of the preceding theorem an A[G]-module V
satisfying conditions (1), (2), and (3) exists, then we write πe({ρv }v∈S ) := V . (Thisis justified by the uniqueness statement of the theorem.) If S consists of a single
place v then we write πe(ρv ) rather than πe({ρv }v∈S ).

LOCAL LANGLANDS IN FAMILIES

51

6.2.3. Remark. We don’t consider here the problem of proving in general thata representation V satisfying conditions (1), (2) and (3) of Theorem 6.2.1 exists,
although we conjecture that it does. (This is Conjecture 1.4.1 of the introduction.)
When n = 2 and p is odd, or when p is a banal prime, this conjecture is a result of
the second author [11], [10].
In the global applications considered in the work of the first author [6, 7], andin subsequent applications, the problem that we will confront will rather be that of
having a smooth G-representation at hand (for a certain ring A), which we wish toshow satisfies the conditions to be πe({ρv }v∈S ) for an appropriate ρ. Thus one of ourgoals in the following subsection is to establish a workable criterion for recognizingπe({ρv }v∈S ) (namely Theorem 6.2.15 below).
The following result shows that the existence of πe({ρv }v∈S ) is equivalent to
the existence of the collection of representations πe(ρv ), and explains the relation
between them. We postpone its proof to the following subsection.
6.2.4. Proposition. In the context of Theorem 6.2.1, the A[G]-module πe({ρv }v∈S )exists if and only if each of the individual A[GLn (Ev )]-modules πe(ρv ) exist. Furthermore, πe({ρv }v∈S ) is isomorphic
to the maximal torsion free quotient of theNtensor product (taken over A) v∈S πe(ρv ).
The following two theorems, whose proofs we again postpone, describe the sensein which the representation πe({ρv }) interpolates the Breuil-Schneider modified local
1
Langlands correspondence over Spec A[ ].
p1
6.2.5. Theorem. Let p be a prime of A[ ], and suppose that p lies on exactly onep1
irreducible component of Spec A[ ]. Then, assuming that πe({ρv }v∈S ) exists, therepis a a κ(p)-linear G-equivariant surjectionOπe(ρv,p ) → κ(p) ⊗A πe({ρv }v∈S ).
v∈SMoreover, if there exists a minimal prime a of A such that ρa is a minimal liftof ρp , then this surjection is an isomorphism.
It seems likely that the above result holds even when p is contained in multiple
1
irreducible components of Spec A[ ]. Nonetheless we are at present only able topprove a somewhat weaker statement:
1
6.2.6. Theorem. Assume that πe({ρv }v∈S ) exists, let p be a prime of Spec A[ ], letpa1 , . . . , ar be the minimal primes of A contained in p, for each i = 1, . . . , r let Vi be
the maximal A-torsion free quotient of πe({ρv }v∈S ) ⊗A A/ai , and denote by W theimage of the diagonal map
Yκ(p) ⊗A πe({ρv }v∈S ) →κ(p) ⊗A/ai Vi .
i

Then there is a a κ(p)-linear G-equivariant surjectionOπe(ρv,p ) → W.
v∈S

52

MATTHEW EMERTON AND DAVID HELM

Moreover, if there exists a minimal prime a of A such that ρa is a minimal liftof ρp , then this surjection is an isomorphism.
6.2.7. Conjecture. Under the hypotheses of Theorem 6.2.6, the map
κ(p) ⊗A πe({ρv }v∈S ) → W
is an isomorphism. In particular the conclusion of Theorem 6.2.5 holds for all p.
Although this conjecture seems difficult to establish in general, we have thefollowing result for small n, which we prove in the next section.
6.2.8. Proposition. Conjecture 6.2.7 holds when n = 2 or n = 3.
In a similar vein, we conjecture:
6.2.9. Conjecture. Assuming that πe({ρv }v∈S ) exists, there is a G-equivariant klinear surjection:Oeπ(ρ)→ k ⊗A πe({ρv }v∈S ).
v∈S

One can also describe the behavior of πe({ρv }v∈S ) under base change. Supposethat B is another ring satisfying Condition 6.1.1, and that f : A → B is a local
homomorphism. If we are given a Galois representation ρv : GEv → GLn (A) wherev does not lie over p, then we may then apply the preceding considerations to
the Galois representations B ⊗A ρv . The following proposition relates πe(ρv ) andπe(B ⊗A ρv ). (We again postpone the proof to the following subsection.)
6.2.10. Proposition. Suppose that for every minimal prime of Spec B, its image pin Spec A is contained in a minimal prime a of A such that ρv,a is a minimal lift of
ρv,p . (For example, suppose that each component of Spec B dominates a componentof Spec A.) Then if πe({ρv }v∈S ) exists, so does πe({B ⊗A ρv }v∈S ), and there is anatural surjection: B ⊗A πe({ρv }v∈S ) → πe({B ⊗A ρv }v∈S ).
We now give some examples illustrating Definition 6.2.2.
6.2.11. Example. Suppose that A = O is the ring of integers in a finite extension
K of Qp . If ρ : GEv → GLn (O) is continuous (for some place v of E that does notlie over p), write ρK := K ⊗O ρ. Then πe(ρ) exists, and is the smooth contragredientto the lattice π(ρK )◦ of Theorem 3.3.2.
6.2.12. Remark. Suppose given A as in Theorem 6.2.1, and a continuous repre1
sentation ρ : GQ` → GL2 (A) for some ` 6= p. Consider a point p ∈ Spec A[ ]. Ifpρp is not of the form χ⊕ | | χ for some character χ of GQ` , then for any minimal
prime a of A, containing p, ρa is necessarily a minimal lift of ρp , and so (assumingthat V := πe(ρ) exists), the surjection of Theorem 6.2.5 is an isomorphism; that is,
Vp is isomorphic to πe(ρp ). On the other hand if ρp does have the form χ⊕ | | χ,then there exist non-minimal lifts of ρp , and so Vp need not a priori be isomorphic
to πe(ρp ). We now give an example showing that it can indeed happen that Vp is
not isomorphic to πe(ρp ).
6.2.13. Example. Suppose that ` and p are distinct, and that ` 6≡ 1 mod p. If
A is as in Theorem 6.2.1, then Ext1Zp [GQ ] (| |, 1) is free of rank 1 over A. Let c`denote a generator of this Ext1 -module, and for any a ∈ A, let ρa : GQ` → GL2 (A)be the rank two representation underlying a · c. One checks that if a is a regular

