ε-FACTORS OF WEIL-DELIGNE REPRESENTATIONS

ON THE ε-FACTORS OF WEIL-DELIGNE REPRESENTATIONS

KAZIM ILHAN IKEDA∗

ABSTRACT. An explicit expression for theε-factorεK((V,N),ψ,dµ) of a representation(V,N) of the Weil-Deligne groupWDK of a local fieldK is given in terms of the non-abelian local class field theory ofK.

CONTENTS

1. Introduction 11.1. Aim of this work 21.2. Notation 42. Deligne–Dwork–Langlands’ Theorem 42.1. Existence and uniqueness ofε-factors of continuous representations of the Weil groupWK of K overC 42.2. Basic properties ofεK(V,ψ,dµ) 53. Weil-Deligne representations ofWDK overC and theirε-factors 63.1. Weil-Deligne representations ofWDK overE 63.2. ε-factors of Weil-Deligne representations overC 74. Admissible representations ofWAK overC and theirε-factors 84.1. Admissible representations of Weil-Arthur groupWAK of K overC 84.2. ε-factors of admissible representations ofWAK overC 95. ε-factors attached to the representations of local absoluteGalois groups overC 95.1. The maximald-abelian extensionKd−ab of K 105.2. Representations of the local Galois groupGK of K overC 105.3. Non-abelian local class field theory 10

5.4. The Haar measure on the compact group(

∇(ϕK)K

)d−ab12

5.5. Main theorems 126. An explicit description ofε-factors of Weil-Deligne representations overC 187. A formula forε-factors of admissible representations ofWAK overC 218. ε-factors of Langlands parameters 228.1. ε-factors of Weil-Deligne homomorphisms ofWDK in G 228.2. Admissible homomorphisms ofWAK into G and theirε-factors 238.3. ε-factors of local Langlands parameters of G 24References 25

1. INTRODUCTION

2000Mathematics Subject Classification.11F80, 11S37.Key words and phrases.Local Galois representations, Weil-Deligne representations, Local constants, non-

abelian local class field theory.∗Member of “Feza Gursey Fizik ve Matematik Arastırma ve Uygulama Merkezi”, Kandilli, Istanbul.

1

2 K. I IKEDA

1.1. Aim of this work. Let ψ : K+ → C

× be a non-trivial additive character ofK, anddµ an additive Haar measure onK+. For a continuous complex representationV of theWeil groupWK of a local fieldK, let εDeligne

K (V,ψ,dµ) denote Deligne’s local constant ofthe representationV of WK with respect toψ anddµ (look at [1] and [15]). In particular,choosing the Haar measuredµψ onK+ self-dual with respect to the additive characterψ of

K+, Langlands’ local constantεLanglandsK (V,ψ) of the representationV of WK with respect

to ψ (look at [11] and [15]) is defined by

εLanglandsK (V,ψ) = εDeligne

K (V ⊗ω1/2,ψ,dµψ),

whereω1/2 : WK → C

× is defined byω1/2(w) =| ArtK(w) |1/2K for everyw ∈ WK . Here,

ArtK : WK → K× denotes the local Artin reciprocity map ofK, which induces the localArtin reciprocity isomorphismArtK∗ : Wab

K∼−→ K× of K. Moreover, introduce

εLanglandsK (s,V,ψ) = εDeligne

K (V ⊗ωs,ψ,dµψ)

for s∈ C.Note that,ε-factors are very important and central in the theory of Artin (and more

generally motivic)L-functions. In fact, ifK denotes a global field, the “completed”L-function LK(s,V) of a representationWK → GL(V) of the Weil groupWK of the globalfield K defined by the product

LK(s,V) = ∏ν∈hK∪aK

LKν (s,Vν)

is convergent fors∈ C satisfying Re(s) ≫ σ , for someσ , and defines a meromorphicfunction in the whole complex plane satisfying the functional equation

LK(s,V) = εK(s,V)LK(1−s,V∗),

whereV∗ is the dual of the representationV of WK , and theε-factor εK(s,V) of the rep-resentationV of WK appearing mysteriously in the functional equation ofLK(s,V) andcontaining the basic information about the representationV of WK overC is defined by theproduct

εK(s,V) = ∏ν∈hK∪aK

εLanglandsKν

(s,Vν ,ψν),

where the product does not depend on the choice of a non-trivial additive characterψ ofA

+K trivial on K, where the local component ofψ at a placeν is denoted byψν , which is a

non-trivial additive character of the local fieldK+ν .

Theroot number WK(V) of the representationV of the Weil groupWK of the global fieldK is defined by the quotient

WK(V) =εK(

12,V)

| εK(12,V) |

.

