R. van Dobben de Bruyn

The Local Langlands Correspondences

Part III Essay, 3 May 2012

Supervisor: Dr T. Yoshida

Trinity College, University of Cambridge

Preface

The main aim of this essay is to state the local Langlands correspondences forGLn, and to define all the objects involved in the statement of the theorem.This work is aimed at graduate students. We will assume knowledge aboutlocal fields, basic (infinite) Galois theory and some understanding of the rep-resentation theory of finite groups. We will develop representation theory fora particular type of infinite topological groups (called locally profinite groups).Some knowledge about commutative algebra might prove useful, but is not re-quired. The language of category theory (“abstract nonsense”) is used freely(especially adjunctions), but one should also be able to read the work withoutany knowledge about such matters.

2

Contents

Introduction 6

1 Locally profinite groups 10

1.1 Topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Smooth representations . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Induced representations . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 Dual representations . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Tate’s Thesis 24

2.1 Haar measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 The additive group of a local field . . . . . . . . . . . . . . . . . 28

2.3 The multiplicative group of a local field . . . . . . . . . . . . . . 31

2.4 Epsilon factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Weil groups 40

3.1 Local Class Field Theory . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Representations of the Weil group . . . . . . . . . . . . . . . . . 44

3.3 L-functions and epsilon factors . . . . . . . . . . . . . . . . . . . 46

3.4 Deligne representations . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Representations of the General Linear Group 52

4.1 Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 L-functions and epsilon factors . . . . . . . . . . . . . . . . . . . 56

5 The Local Langlands Correspondences 60

References 62

4

Introduction

Class field theory provides a description of the abelian extensions of a field K(global or local) in terms of its own arithmetic. The Langlands programmeis a series of conjectures generalising class field theory to include informationabout non-abelian extensions as well. Like class field theory, the Langlandsprogramme has global and local versions.

In this essay, we will only be concerned with the local case. Then an importantobject associated to a field is the Weil group, which is defined in Chapter 3.It is a dense subgroup of the absolute Galois group of the field, and local classfield theory gives an isomorphism between the abelianisation of the Weil groupand the multiplicative group Kˆ.

This isomorphism has important consequences for representation theory of thetwo groups. Representations of the absolute Galois group arise for instancenaturally from elliptic curves (as Tate modules) or more generally via etalecohomology. Such representations give (by restriction) a representation of theWeil group, and if the representation is 1-dimensional (i.e. a character), thenclass field theory asserts that it corresponds to a character of Kˆ.

More generally, the Langlands correspondence asserts that n-dimensional rep-resentations of the Weil group correspond to representations of GLnpKq in anatural way. This naturality is expressed by associating to representations ofthe Weil group, as well as representations of GLnpKq, a function called the L-function. However, it turns out that this L-function does not contain as muchinformation for higher-dimensional representations as it does for 1-dimensionalones. Therefore, one also defines ε-factors on both sides.

The local Langlands correspondence then says that there is a bijection betweenthe set GnpKq of isomorphism classes of a particular kind of n-dimensionalrepresentations of the Weil group and the set AnpKq of isomorphism classesof irreducible representations of GLnpKq. This bijection is natural in manyways; traditionally one especially emphasises that it preserves L-functions andε-factors.

The local Langlands correspondence was turned into a theorem in 2001 by Harrisand Taylor [6], and independently in 2002 by Henniart [7]. A much more generalversion, where GLn is replaced by a more general algebraic group, is still an openproblem (if only because it not clear what the precise formulation has to be).

The main aim of this essay is to state the local Langlands conjecture in theway it was proven by Harris-Taylor and Henniart. This involves defining L-functions and ε-factors on both the Weil group side and the GLnpKq-side of thecorrespondence. For the 1-dimensional case, this is the local theory from Tate’sthesis [16]. For higher dimensions, we only state the results, and refer the readerto the literature for the proofs. We note that one of the proofs requires a globalargument.

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Notation

Throughout this essay, K will denote a non-archimedian local field, that is, afinite extension of either Qp for some prime p, or of FqppT qq for some finite fieldFq. We fix a separable algebraic closure K of K, and write GK for the absoluteGalois group of K.

We write OK for the ring of integers in K, pK for the (unique) prime ideal inOK and vK for the valuation of K. The valuation is a surjective homomorphismKˆ Ñ Z, and the valuation of 0 is defined to be `8.

The residue field OK{pK is denoted k, and its number of elements is denotedq. The characteristic of k is denoted p. The absolute value } ¨ }K on K isnormalised via the rule }x}K “ q´vKpxq. When the field K is clear, we will dropthe subscript from the notations } ¨ }K and pK .

A basis for the open neighbourhoods around 0 on K is given by the sets pn forn P Z, and K is the union of these sets.

All finite extensions L{K will be assumed to lie within K. This way, we identifyGL with a subgroup of GK . If L is not normal, such an identification is onlycanonical up to an inner automorphism of GK .

The group Kˆ fits in the short exact sequence

1 Ñ OˆK Ñ KˆvKÝÑ ZÑ 1,

which splits by projectivity of Z, i.e. by the choice of a uniformiser. A basis forthe open neighbourhoods of the identity is given by the sets

UnK “ 1` pn,

for n ą 0. We write UK for OˆK , and we observe that

UK{U1K – kˆ,

whereas (for i ą 0):U iK{U

i`1K – k.

The group UK is the unique maximal compact open subgroup of Kˆ. Therefore,in contrast with the additive case, Kˆ is not the union of its compact opensubgroups.

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Acknowledgements

Most of this essay owes its presentation to [3]. The most important other worksthat were used are [17], [14] and [16]. For the statement of the theorems inChapter 4, I turned to [4], where also the proofs can be found. For a completelist of the used literature, see the References.

Besides the literature, I want to express my gratitude towards all the people whohelped me with this essay. First of all, I would like to thank the essay setter andsupervisor Dr T. Yoshida for the useful explanations he gave, both motivational(“why do we want to study this?”) and technical. Secondly, I want to thank theother person taking the essay, for the interesting discussions we had about thecontents, of which I wish there were more. Also, I would like to thank my goodfriend and formerly fellow student Johan Commelin for proofreading the essay.His suggestions ranged from typographical to mathematical ones, and made animprovement on the entire work. Finally, I want to thank my good friend andfellow Part III student Manuel Barenz for the advanced LATEX-related tips andtricks he taught me.

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1 Locally profinite groups

In this chapter, we will set up some of the basics we need in further chapters.We will develop the theory of locally profinite groups, and discuss some aspectsof their representation theory.

This chapter is based on [3], although we explain the basics in more depth here.On the other hand, we cover only the necessary material.

1.1 Topological groups

We recall the following definition:

Definition 1.1.1. A topological group is a group G endowed with a topologysuch that the maps

GˆGÑ G GÑ G

px, yq ÞÑ xy, x ÞÑ x´1

are continuous.

Remark 1.1.2. In terms of abstract nonsense, it is a group object in the cate-gory Top.

Lemma 1.1.3. Let G be a topological group. Then G is Hausdorff if and onlyif it is T1.

Proof. Clearly every Hausdorff space is T1. Conversely, letG be a T1 topologicalgroup. Since both multiplication and inversion are continuous, so is the map

GˆGÑ G

px, yq ÞÑ xy´1.

Since G is T1, the singleton t1u is closed, hence so is its inverse image underthe above map. This is the diagonal, hence G is Hausdorff.

Lemma 1.1.4. Let G be a topological group, and let H be a subgroup. Then theclosure H of H is also a subgroup. If furthermore H is normal, then so is H.

Proof. By a standard theorem of topology, the closure H ˆH of HˆH Ď GˆGis the product H ˆH. The map

f : GˆGÑ G

px, yq ÞÑ xy

is continuous, so fpH ˆHq Ď fpH ˆHq Ď H. Hence, H is closed under multi-plication, and a similar argument shows that H is closed under inversion.

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If H is normal, then gHg´1 “ H for all g P G. If X Ď G is any closed setcontaining H, then also the set gXg´1 is closed and contains H. Hence,

H “č

XĚHclosed

X “č

XĚHclosed

gXg´1 “ gHg´1.

Remark 1.1.5. A quotient of G by a subgroup H is T1 (or Hausdorff) if andonly if H is closed. In particular, the quotient of G by the closure of t1u is T1.

From now on, we will assume that all topological groups are T1, unless other-wise specified. For that reason, we will only ever consider quotients by normalsubgroups that are closed.

An important example is given by the following construction.

Definition 1.1.6. If G is a topological group, then the T1-abelianisation is thequotient of G by the closure of the commutator subgroup. It is denoted Gab.

Remark 1.1.7. Note that the T1-abelianisation is T1 and abelian (hence thename). Hence, it is also Hausdorff, so we could also call it the T2-abelianisation.

Lemma 1.1.8. Let G be a topological group. Then the T1-abelianisation Gab

satisfies the following universal property:

For every continuous homomorphism f : GÑ A into an abelian, T1 topologicalgroup A factors uniquely through

G A

Gab

f

π fab

where π : GÑ Gab is the natural projection.

Proof. Because A is abelian, we necessarily have rG,Gs Ď ker f . Because A isT1 and f is continuous, the kernel of f is closed. Hence, we are done by thefundamental theorem of homomorphisms.

Corollary 1.1.9. There is a natural isomorphism

HompG,Aq – HompGab, Aq.

That is, the T1-abelianisation functor TopGp Ñ T1-Ab is left adjoint to theinclusion functor. In particular, it is right exact.

Definition 1.1.10. A topological group G is locally profinite if every neigh-bourhood of the identity contains a compact open subgroup of G.

Lemma 1.1.11. Let G be a locally profinite group. If U Ď G is a neighbourhoodof the identity, then it contains a compact open normal subgroup of G.

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Proof. By the definition of locally profinite groups, we can w.l.o.g. assume thatU is a compact open subgroup. Then all its conjugates xUx´1 are open as well,hence so is the intersection

U 1 “č

xPG

xUx´1.

Clearly, U 1 is an open normal subgroup contained in U . It is compact since anyopen subgroup of the profinite group U is compact.

An important class of locally profinite groups is given by the following lemma.

Lemma 1.1.12. Every profinite group is locally profinite.

Proof. Assume G “ limÐÝ

Gα, where α runs over some directed set A, and all

groups Gα are finite. Let U be a neighbourhood of the identity, and w.l.o.g.assume that U is open. Let V Ď

ś

αPAGα be an open set with V XG “ U .

By the definition of the product topology onś

αPAGα, V contains an openneighbourhood of the identity of the form

ś

αPA Uα, where Uα “ Gα for almostall α P A. Then set V 1 “

ś

αPA U1α, where

U 1α “

"

Gα if Uα “ Gα,t1u if Uα Ĺ Gα.

Clearly, V 1 Ď V , and V 1 is an open (normal) subgroup ofś

αPAGα. Hence, theintersection U 1 “ V 1 XG is an open subgroup contained in U . In particular, U 1

is closed, hence compact because G is.

Lemma 1.1.13. Let G be a locally profinite group, and H a closed subgroup.Then H is locally profinite. If furthermore H is normal, then G{H is locallyprofinite.

Proof. This can be checked easily.

Lemma 1.1.14. Let G be a locally profinite group. Then G is locally compact.Furthermore, G is compact if and only if it is profinite.

Proof. The open neighbourhood G of the identity contains some open compactsubgroup. Hence, G is locally compact.

If G is profinite, then clearly G is compact. Conversely, suppose G is compact.

By Lemma 1.1.11, a basis for the open neighbourhoods of the identity is givenby the open normal subgroups. If we put

G1 “ limÐÝ

UŸGopen

G{U,

then there is a natural homomorphism f : G Ñ G1. It is continuous since thecomposition with any projection G1 Ñ G{U is. Moreover, since G is compactand G1 is Hausdorff, f is a closed map.

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On the other hand, let V Ď G1 be an open set of the form π´1U ptxuq for some

open normal subgroup U Ď G and some x P G{U , where πU : G1 Ñ G{U denotesthe projection. Then choosing some y P G with y ” x mod U gives an elementwith fpyq P V . Since a basis of the topology on G1 is given by the sets V of thisform, this shows that the image of f is dense.

Finally, f is injective: if x P G satisfies fpxq “ 1, then x P U for all opennormal U Ď G. However, G is Hausdorff, so if x ‰ 1, there exists an openneighbourhood of 1 which does not contain x. This is impossible, so x “ 1.

Since f is a closed map with dense image, it is surjective. Since f is a bijectiveclosed (continuous) map, it is a topological isomorphism.

This justifies the definition of locally profinite groups, as we conclude:

Corollary 1.1.15. A topological group G is locally profinite if and only if everyopen neighbourhood of the identity contains an open subgroup of G that is aprofinite group.