LOCAL LANGLANDS IN FAMILIES

53

element, then V := πe(ρa ) exists, and in fact is isomorphic to StA (the Steinberg
representation of GL2 (Q` ) with coefficients in A); in particular, it is independent
1of the regular element a. Note that if a ∈ p ∈ Spec A[ ] (i.e. the regular functionpassociated to a vanishes at p), then ρa,p := κ(p) ⊗ ρa is split and hence unramified,
and thus πe(ρa,p ) is a non-split extension of Steinberg by trivial. In particular, atsuch a point p, Vp fails to be isomorphic to πe(ρa,p ).
6.2.14. Example. Suppose that ` and p are distinct, p is odd, and ` 6≡ ±1 mod p.
Let A be the ring W (k)[[a, b]]/ab. Fix a Frobenius element Fr of GQ` and a generator
σ of the p-power inertia in GQ` . Let ρ be the representation of GQ` such that ρ istrivial on the prime-to-p inertia in GQ` , and such that:q(1 + a) 0
ρ(Fr) =
011 b
ρ(σ) =.
0 1
It is not hard to show in this situation that V := πe(ρ) exists. On the other hand,
V is principal series at the minimal prime ideal hbi of A, and Steinberg along theminimal prime ideal hai of A. In particular the Iwahori invariants of V have different
ranks along these two branches, showing that V cannot be free as an A-module.
We conclude with a “recognition theorem” that is useful for verifying that agiven A[G]-module is isomorphic to πe({ρv }v∈S ). As with the other results of thissection, we defer its proof to the next subsection.
6.2.15. Theorem. Let V be an admissible smooth A[G]-module, such that thesmooth dual of V /mV is essentially AIG, and suppose that there exists a Zariski
1
dense subset Σ of Spec A[ ] such that:p(1) For all v ∈ S, and all p in Σ, there exists a minimal prime a of A such that
ρv,a is a minimal lift of ρv,p .
(2) For each point p of Σ there exists an isomorphism:O∼κ(p) ⊗A V −→πe(ρv,p ).
v∈S(3) The diagonal map:
V →

YO

π
e(ρv,p ).

p∈Σ v∈S

is an injection.
Then V satisfies conditions (1), (2), and (3) of Theorem 6.2.1; that is, πe({ρv }v∈S )exists and is isomorphic to V .
6.3. The proofs of Theorem 6.2.1 and some related results. We develop aseries of deductions involving the various conditions of Theorem 6.2.1. These will
be used not only to prove Theorem 6.2.1, and the other outstanding results fromthe preceding subsection, but also to provide a criterion for verifying the conditionsof Theorem 6.2.1, which will be useful in applications.
If A is a local ring with residue field K, and V is a representation of F over A,
we let V denote the representation V ⊗A K.

54

MATTHEW EMERTON AND DAVID HELM

6.3.1. Lemma. Let A be a Noetherian W (k)-algebra that is a local ring with residue
field K. If V is an admissible smooth representation of G over A, then V
dimensional over K if and only if V (n) is a cyclic A-module.

(n)

is 1-

(n)

is one-dimensional then V (n) is a finitely
Proof. Theorem 3.1.14 shows that if V
generated A-module, and so the lemma follows from Nakayama’s lemma together(n)∼with the isomorphism V (n) ⊗A K −→ V .

6.3.2. Lemma. Let A be a Noetherian W (k)-algebra that is a local ring with residue
field K in which all `i are invertible. If V is an admissible smooth representationof G over A, then the following are equivalent:
(n)

6= 0.
(1) For any non-zero quotient K[G]-module W of V , one has W(2) For any non-zero quotient A[G]-module W of V , one has W (n) 6= 0.
(3) J(V ) generates V over K[G].
(4) J(V ) generates V over A[G].
Proof. It is clear that (2) implies (1), as any quotient of V is also a quotient of V .
Suppose that (1) holds and that W is a quotient of V with W (n) = 0. Then(n)W
= 0, and so (1) implies that W = 0. Then W = 0 by Nakayama’s Lemma, so(1) implies (2).
(n)

(n)

If W is a quotient of V , then we have that W
is a quotient of V
since thederivative functor is exact. If we let U denote the K[G]-submodule of V generated(n)by J(V ), we see that W
vanishes if and only if W is a quotient of V /U . Thus (1)and (3) are equivalent.
Clearly (4) implies (3), since J(V ) is the image of J(V ) in V . Conversely, suppose
J(V ) generates V over K[G]. Since J(V ) maps surjectively onto J(V ), Lemma 2.1.7
implies that J(V ) generates V over A[G].