SettingWKν (Vν ,ψν) =εLanglandsKν

(V,ψν )

|εLanglandsKν

(V,ψν )|for eachν ∈ hK ∪aK , the global root numberWK(V)

of V is the product of local root numbersWKν (Vν ,ψν) of Vν with respect toψν as

WK(V) = ∏ν∈hK∪aK

WKν (Vν ,ψν).

ON THE ε-FACTORS OF WEIL-DELIGNE REPRESENTATIONS 3

Root numbers andε-factors are very important and interesting objects, for example aweaker version of Birch and Swinnerton-Dyer Conjecture, which is called the Parity Con-jecture claims that, for any elliptic curveE overQ, the congruence

ords=1L(s,E)≡ rankE(Q) (mod 2)

should hold, where the parity of the analytic rank is connected with the root number ofE; that is, the sign of the functional equation ofL(s,E). Thus, it is important to have anexplicit expression or a formula forε-factors.

By Deligne, Dwork, and Langlands’ fundamental theorem (look at [1, 11, 15]), ifdim(V) = 1, then there is an explicit expression of the localε-factor εLanglands

K (V,ψ) ofV with respect toψ in terms of the local Artin reciprocity isomorphismArtK∗ : Wab

K∼−→K×

of the local fieldK. On the other hand, if dim(V) 6= 1, there is unfortunatelyno knownex-plicit formula for theε-factor1 εLanglands

K (V,ψ) of V with respect toψ or εDeligneK (V,ψ,dµ)

of V with respect toψ anddµ .Following Remark 5.6 of [4], the first aim of this paper, is to get an explicit expression

for the localε-factor εLanglandsK (V,ψ) of a continuous semisimple representationV of WK

with respect toψ in terms of the non-abelian local class field theory developed in [7, 8]; orequivalently, by the Laubie’s theory [12] (also look at [6]). The second aim is to get explicitexpressions for the localε-factor εLanglands

K ((V,N),ψ) of a Weil-Deligne representation(V,N) of the Weil-Deligne groupWDK of K with respect toψ and for the localε-factorεLanglands

K (ϕ,ψ) of an admissible representationϕ of the Weil-Arthur groupWAK of K withrespect toψ.

The paper is organized as follows. Parts 2 and 3, 4 are preliminary in nature. Moreprecisely, following [1, 11, 15], in Part 2, we shall recall Deligne, Dwork, and Langlands’fundamental theorem on the existence and uniqueness of local ε-factors of continuousrepresentations ofWK and their basic properties. In Part 3, first we shall briefly reviewWeil-Deligne representations of the groupWDK over a fieldE of characteristic 0, and thenintroduce the localε-factors of Weil-Deligne representations ofWDK overC. Next, inPart 4, we shall first recall admissible representations of Weil-Arthur groupWAK of K and,after discussing the relationship between Weil-Deligne representations ofWDK overC andadmissible representations ofWAK overC, defineε-factors of admissible representationsof WAK overC. Parts 5 and 6, 7, occupy the major portion of the paper, wherewe providean explicit description of the localε-factor εLanglands

K (V,ψ) of a continuous representa-tion V of GK with respect to a fixed non-trivial additive characterψ : K+ → C

× of K interms of the non-abelian local class field theory developed in the papers [7, 8, 6, 12]. Themain idea is to introduce an integral expression IK(V,ψ) similar to the integral expression∫UK

χ∗(Art−1K (y))−1ψ(y/c)dµψ(y) appearing in the formula for the abelian local constant

1The lack of such an explicit formula has been pointed out by Tate, in part(3.4) of [15], as follows:

“... For this there is at present only an existence theorem (see below), no explicit formula.This lack is not surprising if we recall that the formulas defining ε in (3.2) make essentialuse of the interpretation ofχ as a quasi-character ofF∗; if we think of χ as a quasi-characterof WF , we have no way to defineε(χ,ψ,dx) without using the reciprocity law isomorphismF∗ ≈Wab

F . In fact it was his idea about “nonabelian reciprocity laws”relating representationsof degreenof WF to irreducible representationsπ of GL(n,F), and the possibility of definingε(π,ψ,dx) for the latter, which led Langlands to conjecture and prove aversion of thefollowing big ...”

4 K. I IKEDA

εTateK (χ ,ψ) ∈ C

× of Tate given by

εTateK (χ ,ψ) = χ∗(Art

−1K (c))

∫UK

χ∗(Art−1K (y))−1ψ(y/c)dµψ(y)

|∫UK

χ∗(Art−1K (y))−1ψ(y/c)dµψ(y) |

,

but which is modeled after the non-abelian local class field theory, and then apply Deligne,Dwork, and Langlands’ theorem to this setting. Finally, in Part 8, which is the last part ofthis work, we shall provide a formula forε-factors of local Langlands parameters

WAK → LG

of a reductive group G defined overK, which is necessary in our future research.