The following result generalises a well-known result about profinite groups:

Lemma 1.1.16. A topological group is locally profinite if and only if it is locallycompact and totally disconnected.

We will not prove this, because we do not need it for our purposes.

1.2 Smooth representations

From here on, G will denote a locally profinite group. All group representations(not necessarily finite-dimensional) will be over C. We denote by pρ, V q therepresentation ρ : GÑ AutpV q.

Remark 1.2.1. Recall that a representation pρ, V q is called semisimple if itsatisfies one of the following equivalent conditions:

• ρ is the direct sum of irreducible representations,• every G-stable subspace W has a G-stable complement W 1:

V –W ‘W 1,

• V is the sum of its irreducible G-subspaces.

Definition 1.2.2. A representation pρ, V q of a locally profinite group G is calledsmooth if the map

¨ : Gˆ V Ñ V

pg, vq ÞÑ gv

is continuous, when V is endowed with the discrete topology.

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Lemma 1.2.3. Let pρ, V q be a representation of the locally profinite group G.Then the following are equivalent:

(1) pρ, V q is smooth;(2) the stabiliser of every element v P V is open;(3) V is the union of the sets V U for U an open subgroup;(4) V is the union of the sets V U for U an open compact normal subgroup.

Proof. Clearly (2) and (3) are equivalent. Since by Lemma 1.1.11 every opensubgroup contains a compact open normal subgroup, (2) and (4) are equivalent.

For every v P V , the inverse image under ¨ : Gˆ V Ñ V of tvu is the set

tpg, wq P Gˆ V : gw “ vu “ď

wPGv

tpg, wq P Gˆ V : gw “ vu.

If we choose for every w P Gv an element gw P G such that gww “ v, thenď

wPGv

tpg, wq P Gˆ V : gw “ vu “ď

wPGv

pgw Stabpvqq ˆ twu.

Since V has the discrete topology, the latter is open in G ˆ V if and only ifStabpvq is open in G.

From property (2), it follows that subrepresentations and quotient representa-tions of smooth representations are again smooth.

Remark 1.2.4. The category of abstract (i.e. not necessarily smooth) repre-sentations of G is equivalent to the category of functors G Ñ C-Vec, where Gis viewed as a one-object category whose morphisms are the elements of G. Itis an abelian category, and we denote it by RepapGq.

The full subcategory of smooth representations is also abelian, and we denoteit ReppGq.

If pρ1, V1q and pρ2, V2q are abstract or smooth representations of G, we writeHomGpρ1, ρ2q for the set of G-homomorphisms from pρ1, V1q to pρ2, V2q.

Lemma 1.2.5. If G is profinite, then any smooth representation is semisimple.

Proof. Let pρ, V q be a smooth representation of G. Any vector v P V is fixedby some compact open normal subgroup U Ď G. Then the subrepresentationgenerated by v is an irreducible representation of the finite group G{U , hencefinite-dimensional. Such a representation is semisimple, hence it is the sum ofits irreducible subrepresentations. We repeat the argument for all v P V , andthe result follows.

Definition 1.2.6. A character χ of G is a continuous homomorphism

χ : GÑ Cˆ

with respect to the usual topology on Cˆ.

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Any character χ defines a one-dimensional representation pχ,Cq.

Proposition 1.2.7. Let χ : G Ñ Cˆ be a homomorphism. Then the followingare equivalent:

(1) χ is a character,(2) the representation pχ,Cq is smooth,(3) the kernel of χ is open.

Proof. Since the kerχ is the stabiliser of any nonzero vector, the equivalence of(2) and (3) follows from the lemma above. It is obvious that (3) implies (1).

Now assume that χ is continuous. Let U be an open neighbourhood of theidentity in Cˆ, and choose U such that it contains no non-trivial subgroup ofCˆ. Since χ is continuous, the set χ´1pUq is open in G, hence it contains acompact open subgroup U 1 of G. Then χpU 1q is a subgroup of Cˆ contained inU , hence it is trivial. Hence, U 1 Ď kerχ, and the kernel of χ contains an openneighbourhood of the identity.

Definition 1.2.8. Given an abstract (i.e. not necessarily smooth) representa-tion pρ, V q of G, we can construct a smooth representation pρ8, V 8q as follows.Let

V 8 “ď

UŸGopen

V U .

Remark 1.2.9. If U is any open normal subgroup, then V U is fixed by G, sincefor every g P G:

gV U “ V gUg´1

“ V U .

Therefore, V 8 is a G-invariant subspace of V , and we define

ρ8 : GÑ AutpV 8q

g ÞÑ ρpgqˇ

ˇ

V8.

It is the maximal smooth subrepresentation of V .

Lemma 1.2.10. The functor pρ, V q ÞÑ pρ8, V 8q is right adjoint to the inclusionfunctor from ReppGq into RepapGq.

Proof. Let pρ, V q be a smooth representation of G and pσ,W q an abstract rep-resentation. Let f : V Ñ W be a G-linear map. If U Ÿ G is an open normalsubgroup, then fpV U q ĎWU . Hence,

fpV q “ f

ˆ

ď

UŸGopen

V U˙

“ď

UŸGopen

fpV U q Ďď

UŸGopen

WU “W8.

Therefore, there is a natural isomorphism

HomGpρ, σq – HomGpρ, σ8q.

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Corollary 1.2.11. In particular, the functor pρ, V q ÞÑ pρ8, V 8q is left exact.

Remark 1.2.12. Abstract representations of G correspond to modules over thegroup algebra CrGs. There is not really a nice way to define smoothness directlyin terms of the group algebra, but one can study ‘smooth modules’ over anotheralgebra, called the Hecke algebra. For details, see [3].

1.3 Induced representations

We write pλG,CrGsq for the regular representation of G. The group homo-morphism λG : G Ñ AutpCrGsq maps an element g P G to the automorphismdefined by g1 ÞÑ gg1. Then λG is given by left translation, i.e.

λGpgq

˜

nÿ

i“1

aigi

¸

“

nÿ

i“1

aiggi.

Similarly, pρG,CrGsq denotes the representation induced by right translation:

ρGpgq

˜

nÿ

i“1

aigi

¸

“

nÿ

i“1

aigig´1.

It corresponds to CrGs as a right module over itself, viewed as a left module byinverting the action.

Definition 1.3.1. Let G be a locally profinite group, and let H be a closedsubgroup. Let pσ,W q be an abstract representation of H. Then define a repre-sentation of G on HomHpλG, σq by

pgfq

˜

nÿ

i“1

aigi

¸

:“ f

˜

nÿ

i“1

aigig

¸

,

for g P G and f P HomHpλG, σq. It is called the abstract induced representation,and is denoted a-IndGHpσq.

Just like in the case for finite groups, it satisfies the following property.

Lemma 1.3.2 (Abstract Frobenius Reciprocity). Let pρ, V q and pσ,W q be ab-stract representations of G and H respectively. There is a natural isomorphism

HomG

´

ρ, a-IndGHpσq¯

– HomH

`

ρˇ

ˇ

H, σ˘

.

Proof. The unit for the adjunction is given by

ηρ : ρÑ a-IndGH`

ρˇ

ˇ

H

˘

v ÞÑ

˜˜

nÿ

i“1

aigi

¸

ÞÑ

˜

nÿ

i“1

aiρpgiqv

¸¸

,

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for all v P V . Observe that ηρpvq is indeed an H-linear map λG Ñ ρ|H , andthat ηρ is G-linear.

The counit for the adjunction is given by

εσ : a-IndGHpσqˇ

ˇ

ˇ

HÑ σ

f ÞÑ fp1q,

for all f P HomHpλG, σq. This map is H-linear.

One checks easily that the maps

HomG

´

ρ, a-IndGHpσq¯

ÐÑ HomH

`

ρˇ

ˇ

H, σ˘

φ ÞÝÑ εσ ˝ φˇ

ˇ

H

a-IndGHpfq ˝ ηρ ÐÝß f

are each others inverses.

Corollary 1.3.3. We have a composite adjunction

ReppGq Õ RepapGq Õ RepapHq.

In particular, if pρ, V q is a smooth representation of G and pσ,W q an abstractrepresentation of H, then

HomG

´

ρ, a-IndGHpσq8¯

– HomH

`

ρˇ

ˇ

H, σ˘

.

Definition 1.3.4. If pσ,W q is a smooth representation of H, then the smoothlyinduced representation IndGHpσq is the smooth representation a-IndGHpσq

8 of G.

Because ReppHq is a full subcategory of RepapHq and the restriction to Hof a smooth representation of G is smooth, the above corollary also gives thefollowing result.

Corollary 1.3.5 (Smooth Frobenius Reciprocity). Let pρ, V q and pσ,W q besmooth representations of G and H respectively. There is a natural isomorphism

HomG

´

ρ, IndGHpσq¯

– HomH

`

ρˇ

ˇ

H, σ˘

.

Remark 1.3.6. Because CrGs is a free CrHs-module, in particular it is projec-tive, so the functor a-IndGH is exact. By abstract nonsense, it is also clear thatIndGH is left exact (being a right adjoint functor). One can in fact show that itis exact, cf. Proposition 2.4 of[3].

A very similar construction is given by the following definition.

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Definition 1.3.7. Let U Ď G be an open subgroup, and let pσ,W q be anabstract representation of U . Then define the compactly induced representationc-IndGU pσq of σ as the representation of G on CrGs bCrUsW by

gpxb wq “ gxb w.

Here, CrGs is viewed as a right CrU s-module in the obvious way, so that thetensor product makes sense.

In other words, it is obtained by extension of scalars from CrU s to CrGs. Inparticular, we get the following two results for free.

Lemma 1.3.8. The functor c-IndGU is exact.

Proof. This is because CrGs is a free CrU s-module, hence in particular flat.

Lemma 1.3.9 (Abstract Frobenius Reciprocity for Compact Induction). Letpρ, V q and pσ,W q be abstract representations of G and U respectively. Thenthere is a natural isomorphism

HomG

´

c-IndGU pσq, ρ¯

– HomU

`

σ, ρˇ

ˇ

U

˘

.

Proof. This is immediate from the adjoint property of extension of scalars.

It turns out that compact induction preserves smoothness:

Lemma 1.3.10. Let pσ,W q be a smooth representation of U . Then c-IndGU pσqis smooth as well.

Proof. Let a1, . . . , an P C, g1, . . . , gn P G and w1, . . . , wn P W be given. Theneach wi is fixed by some open subgroup Ui of U . Hence, the element

nÿ

i“1

paigi b wiq

is fixed by the intersection of the Ui, which is an open subgroup of U , hencealso of G (since U is open in G).

Corollary 1.3.11 (Smooth Frobenius Reciprocity for Compact Induction). Letpρ, V q and pσ,W q be smooth representations of G and U respectively. Then thereis a natural isomorphism

HomG

´

c-IndGU pσq, ρ¯

– HomU

`

σ, ρˇ

ˇ

U

˘

.

Remark 1.3.12. Note that compact induction is left adjoint to restriction,while smooth induction is right adjoint to restriction.

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Remark 1.3.13. We can identify HomU pλG, σq8 with the set of maps f : GÑ

W satisfyingfphgq “ σphqfpgq, h P U, g P G,

and we can identify CrGsbCrUsW with the subset consisting of maps with finitesupport modulo U . As such, we get a natural G-linear injection

c-IndGU pσq ÝÑ IndGU pσq

mapping an element of the form g bw to the U -linear map CrGs ÑW definedby g ÞÑ w. It is an isomorphism if and only if rG : U s is finite.

In [3] as well as in [14], this is used as the definition of Ind and c-Ind. Also, thedefinition of c-Ind can then be extended to closed subgroups H, by replacingfinite support by compact support in Remark 1.3.13 above. Note that if H isopen, then HzG is discrete, so a subset of HzG is compact if and only if it isfinite.

1.4 Dual representations

Definition 1.4.1. Let pρ, V q be an abstract representation of G. Then theabstract dual representation pρ˚, V ˚q is the representation of G on the dualspace

V ˚ :“ HomCpV,Cq

given by

ρ˚pgqf :“

ˆ

x ÞÑ fpρpg´1qxq

˙

.

One easily checks that this is indeed a representation of G.

Definition 1.4.2. Let pρ, V q and pσ,W q be abstract representations of G, andlet φ : ρÑ σ be a G-linear map. Then define the map

φ˚ : ρ˚ Ñ σ˚

f ÞÑ f ˝ φ.

Then φ˚ is a G-linear map. The construction pρ, V q ÞÑ pρ˚, V ˚q becomes afunctor D : RepapGq

op Ñ RepapGq.

Lemma 1.4.3. Let pρ, V q be an abstract representation of G. There is a naturalG-linear map

evρ : ρ ÝÑ ρ˚˚

v ÞÝÑ pf ÞÑ pfpvqqq.