6.3.3. Lemma. Let A be a local ring satisfying Condition 6.1.1. If V is an admis(n)sible smooth representation of G over A, and if Va is nonzero for each minimal(n)prime a of A, then if V
is a cyclic A-module, it is in fact free of rank 1 over A.
Proof. Since (V (n) )a = (Va )(n) , we have that (V (n) )a is nonzero for all a. Our
hypotheses on A imply that A injects into the product of the fields Aa ; it follows
that the annihilator of V (n) in A is the zero ideal. The lemma follows.

6.3.4. Proposition. Let A be a local ring satisfying Condition 6.1.1, let V be anadmissible smooth representation of G over A, and suppose that (Va )(n) is nonzero(n)for each minimal prime a of A, that V
is one-dimensional, and that for any(n)non-zero quotient k[G]-module W of V , one has W
6= 0. Then:(1) V (n) is free of rank 1 over A.
(2) J(V ) generates V over A[G].
(3) EndA[G] (V ) = A.
Proof. The first claim follows immediately from Lemmas 6.3.1 and 6.3.3. The
second is a consequence of Lemma 6.3.2.
By Proposition 3.1.16, the natural map A → EndA[Pn ] (J(V )) is an isomorphism.
This latter map factors as the composition
A → EndA[G] (V ) → EndA[Pn ] (J(V )),LOCAL LANGLANDS IN FAMILIES

55

and restriction of endomorphisms from V to J(V ) is injective because J(V ) generates V . Thus EndA[G] (V ) = A.

The following result gives some equivalent formulations of the hypotheses on V
appearing in the preceding proposition.
6.3.5. Lemma. Let K be a field in which all `i are invertible. If V is an admissible
smooth representation of G over K, then the following are equivalent:
(1) V

(n)

is one-dimensional, and for any non-zero quotient K[G]-module W

of V , one has W
W

(n)

(n)

6= 0 (and hence V(n)is isomorphic to W(n)

, so thatis again one-dimensional).

(n)

(2) V
is one-dimensional, and J(V ) generates V over K[G].
(3) V is of finite length (and hence has a cosocle), cosoc(V ) is absolutely irre(n)(n)
ducible, and V, with both being non-zero.
is isomorphic to cosoc(V )∨(4) The smooth K-dual V of V is essentially AIG.
Proof. The equivalence of conditions (1) and (2) follows from Lemma 6.3.2 (appliedwith A = K and V = V ).
over K[G], and hence of finite
If condition (2) holds, then V is finitely generatedL
length, by [17], II.5.10. Write cosoc(V ) = j W j , where each W j is irreducible.
Condition (1) (which also holds, since it is equivalent to condition (2), as we have(n)already observed) shows that W jposition:V(n)is one-dimensional for each j. Since the com-(n) ∼ M (n)→ cosoc(V )Wj−→jis surjective (the derivative functor is exact), we see that in fact there is only one
summand, and hence that cosoc(V ) is irreducible. Proposition
6.3.4 (applied with
A = K and V = W ) then implies that EndG cosoc(V ) ) = K, and hence thatcosoc(V ) is in fact absolutely irreducible. Thus (2) implies (3).
(n)If condition (3) holds, then by assumption cosoc(V )is nonzero. It is there(n)fore one-dimensional by Theorem 3.1.15. Thus V
is one-dimensional, giving thefirst half of condition (1). The second half of condition (1) follows from the fact
(n)that cosoc(V ) is irreducible, and satisfies cosoc(V )6= 0. Thus (3) implies (1).
∨∨Now consider the smooth dual V . If V is finite length, the socle of V is the∨smooth dual of the cosocle of V . In particular soc(V ) is absolutely irreducible∨ (n)∨and generic if and only if cosoc(V ) is. Moreover, the map soc(V )→ (V )(n)(n)(n)
is dual to the map V→ cosoc(V )so that one is an isomorphism if and only
if the other is. Thus (3) is equivalent to (4).

6.3.6. Lemma. If A is a reduced Noetherian W (k)-algebra, and if V is an admissible
A[G]-module, then the following are equivalent:(1) V is A-torsion free, i.e. every associated prime of V is a minimal primeof A.
(2) The natural map V → V ⊗A K(A) is an injection, where K(A) is the productover minimal primes a of A of the fields Aa .

56

MATTHEW EMERTON AND DAVID HELM

(3) The natural map
V →Y(V /aV )tfais injective, where a runs over the minimal primes of A and (V /aV )tf isthe maximal A/a-torsion free quotient of V /aV .
If these equivalent conditions hold, then for any Zariski dense set of primes Σ ofSpec A, the map
YV →V ⊗A κ(p)p∈Σ

is injective.
Proof. If one replaces V with a finitely generated A-module M in conditions (1), (2)and (3), then the equivalence of these conditions is standard commutative algebra.
On the other hand, if U is a sufficiently small compact open subgroup of G, thenV U is a direct summand of V that is finitely generated as an A-module, and V is
the union of the V U .
On the other hand, it is easy to see that each of conditions (1), (2) and (3)holds for V if, and only if, it holds with V U in place of V for each sufficiently small
compact open subgroup U of V . As (1), (2) and (3) are equivalent condtions oneach such V U , they are equivalent conditions on V .
Now let Σ be a Zariski dense set of primes of A, suppose that the equivalentconditions (1), (2), and (3) hold, and suppose that x is an element of V that maps
to zero in V ⊗A κ(p) for all p ∈ Σ. Choose a compact open subgroup U of G fixing
x; then V U is finitely generated over A, and x maps to zero in V U ⊗A κ(p) forall p in Σ. It follows that the support of x (considered as an element of V U ) isa closed subset of Spec A contained in the complement of Σ. In particular thatthe annihilator of x is not contained in any minimal prime of A, contradicting
condition (1).