1.2. Notation. All through this workK denotes a local field with finite residue-class fieldκK = OK/pK of qK =: q= pf elements, whereOK denotes the ring of integers inK withthe unique prime idealpK . As usual, the unit group ofK is denoted byUK and the higherunit groups ofK by U i

K , where 0≤ i ∈ Z. The normalized discrete valuation onK isdenoted byνK : K → Z∪{∞} and the corresponding absolute value onK is denoted by| • |K : K → R≥0. In this text, we fix a separable closureKsep of K once and for all. Theabsolute Galois group ofK and the absolute Weil group ofK are denoted byGK andWK

respectively. The groupGK is equipped with its Krull topology, andWK is topologized bydeclaring the inertia subgroupIK of WK to be open, whileIK is equipped with the profinitetopology. Thus,IK is a compact group. The higher ramification subgroups ofGK in uppernumbering are denoted byGv

K , where−1≤ v∈ R. Note that,G0K = IK , the inertia group

of K. The higher ramification subgroupsWvK of WK are defined byWv

K = WK ∩GvK , for

−1 ≤ v ∈ R. Then,W0K is the inertia subgroupIK of WK . So, it follows thatWv

K = GvK

for every 0≤ v ∈ Z. For a finite separable extensionL of the local fieldK, the equality

WvK ∩WL = W

ψL/K(v)L holds forv≥ 0, whereψL/K : R≥−1 → R≥−1 is the Hasse-Herbrand

function of the extensionL/K. We shall denote the local Artin reciprocity map ofK by

ArtK : WK → K×,

and the local Artin reciprocity isomorphism ofK by

ArtK∗ : WabK

∼−→ K×.

Finally, letWDK denote the Weil-Deligne group ofK.The collection of all isomorphism classes of continuous complex semisimple represen-

tations ofWK is denoted by M(WK), and the group of all virtual representations ofWK

by R(WK). As usual, R0(WK) denotes the subgroup of R(WK) consisting of all virtualrepresentations ofWK of degree 0.

2. DELIGNE–DWORK–LANGLANDS’ T HEOREM

In this part, we shall first recall the Theorem of Deligne, Dwork, and Langlands aboutthe existence and uniqueness ofε-factors of continuous representations of the Weil groupWK of K, and then list their basic properties. The main reference for this part is [15].

2.1. Existence and uniqueness ofε-factors of continuous representations of the Weilgroup WK of K over C. A fundamental result of Deligne, Dwork, and Langlands states,for each local fieldK, the existence of auniqueassignment

εLanglandsK : (V,ψ) εLanglands

K (V,ψ)

which associates with each pair(V,ψ) consisting of a continuous semisimple representa-tion V of WK overC and a non-trivial additive characterψ : K+ → C

× of K, a non-zero


numberεLanglandsK (V,ψ) ∈ C

×, called Langlands’ localε-factor (i.e. Langlands’ local con-stant) of the pair(V,ψ), such that:

(i) If V andV ′ are equivalent representations ofWK overC, then

εLanglandsK (V,ψ) = εLanglands

K (V ′,ψ);

(ii) If V is a 1-dimensional continuous complex representation ofWK , andχ : WK →C

× is the corresponding quasi-character, then the equality

εLanglandsK (V,ψ) = εTate

K (χ ,ψ)

is satisfied. Here, the numberεTateK (χ ,ψ) ∈ C

× is the abelian local constant ofTate, which is defined by

εTateK (χ ,ψ) = χ∗(Art

−1K (c))

∫UK

χ∗(Art−1K (y))−1ψ(y/c)dµψ(y)

|∫UK

χ∗(Art−1K (y))−1ψ(y/c)dµψ(y) |

,

whereχ∗ : WabK → C

× is the quasi-character ofWabK corresponding to the quasi-

characterχ of WK , dµψ is the additive Haar measure onK+ self-dual with respectto the additive characterψ of K+, andc ∈ K× such thatνK(c) = a(χ)+ n(ψ).Here,a(χ) denotes the exponent of the Artin conductorfArtin(χ) of χ andn(ψ)the largest integern such thatψ(π−nOK) = 1;

(iii) If E is any finite separable extension ofK, then the function

εLanglandsE : (V,ψ ◦TrE/K) εLanglands

E (V,ψ ◦TrE/K)

which associates with each pair(V,ψ ◦TrE/K) consisting of a semisimple repre-sentationV of WE overC and an additive characterψ : K+ → C

× of K, a non-zeronumberεLanglands

E (V,ψ ◦TrE/K) ∈ C

× is “inductive in degree 0 overK”. That is,(a) The map

εLanglandsE (−,ψ ◦TrE/K) : M(WE)→ C

×

is “additive”. Namely, it satisfies

εLanglandsE (V,ψ ◦TrE/K) = εLanglands

E (V ′,ψ ◦TrE/K)εLanglandsE (V ′′,ψ ◦TrE/K),

for each short exact sequence

0→V ′ →V →V ′′ → 0

of semisimple representations ofWE overC;(b) For every tower of finite separable extensionsK ⊆ E ⊆ E′,

εLanglandsE (IndWE

WE′V,ψ ◦TrE/K) = εLanglands

E′ (V,ψ ◦TrE′/K)

for everyV ∈ R0(WE′).