It is injective, and it is an isomorphism if and only if V is finite-dimensional.

Proof. The map is clearly C-linear. If g P G and v P V are given, then for allf P HomCpV,Cq we have

evρpρpgqvqpfq “ fpρpgqvq “ pρ˚pg´1qfqpvq “ evρpvqpρ˚pg´1qfq,

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so evρpρpgqvq is given by

f ÞÑ evρpvqpρ˚pg´1qfq,

which is the definition of ρ˚˚pgq evρpvq. Hence, evρ is G-linear.

The final statement is a well-known result from linear algebra.

Lemma 1.4.4. Let pρ, V q and pσ,W q be abstract representations of G. Thenthere is a natural isomorphism

HomGpρ, σ˚q – HomGpσ, ρ

˚q.

Proof. Define maps

HomGpρ, σ˚q ÐÑ HomGpσ, ρ

˚q

φ ÞÝÑ φ˚ ˝ evσ

ψ˚ ˝ evρ ÐÝß ψ.

We can identify any φ P HomGpρ, σ˚q with the C-bilinear map

fφ : V ˆW Ñ Cpv, wq ÞÑ φpvqpwq.

Similarly, any ψ P HomGpσ, ρ˚q can be identified with the C-bilinear map

gψ : W ˆ V Ñ Cpw, vq ÞÑ ψpwqpvq.

Under this correspondence, the first map is given by

fφ ÞÑ

ˆ

pw, vq ÞÑ fφpv, wq

˙

,

and the second by

gψ ÞÑ

ˆ

pv, wq ÞÑ gψpw, vq

˙

,

and it is clear that the maps are each others inverses.

Remark 1.4.5. The lemma shows that the functor D : RepapGqop Ñ RepapGq

is right adjoint to the functor Dop : RepapGq Ñ RepapGqop.

Definition 1.4.6. Let pρ, V q be a smooth representation of G. Then define thesmooth dual representation pρ, V q as ppρ˚q8, pV ˚q8q.

Corollary 1.4.7. Let pρ, V q and pσ,W q be smooth representations of G. Thenthere is a natural isomorphism

HomGpρ, σq – HomGpσ, ρq.

Proof. This follows from the lemma, together with the adjunction property ofthe functor pτ,Xq ÞÑ pτ8, X8q.

20

Example 1.4.8. Let χ be a character of G. Then pχ,Cq is a one-dimensionalrepresentation, and we can identify the dual space C˚ with C via f ÞÑ fp1q.Then χ˚ is given by the homomorphism

χ˚ : GÑ Cg ÞÑ χpg´1q.

This is obviously a character as well, so χ˚ is already smooth. Hence, χ “ χ˚

is given by g ÞÑ χpg´1q.

Lemma 1.4.9. Let pρ, V q be a smooth representation of G, and let U Ď G be acompact open subgroup. Then the map V U Ñ pV U q˚ given by f ÞÑ f |V U is anisomorphism.

Proof. Note that V U is the set of functions f : V Ñ C that satisfy

fpρph´1qvq “ fpvq

for all v P V . That is, f is U -linear, when we equip C with trivial U -action, so

V U “ HomU pρ|U , 1U q.

Also, any C-linear map pV U q˚ Ñ C is automatically U -linear, since the actionon both sides is trivial. Therefore,

pV U q˚ “ HomU pVU , 1U q.

The representation pρ|U , V q of U is semisimple since U is profinite (Lemma1.2.5). Hence, it composes as a direct sum of irreducible U -representations.

Therefore any U -homomorphism f : ρ|U Ñ 1U is trivial on the irreducible com-ponents that are not isomorphic to 1U . The components that are isomorphic to1U are exactly the ones contained in V U , and the result follows.

Remark 1.4.10. From now on, we will identify pV U q˚ with the subset V U ofV .

Corollary 1.4.11. We can identify V withď

UŸGopen

compact

`

V U˘˚.

Proof. This is clear from the definition of V .

Lemma 1.4.12. Let pρ, V q, pσ,W q and pτ,Xq be smooth representations of G,and let φ : ρÑ σ and ψ : σ Ñ τ be G-linear maps. Then the sequence

VφÝÑW

ψÝÑ X

is exact if and only if for every compact open subgroup U Ď G the sequence

V UφÝÑWU ψ

Ñ XU

is exact.

21

Proof. If U Ď G is a compact open subgroup, then every smooth representationof U is the direct product of some family of irreducible representations. Fur-thermore, V U is the direct sum over the subfamily of trivial representations.Therefore,

φ´1pWU q Ď V U

andφpV q XWU “ φpV U q.

If the sequence V ÑW Ñ X is exact, therefore so is the sequence

V U ÑWU Ñ XU .

The other implication is trivial, as W is the union of the sets WU for U Ď Gopen compact.

Definition 1.4.13. A smooth representation pρ, V q of G is called admissible iffor every compact open subgroup U Ď G the space V U has finite dimension.

Proposition 1.4.14. Let pρ, V q be a smooth representation of G. Then theimage of the G-linear map

ev8ρ : ρÑ pρq˚

v ÞÑ

ˆ

f ÞÑ fpvq

˙

is inside ˇρ. Furthermore, the map ev8ρ is injective, and it is an isomorphism ifand only if ρ is admissible.

Proof. The first assertion follows from the universal property of p´q8 (cf. theproof of Lemma 1.2.10). By Corollary 1.4.11, we have

ˇV “ď

UŸGopen

compact

`

V U˘˚˚

.

The map

ev8ρ :ď

UŸGopen

compact

V U Ñď

UŸGopen

compact

`

V U˘˚˚

is the obvious one, so we are done by Lemma 1.4.12.

Proposition 1.4.15. The functor ReppGqop Ñ ReppGq given by pρ, V q ÞÑ

pρ, V q is exact.

Proof. Let 0 Ñ V ÑW Ñ X Ñ 0 be a short exact sequence of representationsof G. Then for each compact open subgroup U Ď G, the sequence

0 Ñ V U ÑWU Ñ XU Ñ 0

is exact (Lemma 1.4.12). By linear algebra, also the sequence

0 Ñ`

XU˘˚Ñ

`

WU˘˚Ñ

`

V U˘˚Ñ 0

is exact. The result then follows from Lemma 1.4.12 and Corollary 1.4.11.

22

Corollary 1.4.16. An admissible representation pρ, V q of G is irreducible ifand only if pρ, V q is irreducible.

23

2 Tate’s Thesis

The reader is not assumed to be familiar with measure theory or Fourier analysis.In fact, the structure of locally profinite groups is so strong that the theory isalmost entirely algebraic, as we will see.

The treatment of this chapter is influenced by sections 3 and 23 of [3], as wellas by Tate’s thesis [16].

2.1 Haar measures

We will focus more on the Haar integral than on the Haar measure, becausewe do not want to worry about the intricacies of measure theory. (Even thedefinition of a measure is quite involved.)

Definition 2.1.1. Let G be a locally profinite group, and let f : G Ñ C be afunction. Then f is smooth if it is locally constant.

We denote by C8pGq the C-vector space of smooth functions f : G Ñ C. Thesubspace of functions that have compact support is denoted C8c pGq.

Definition 2.1.2. A smooth function f : GÑ C is called positive if fpgq P Rě0

for all g P G. We write f ľ 0.

Example 2.1.3. Let U Ď G be any compact open subset. Then the indicatorfunction IU is the function

IU pgq “"

1 if g P U,0 else.

It is locally constant since U is both open and closed, and it has compact supportsince U is compact.

Lemma 2.1.4. The space C8c pGq is spanned by the functions IgU for g P G andU Ď G a compact open subgroup.

Proof. Let f P C8c pGq. Since f is locally constant, for each g P G there existsa compact open subgroup Ug Ď G such that fpgUgq “ fpgq. Let S “ tg P G :fpgq ‰ 0u be the support of f . The open sets gUg (for g P S) cover the compactset S, so there is a finite subset g1, . . . , gn of S such that

S “nď

i“1

giUgi .

Set U to be the intersection of the Ugi and Ug for some g R S (if such an elementexists).

Then fpghq “ fpgq for all g P G and all h P U , and f is essentially a mapG{U Ñ C with compact support. Since the topology on G{U is discrete, f infact has finite support modulo U , and the result follows.

24

Remark 2.1.5. Similarly, one can show that the space C8c pGq is spanned bythe functions IUg for g P G and U Ď G a compact open subgroup.

We have a representation pρG, C8c pGqq of G by translation on the right:

pρGpgqfqpxq :“ fpxgq

for g P G, f P C8c pGq and x P G. The proof of the lemma shows that thisrepresentation is smooth.

Definition 2.1.6. A right Haar integral on G is a G-linear map I : ρG Ñ 1Gsuch that Ipfq P Rě0 for all f ľ 0.

Proposition 2.1.7. There exists a right Haar integral I : ρG Ñ 1G. Moreover,if I 1 is another right Haar integral, then there exists a constant c P Rą0 suchthat I “ cI 1.

Proof. For every compact open subgroup, the functions tIUg : g P UzGu span a

G-subspace VU of C8c pGq. It is isomorphic to c-IndGU p1U q by Remark 1.3.13, soby Frobenius reciprocity we have

dim HomGpVU , 1Gq “ dim HomU p1U , 1U q “ 1. (1)

On the other hand, Remark 2.1.5 asserts that

C8c pGq “ď

UĎGopen

compact

VU . (2)

Now fix some compact open subgroup U0. Then define I : C8c pGq Ñ C as

IUg ÞÑrU : U X U0s

rU0 : U X U0s.

This map clearly has the desired properties, and it is unique up to scaling by apositive constant because of (1) and (2).

We also have a representation pλG, C8pGqq of G by translation on the left:

pλGpgqfqpxq :“ fpg´1xq

for g P G, f P C8pGq and x P G. Once again, C8c pGq is a G-subspace of C8pGq.

Definition 2.1.8. A left Haar integral on G is a G-linear map I : λG Ñ 1Gsuch that Ipfq P Rě0 for all f ľ 0.

Definition 2.1.9. Denote the underlying set of the group G by X. Endow theset X with the group structure Gop as the unique group structure making thebijection

GÑ Gop

g ÞÑ g´1

25

an isomorphism. That is, in Gop, the multiplication map ˚ is given by

pg1, g2q ÞÑ g1 ˚ g2 :“ g2g1.

Giving Gop the topology of G makes the isomorphism G Ñ Gop a topologicalone, since inversion onG is a homeomorphism. In particular, Gop is a topologicalgroup.

Remark 2.1.10. We now have a representation pρGop , C8c pXqq given by

pρGoppgqfqpxq :“ fpx ˚ gq “ fpgxq.

Under the topological isomorphism G – Gop, this representation correspondsto the representation pλG, C

8c pXqq.

We use the set X in the notation here to emphasise that the underlying vectorspaces of ρGop and λG are the same. The notation C8c pG

opq would have beenambiguous, since it is not clear whether we identify C8c pGq with C8c pG

opq viag ÞÑ g or via g ÞÑ g´1. We use the former.

Proposition 2.1.11. There exists a left Haar integral I : λG Ñ 1G. Moreover,if I 1 is another left Haar integral, then there exists a constant c P Rą0 such thatI “ cI 1.

Proof. A left Haar integral on G corresponds to a right Haar integral on Gop.Hence, the result follows from Proposition 2.1.7.

Remark 2.1.12. By construction (see the proof of Proposition 2.1.7), there issome compact open subgroup U1 with IpIU1q “ 1 for some left Haar integral I.If U Ď G is any compact open subgroup, then U 1 :“ U X U1 is also a compactopen subgroup, and

1 “ IpIU1q “

ÿ

hPU1{U 1

IpIhU 1q “ rU1 : U 1sIpIU 1q.

Hence, IpIU 1q “ 1rU1:U 1s , and a similar argument shows that

IpIU q “rU : U 1s

rU1 : U 1s.

In particular, we conclude that

IpIU q ą 0,

for any left Haar integral I on G.

Definition 2.1.13. Let I be a left Haar integral. Then the induced left Haarmeasure is the map

µG : tU Ď G : U compact openu Ñ Rě0

U ÞÑ IpIU q.

26

One can show that this ‘is’ a measure, but we will not even state the definitionof a measure. From now on, we will fix some left Haar measure µG. In spe-cific examples, we might force a condition on µG to hold, defining the measurecompletely.

If f P C8c pGq is a smooth function of compact support, then we denote Ipfq by

ż

G

fpgq dµGpgq.

Sometimes the variable g is dropped, so the notation becomesż

G

f dµG.

If U Ď G is a compact open subset, then we will writeż

U

fpgq dµGpgq :“

ż

G

IUfpgq dµGpgq.