6.3.7. Lemma. Let A be a reduced Noetherian W (k)-algebra, and let V1 and V2 be
two admissible smooth A[G]-modules such that:(n)(1) For each i, Vi is free of rank one over A.
(2) For each i, Vi is generated by J(Vi ) as an A[G]-module.
(3) There exists a Zariski dense set Σ of primes of A such that for all p ∈ Σ,
(V1 )p is isomorphic to (V2 )p , as Ap [G]-modules.
(4) The natural map:YVi →Vi ⊗A κ(p)p∈Σis injective for each i. (This is automatic if Vi is A-torsion free.)∼

Then there is an A-linear G-equivariant isomorphism V1 −→ V2 .
Proof. Let K0 be the product over p ∈ Σ of the residue fields κ(p). Condition (3)gives us an isomorphismYY∼
V1 ⊗A κ(p) −→V2 ⊗A κ(p).
ppThere are natural injections:V i ⊗A K 0 →Yp

Vi ⊗A κ(p)

LOCAL LANGLANDS IN FAMILIES

57

for each i; these are not in general isomorphisms. However, for sufficiently small
compact open subgroups U of G, ViU is a direct summand of Vi that is finitely
generated as an A-module. For each such U , the induced map:YU
ViU ⊗A K0 →Vi ⊗A κ(p)p

is an isomorphism. Passing to the limit on both
Q sides, we obtain an isomorphism of
Vi ⊗A K0 with the space of smooth elements of p Vi ⊗A κ(p). Thus the isomorphism:YY∼
V1 ⊗A κ(p) −→V2 ⊗A κ(p)pp0induces an isomorphism of V1 ⊗A K with V2 ⊗A K0 .
Moreover, by Condition (4), Vi embeds in Vi ⊗A K0 for each i. We may thus
regard V1 and V2 as submodules of V1 ⊗A K0 .
(n)
By (1), (V1 ⊗A K0 )(n) is free of rank one over K0 , and Vi is a free A-submodule0 (n)of (V1 ⊗A K )for each i. There thus exists an element c of (K0 )× such that(n)(n)
cV2 and V1 coincide as submodules of (V1 ⊗A K0 )(n) . It follows that cJ(V2 ) andJ(V1 ) coincide as submodules of J(V1 ⊗A K0 ). Since V1 and V2 are generated by
J(V1 ) and J(V2 ) over A[G], we must have V1 = cV2 ; in particular V1 and V2 are
isomorphic.

We can now prove the uniqueness claim of Theorem 6.2.1.
6.3.8. Proposition. Let A be a local ring satisfying Condition 6.1.1, and let V1 andV2 be two admissible smooth A[G]-modules. Suppose that:(1) The Vi are A-torsion free.
(n)(2) For each minimal prime a of A, (Vi )a is nonzero.
(3) For each i, V i satisfies the equivalent conditions of Lemma 6.3.5.
(4) For each minimal prime a of A, there is a G-equivariant isomorphism∼(V1 )a −→ (V2 )a .
Then there is an A-linear G-equivariant isomorphism V1 ∼= V2 (which, by part (3) of
Proposition 6.3.4, is uniquely determined up to multiplication by an element of A× ).
(n)

Proof. By part (1) of Proposition 6.3.4, we have that Vi
is free of rank 1 over
A for each i. As the minimal primes of A are dense in Spec A, it thus follows by
Lemma 6.3.7 that V1 is isomorphic over A[G] to V2 .

The purpose of our next collection of results, which are rather technical, is toallow us to make a tensor factorization in the context of Theorem 6.2.1, and hence
work with one Ev at a time.
6.3.9. Proposition. Let K be a field in which all `i are invertible. If V is anadmissible smooth representation of G over K satisfying the equivalent conditions ofLemma 6.3.5, there exist admissible smooth representations V v of Gv (v ∈ S), eachindividually satisfying the equivalent conditionsNof Lemma 6.3.5 (with G replaced by
Gv ), together with a G-equivariant surjection v V v → V .
Proof. We proceed by induction on the cardinality s of S. In the case when s = 1
there is nothing to prove, and so we assume that s > 1, and write S = v1 , . . . , vs ,G0 = Gv2 ×· · ·×Gvs , so that G = Gv1 ×G0 . Since cosoc(V ) is absolutely irreducible,

58

MATTHEW EMERTON AND DAVID HELM∼

there is an isomorphism cosoc(V ) −→ π v1 ⊗ π 0 , where π v1 (resp. π 0 ) is a genericabsolutely irreducible representation of Gv1 (resp. G0 ).
Since V is of finite length, we may and do choose a quotient W of V which is
maximal with respect to the following property: there is a surjective map
0

φ : V v1 ⊗ V → W ,0where V v1 and V each satisfy the equivalent conditions of Lemma 6.3.5 (withrespect to Gv1 and G0 respectively). Since cosoc(V ) satisfies these conditions, we
see that W 6= 0. Thus cosoc(V ) is a quotient of W , and hence V(n)is isomorphic(n)to W , as both are one-dimensional.
Let U be the kernel of the quotient map V → W , and suppose that U is non-zero.
Extending scalars if necessary, we then may find a non-zero absolutely irreducible
00quotient θv1 ⊗θ of U , where θv1 (resp. θ ) is an absolutely irreducible representation
0of Gv1 (resp. G0 ). If we let T denote the kernel of the quotient map U → θv1 ⊗ θ ,and if we write X := V /T , then there is a short exact sequence0

0 → θv1 ⊗ θ → X → W → 0,

(6.1)

which we may pullback via φ to obtain a short exact sequence0

0

0 → θv1 ⊗ θ → Y → V v1 ⊗ V → 0.