2.2. Basic properties ofεK(V,ψ,dµ). Let ψ : K+ → C

× be a fixed non-trivial additivecharacter ofK, anddµψ the additive Haar measure onK+ self-dual with respect to theadditive characterψ of K+. Then (look at [15]):

(i) εLanglandsK (V,ψa) = (detV)(a)εLanglands

K (V,ψ), for everya∈ K×. Here, fora∈ K×,the additive characterψa : K+ → C

× of K is defined byψa(y) = ψ(ay) for everyy∈ K+;

(ii) εLanglandsK (V ⊗ωs,ψ) = q−s(n(ψ)dim(V)+a(V))

K εLanglandsK (V,ψ), whereqK is the num-

ber of elements of the finite residue-class fieldκK of K. Fors∈ C, ωs : WK → C

×

is the quasi-character defined byωs(w) =| ArtK(w) |sK for everyw∈WK ;

6 K. I IKEDA

(iii) εDeligneK (V,ψ,dµ) = (dµ

dµ ψ)dimVεLanglands

K (V⊗ω− 12,ψ), for any additive Haar mea-

suredµ onK+.

For s∈ C, define

εLanglandsK (s,V,ψ) = εDeligne

K (V ⊗ωs,ψ,dµψ)

for s∈ C. Thus, the identityεLanglandsK (s,V,ψ) = εLanglands

K (V ⊗ωs− 12,ψ) holds fors∈ C.

3. WEIL-DELIGNE REPRESENTATIONS OFWDK OVERC AND THEIR ε -FACTORS

In this part, we shall review theε-factors of Weil-Deligne representations ofWDK overE, whereE is a field of characteristic 0.

3.1. Weil-Deligne representations ofWDK over E. Recall that, ann-dimensional Weil-Deligne representation ofWDK overE is a pair(V,N) consisting of:

(i) a continuous representationρ : WK → GL(V) of WK on ann-dimensional linearspaceV overE, where GL(V) is equipped with the discrete topology;

(ii) a nilpotentE-endomorphismN of V, such that

ρ(w)Nρ(w)−1 =|| w ||K N,

for everyw ∈ WK . Here,|| w ||K=| ArtK(w) |K= qνK(ArtK(w))K = qvK(w)

K for everyw∈WK .

Moreover, a Weil-Deligne representation(V,N) of WDK overE is calledΦ-semisimple, ifρ : WK → GL(V) is a semisimple representation ofWK onV overE in the usual sense.

Given any two Weil-Deligne representations(V,N) and (V ′,N′) of WDK over E, anintertwining operatorf : V → V ′ satisfying f ◦N = N′ ◦ f is called a morphism of Weil-Deligne representations from(V,N) to (V ′,N′). If the intertwining operatorf : V → V ′

satisfying f ◦N = N′ ◦ f is an equivalence of representationsV andV ′, then(V,N) and(V ′,N′) are said to be isomorphic or equivalent. Therefore, we can define the category ofWeil-Deligne representations ofWDK overE, with ⊕, ⊗, and the dual given by

(V,N)⊕ (V ′,N′) = (V ⊕V ′,

(N 00 N′

)), (V,N)⊗ (V ′,N′) = (V ⊗V ′,N⊗1+1⊗N′),

and(V,N)∨ = (V∨,−N∨)

respectively, for any Weil-Deligne representations(V,N) and(V ′,N′) of WDK overE.For (V,N) any Weil-Deligne representation ofWDK overE, let (V,N)ss denote theΦ-

semisimplification of the representation(V,N) of WDK overE, which is defined, followingTate [15], by

(V,N)ss= (Vu−vK ,N),

whereu∈ End(V) is the unique unipotent automorphism ofV commuting withN andWK

and such thatexp(aN)wu−vK(w)

is a semisimple automorphism ofV for everya ∈ E and everyw ∈ WK − IK . Note that(V,N) is Φ-semisimple if and only if(V,N) = (V,N)ss.