Now if g P G is given, then the map

C8c pGq Ñ C

f ÞÑ

ż

G

fpxgq dµGpxq

is a left Haar integral, hence there exists an element δGpgq P Rą0 such that

δGpgq

ż

G

fpxgq dµGpxq “

ż

G

fpxq dµGpxq.

The last expression can also be given asż

G

fpxgq dµGpxgq,

so dµGpxgq “ δGpgq dµGpxq.

Definition 2.1.14. The modulus of G is the map GÑ Cˆ given by g ÞÑ δGpgq.

Remark 2.1.15. If U Ď G is a compact open subgroup, then the indicatorfunction IU is invariant under left translation by U . The integral

ż

G

IU pxq dµGpxq

is nonzero by Remark 2.1.12, which shows that δGphq “ 1 for all h P U .

Lemma 2.1.16. The modulus of G is a character that is independent of thechoice of Haar measure.

Proof. It is clearly a homomorphism. If U Ď G is any compact open subgroup,then U Ď ker δG. Hence, the kernel of δG is open, so δG is a character byProposition 1.2.7. The modulus clearly does not depend on µG.

27

Remark 2.1.17. If f P C8c pGq is a smooth function of compact support, thenδ´1G f is smooth and of compact support as well. One can easily check that

f ÞÑ

ż

G

δ´1G f dµG

is a right Haar integral.

Definition 2.1.18. A locally profinite group G is called unimodular if any leftHaar integral is also a right Haar integral.

Lemma 2.1.19. A locally profinite group G is unimodular if and only if δG “ 1.

Proof. Clear from the previous remark and the definition of the modulus.

Example 2.1.20. If G is compact (i.e. profinite), then δG is trivial on anycompact open subgroup U Ď G. In particular, δG is trivial on G, so G isunimodular.

Lemma 2.1.21. Let χ be a character of G. Let U be a compact open subgroupof G. Then

ż

U

χ dµG “

"

µGpUq if χˇ

ˇ

U“ 1U ,

0 else.

Proof. Note that χ is a smooth function by our assumptions on characters.Furthermore, U is compact, so the integral exists.

Since χ is a character, there is a normal compact open subgroup U 1 Ď G onwhich χ is the trivial character. The quotient G1 “ U{pU XU 1q is a finite group,and there is a unique character ρ on G1 such that

χˇ

ˇ

U“ ρ ˝ π,

where π : U Ñ G1 is the canonical projection. Then the integral is a sum

ÿ

gPG1

µGpU X U1qρpgq,

and the result follows from Schur orthogonality for finite groups.

2.2 The additive group of a local field

In this section and the next, we will cover a simple version of what is basically(the local part of) Tate’s Thesis. The original thesis can be found in [16], butwe will follow the treatment of [3].

Observe that K is a locally profinite group. A basis for the open neighbourhoodsof the zero element is given by the sets pn for n P Z. Furthermore, K is theunion of its compact open subgroups, since these are just the pn.

28

Lemma 2.2.1. Let µK be a left Haar measure on K. Let a P Kˆ. Then

dµKpaxq “ }a} dµKpxq.

Proof. Let n P Z be given. Then

ż

K

Ipnpxq dµKpaxq “ż

K

Ipnpa´1xq dµKpxq “

ż

K

Iapnpxq dµKpxq.

The latter equals µKppn`vKpaqq, which is q´vKpaq times the size of µppnq. Since

the Haar integral is translation invariant, this gives

ż

K

Ib`pnpxq dµKpaxq “ }a}

ż

K

Ib`pn dµKpxq

for all b P K, n P Z. The functions Ib`pn span C8c pKq by Lemma 2.1.4.

Definition 2.2.2. Let K be a local field. Then the character group of K is thegroup (under multiplication) of characters ψ : K Ñ Cˆ. It is denoted pK.

If ψ P pK is a nontrivial character, then the kernel of ψ is open, so it containsthe set pn for some n P Z.

Definition 2.2.3. Let ψ P pK be a nontrivial character. The least integer n P Zsuch that pn Ď kerψ is called the level of ψ.

Definition 2.2.4. If ψ is a character, then so is the map x ÞÑ ψpaxq for anya P K. It is denoted aψ.

Remark 2.2.5. If ψ is a nontrivial character of level n, and a P Kˆ is a nonzeroelement, then the character aψ has level n´ vKpaq.

Proposition 2.2.6. Let ψ P pK be a nontrivial character. Then the map

fψ : K Ñ pK

a ÞÑ aψ

is an isomorphism.

Proof. It is clearly a homomorphism. If a P K is such that fψpaq “ 1, then inparticular ψpaxq “ 1 for all x P K. But that forces a “ 0, as otherwise ψ “ 1.

Now let ϕ be any nontrivial character ofK, say of levelm. Let π be a uniformiserof K, and let n be the level of ψ. Then the character πn´mψ is trivial on pm,as is ϕ.

The characters on pm´1 that are trivial on pm correspond bijectively to thecharacters on pm´1{pm – k. The q ´ 1 characters uπn´mψ for u P UK{U

1K are

all distinct, so by a counting argument they exhaust all characters on pm´1 thatare trivial on p. Hence, there is some u1 P UK such that u1π

n´mψ is identicalto ϕ on pm´1.

29

Proceeding inductively, we get a sequence u1, u2, . . . of elements in UK such thatuiπ

m´nψ and ϕ agree on pm´i and such that ui ” uj mod pj for i ą j. It isclear that the limit u of this sequence satisfies uπm´nψ “ ϕ.

We now fix not only a left Haar measure dµ on K (we drop the subscript Kfrom the notation), but also a nontrivial character ψ of K of level m.

Remark 2.2.7. If Φ P C8c pKq, then for any ξ P K the function Φpxqψpξxq islocally constant since the kernel of ψ is open. Also, the function Φpxqψpξxq hascompact support, since Φ has compact support.

Definition 2.2.8. Given Φ P C8c pKq. Then the Fourier transform (with re-spect to dµ and ψ) of Φ is the function K Ñ C defined by

Φpξq “

ż

K

Φpxqψpξxq dµpxq.

The integral is well defined by Remark 2.2.7.

Lemma 2.2.9. Let Φ P C8c pKq, and let a P K. Write Φ1pxq “ Φpx´ aq. Then

Φ1pξq “ aψpξqΦpξq.

Proof. We compute

Φ1pξq “

ż

K

Φpx´ aqψpξxq dµpxq

“

ż

K

Φpxqψpξpx` aqq dµpxq “ aψpξqΦpξq.

Lemma 2.2.10. Let Φ P C8c pKq, and let a P K. Let Φ1 “ aψ ¨ Φ. ThenΦ1 P C8c pKq, and

Φ1pξq “ Φpξ ` aq.

Proof. The first assertion follows from Remark 2.2.7. We compute

Φ1pξq “

ż

K

aψpxqΦpxqψpξxq dµpxq

“

ż

K

Φpxqψppξ ` aqxq dµpxq “ Φpξ ` aq.

Lemma 2.2.11. Let n P Z be given. Then

Ipn “ µppnqIpm´n .

Proof. Let ξ P K be given. We have

ż

G

Ipnpxqψpξxq dµpxq “ż

pnψpξxq dµpxq.

When ξ P pm´n, the character ξψ is trivial on pn, and the result follows.

30

Hence assume ξ R pm´n, and set i “ vKpξq. Then i ă m´ n, andż

pnψpξxq dµpxq “

ż

pn`iψpxq dµpxq.

This is zero by Lemma 2.1.21, since ψ|pn`i is a nontrivial character on pn`i.

Proposition 2.2.12. For Φ P C8c pKq, we have Φ P C8c pKq. Furthermore,there exists a number c P Rą0 (depending on dµ and ψ) such that

ˆΦpxq “ cΦp´xq

for all Φ P C8c pKq, x P K.

Proof. We take c “ µpOKq2q´m. Firstly, let Φ “ Ipn for some n P Z. Then

Lemma 2.2.11 shows that Φ P C8c pKq, and we compute

ˆΦ “ µppnqIpm´n “ µppnqµppm´nqIpm .

Because µppnq “ q´nµpOKq and µppm´nq “ qn´mµpOKq, the result holds forΦ “ Ipm .

Now consider the function Φ1 “ Ia`pn for a P K, n P Z. By Lemma 2.2.9, we

have Φ1 “ aψ ¨ Φ, where Φ “ Ipn . Then Lemma 2.2.10 asserts that Φ1 P C8c pKq,and that

ˆΦ1pξq “

ˆΦpξ ` aq.

Therefore, the result also holds for the function Ia`pn . Since these functionsspan C8c pKq by 2.1.4, we are done.

Remark 2.2.13. From now on, we will w.l.o.g. assume that µpOKq “ qm2 . This

determines the Haar measure completely, and this measure is called self-dual.

Theorem 2.2.14 (Fourier inversion formula). Let Φ P C8c pKq. Then

ˆΦpxq “ Φp´xq,

for all x P K.

2.3 The multiplicative group of a local field

Firstly, note that Kˆ is a locally profinite group in the topology induced fromK. This implies in particular that the set C8c pK

ˆq can be identified with thesubset of C8c pKq consisting of functions that vanish at the origin.

Definition 2.3.1. Let χ be a character of Kˆ. Then χ is called unramified ifits restriction to UK is trivial. Otherwise, χ is called ramified. This terminologywill be explained in Remark 3.2.3 and Remark 3.2.4.

We will fix a character χ of Kˆ and a Haar measure µˆ on Kˆ.

31

Remark 2.3.2. Let π P p be a uniformiser. Then the set πnUK “ pnzpn`1

does not depend on π. It will be denoted Sn, and it is a compact open subsetof Kˆ. It satisfies

ISn “ Ipn ´ Ipn`1 .

In particular, if Φ P C8c pKq is a smooth function of compact support, then ISnΦis also smooth and of compact support. Since it vanishes at 0, we have

ISnΦ P C8c pKˆq.

Since Φ has compact support, there is some m P Z such that Φ vanishes outsidepm. Hence, for all j ă m, it holds that

ISjΦ “ 0.

Definition 2.3.3. If Φ P C8c pKq and a P Kˆ are given, then we write aΦ forthe function x ÞÑ Φpa´1xq.

In this notation, it follows that ISn “ πnIUK . More generally, if U Ď K is anycompact open set, and Φ is the indicator function of U , then aΦ is the indicatorfunction of aU .

Lemma 2.3.4. The space C8c pKq is spanned by C8c pKˆq and IOK .

Proof. This can be proven using explicit knowledge about C8c pKq and C8c pKˆq,

cf. Lemma 2.1.4. However, a much easier argument goes as follows:

We have identified C8c pKˆq with the subset of C8c pKq of functions that vanish

at 0. That is, we have a short exact sequence of C-vector spaces

0 Ñ C8c pKˆq Ñ C8c pKq

φÝÑ CÑ 0,

where φ is given by Φ ÞÑ Φp0q. Picking any function Φ P C8c pKq that does notvanish at 0 induces a splitting, and IOK is such a function.

Definition 2.3.5. Let Φ P C8c pKq. Then we define the formal Laurent series

ZpΦ, χ,Xq “ÿ

nPZznpΦ, χq X

n P CppXqq,

where

znpΦ, χq “

ż

Sn

Φpxqχpxq dµˆpxq.

Note that Φ ¨ χ is a smooth function with compact support, so the integral isdefined. Note also that znpΦ, χq “ 0 for n sufficiently small, so the series isindeed a Laurent series.

Definition 2.3.6. Let Φ P C8c pKq. Then the zeta function of Φ (with respectto χ) is the function

ζpΦ, χ, sq “ ZpΦ, χ, q´sq.

For now, the zeta function is only a formal Laurent series in q´s. However, wewill show that it is actually a rational function in q´s, so it defines a bona fidemeromorphic function on C in the variable s.

32

Remark 2.3.7. It is clear that the function

Z : C8c pKq Ñ CppXqqΦ ÞÑ ZpΦ, χ,Xq

is C-linear.

Lemma 2.3.8. Let Φ P C8c pKq and a P Kˆ. Then

ZpaΦ, χ,Xq “ χpaqXvKpaqZpΦ, χ,Xq.

Proof. Write m “ vKpaq. For any n P Z, we compute

znpaΦ, χq “

ż

Sn

Φpa´1xqχpxq dµˆpxq

“

ż

Sn´m

Φpyqχpayq dµˆpayq

“ χpaq

ż

Sn´m

Φpyqχpyq dµˆpyq “ χpaqzn´mpΦ, χq,

and the result follows.

Lemma 2.3.9. The image of C8c pKˆq under Z is CrX,X´1s.

Proof. Let Φ P C8c pKˆq. Then Φ vanishes at 0, hence also in some open

neighbourhood pn of 0. Therefore, the integral

ż

Sm

Φ ¨ χ dµˆ

vanishes for m ě n, so ZpΦ, χ,Xq is a Laurent polynomial in X.