(6.2)

Applying the n-th derivative functor to (6.1), (and recalling that the surjections
V

(n)

→X

(n)

→W

(n)

are in fact isomorphisms,) we obtain an isomorphism:
0

(n)

0

∼

θv1 ⊗K (θ )(n) −→ (θv1 ⊗K θ )(n) = 0.
0(n)Hence either θv1 = 0 or (θ )(n) = 0. Also, we conclude that θv1 ⊗ θ cannot be aquotient of V , and hence cannot be a quotient of X. Thus (6.1) is non-split, andhence (6.2) is also non-split (since φ is surjective).
The non-split short exact sequence (6.2) corresponds to a non-trivial element of
0
0
Ext1G (V v1 ⊗ V , θv1 ⊗ θ ), which by the Künneth formula admits the description
00Ext1G (V v1 ⊗ V , θv1 ⊗ θ )
∼

0

0

0

0

−→ HomGv1 (V v1 , θv1 ) ⊗ Ext1G0 (V , θ ) ⊕ Ext1Gv1 (V v1 , θv1 ) ⊗ HomG0 (V , θ ).
Now by assumption, the nth derivative (as a GLn (Ev1 )-module) of any non-zero
quotient of V v1 is a non-zero space, while the nth derivative (as a G0 -module)
0of any non-zero quotient of V is a non-zero space. Thus if (θv1 )(n) = 0, thenHomG1 (V v1 , θv1 ) = 0, and thus Y corresponds to a non-trivial element of the tensor
0 0
product Ext1G1 (V v1 , θv1 ) ⊗ HomG0 (V , θ ). Concretely, this means we may form a00
non-trivial extension E1 of V v1 by θv1 , and find a non-zero map ψ : V → θ (which0is then surjective, since θ is irreducible), so that Y is obtained as the pushforward0of E1 ⊗ V via the map
00

id ⊗ ψ : θv1 ⊗ V → θv1 ⊗ θ .
0Thus Y , and hence X, is a quotient of E1 ⊗ V , contradicting the maximality of0
W . If instead we had (θ )(n) = 0, then we would similarly conclude that X may

LOCAL LANGLANDS IN FAMILIES

59
0

0

be written as a quotient of V 1 ⊗ E2 , for some non-trivial extension E2 of V by θ ,again contradicting the maximality of W .
From these contradictions we conclude that in fact U = 0, and thus that V = W .
0
Thus we may write V as a quotient of V v1 ⊗ V as above. Applying the inductive0
hypothesis to V , the proposition follows.
6.3.10. Corollary. If V is an admissible smooth representation of G over a local ring A satisfying Condition 6.1.1, such that V := V /mV satisfies the equivalent conditions of Lemma 6.3.5, and if S 0 ⊂ S is any subset, then the GS 0 00
representation V (n),S\S /mV (n),S\S satisfies the conditions of Lemma 6.3.5 (withrespect to A[GS 0 ]).
N
Proof. Choose a surjection v∈S V v → V satisfying the conditions of the preceding(n)proposition. Since V v

is one-dimensional for each v (and so in particular for each(n),S\S 00v ∈ S \ S ), applying the exact functor V 7→ V
yields a surjection
O
O
0
0

0
0
(n),S\S(n),S\S∼∼V v −→Vv→V−→ V (n),S\S /mV (n),S\S .
v∈S 0

v∈S

The lemma follows.



We now return to the setting of the previous subsection. That is, for each v is S
we are given a representation ρv : GEv → GLn (A). The above results allow us toestablish Theorem 6.2.1 and Proposition 6.2.4 more or less immediately.
Proof of Theorem 6.2.1. Suppose we have V1 , V2 satisfying conditions (1), (2),and (3) of Theorem 6.2.1. Then for all minimal primes a of A, we have a κ(a)∼linear G-equivariant isomorphism (V1 )a −→ (V2 )a . Thus V1 and V2 satisfy all of(n)the hypotheses of Proposition 6.3.8, and are therefore isomorphic. Moreover, V1
is cyclic as an A-module by Lemma 6.3.1. Let x be any element of V1 whose(n)(n)
image in V1 generates V1 as an A-module, and let W be the A[G]-submodule(n)of V1 generated by x. Then the image of W (n) in V1 contains the image of x,(n)and hence is all of V1 . It follows that J(W ) is equal to all of J(V1 ), and hence,by Proposition 6.3.4, part (2), we must have W = V1 . Thus V1 is cyclic as an
A[G]-module. Finally, EndA[G] (V1 ) is isomorphic to A by Proposition 6.3.4, part(3).

Proof of Proposition 6.2.4. Suppose that for each v, we have a representation πe(ρv )
satisfying conditions (1), (2), and (3) of Theorem 6.2.1 for ρv . Then it is clear thatthe maximal A-torsion free part of the tensor product over all v of πe(ρv ) satisfies
the conditions of Theorem 6.2.1 for the collection {ρv }.
Conversely, suppose we have a representation πe({ρv }v∈S ) satisfying the hypotheses of Theorem 6.2.1 for the collection {ρv }. Then for any minimal prime a of A,
we have an isomorphism:O∼πe({ρv }v∈S ) ⊗A κ(a) −→πe(ρv,a ).
v∈S
0Fixing a place v, and taking derivatives at all v 6= v, we obtain an isomorphism:(n),S\{v}(eπ ({ρv }v∈S ))a∼−→ πe(ρv,a ).