A Φ-semisimple Weil-Deligne representation(V,N) of WDK overE is:

– Irreducible if and only ifV is an irreducible representation ofWK over E andN = 0;


– Indecomposable if and only if(V,N) is a “Steinberg representation”; that is,(V,N)has the form(Vo,No)⊗Sp(d), where Sp(d) denotes the special representation ofWDK overE for some 1≤ d∈Z, and(Vo,No) denotes an irreducible Weil-Delignerepresentation ofWDK overE (soVo is an irreducible representation ofWK overE andNo = 0).

Moreover, anyΦ-semisimple Weil-Deligne representation(V,N) of WDK overE is a finitedirect sum

(V,N) = ((V1,N1)⊗Sp(d1))⊕·· ·⊕ ((Vr ,Nr)⊗Sp(dr))

of indecomposable Weil-Deligne representations(Vj ,Nj)⊗Sp(d j) of WDK overE, whereVj is an irreducible representation ofWK overE, Nj = 0, and Sp(d j) is the special repre-sentation ofWDK overE for some 1≤ d j ∈ Z, for eachj = 1, · · · , r for some 1≤ r ∈ Z.

3.2. ε-factors of Weil-Deligne representations overC. Recall that, Langlands’ localε-factor εLanglands((V,N),ψ) of a Weil-Deligne representation(V,N) of WDK over E ⊆ C

with respect to a non-trivial additive characterψ : K+ → C

× is defined by

(3.1) εLanglands((V,N),ψ) = εLanglands(V,ψ).det(−Φ |V IK /V

IKN).

Here,VN denotes the kernel ker(N) of the monodromy operatorN onV and

V IKN = {v∈VN | ρ(w)(v) = v,∀w∈ IK}.

Note that, as a function of(V,N), εLanglands((V,N),ψ) is not additive. Moreover, define

(3.2) εLanglands((V,N),ψ; t) = εLanglands((V,N),ψ).ta((V,N)),

and

(3.3)εLanglands(s,(V,N),ψ) = εLanglands((V,N),ψ; t) |t=q−s

= εLanglands((V,N),ψ).q−s.a((V,N)),

for s∈ C, where the exponenta((V,N)) of the Artin conductorfArtin((V,N)) of the repre-sentation(V,N) of WDK is defined by

a((V,N)) = a(V)+dim(V IK )−dim(V IKN ).

Recall that, the exponenta(V) of the the Artin conductorfArtin(V) of the representationVof WK overE has an explicit expression (also look at [5]) in terms of higher ramificationsubgroupsWs

K of WK as

a(V) =

∫ ∞

−1codim(VWs

K )ds.

It is well known that

(3.4) εLanglands((V,N),ψ) = εLanglands((V,N)ss,ψ)

anda((V,N)) = a((V,N)ss).

So, assume that(V,N) is aΦ-semisimple representation ofWDK overE ⊆ C. Then,V is asemisimple representation ofWK in the usual sense. Let

V =V1⊕·· ·⊕Vk

be the decomposition of the representationV of WK into irreducible representations ofWK .Then, by the additivity ofεLanglands(V,ψ) as a function ofV,

εLanglands(V,ψ) = εLanglands(V1,ψ) · · ·εLanglands(Vk,ψ).

8 K. I IKEDA

Recall that, an irreducible representationV of WK has the formVGal⊗ωs, whereVGal isa Galois type representation ofWK and, for somes∈ C, ωs is the 1-dimensional represen-tation ofWK defined byωs(w) =|| w ||sK , for everyw ∈ WK . Therefore, Langlands’ localε-factorεLanglands(V,ψ) of an irreducible representationV of WK with respect toψ has theform

(3.5)εLanglands(V,ψ) = εLanglands(VGal⊗ωs,ψ)

= εLanglands(VGal,ψ)q−s.[a(VGal)+n(ψ)dim(VGal)]K .

From this identity, it clearly follows that, we have to studyLanglands’ localε-factors ofGalois type representations ofWK (i.e. continuous representations ofGK) with respect toψ.

4. ADMISSIBLE REPRESENTATIONS OFWAK OVERC AND THEIR ε -FACTORS

For automorphic purposes, it is convenient to replaceΦ-semisimple Weil-Deligne rep-resentations ofWDK overC with the admissible representations of the Weil-Arthur groupWAK =WK ×SL(2,C) of K overC.

4.1. Admissible representations of Weil-Arthur group WAK of K over C. Recall that,ann-dimensional representation

ϕ : WAK → GL(n,C)

of WAK overC is called admissible, if (i)ϕ is trivial on an open subgroup of the inertiagroupIK of K; (ii) ϕ(ΦK) is semisimple; and (iii)ϕ |SL(2,C) is an algebraic homomorphism.Two n-dimensional admissible representationsϕ1 andϕ2 of WAK overC are equivalent ifϕ1 andϕ2 are GL(n,C)-conjugate.