Since χ is a character, it has an open kernel; say that U iK Ď kerχ. Let Φ0 bethe indicator function of U iK . Note that

U iK X Sm “

"

U iK m “ 0,∅ m ‰ 0.

Hence, using that χ is trivial on U iK , we get

ż

Sm

Φ0 ¨ χ dµˆ “

ż

UiKXSm

χ dµˆ “

"

µˆpU iKq m “ 0,0 m ‰ 0.

It follows that ZpΦ0, χ,Xq is a nonzero constant Laurent polynomial. We aredone by Lemma 2.3.8.

Lemma 2.3.10. Let Φ be the indicator function of OK . Then

1

µˆpUKqZpΦ, χ,Xq “

"

p1´ χpπqXq´1 if χ is unramified,0 if χ is ramified.

33

Proof. For n ă 0, we have Sn XOK “ ∅, so

znpΦ, χq “

ż

Sn

Φ ¨ χ dµˆ “ 0.

For n ě 0, we have Sn Ď OK , so

znpΦ, χq “

ż

Sn

Φ ¨ χ dµˆ “

ż

Sn

χpxq dµˆpxq

“

ż

S0

χpπnyq dµˆpπnyq “ χpπqnż

UK

χpyq dµˆpyq.

Hence,

ZpΦ, χ,Xq “

ˆż

UK

χ dµˆ˙ 8

ÿ

n“0

pχpπqXqn

“

ˆż

UK

χ dµˆ˙

1

1´ χpπqX.

The integral is µˆpUKq if χ is unramified and 0 if χ is ramified.

Corollary 2.3.11. Let Φ P C8c pKq. There exists a constant s0 P R such thatthe integral

ż

KˆΦpxqχpxq }x}s dµˆpxq

converges absolutely and uniformly on compact subsets of the right half planetRe s ą s0u. On this right half plane, the integral is equal to ζpΦ, χ, sq.

Proof. The key observation is that

ż

KˆΦpxqχpxq }x}s dµˆpxq “

ÿ

nPZq´ns

ż

Sn

Φpxqχpxq dµˆpxq,

which is equal to ZpΦ, χ, q´sq as a formal Laurent series in q´s. For all functionsΦ P C8c pK

ˆq the result follows since the sum is finite.

For the function IOK , we take s0 “lnpχpπqq

ln q . Then the series

8ÿ

n“0

pχpπqq´sqn

converges absolutely and uniformly on compact subsets of tRe s ą s0u, and theresult follows from the computations in Lemma 2.3.10.

The vector space C8c pKq is spanned by C8c pKˆq and IOK , so we are done.

Remark 2.3.12. Some authors use the integral above as the definition of thezeta function. In Chapter 4, we will do something similar for functions onMnpKq with respect to a representation of GLnpKq.

34

Proposition 2.3.13. The image of the map Z is equal to PχpXq´1CrX,X´1s,

where

PχpXq “

"

1´ χpπqX if χ is unramified,1 if χ is ramified.

Proof. If χ is ramified, then IOK is mapped to 0 under Z. We are then done byLemma 2.3.4 and Lemma 2.3.9.

If χ is unramified, then the same two lemmata imply that the image of Z isthe linear subspace of CppXqq spanned by CrX,X´1s and p1´ χpπqXq´1. Theresult follows by linear algebra. Alternatively, one can use Lemma 2.3.8 toconclude.

Definition 2.3.14. Define the L-function associated to the character χ as

Lpχ, sq “ Pχpq´sq´1.

2.4 Epsilon factors

Throughout this section, we will fix a nontrivial character χ of Kˆ and aninteger m P Zą0 such that UmK Ď kerχ. We will also fix a nontrivial character ψof K, and we let µ be the self-dual Haar measure of K, cf. Remark 2.2.13. Wewill implicitly fix a Haar measure µˆ of Kˆ, just like in the previous section.

Proposition 2.4.1. Denote by Λ the CpXq-vector space of C-linear functionsλ : C8c pKq Ñ CpXq satisfying

λpaΦq “ χpaqXvKpaqλpΦq,

for all a P Kˆ and all Φ P C8c pKq. Then Λ has dimension 1 over CpXq.

Proof. We will show that any λ P Λ is uniquely determined by λpIUmK q. To beprecise, consider the CpXq-linear map

evIUmK

: Λ Ñ CpXq

λ ÞÑ λpIUmK q.

We will show that it is injective. Thus, let λ P Λ be given such that λpIUmK q “ 0.

If a P UmK , then χpaq “ 1 and vKpaq “ 1. Therefore, the condition on λ forces

λpIaUnK q “ λpIUnK q

whenever n ě m. Hence, for n ě m, we have

λpIUmK q “ÿ

aPUmK {UnK

λpIaUnK q “ qn´m λpIaUnK q.

Therefore, λpIUnK q “ 0 for all n ě m. It follows that λpIaUnK q “ 0 for all a P Kˆ

and all n ě m.

35

All functions of the form IbUiK and all functions of the form IbUK for b P Kˆ

and i P Zą0 are finite linear combination of the functions IaUnK as above. Theyspan C8c pK

ˆq by Lemma 2.1.4, so λ vanishes on C8c pKˆq.

Hence, λ factors through the map φ : C8c pKq Ñ C of the proof of Lemma 2.3.4.That is, the value of λpΦq only depends on Φp0q. But for any a P Kˆ it holdsthat paΦqp0q “ Φp0q, so the condition on λ gives

λpΦq “ λpaΦq “ χpaqXvKpaqλpΦq,

for all a P Kˆ. This immediately forces λpΦq “ 0 for all Φ P C8c pKq. Hence,λ “ 0, and evIUm

Kis injective.

Theorem 2.4.2. There is a unique cpχ, ψ,Xq P CpXq such that

ZpΦ, χ, 1qX q “ cpχ, ψ,XqZpΦ, χ,Xq

for all Φ P C8c pKq.

Proof. The vector space Λ of the previous proposition has dimension 1 overCpXq. The function

Φ ÞÑ ZpΦ, χ,Xq

is an element of Λ by Lemma 2.3.8. It is nonzero by Lemma 2.3.9.

If Φ P C8c pKq and a P Kˆ are given, then for all ξ P K it holds that

xaΦpξq “

ż

K

Φpa´1xqψpξxq dµpxq

“

ż

K

Φpyqψpaξyq dµpayq.

By Lemma 2.2.1, the latter is equal to

}a}

ż

K

Φpyqψpaξyq dµpyq “ }a} Φpaξq.

This givesxaΦ “ }a} a´1Φ.

Hence, for all Φ P C8c pKq, a P Kˆ, it holds that

ZpxaΦ, χ, 1qX q “ }a} Zpa

´1Φ, χ, 1qX q

“ }a} χpa´1qpqXq´vKpa´1qZpΦ, χ, 1

qX q

“ χpaqXvKpaqZpΦ, χ, 1qX q.

This shows that the function

Φ ÞÑ ZpΦ, χ, 1qX q

is also an element of Λ. It is again nonzero by Lemma 2.3.9. The result thenfollows since Λ has dimension 1 over CpXq.

36

Definition 2.4.3. We put γpχ, s, ψq “ cpχ, ψ, q´sq.

Corollary 2.4.4. We have the functional equation

ζpΦ, χ, 1´ sq “ γpχ, s, ψqζpΦ, χ, sq,

for all Φ P C8c pKq.

Proposition 2.4.5. The function γpχ, s, ψq satisfies the functional equation

γpχ, s, ψqγpχ, 1´ s, ψq “ χp´1q.

Proof. The Fourier inversion formula gives

ˆΦpxq “ Φp´xq

for all Φ P C8c pKq, x P K. Hence, for all n P Z, we have

znpˆΦ, χq “

ż

Sn

Φp´xqχpxq dµˆpxq

“

ż

Sn

Φpyqχp´yq dµˆp´yq

“ χp´1q} ´ 1} znpΦ, χq “ χp´1qznpΦ, χq.

Hence, ZpˆΦ, χ,Xq “ χp´1qZpΦ, χ,Xq, so

ζpˆΦ, χ, sq “ χp´1qζpΦ, χ, sq.

On the other hand, the previous corollary gives

ζpˆΦ, χ,Xq “ γpχ, 1´ s, ψqζpΦ, χ, 1´ sq

“ γpχ, 1´ s, ψqγpχ, s, ψqζpΦ, χ, sq,

which gives the result.

Definition 2.4.6. Define the function

ΞpΦ, χ, sq “ζpΦ, χ, sq

Lpχ, sq.

It is a rational function in q´s by Proposition 2.3.13.

Definition 2.4.7. Define the function

εpχ, s, ψq “ γpχ, s, ψqLpχ, sq

Lpχ, 1´ sq.

Corollary 2.4.8. The function ΞpΦ, χ, sq satisfies the functional equation

ΞpΦ, χ, 1´ sq “ εpχ, s, ψq ΞpΦ, χ, sq.

Proof. This is immediate from Corollary 2.4.4.

37

Corollary 2.4.9. The function εpχ, s, ψq satisfies the functional equation

εpχ, s, ψqεpχ, 1´ s, ψq “ χp´1q.

Furthermore, εpχ, s, ψq is of the form aq´ns for some a P Cˆ, n P Z.

Proof. The functional equation follows from Proposition 2.4.5.

Pick some Φ for which ΞpΦ, χ, sq ‰ 0. Then the previous corollary shows thatεpχ, s, ψq is a Laurent polynomial in q´s. Similarly, εpχ, 1 ´ s, ψq is a Laurentpolynomial in q´s. The functional equation asserts that εpχ, s, ψq is an invertibleelement in Crq´s, qss, so it is a monomial.

38

3 Weil groups

The aim of this chapter is to define the L-function and ε-factors associated tocertain types of representations of the Weil group of K.

We will start with an axiomatic description of the Weil group. The reader whois not familiar with local class field theory can take the theorems on faith; wewill only give a brief summary. A good reference on class field theory is forinstance [13] or [1].

After that, we will move on to discuss several aspects of the representationtheory of the Weil group. We will define L-functions and ε-factors associated tofinite-dimensional semisimple representations of the Weil group.

Finally, we introduce the notion of Deligne representations. We will extend thedefinitions of L-functions and ε-factors to all semisimple Deligne representationsof the Weil group.

The presentation of this chapter is guided by [3]. In some cases we will use theslightly more general approach of [17].

3.1 Local Class Field Theory

We denote by GK the absolute Galois group of K. The maximal unramifiedextension ofK is denotedKnr , and its corresponding subgroup ofGK is denotedIK . It is called the inertia subgroup of GK . We can identify GalpKnr{Kq withGalpk{kq.

There is an isomorphism φ : ZÑ Galpk{kq “ GalpKnr{Kq mapping an elementpanqnPZą0

to the automorphism of k the inverse of which is defined by

φ ppanqnq´1 ˇ

ˇ

Fnq“ px ÞÑ xanq.

The image of 1 under this isomorphism is called the geometric Frobenius sub-stitution, and its inverse is called the arithmetic Frobenius substitution. Theformer one is more important for our purposes, and is denoted ΦK . Any ele-ment of GK whose restriction to Knr is ΦK is called a Frobenius element (overK). Usually, we denote such an element by ΦK as well.

By Galois theory, there is a short exact sequence

1 Ñ IK Ñ GK Ñ ZÑ 1,

where we identify Z with GalpKnr{Kq as above.

Definition 3.1.1. The Weil group WK of K is the subgroup of GK that is theinverse image of Z Ď Z in the above short exact sequence. The map WK Ñ Zis called the valuation on WK , and is denoted vK .

We also write } ¨ }K “ q´vKp¨q, and drop the subscript from the notation whenit is clear what field we are working over.

40

Remark 3.1.2. Note that WK consists of the elements of GK which act like apower of ΦK on Knr. There is a short exact sequence

1 Ñ IK ÑWKvKÝÑ ZÑ 1. (3)

Furthermore, if K Ď L is another algebraic extension (inside K), then we havea commutative diagram

1 IL WL Z 1

1 IK WK Z 1,

vL

vK

since every element of GL that acts on Lnr as a power of Frobenius also acts onKnr Ď Lnr as a power of Frobenius.

Remark 3.1.3. Note that the Weil group of K, just like the absolute Galoisgroup, depends on the chosen separable closure of K. As such, it is only definedup to an inner automorphism of GK .

Remark 3.1.4. Note also that a choice of a Frobenius element Φ PWK inducesa right splitting of the exact sequence (3), and hence a decomposition of WK asa semi-direct product

WK “ IK ¸ Z.As a set, we can identify WK with IK ˆ Z; this depends on our choice of Φ.

Definition 3.1.5. The topology on WK is given by the product topology ofIK ˆ Z as above, where IK has its natural profinite topology, and Z has thediscrete topology.