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MATTHEW EMERTON AND DAVID HELM

Moreover (eπ ({ρv }v∈S ))(n),S\{v} is A-torsion free, and (by Corollary 6.3.10) satisfies
condition (3) of Theorem 6.2.1. Thus πe({ρv }v∈S )(n),S\{v} is isomorphic to πe(ρv )(so in particular the latter exists).

We now turn to Theorems 6.2.5 and 6.2.6. Once we have established these,
Proposition 6.2.10 will be an easy consequence. We first need the following lemma:
6.3.11. Lemma. Let A be a normal Qp -algebra that is an integral domain withfield of fractions K, and let (ρ0 , N ) be a Frobenius-semisimple Weil–Deligne representation over A that splits (over K) as a direct sum of absolutely indecomposable
Weil–Deligne representations Spρi ,ni . There exist characters χi : WE → A× suchthat ρi ⊗K χi is defined over a finite extension K0 of Qp contained in A.
Proof. By Lemma 4.1.5 we know that such characters χi exist with values in K× ;it suffices to show that they take values in A× . Let O be the localization of A at aheight one prime. Then ρ0 ⊗A K is O-integral, so each ρi is O-integral as well. Thus
det ρi is a character with values in O× . Since this is true for all O, det ρi takes
values in A× . Moreover, (det ρi ) ⊗K χi takes values in K0× , and K0 is containedin A, so some power of χi takes values in A× . But then χi must take values in A×
as well since A is normal.

6.3.12. Lemma. Let A be a local ring satisfying Condition 6.1.1, let {ρv } be acollection of representations GEv → GLn (A), and suppose that πe({ρv }v∈S ) exists.
Then, for each minimal prime a of A, the representation πe({ρv ⊗A A/a}v∈S ) exists,
and is isomorphic to the maximal A/a-torsion free part of πe({ρv }v∈S ) ⊗A A/a.
Proof. It is straightforward to see that πe({ρv }v∈S ) ⊗A A/a satisfies conditions (2)and (3) of Theorem 6.2.1, so its maximal A/a-torsion free quotient does as well.
This quotient also satisfies condition (1) of Theorem 6.2.1 by construction.
Proof of Theorem 6.2.5. By Proposition 6.2.4, it suffices to consider the case when S
has only one element v. By Lemma 6.3.12 we may assume A is a domain with fieldof fractions K. Fix an algebraic extension K0 of K such that K0 contains a squareroot of `, where ` is the residue characteristic of v, and such that the Frobeniussemisimple Weil–Deligne representation associated to ρv ⊗A K0 splits as a direct
sum of absolutely indecomposable Weil–Deligne representations Spρi ,Ni over K0 .
Let K0 be the maximal subfield of K0 that is algebraic over Qp .
Let A0 be the integral closure of Ap in K0 , and let p0 be a prime ideal of A0 over p.
Then, by Lemma 6.3.11, there exist characters χi , with values in (A0p0 )× , such thatfor each i, ρi ⊗ χ−1
is defined over K0 .
i
Let πi be the admissible representation of GLn (Ev ) over K0 that corresponds
to ρi ⊗ χi under the unitary local Langlands correspondence. Then for each i,(Stπi ,Ni ⊗K0 K0 ) ⊗ (χi ◦ det) corresponds to Spρi ,Ni under unitary local Langlands.
Without loss of generality, we assume that the representations πi are ordered sothat for all i < j, (Stπi ,Ni ⊗K0 κ(p0 )) ⊗ χi does not precede (Stπj ,Nj ⊗K0 κ(p0 )) ⊗ χj .
(It is then also true that (Stπi ,Ni ⊗K0 K0 )⊗χi does not precede (Stπj ,Nj ⊗K0 K0 )⊗χj .)Let M be the smooth A0p0 -linear dual of the module:On−1
GL (E )
(Stπi ,Ni ⊗K0 A0p0 ) ⊗ χi ,(| | ◦ det)− 2 IndQ n viLOCAL LANGLANDS IN FAMILIES

61

where Q is a suitable block upper triangular parabolic subgroup of GLn (Ev ). Then,by construction, M ⊗A0p0 K0 is isomorphic to πe(ρ ⊗A K0 ). Moreover, because of ourassumptions on the ordering of the πi , the smooth κ(p0 )-dual of M/p0 M is essentially AIG by Corollary 4.3.3, and hence J(M ) generates M as an A0p0 [G]-module.
Moreover M (n) is free of rank one over A0p0 by Corollary 3.1.13. Finally, M is A0p0 torsion free by construction (in fact, M is free over A0p0 .) Thus by Lemma 6.3.7, M
is isomorphic to πe(ρv ) ⊗A A0p0 .
The injection of Theorem 4.5.7 yields a surjection:πe(ρv ⊗A κ(p0 )) → M ⊗A0p0 κ(p0 )that is an isomorphism if, and only if, ρv ⊗A K is a minimal lift of ρv ⊗A κ(p). Taking
smooth duals and applying Lemma 3.2.5, we see that this surjection descends tothe desired surjectionπe(ρv,p )) → πe(ρv ) ⊗A κ(p).
Proof of Theorem 6.2.6. As above, it suffices by Proposition 6.2.4 to consider thecase when S has only one element v. By Lemma 6.3.12, for each minimal prime aof A containing p we have a surjection:πe(ρv ) ⊗A κ(p) → πe(ρv ⊗A A/a) ⊗A κ(p).
Then W is the image of πe(ρv ) ⊗A κ(p) in the productYπe(ρv ⊗A A/a) ⊗A κ(p).
a