Now, Φ-semisimple Weil-Deligne representations ofWDK over C correspond to theadmissible representations of the Weil-Arthur groupWAK = WK ×SL(2,C) of K overCvia a bijection between equivalence classes ofΦ-semisimple Weil-Deligne representations(V,N) of WDK overC and GL(n,C)-conjugacy classes of admissible representations of theWeil-Arthur groupWAK = WK ×SL(2,C) of K overC. One direction of this bijection isdefined, for any admissible representationϕ : WAK → GL(n,C) of WAK overC, by

ϕ (ρϕ ,Nϕ),

where (ρϕ ,Nϕ) is the Φ-semisimple Weil-Deligne representation ofWDK over C de-

fined by Nϕ = dϕ((

0 10 0

)), wheredϕ : sl(2,C) → gl(n,C) is the derivative of the

n-dimensional Lie group representationϕ : SL(2,C)→ GL(n,C) of SL(2,C) overC, andρϕ : WK → GL(n,C) is defined by

ρϕ(w) = ϕ(w)ϕ

((qvK(w)/2

K 0

0 q−vK(w)/2K

)),

for everyw∈WK . For the reverse direction, for anyΦ-semisimple Weil-Deligne represen-tation(V,N) of WDK overC, define the assignment

(V,N) ϕ(V,N),

whereϕ(V,N) : WAK → GL(V) is the admissible representation ofWAK overC, which isdefined by the uniquesl2-triple (e, f ,h) determined by the nilpotent endomorphismN of


V coming from(V,N), by

ϕ(V,N)(w) = exp

(−v(w)

2logqK .h

)ρ(w),

for eachw∈WK , and by the representation of SL(2) onV overC determined by(e, f ,h).Moreover, under this bijective correspondence(V,N) ϕ(V,N), if the Φ-semisimple Weil-Deligne representation(V,N) of WDK overC is:

– Irreducible, that isV is an irreducible representation ofWK overC andN = 0, thenthe corresponding admissible representationϕ(V,N) of WAK overC is V ⊗1;

– Indecomposable, that is(V,N) = (Vo,No)⊗Sp(d), whereVo is an irreducible rep-resentation ofWK overC andNo = 0, and Sp(d) is the special representation ofWDK overC, then the corresponding admissible representationϕ(V,N) of WAK over

C is Vo⊗V(d), whereV(d) is the unique irreducible representation of SL(2,C) ofdimensiond overC

Thus, if (V,N) is anyΦ-semisimple Weil-Deligne representation ofWDK overC, the cor-responding admissible representationϕ(V,N) of WAK overC is

ϕ(V,N) = (V1⊗V(d1))⊕·· ·⊕ (Vr ⊗V(dr )),

where

(V,N) = ((V1,N1)⊗Sp(d1))⊕·· ·⊕ ((Vr ,Nr)⊗Sp(dr))

is the finite direct sum decomposition of(V,N) into indecomposable Weil-Deligne repre-sentations(Vj ,Nj)⊗Sp(d j) of WDK overC, whereVj is an irreducible representation ofWK overC, Nj = 0, and Sp(d j) is the special representation ofWDK overC correspondingto the unique irreducible representationV(d j ) of SL(2,C) of dimensiond j overC for some1≤ d j ∈ Z, for eachj = 1, · · · , r for some 1≤ r ∈ Z.

4.2. ε-factors of admissible representations ofWAK over C. Let ϕ : WAK → GL(V)be an admissible representation of the Weil-Arthur groupWAK of K on ann-dimensionalspaceV overC. Then we set

(4.1) εLanglands(ϕ,ψ) = εLanglands((ρϕ ,Nϕ),ψ),

(4.2) εLanglands(ϕ,ψ; t) = εLanglands((ρϕ ,Nϕ),ψ; t),

and

(4.3) εLanglands(s,ϕ,ψ) = εLanglands(s,(ρϕ ,Nϕ),ψ)

for s∈ C, and further set

a(ϕ) = a((ρϕ ,Nϕ)) andfArtin(ϕ) = fArtin((ρϕ ,Nϕ)).

5. ε -FACTORS ATTACHED TO THE REPRESENTATIONS OF LOCAL ABSOLUTEGALOIS

GROUPS OVERC

In this part, we shall describe Langlands’ localε-factor of a continuous representationof the absolute Galois groupGK of the local fieldK with respect to an additive characterψ : K+ → C

× of K.