Proposition 3.1.6. The topology on WK does not depend on the choice of aFrobenius element. Furthermore, WK is a topological group, and the inclusion

ιK : WK Ñ GK

is continuous and has dense image.

Proof. By the definition of product topology, using that Z has the discretetopology, we see that the topology on WK is the coarsest topology satisfyingthe following properties:

• IK is an open subgroup endowed with its natural topology;• a set U ĎWK is open if and only if ΦU is open.

It follows that the topology is invariant under translation by elements of WK .Hence, it is also the coarsest translation-invariant topology in which IK is openand has its natural topology. This description is independent of Φ.

We will show continuity of multiplication. Inversion can be treated similarly,and is left as an exercise to the reader. We consider the map

pIK ¸ Zq ˆ pIK ¸ Zq Ñ IK ¸ Z

41

that is given by multiplication. Explicitly, it is given by

ppx,Φnq, py,Φmqq ÞÑ pxΦnyΦ´n,Φn`mq.

By the universal property of the product topology, it suffices to show that themaps

pIK ¸ Zq ˆ pIK ¸ Zq Ñ IK

px,Φnq, py,Φmq ÞÑ xΦnyΦ´n

and

pIK ¸ Zq ˆ pIK ¸ Zq Ñ Zpx,Φnq, py,Φmq ÞÑ Φn`m

are continuous. The first one is continuous since conjugation in GK is continu-ous. The second one is continuous since it factors through ZˆZÑ Z, which iscontinuous because the topology on both sides is discrete.

Let U Ď GK be an open subgroup, and write H “ U XWK . Then H{pIK XHqis a subgroup of Z, say it is nZ for some n P Zě0. Then pick h P H withh ” n mod IK XH. Then clearly

H “ pH X IKqxhy,

and the latter is an open set of WK since it is a union of Φ-translates of opensubsets of IK . This implies that the map ιK : WK Ñ GK is continuous. It isclear that it has dense image.

Lemma 3.1.7. Let L{K be a finite extension, such that L Ď K. Then

WL “ GL XWK .

The topology on WL coincides with the subspace topology in WK , and WL isopen in WK .

Proof. The inclusion WL Ď GL X WK is obvious. For the reverse inclusionand the comparison of topologies, we will treat the cases where L{K is eitherunramified or totally ramified. The general case follows since every extensionL{K admits an intermediate field M such that M{K is unramified and L{M istotally ramified.

If L{K is unramified, then IK is contained in GL. On the quotients GL{IL – Zand GK{IK – Z, the map induced by GL Ď GK is multiplication by n, wheren “ rL : Ks. The reverse inclusion follows from the fact that the inverse image

in Z of Z under the multiplication by n map is Z.

If ΦK P WK is a Frobenius element over K, then ΦnK P WL is a Frobeniuselement over L. Under the identification WK “ IK ¸ Z, the subgroup WL

corresponds to the subgroup IK¸nZ. Hence, the topology onWL is the subspacetopology of WK , and WL is open in WK .

42

If L{K is totally ramified, then GL X IK “ IL. On the quotients GL{IL – Zand GK{IK – Z, the map induced by GL Ď GK is the identity. Hence, theequality WL “ GL XWK is obvious.

Any Frobenius element ΦK P WK over K is also a Frobenius element over L.Under the identification WK “ IK ¸ Z, the subgroup WL corresponds to thesubgroup IL ¸ Z. Hence, the topology on WL is the subspace topology of WK ,and WL is open in WK .

Lemma 3.1.8. Let L{K be a finite extension, such that L Ď K. Then thecanonical map WK{WL Ñ GK{GL is a bijection, and WL Ď WK is normal iffL{K is Galois.

Proof. The canonical map WK{WL Ñ GK{GL is injective by the previouslemma. Furthermore, the topology on GK{GL is discrete since L{K is finite.Hence, the map above is surjective since the image is dense (Proposition 3.1.6).

If L{K is Galois, then GL Ď GK is normal, hence WL “ GLXWK is normal inWK .

Conversely, if WL is normal in WK , let g P GK be given. Let g0 P WK be suchthat g´1

0 g P GL (it exists by the first part of the lemma). Write x “ g´10 g, so

that g “ g0x. ThengGLg

´1 “ g0GLg´10 ,

and the latter is clopen since GL is. It contains WL since WL is normal in WK .But GL is the closure of WL in GK , since GL is closed in GK and WL is densein GL. Hence GL Ď gGLg

´1, so they are equal and GL is normal in GK .

We are now ready to state the results from class field theory, without proofs.

Theorem 3.1.9 (Local Class Field Theory). There exists a unique continuousgroup homomorphism

ArtK : WK Ñ Kˆ

such that the following properties hold:

• The induced map ArtabK is a topological isomorphism

ArtabK : W abK

„ÝÑ Kˆ;

• The valuations on WK and Kˆ coincide;• If L{K is a finite extension contained in K, then the diagram

WL Lˆ

WK Kˆ

ArtL

NL{K

ArtK

commutes;

43

• If L{K is a finite extension contained in K, then the diagram

WK Kˆ

WL Lˆ

ArtK

verL{K

ArtL

commutes.

3.2 Representations of the Weil group

Lemma 3.2.1. Let pρ, V q be an irreducible smooth representation of WK . ThenV is finite-dimensional.

Proof. If v P V is any nonzero element, then there is an open subgroup U ĎWK

fixing v; assume w.l.o.g. that U Ď IK . Then there exists an open U 1 Ď GK withU “ U 1 X IK . Let L{K be the (finite) field extension corresponding to U 1, andassume w.l.o.g. that L{K is (finite) Galois.

Now the subgroup U ĎWK is the intersection of the normal subgroups IK andU 1 XWK , hence it is normal (in WK) as well. Since its conjugates xUx´1 “ Ufix the elements ρpxqv, we see that U in fixes the subspace V 1 spanned by theelements ρpxqv. But since pρ, V q is irreducible, we have V 1 “ V , so U fixes allof V . Hence, U Ď ker ρ.

If we choose a Frobenius element Φ PWK , then we get a decomposition

WK “ IK ¸ Z.

Hence, Φ acts on IK by conjugation, hence also on the finite group IK{U .Therefore, some positive power Φn acts trivially on IK{U . In particular, theconjugation action of Φn on ρpIKq, is trivial, i.e. ρpΦnq commutes with ρpIKq.Hence, ρpΦnq actually commutes with all of ρpWKq, so by Schur’s lemma (see[3, Lemma 2.6]), it acts on V as a scalar.

Hence, for every w P V the space spanned by tρpΦmqw : m P Zu is finite-dimensional. Letting w range over a basis of the finite-dimensional vector spacespanned by tρpxqv : x P IKu, we get the result.

Definition 3.2.2. A character χ of WK (or of GK) is called unramified if it istrivial on IK . Otherwise, χ is called ramified.

Remark 3.2.3. Any character χ of Kˆ gives a character χ ˝ ArtK of WK .Moreover, χ is unramified if and only if χ ˝ArtK is unramified.

Remark 3.2.4. An intermediate fieldK Ď L Ď K defines a rL : Ks-dimensionalrepresentation of GK over the field K. The extension L{K is unramified if andonly if IK acts trivially on L; hence the terminology.

44

Definition 3.2.5. Let n P Zą0. Then we write Gssn pKq for the set of isomor-

phism classes of n-dimensional semisimple smooth representations of WK . Wewrite GsspKq for the union of the sets Gss

n pKq.

Definition 3.2.6. Consider the free abelian group Z‘GsspKq generated by the set

GsspKq. Let A be the subgroup generated by elements of the form rσs´rρs´rτ sfor every short exact sequence

0 Ñ ρÑ σ Ñ τ Ñ 0.

The quotient Z‘GsspKq{A is called the Grothendieck group of WK , and it is

denoted RpWKq.

Every semisimple smooth representation of WK is the direct sum of irreduciblesubrepresentations. Therefore, the Grothendieck group of WK is isomorphic tothe free abelian group generated by isomorphism classes of irreducible smoothrepresentations of WK .

Lemma 3.2.7. Let A be an abelian group, and let φ : GsspKq Ñ A be a function.Then φ can be extended to a homomorphism φ : RpWKq Ñ A if and only ifφpσq “ φpρq ` φpτq for every short exact sequence

0 Ñ ρÑ σ Ñ τ Ñ 0.

Proof. This is clear from the definition of RpWKq.

Definition 3.2.8. A function φ : GsspKq Ñ A satisfying one of the equivalentconditions in the lemma above is called additive.

Example 3.2.9. Let L{K be a finite extension contained in K. Then the maps

GsspKq Ñ RpWLq

pρ, V q ÞÑ”

ρˇ

ˇ

WL

ı

and

GsspLq Ñ RpWKq

pρ, V q ÞÑ”

IndWK

WLρı

are both additive, so they define homomorphisms

RpWKq Õ RpWLq.

We denote these homomorphisms by ResL{K and IndL{K respectively.

Remark 3.2.10. In the above example, one needs to check that the inductionof a semisimple representation is again semisimple. The proof is similar to thecase of finite groups, and it is carried out in Lemma 2.7 of [3].

Example 3.2.11. The map dim: GsspKq Ñ Z is clearly additve, so it defines ahomomorphism

dim: RpWKq Ñ Z.The kernel of this homomorphism is denoted R0pWKq.

45

Proposition 3.2.12. The group R0pWKq is generated by elements of the form

IndL{Kprχs ´ rχ1sq,

where L{K is a finite extension and χ, χ1 are characters of WL.

Proof. This is basically a version of Brauer’s induction theorem for WK insteadof a finite group. We refer to Proposition 2.3.1 of [17] or Lemma 30.1.1 of [3]for a proof.

Corollary 3.2.13. The group RpWKq is generated by elements of the formIndL{Krχs, where L{K is a finite extension and χ is a character of WL.

Proof. The element r1WKs is surely of the required form (taking L “ K). The

result follows since RpWKq is spanned by R0pWKq and r1WKs.

Definition 3.2.14. Let A be an abelian group. A family λL : RpWLq Ñ A ofmaps (where L ranges over the finite extensions of K) is called additive over Kif for each L{K the map λL is a homomorphism.

Definition 3.2.15. Let A be an abelian group. A family λL : RpWLq Ñ A ofmaps is called inductive in degree 0 if it is additive over K and for each towerK Ď L ĎM of finite extensions the diagram

R0pWM q

A

R0pWLq

λM

IndM{L

λL

commutes.

Remark 3.2.16. By Proposition 3.2.12, a family that is inductive in degree 0is uniquely determined by its values on rχs P RpWLq for L{K finite and χ acharacter of WL.

3.3 L-functions and epsilon factors

The definition of the L-function is not too involved. We will not prove theexistence of ε-factors (Theorem 3.3.7). The proof can be found in [3]. It requiresa global argument, which can be found in Tate’s Thesis [16].

Note that if pρ, V q is a representation of WK , then V IK is fixed by WK , sinceIK is normal in WK . Furthermore, any two Frobenius elements in WK differby an element of IK , hence their actions on V IK are identical. That is, theautomorphism

ρpΦqˇ

ˇ

V IK

does not depend on the choice of Frobenius element Φ PWK .

46

Definition 3.3.1. Let pσ, V q be a finite-dimensional, semisimple smooth repre-sentation of WK . Let Φ PWK be a Frobenius element over K. Then define

Zpσ, tq “ det`

1´ σpΦqˇ

ˇ

V IKt˘´1

.

The L-function associated to pσ, V q is defined as

Lpσ, tq “ Zpσ, q´sq.

Example 3.3.2. If χ is a character of WK , then we have two cases:

• If χ is unramified, then CIK “ C, so

Lpσ, sq “`

1´ χpΦqq´s˘´1

.

• If χ is ramified, then CIK “ 0, so

Lpσ, sq “ 1.

Example 3.3.3. If pσ, V q is an irreducible representation of WK of (finite)dimension at least 2, then V IK is a subrepresentation. Hence, either V IK “ 0or V IK “ V .

Suppose that V IK “ V , then IK Ď kerσ, so V is an irreducible representationof the quotient WK{IK “ Z. But then any eigenvector of σp1q spans a one-dimensional subrepresentation, which contradicts irreducibility of pσ, V q. Hence,V IK “ 0, and

Lpσ, sq “ 1.

Lemma 3.3.4. If pσ1, V1q, pσ2, V2q are finite-dimensional, semisimple smoothrepresentations of WK , then

Lpσ1 ‘ σ2, sq “ Lpσ1, sqLpσ2, sq.