By Theorem 6.2.5 we also have surjections:fa : πe(ρv,p ) → πe(ρv ⊗A A/a) ⊗A κ(p)for all minimal primes a of A contained in p. This gives a diagonal map:Yπe(ρv,p ) →πe(ρv ⊗A A/a) ⊗A κ(p).
a0

Let W be the image of this map. It suffices to show that W 0 is isomorphic to W .
The spaces W (n) and (W 0 )(n) are one-dimensional κ(p)-subspaces of
Y
(n)
π
e(ρv ⊗A A/a) ⊗A κ(p)athat project isomorphically onto each factor. There thus exists for each a a scalar
ca in κ(p)× such that c(W 0 )(n) coincides with W (n) as subspaces of
Y
(n)
π
e(ρv ⊗A A/a) ⊗A κ(p,awhere c is the automorphism of this product given by multiplication by ca on thefactor corresponding to a.
As the K-duals of W ⊗κ(p) K and W 0 ⊗κ(p) K are essentially AIG, this impliesthat W and cW 0 coincide, and thus W and W 0 are isomorphic.
Proof of Proposition 6.2.8. As usual, we invoke Proposition 6.2.4 to reduce to thecase where S has a single element v. As in the proof of Theorem 6.2.6, let W be

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MATTHEW EMERTON AND DAVID HELM

the image of κ(p) ⊗A πe(ρv ) under the diagonal map
Yκ(p) ⊗A πe(ρv ) →κ(p) ⊗A/ai Vi ,iwhere a1 , . . . , ai are the minimal primes of A contained in p and, for each i, Vi is
the maximal A-torsion free quotient of πe(ρv ) ⊗A A/ai .
As πe(ρv ) embeds in the product of the Vi , every Jordan–Hölder constituentof κ(p) ⊗A πe(ρv ) is isomorphic to a Jordan–Hölder constituent of κ(p) ⊗A/ai Vi
for some i, and hence to a Jordan–Hölder constituent of W . In particular, every
Jordan–Hölder constituent of the kernel of the map
κ(p) ⊗A πe(ρv ) → W
is a Jordan–Hölder constituent of κ(p) ⊗A πe(ρv ) that appears with multiplicity atleast two. Since the smooth dual of κ(p)⊗A πe(ρv ) is essentially AIG, Corollary 4.3.11
above shows that no such Jordan–Hölder constituent can exist when n = 2 or 3. 
Proof of Proposition 6.2.10. By Proposition 6.2.4 we may assume S consists ofa single element. For any minimal prime b of B, we have by Theorem 6.2.6 asurjection:πe(ρv ) ⊗A κ(f −1 (b)) → πe(ρv,f −1 (b) )and hence (after a base change) a surjection:πe(ρv ) ⊗A κ(b) → πe(ρv ⊗A κ(b)).
Let V be the image of the composed map:YY
πe(ρv ) ⊗A B →πe(ρv ) ⊗A κ(b) →πe(ρv ⊗A κ(b)).
bbOne easily verifies that V satisfies conditions (1), (2) and (3) of Theorem 6.2.1 forthe representation ρv ⊗A B over B.
We now turn to the proof of Theorem 6.2.15. This will require several preliminarylemmas.
6.3.13. Lemma. Suppose that Theorem 6.2.15 holds when S has only one element.
Then Theorem 6.2.15 holds for an arbitrary finite set S.
Proof. Suppose we have established Theorem 6.2.15 in the case in which S has only
one element. We can then establish the general case as follows: suppose V satisfies
the conditions of Theorem 6.2.15 for the collection {ρv }. If we fix a place v ∈ S,
then V (n),S\{v} satisfies the conditions of Theorem 6.2.15 for the representation
ρv . Thus V (n),S\{v} is isomorphic to πe(ρv ). It follows by Proposition 6.2.4 thatπe({ρv }v∈S ) exists and is isomorphic to the maximal torsion-free quotient of thetensor product of the representations V (n),S\{v} .
For any prime p of A lying over a prime of Σ, we have an isomorphism:O∼
V ⊗A κ(p) −→V (n),S\{v} ⊗A κ(p).
v∈S

It thus follows by Lemma 6.3.7 that V and πe({ρv }v∈S ) are isomorphic, as required.

6.3.14. Proposition. Let A be a local ring satisfying Condition 6.1.1, let ρv be an ndimensional representation of GEv over A, and let V be an admissible A[G]-module
such that:LOCAL LANGLANDS IN FAMILIES