10 K. I IKEDA

5.1. The maximal d-abelian extensionKd−ab of K. Let Kd−ab denote the maximal “d-abelian” extension ofK insideKsep. That is,Kd−ab is thed-abelian closure ofK insideKsep defined as the union of all solvable extensions ofK insideKsep of derived length atmostd. Note that,Kd−ab is a Galois extension ofK whose Galois group Gal(Kd−ab/K)

is the maximald-abelian Hausdorff quotient (that is, the “d-abelianization”)GK/G[d]K of

the absolute Galois groupGK of K, whereG[d]K denotes the closure of thed-th derived

subgroup ofGK . Note that, ford ≤ d′, Kd−ab⊆ Kd′−ab and there exists a morphism

ResKd′−ab

Kd−ab : Gal(Kd′−ab/K)→ Gal(Kd−ab/K)

defined by restriction toKd−ab.

5.2. Representations of the local Galois groupGK of K over C. Given a continuousrepresentation

ρ : GK → GL(V)

of the absolute Galois groupGK of the local fieldK on ann-dimensional vector spaceVoverC. As im(ρ) is a finite subgroup of GL(V), there exists a finite Galois extensionKρ/Ksatisfying Gal(Ksep/Kρ) = ker(ρ) and a faithful representation

ρ∗ : Gal(Kρ/K)→ GL(V)

of Gal(Kρ/K) onV overC. The group Gal(Kρ/K) is solvable. Denote the derived lengthof the group Gal(Kρ/K) by ∂K(ρ). For any integerd satisfying∂K(ρ) ≤ d, there exists anatural continuous homomorphism

ResKd−ab

Kρ : Gal(Kd−ab/K)→ Gal(Kρ/K)

defined by restriction toKρ , which in return defines a representation

ρd−ab= ρ∗ ◦ResKd−ab

Kρ : Gal(Kd−ab/K)ResK

d−abKρ

−−−−−→ Gal(Kρ/K)ρ∗−→ GL(V)

of Gal(Kd−ab/K) onV overC. Finally, for ∂K(ρ)≤ d ≤ d′, the following triangle

(5.1) Gal(Kd′−ab/K)ρd′−ab

&&MMMMMMMMMMM

ResKd′−ab

Kd−ab

��

GL(V)

Gal(Kd−ab/K)

ρd−ab

88pppppppppp

is commutative.

5.3. Non-abelian local class field theory.Following [7, 8] and [12] closely, fix an ex-tensionϕK of the Frobenius automorphismFrK of Knr to Ksep. Namely, fix aLubin-Tatesplitting ϕK overK. Thenon-abelian local class field theory of K (in the sense of Koch),constructed in [7, 8, 12], establishes an algebraic and topological isomorphism

ΦΦΦ(ϕK)K : GK

∼−→ ∇(ϕK)

K


between the absolute Galois groupGK of the local fieldK and a certain topological group

∇(ϕK)K , which depends only onK and on the Lubin-Tate splittingϕK overK. The construc-

tion of the topological group∇(ϕK)K involves the theory ofAPF-extensions ofK and the

fields of norms construction of Fontaine and Wintenberger.

Moreover, the isomorphismΦΦΦ(ϕK)K : GK

∼−→ ∇(ϕK)

K is called thenon-abelian local reci-procity map of K, because it is “natural” in the sense that the non-abelian analogues ofthe abelian local class field theoretic properties, such as “existence”, “functoriality”, and acertain “ramification theoretic” property, are all satisfied (look at Section 8 of [6] together

with [8] for detailed account). Following [4], denote the inverseΦΦΦ(ϕK)K

−1: ∇(ϕK)

K∼−→ GK of

the non-abelian local reciprocity isomorphismΦΦΦ(ϕK)K : GK

∼−→ ∇(ϕK)

K by

{•,K}ϕK : ∇(ϕK)K

∼−→ GK ,

and call it thenon-abelian local norm-residue symbol of K.The non-abelian local reciprocity mapΦΦΦ(ϕK)

K : GK∼−→ ∇(ϕK)

K of K induces an isomor-phism

ΦΦΦ(ϕK)K

d−ab: Gd−ab

K∼−→(

∇(ϕK)K

)d−ab,

with inverse

{•,K}d−abϕK

:(

∇(ϕK)K

)d−ab ∼−→ Gd−ab

K ,

for eachd = 1,2,3, · · · . If d = 1, thenΦΦΦ(ϕK)K

1−ab= ArtK∗ : Gab

K∼−→(

∇(ϕK)K

)ab= K× is

the local Artin reciprocity isomorphism ofK “in Galois form” with inverse{•,K}1−abϕK

=

(•,K) :(

∇(ϕK)K

)ab= K× ∼

−→ GabK , which is the norm-residue symbol ofK. Here,K× is the

profinite completion ofK×.