Proof. Observe that pV1 ‘ V2qIK “ V IK1 ‘ V IK2 . Hence, the characteristic poly-

nomial of pσ1‘σ2qpΦq|pV1‘V2qIK is the product of the characteristic polynomials

of σ1pΦq|V IK1and σ2pΦq|V IK2

. Hence,

Zpσ1 ‘ σ2, tq “ Zpσ1, tqZpσ2, tq,

and the result follows.

Remark 3.3.5. We could have defined the L-function of irreducible represen-tations by the two examples above, extending to all semisimple representationsusing Lemma 3.3.4, cf. [3]. This is also why we only define the L-function forsemisimple representations: we can reduce most statements to the irreduciblecase.

Definition 3.3.6. If L{K is a finite extension within K, and ψ is a nontrivialcharacter of K, then we put ψL “ ψ ˝ TrL{K .

47

Theorem 3.3.7. Let ψ be a nontrivial character of K. Then there exists aunique family of functions

λL : RpWLq Ñ Crq´s, qssˆ

that is inductive in degree 0 and satisfies

λLprχ ˝ArtLsq “ εpχ, s, ψLq

whenever L{K is finite and χ is a character of Lˆ.

Proof. See Theorem 29.4 of [3].

Remark 3.3.8. Uniqueness is clear, since characters on WL correspond to char-acters on Lˆ (Theorem 3.1.9), and a family that is inductive in degree 0 isuniquely determined by its values on characters.

Definition 3.3.9. Let ψ be a nontrivial character of K, let L{K be finite andlet pρ, V q be a semisimple smooth representation of WL. We write

εpρ, s, ψLq :“ λLprρsq,

where λL is the map from the theorem. In particular, if χ is a character of Lˆ,we have

εpχ ˝ArtL, s, ψLq “ εpχ, s, ψLq.

Corollary 3.3.10. Let ψ be a nontrivial character of K, and let L{K be finite.Then the ε-factor satisfies

εpρ1 ‘ ρ2, s, ψLq “ εpρ1, s, ψLqεpρ2, s, ψLq,

whenever pρ1, V1q and pρ2, V2q are semisimple smooth representations of WL.

Proposition 3.3.11. Let pρ, V q be a semisimple smooth representation of WK .Then the ε-factor satisfies the functional equation

εpρ, s, ψqεpρ, 1´ s, ψq “ det ρp´1q,

where the character det ρ of WK is viewed as a character of Kˆ via ArtK .

Proof (sketch). One checks that the family of functions

µL : RpWLq ÝÑ Crq´s, qssˆ

σ ÞÝÑ detσp´1qpεpσ, 1´ s, ψLqq´1

is also inductive in degree 0. The result then follows from the functional equationfor Kˆ (Corollary 2.4.9).

3.4 Deligne representations

We only give the definitions. For motivation, see for instance section 32 of [3].

48

Definition 3.4.1. A Deligne representation of WK is a triple pρ, V,Nq wherepρ, V q is a finite-dimensional smooth representation of WK , and N P EndCpV qis a nilpotent endomorphism satisfying

ρpgqNρpgq´1 “ }g}N,

for all g PWK .

Definition 3.4.2. A Deligne representation pρ, V,Nq is called semisimple ifpρ, V q is semisimple. We write Gn for the set of isomorphism classes of n-dimensional semisimple Deligne representations of WK .

Example 3.4.3. Any smooth representation pρ, V q ofWK gives rise to a Delignerepresentation by setting N “ 0. Hence, we can identify Gss

n pKq with a subsetof GnpKq.

Example 3.4.4. Let V “ Cn with standard basis e1, . . . , en. Let ρ be therepresentation defined by

ρpgqei “ }g}i´1 ¨ ei.

Let N be the nilpotent element defined by

Nei “

"

ei`1 i ă n,0 i “ n.

Then for g PWK and i P t1, . . . , n´ 1u, it holds that

ρpgqNρpgq´1ei “ }g}1´iρpgqNei

“ }g}1´iρpgqei`1

“ }g}ei`1 “ }g}Nei,

so pρ, V,Nq is indeed a Weil deligne representation. It is denoted Sppnq, and itis semisimple since ρ – 1WK

‘ } ¨ } ‘ . . .‘ } ¨ }n´1.

Definition 3.4.5. Given a Deligne representation pρ, V,Nq, define the dualrepresentation

­pρ, V,Nq :“ pρ, V ,´Nq.

Definition 3.4.6. Given Deligne representations pρi, Vi, Niq for i P t1, 2u, define

pρ1, V1, N1q ‘ pρ1, V1, N1q :“ pρ1 ‘ ρ2, V1 ‘ V2, N1 ‘N2q

and

pρ1, V1, N1q b pρ1, V1, N1q :“ pρ1 b ρ2, V1 b V2, N1 b IV2 ` IV1 bN2q.

Remark 3.4.7. Given a Deligne representation pρ, V,Nq of WK , the subspaceVN :“ kerN is a WK-subspace, since for every g P WK , v P V the nilpotentelement N acts on ρpgqv as }g}´1ρpgqNρpgq´1.

Definition 3.4.8. Let pρ, V,Nq be a Deligne representation of WK . Then define

Lppρ, V,Nq, sq :“ LpρN , sq,

49

and

εppρ, V,Nq, s, ψq :“ εpρ, s, ψqLpρ, 1´ sq

Lpρ, sq

LpρN , sq

Lpρ´N , 1´ sq.

Remark 3.4.9. If pρ, V,Nq comes from an ordinary smooth representationpρ, V q, i.e. if N “ 0, then VN “ V , and

Lppρ, V,Nq, sq “ Lpρ, sq;

εppρ, V,Nq, s, ψq “ εpρ, s, ψq.

In that sense, the L-functions and ε-factors above are really an extension of theprevious definitions.

50

4 Representations of the General Linear Group

In this chapter, we will discuss some aspects of the representation theory ofGLnpKq, where K is a non-archimedian local field. We will write Gn for thegroup GLnpKq.

The treatment of this chapter is inspired by [14], and for the proofs we refer to[4].

4.1 Parabolic subgroups

Firstly, we will put a topology on Gn, making it a locally profinite group.

Definition 4.1.1. The topology on the matrix ring MnpKq is the product topol-

ogy, where we identify MnpKq with Kn2

.

Clearly, the product topology makes Kn2

a topological group. Hence, addi-tion and additive inversion in MnpKq are continuous. The more important factwe need is that multiplication and multiplicative inversion are continuous onGLnpKq. These can, however, easily be deduced from the fact that multiplica-tion and multiplicative inversion are continuous on K.

It is also not hard to see that Gn is locally profinite. If U Ď Gn is any openneighbourhood of the identity, then 1`pmMnpOKq Ď U for m sufficiently large.The set 1` pmMnpOKq is an open subgroup of Gn, and it is clearly compact.

Remark 4.1.2. In general, in a topological ring R, taking inverses is not neces-sarily continuous. Therefore, the topology given above generally does not makeGLnpRq into a topological group.

Definition 4.1.3. A composition of the integer m is a sequence pa1, . . . , arq ofpositive integers such that

m “

rÿ

i“1

ai.

Two compositions pa1, . . . , arq, pb1, . . . , bsq of m are called equivalent if r “ sand there exists a permutation σ P Sr such that

paσp1q, . . . , aσprqq “ pb1, . . . , bsq.

An equivalence class for this equivalence relation is called a partition of m.

Some authors (e.g. [14]) use the word (ordered) partition instead of composition.

Definition 4.1.4. Let α “ pa1, . . . , arq be a composition of n. Then define thefollowing groups.

52

Gα :“

$

’

’

’

&

’

’

’

%

¨

˚

˚

˚

˝

A1 0 . . . 00 A2 . . . 0...

.... . .

...0 0 . . . Ar

˛

‹

‹

‹

‚

P Gn

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

A1 P Ga1A2 P Ga2

...An P Gar

,

/

/

/

.

/

/

/

-

Pα :“

$

’

’

’

&

’

’

’

%

¨

˚

˚

˚

˝

A1 ˚ . . . ˚

0 A2 . . . ˚

......

. . ....

0 0 . . . Ar

˛

‹

‹

‹

‚

P Gn

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

A1 P Ga1A2 P Ga2

...An P Gar

,

/

/

/

.

/

/

/

-

Uα :“

$

’

’

’

&

’

’

’

%

¨

˚

˚

˚

˝

Ia1 ˚ . . . ˚

0 Ia2 . . . ˚

......

. . ....

0 0 . . . Iar

˛

‹

‹

‹

‚

P Gn

,

/

/

/

.

/

/

/

-

.

The groups Pα are called the standard parabolic subgroups of Gn.

Remark 4.1.5. Clearly, Pα “ UαGα, and Uα is normal in Pα. Hence,

Pα{Uα – Gα, Pα – Uα ¸Gα,

as abstract groups.

Note also that the three groups defined above are closed subgroups of Gn.

Example 4.1.6. Two easy partitions of n are given by pnq and p1, . . . , 1q. Ifα “ pnq, then Gα “ Pα “ Gn, and Uα “ tInu. If α “ p1, . . . , 1q, then Gαconsists of the invertible diagonal matrices, Pα consists of the invertible uppertriangular matrices, and Uα consists of the upper triangular matrices with only1 on the diagonal. In the latter case, Pα is the standard Borel subgroup of Gn.

Definition 4.1.7. Let pρ, V q be a smooth representation of Gn, and let α “pa1, . . . , arq be a composition of n. Then define

V pUαq “ spantv ´ ρpuqv : v P V, u P Uαu.

It is clearly a Uα-subspace of V , and in fact it is Pα-stable: if u P Uα, p P Pαand v P V , then

ρppq pv ´ ρpuqvq “ ρppqv ´ ρppup´1qρppqv,

and pup´1 P Uα since Uα is normal in Pα.

In particular, the quotient V {V pUαq has a natural representation of Pα, whichis trivial on Uα. Therefore, it defines a representation of Gα on V {V pUαq, whichis denoted pρUα , VUαq.

Let φ : ρ Ñ σ be a Gn-homomorphism between two smooth representationspρ, V q, pσ,W q of Gn. Then for each v P V and each u P Uα, we have

φpv ´ ρpuqvq “ φpvq ´ φpρpuqvq “ φpvq ´ σpuqφpvq,

53

so φ maps V pUαq into W pUαq. Hence, φ induces a Gα-homomorphism

φUα : ρUα Ñ σUα

on the quotient. It is clearly functorial.

Definition 4.1.8. The functor ReppGnq Ñ ReppGαq given by

pρ, V q ÞÑ pρUα , VUαq

is called the Jacquet functor.

Lemma 4.1.9. Let U Ď Uα be a compact subgroup. Let pρ, V q be a smoothrepresentation of Uα, and let V 1 be a subspace stable under the action of Uα.Then

V 1 X V pUq “ V 1pUq.

Proof. By Lemma 1.1.14, U is a profinite group, since it is compact. Now therepresentation V of U is semisimple by Lemma 1.2.5. Hence, V is the directsum of irreducible representations. The result follows from

à

i

VipUq “

˜

à

i

Vi

¸

pUq,

as V 1 is a direct sum over a subset of the irreducible summands of V .

Corollary 4.1.10. Let pρ, V q be a smooth representation of Uα, and let V 1 bea subspace stable under the action of Uα. Then

V 1 X V pUαq “ V 1pUαq.

Proof. Note that Uα is the union of the compact subgroups

Umα :“

$

’

’

’

&

’

’

’

%

¨

˚

˚

˚

˝

Ia1 A1,2 . . . A1,r

0 Ia2 . . . A2,r

......

. . ....

0 0 . . . Iar

˛

‹

‹

‹

‚

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Ai,j P pmpj´iqMpai, aj ,OKq

,

/

/

/

.

/

/

/

-

,

for m P Z. Hence,V pUαq “

ď

iPZV pUmα q,

and the result follows from the lemma.

Proposition 4.1.11. The Jacquet functor is additive and exact.

Proof. Additivity is clear. Let pρ, V q, pσ,W q and pτ,Xq be smooth representa-tions of Gα, and suppose we have a short exact sequence

0 ÝÑ ρfÝÑ σ

gÝÑ τ ÝÑ 0.

54

Then we get a sequence

0 ÝÑ V pUαq ÝÑW pUαq ÝÑ XpUαq ÝÑ 0,

which we will prove to be exact.

We can interpret V as a subrepresentation of W , and then exactness at W pUαqfollows from the corollary: if w PW pUαq satisfies gpwq “ 0, then w is in W pUαqand in ker g “ V , hence w P V pUαq.

Clearly, the restriction of f to V pUαq is still injective, so exactness at V pUαq isobvious. If x P X and u P Uα are given, then pick w PW with gpwq “ x. Then

gpw ´ σpuqwq “ x´ τpuqx,

so all generators of XpUαq are in the image. Hence, the sequence is exact.

The result now follows from the snake lemma, applied to the diagram

0 V pUαq W pUαq XpUαq 0

0 V W X 0,

using that the vertical maps are injective.