63(1) V is torsion free over A.
(2) The smooth dual of V /mV is essentially AIG.
1(3) There exists a Zariski dense set of primes Σ in Spec A[ ] such that for eachpprime p ∈ Σ, V ⊗A κ(p) is isomorphic to πe(ρv,p ),Then V satisfies conditions (1), (2), and (3) of Theorem 6.2.1 with respect to ρv ;that is, πe(ρv ) exists and is isomorphic to V .
Proof. Note that conditions (1) and (3) of Theorem 6.2.1 are immediate from the
hypotheses. It thus suffices to construct, for each minimal prime a of A, an isomor∼
phism Va −→ πe(ρv,a ).
Fix a minimal prime a of A. The map V → (V /aV )tf becomes an isomorphism
after localizing at any prime p of A that contains a but no other minimal prime of
A. Thus, replacing V with (V /aV )tf , A with A/a, and Σ with the set of p in Σthat contain a but no other minimal prime of A, we reduce to the case where A isa domain with field of fractions K.
Let K0 be an algebraic extension of K such that K0 contains a square root of `,where ` is the residue characteristic of v, and such that the Frobenius-semisimple
Weil–Deligne representation associated to ρv ⊗A K0 splits as a direct sum of absolutely indecomposable Weil–Deligne representations Spρi ,Ni over K0 . Let K0 be themaximal subfield of K0 that is algebraic over Qp .
1
Let A0 be the integral closure of A[ ] in K0 . By Lemma 6.3.11, there existpcharacters χi , with values in (A0 )× , such that for each i, ρi ⊗ χi is defined over K0 .
Let πi be the admissible representation of GLn (Ev ) over K0 that corresponds
to ρi ⊗ χi under the unitary local Langlands correspondence. Then for each i,(Stπi ,Ni ⊗K0 K0 ) ⊗ (χi ◦ det) corresponds to Spρi ,Ni under unitary local Langlands.
Without loss of generality, we assume that the representations πi are ordered sothat for all i < j, (Stπi ,Ni ⊗K0 K0 ) ⊗ (χi ◦ det) does not precede (Stπj ,Nj ⊗K0 K0 ) ⊗(χj ◦ det). Then, for all p in an open dense subset U1 of Spec A0 , and all i < j,(Stπi ,Ni ⊗K0 κ(p)) ⊗ (χi ◦ det) does not precede (Stπj ,Nj ⊗K0 κ(p)) ⊗ (χj ◦ det).
Let M be the smooth A0 -linear dual of the module:On−1
GL (E )
(Stπi ,ni ⊗K0 A0 ) ⊗ χi ,(| | ◦ det)− 2 IndQ n v
iwhere Q is a suitable block upper triangular parabolic subgroup of GLn (Ev ). Let
U2 be the open dense subset of Spec A0 consisting of those p0 such that ρv ⊗A K0 is
a minimal lift of ρv ⊗A κ(p0 ). Then for all p0 ∈ U2 , M ⊗A0 κ(p0 ) is isomorphic toπe(ρv ⊗A κ(p0 )).
Now let M 0 be the A0 [G]-submodule of M generated by J(M ). Then (M 0 )(n)is isomorphic to M (n) , and the latter is locally free of rank one over A0 by Corollary 3.1.13. The module M 0 is A0 -torsion free, as it is contained in the free A0 module M . Moreover, for all p0 ∈ U1 ∩ U2 , we have isomorphisms:∼∼

M 0 ⊗A0 κ(p0 ) −→ M ⊗A0 κ(p0 ) −→ πe(ρv ⊗A κ(p0 )).
Set
M 00 = M 0 ⊗A0 (M 0 )(n)Then (M 00 )(n) is free of rank one over A0 .

−1

.

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MATTHEW EMERTON AND DAVID HELM

Let Σ0 be the set of primes of A0 lying over primes in Σ. Then Σ0 ∩ U1 ∩ U2 is∼dense in Spec A0 , and we have isomorphisms: V ⊗A κ(p0 ) −→ M 00 ⊗A0 κ(p0 ) for all000
p ∈ Σ ∩ U1 ∩ U2 . It follows by Lemma 6.3.7 that V ⊗A A is isomorphic to M 00 . In
particular V ⊗A K0 is isomorphic to πe(ρv ⊗A K0 ), and hence V ⊗A K is isomorphic
to πe(ρv ⊗A K), as required.
Proof of Theorem 6.2.15. By Lemma 6.3.13 it suffices to consider the case where
S has a single element. Let V tf be the maximal A-torsion free quotient of V , andlet V tor be the kernel of the map V → V tf . Then V tor /mV tor embeds in V /mV ,and the latter has finite length by Lemma 6.3.5. It follows that V tor /mV tor is
finitely generated as an A[G]-module, and (since V is m-adically separated), thus
V tor is finitely generated as well. In particular its support is a closed subset Z ofSpec A. Let U be the complement of Z in Spec A. Then for all p in Σ ∩ U , we haveisomorphisms:∼∼πe(ρv,p ) −→ V ⊗A κ(p) −→ V tf ⊗A κ(p).
It follows by Proposition 6.3.14 that V tf satisfies conditions (1), (2), and (3) of
Theorem 6.2.1; in particular πe(ρv ) exists and is isomorphic to V tf . Thus, by Thetf
orem 6.2.5, V ⊗A κ(p) is isomorphic to πe(ρv ⊗A κ(p)) for all p for which there
exists a minimal prime a of A such that ρv,a is a minimal lift of ρv,p . In particular
this holds for all p in Σ. Thus we have an isomorphism:∼

V ⊗A κ(p) −→ V tf ⊗A κ(p)for all p ∈ Σ; by Lemma 6.3.7 it follows that V is isomorphic to V tf , and hence
that V satisfies conditions (1), (2), and (3) of Theorem 6.2.1.

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(Matthew Emerton) Mathematics Department, University of Chicago, 5734 S. University Ave., Chicago, IL 60637(David Helm) Mathematics Department, University of Texas at Austin, 1 University
Station C1200, Austin, TX, 78712
E-mail address, Matthew Emerton: emerton@math.uchicago.edu
E-mail address, David Helm: dhelm@math.utexas.edu