Lemma 4.1.12. Let pρ, V q and pσ,W q be smooth representations of Gn and Gαrespectively. There is a natural isomorphism

HomGn

´

ρ, IndGnPα pσq¯

– HomGα pρUα , σq .

Proof. By smooth Frobenius reciprocity, there is a natural isomorphism

HomGn

´

ρ, IndGnPα pσq¯

– HomPα

´

ρˇ

ˇ

Pα, σ¯

,

where σ is viewed as a representation of Pα by making it act trivially on Uα.But then any homomorphism f : ρ|Pα Ñ σ factors as

V W

V {V pUαq

f

π

where π : V Ñ V {V pUαq is the natural projection. That is, there is a naturalisomorphism

HomPα

´

ρˇ

ˇ

Pα, σ¯

– HomGα pρUα , σq .

Definition 4.1.13. A smooth irreducible representation pρ, V q of Gn is calledcuspidal if ρUα “ 0, for all α ‰ pnq.

55

Corollary 4.1.14. A smooth irreducible representation pρ, V q of Gn is cuspidalif and only if it is not a subrepresentation of IndGnPα pσq for a smooth represen-tation pσ,W q of Gα, for any α ‰ pnq.

Remark 4.1.15. Some authors (e.g. [4]) use the term absolutely cuspidal in-stead of cuspidal. Also the term supercuspidal is sometimes used.

Given (abstract) representations σi of Gai for i P t1, . . . , ru, we get a representa-tion σ “ σ1b . . .b σr of Gα. Since Gα is topologically isomorphic to the groupG1 ˆ . . .ˆGr, the product of open subgroups of the Gi is an open subgroup ofGα. Therefore, if σ1, . . . , σr are smooth, so is σ.

Definition 4.1.16. Let σ1, . . . , σr be smooth representations of Ga1 , . . . , Garrespectively. View σ “ σ1 b . . .b σr as a representation of Pα that is trivial onUα. Then we obtain a representation of Gn by considering the representation

σ1 ˆ . . .ˆ σr :“ IndGnPα pσq.

It is smooth by definition of Ind, since σ is smooth.

Remark 4.1.17. From Lemma 4.1.12, we get a natural isomorphism

HomGn pρ, σ1 ˆ . . .ˆ σrq – HomGα pρUα , σ1 b . . .b σrq ,

whenever pρ, V q is a smooth representation of Gn.

4.2 L-functions and epsilon factors

This is only an outline, and we do not include any of the proofs. A full accountcan be found in [4], sections 3-5 of chapter 1 (Local Theory).

Throughout this section, G will denote the group Gn for some n P Zą0. Wefix a left Haar measure µˆ on G. All representations of G are assumed to beadmissible.

We will write A for the ring MnpKq of nˆn-matrices. It is a locally profinitegroup, cf. Definition 4.1.1. If ψ is a character of K, we will write ψA for thecharacter ψ ˝ Tr of MnpKq.

Definition 4.2.1. Let pπ, V q be an irreducible smooth representation of G.Then define Cpπq to be the subspace of MappG,Cq spanned by the functions

γvbv : GÑ Cg ÞÑ vpπpgqvq.

Lemma 4.2.2. The map V b V Ñ Cpπq given by vb v ÞÑ γvbv is C-linear andsurjective.

56

Proof. This is because for each g P G the map

V b V Ñ Ccb c ÞÑ vpπpgqvq

is linear. Surjectivity is by definition.

Definition 4.2.3. A function f : G Ñ C of the form γvbv is called a (matrix)coefficient of π.

Lemma 4.2.4. Let f be a coefficient of π. Then the map f given by g ÞÑ fpg´1q

is a coefficient of π.

Proof. Assume that f “ γvbv. Since π is admissible, the map ev8π : V Ñ ˇV isan isomorphism. Then the map

γev8π pvqbv: GÑ Cg ÞÑ ev8π pvqpπpgqvq

coincides with f , since

ev8π pvqpπpgqvq “ pπpgqvqpvq “ vpπpg´1qvq,

for every g P G, by the definition of π.

Definition 4.2.5. Let µ be a left Haar measure on G, and let ψ be a nontrivialcharacter of K. Then define the Fourier transform Φ of any Φ P C8c pAq as

Φpξq “

ż

A

ΦpxqψApξxq dµpxq.

Proposition 4.2.6. Assume Φ P C8c pAq. Then Φ P C8c pAq. Furthermore,there exists a left Haar measure µA on A (depending on ψ) such that

ˆΦpxq “ Φp´xq

for all x P A and all Φ P C8c pAq.

Proof. Omitted. The proof is analogous to that of Proposition 2.2.12.

Definition 4.2.7. Let pπ, V q be an irreducible smooth representation of G, letΦ P C8c pAq be a smooth function and let f P Cpπq be a coefficient of π. Thenthe zeta function of Φ with respect to π and f is

ζpΦ, f, sq :“

ż

G

Φpxqfpxq}detx}s dµˆpxq.

Theorem 4.2.8. Let pπ, V q be an irreducible smooth representation of G. Thenthere exists a constant s0 P R such that the integral

ż

G

Φpxqfpxq}detx}s dµˆpxq

57

converges absolutely and uniformly on compact subsets of tRe s ą s0u, for allfunctions Φ P C8c pAq and all coefficients f of π. Moreover, on that half plane,the function is a rational function in q´s.

Proof. See Theorem 3.3(1) of [4].

Remark 4.2.9. Similarly to Remark 2.3.7, we get a C-linear map

Z : C8c pAq b Cpπq Ñ Cpq´sqΦ ÞÑ ζpΦ, f, sq.

Proposition 4.2.10. There is a unique polynomial Pπ P CrXs with Pπp0q “ 1such that the image of the map Z above is equal to

Pπpq´sq´1 Crq´s, qss.

Proof. See Theorem 3.3(2) of [4].

Definition 4.2.11. Let pπ, V q be an irreducible smooth representation of G.Then define the L-function associated to π as

Lpπ, sq “ Pπpq´sq´1.

Remark 4.2.12. Any two Haar measures µˆ on G differ by a scalar. Therefore,the image of the map Z does not depend on the chosen Haar measure. Hence,the L-series of π does not depend on any choice.

Definition 4.2.13. Define the function

ΞpΦ, f, sq “ζpΦ, f, s` 1

2 pn´ 1qq

Lpπ, sq.

It is a rational function in q´s by Proposition 4.2.10.

Theorem 4.2.14. Let ψ be a nontrivial character of K, let µA be the self-dual Haar measure with respect to ψ, and let pπ, V q be an irreducible smoothrepresentation of G. Then there exists a unique function γpπ, s, ψq P Cpq´sqsuch that

ζpΦ, f , n´ sq “ γ`

π, s´ 12 pn´ 1q, ψ

˘

ζpΦ, f, sq. (4)

Proof. This is essentially Theorem 3.3(4) of [4].

Remark 4.2.15. The theorem in [4] actually proves the functional equation

ΞpΦ, f , 1´ sq “ εpπ, s, ψq ΞpΦ, f, sq, (5)

where εpπ, s, ψq is defined as

εpπ, s, ψq “ γpπ, s, ψqLpπ, sq

Lpπ, 1´ sq.

It is easy to check that equations (4) and (5) are equivalent, analogously toCorollary 2.4.8.

58

Analogously to Proposition 2.4.5 and Corollary 2.4.9, we deduce:

Proposition 4.2.16. The function γpπ, s, ψq satisfies the functional equation

γpπ, s, ψqγpπ, s, ψq “ ωπp´1q,

where ωπ is the central character of π.

Corollary 4.2.17. The function εpπ, s, ψq satisfies the functional equation

εpπ, s, ψqεpπ, s, ψq “ ωπp´1q.

Furthermore, εpπ, s, ψq is of the form aq´ns for some a P Cˆ, n P Z.

59

5 The Local Langlands Correspondences

With all the machinery in place, we can finally get to the theorem that is thelocal Langlands correspondence for GLnpKq.

Write AnpKq for the set of isomorphism classes of irreducible admissible smoothrepresentations of GLnpKq. Recall that we write GnpKq for the set of isomor-phism classes of n-dimensional semisimple Deligne representations of WK .

Theorem 5.0.1 (The Local Langlands Correspondences). There is a uniquefamily of bijections

πn : GnpKq Ñ AnpKqsuch that π1 is given by class field theory, and

Lpπpρq ˆ πpσq, sq “ Lpρb σ, sq

εpπpρq ˆ πpσq, s, ψq “ εpρb σ, s, ψq

for all ρ P GnpKq, σ P GmpKq.

Proof. This is hard. The first proof was given by Harris and Taylor [6] in 2001,and independently by Henniart [7] in 2002 (building on an earlier paper byHarris [5]).

Remark 5.0.2. A proof for n “ 2 was known earlier ([11], [12]), and there is atextbook [3] about the case.

Remark 5.0.3. A new approach for the proof of the local Langlands correspon-dence was given by Scholze in [15].

Remark 5.0.4. The map π1 is given by class field theory. Explicitly, this meansthat

πpχ ˝ArtKq “ χ

for all characters χ of Kˆ. We will at least check the following.

Lemma 5.0.5. The correspondence π1 satisfies the conditions

Lpπ1pχq ˆ π1pχ1q, sq “ Lpχb χ1, sq

εpπ1pχq ˆ π1pχ1q, s, ψq “ εpχb χ1, s, ψq

for all characters χ, χ1 of WK , as promised by the theorem.

Proof. Note that for characters χ, χ1 of WK , the tensor product χb χ1 is againa character. Furthermore, π1pχq ˆ π1pχ

1q is none other than π1pχ b χ1q. Ittherefore suffices to prove

Lpχ ˝ArtK , sq “ Lpχ, sq

εpχ ˝ArtK , s, ψq “ εpχ, s, ψq

for all characters χ of Kˆ. The first one follows from the computations inProposition 2.3.13 and Example 3.3.2, and the second one is by definition.

60

References

[1] E. Artin, J Tate, Class Field Theory (2nd Revised edition). AMS, 2008.

[2] A. Borel, Automorphic L-functions, in: Automorphic Forms, Representa-tions and L-functions (A. Borel, W. Casselman eds.). Proc. Symp. PureMath. 33(2), AMS, 1979, p. 27-61.

[3] C.J. Bushnell, G. Henniart, The Local Langlands Conjecture for GL(2).Grundlehren der Math. Wiss. 335, Springer, 2006.

[4] R. Godement, H. Jacquet, Zeta Functions of Simple Algebras. LectureNotes in Math. 260, Springer, 1972.

[5] M. Harris, Supercuspidal Representations in the Cohomology of Drinfel’dUpper Half Spaces: Elaboration of Carayol’s Program. Invent. Math. 129,1997, p. 75-120.

[6] M. Harris, R. Taylor, The Geometry and Cohomology of Some SimpleShimura Varieties. Ann. of Math. Studies 151, Princeton UP, 2001.

[7] G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n)sur un corps p-adique. Invent. Math. 139, 2002, p. 439-455.

[8] G. Henniart, Une characterisation de la correspondance de Langlands localepour GL(n). Bull. SMF 130(4), 2002, p. 587-602.

[9] H. Jacquet, Principal L-functions of the Linear Group, in: AutomorphicForms, Representations and L-functions (A. Borel, W. Casselman eds.).Proc. Symp. Pure Math. 33(2), AMS, 1979, p. 63-86.

[10] H. Jacquet, I.I. Piatetski-Shapiro, J.A. Shalika, Rankin-Selberg Convolu-tions. Amer. J. Math. 105, 1983, p. 367-464.

[11] P.C. Kutzko, The Langlands conjecture for GL(2) of a local field. Annalsof Math. 112, 1980, p. 381-412.

[12] P.C. Kutzko, The exceptional representations of GL(2). Compositio Math.51, 1984, p. 3-14.

[13] J. Neukirch, Class Field Theory. Springer, 1986.

[14] F. Rodier, Representations de GL(n,k) ou k est un corps p-adique.Seminaire Bourbaki 1981-1982, exp. 587, p. 201-218. Available online atwww.numdam.org.

[15] P. Scholze, The Local Langlands Correspondence for GL(n) over p-adicFields. 2010. Available online at www.math.uni-bonn.de/people/scholze/.

[16] J. Tate, Fourier Analysis in Number Fields and Hecke’s Zeta Functions,Thesis, in: Algebraic Number Theory (J.W.S. Cassels, A. Frohlich eds.).Academic Press, 1967, p. 305-347.

[17] J. Tate, Number Theoretic Background, in: Automorphic Forms, Represen-tations and L-functions (A. Borel, W. Casselman eds.). Proc. Symp. PureMath. 33(2), AMS, 1979, p. 3-26.

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