There exist a dense subgroupZ

∇(ϕK)K of ∇(ϕK)

K such that

ΦΦΦ(ϕK)K (WK) = Z

∇(ϕK)K ,

a subgroup1∇(ϕK)K

0of

Z

∇(ϕK)K satisfying the equality

ΦΦΦ(ϕK)K

(G0

K

)=ΦΦΦ(ϕK)

K

(W0

K

)= 1∇(ϕK)

K

0,

and subgroups

(1∇(ϕK)

K

d−ab)0

ofZ

∇(ϕK)K

d−absatisfying the equality

ΦΦΦ(ϕK)K

d−ab(Gd−ab

K0)=ΦΦΦ(ϕK)

K

d−ab(Wd−ab

K0)=

(1∇(ϕK)

K

d−ab)0

,

for d = 1,2,3, · · · . Note that, the natural continuous homomorphismWd−abK → Gd−ab

K is aninjection with dense image, and the higher ramification subgroupsWd−ab

Kv

of Wd−abK are

defined byWd−abK

v=Wd−ab

K ∩Gd−abK

vfor −1≤ v∈ R. In particular, forv= 0, Wd−ab

K0=

Gd−abK

0, asGd−ab

K0=(

GK/G[d]K

)0= IK/G(d)

K for d = 1,2,3, · · · . Note that, ifd = 1, then

ΦΦΦ(ϕK)K

1−ab= ArtK∗ : Wab

K0 ∼−→

(1∇(ϕK)

K

ab)0

=UK .

The following two tables summarize the abelian and the non-abelian local class fieldtheories ofK :

12 K. I IKEDA

Non-abelian local C.F.T. (ϕK fixed)

GK ∇(ϕK)K

WK Z

∇(ϕK)K

W0K 1∇(ϕK)

K

0

WδK , δ ∈ (i −1, i] 1∇(ϕK)

K

i

ab−→

Abelian local class field theory

GabK K×

WabK K×

WabK

0UK

WabK

δ, δ ∈ (i −1, i] U i

K

By [10], if E/K is any finite Galois extension, then there exists a finite unramifiedextensionE′/E such thatE′/K is “compatible” with respect to the Lubin-Tate splittingϕK

overK (that is,E′ sits in the towerK ⊆ E′ ⊂ Kϕd′K

, whered′ = f (E′/K)) and the following

diagram

(5.2) GE′

ΦΦΦ(ϕE′ )

E′

∼//

inc.

��

∇(ϕE′ )

E′

��

N∞

E′/K

��

GEΦΦΦ(ϕK )

K

∼//

inc.

��

ΦΦΦ(ϕK)K (GE)

��

GK∼

ΦΦΦ(ϕK )K

// ∇(ϕK)K

is commutative (look at the proof of the commutativity of thediagram (7.12) in [7]).

5.4. The Haar measure on the compact group(

∇(ϕK)K

)d−ab. LetdµGK denote the unique

Haar measure on the absolute Galois groupGK of the local fieldK. Likewise, letdµGd−abK

denote the unique Haar measure onGd−abK for d = 1,2,3, · · · . Clearly, the Haar measure

dµGK on GK induces a Haar measure on thed-abelianizationGd−abK of GK , which is noth-

ing butdµGd−abK for d = 1,2,3, · · · .

The non-abelian local reciprocity mapΦΦΦ(ϕK)K : GK

∼−→ ∇(ϕK)

K of K and thed-abelianized

reciprocity isomorphismΦΦΦ(ϕK)K

d−ab: Gd−ab

K∼−→(

∇(ϕK)K

)d−abfor eachd = 1,2,3, · · · define

a Haar measuredµ∇K on ∇(ϕK)K and a Haar measuredµ∇d−ab

K on(

∇(ϕK)K

)d−abfor each

d = 1,2,3, · · · , respectively.Now, assume thatE/K is a finite and separable extension. Then, the identity

(GK : GE)dµGK = dµGE

together with the commutative diagram (5.2) yields

(∇(ϕK)K : N

∞E )dµ∇K = dµ∇E ,

whereN ∞E =ΦΦΦ(ϕK)

K (GE).

5.5. Main theorems. Fix a non-trivial additive characterψ : K+ → C

× of the local fieldK.

Let

ρ : GK → GL(V)


be an irreducible continuous representation of the absolute Galois groupGK of K on ann-dimensional vector spaceV overC. Following Section 5.2, for any integerd satisfying∂K(ρ)≤ d, let

ρd−ab : Gd−abK → GL(V)

denote the canonical representation ofGd−abK onV defined byρ : GK → GL(V). Let

χρd−ab : Gd−abK

ρd−ab

−−−→ GL(V)Tr−→ C

be the character of the representationVρd−ab of Gd−abK , and

detρd−ab : Gd−abK

ρd−ab

−−−→ GL(V)det−→ C

×

be the determinant of the representationVρd−ab of Gd−abK . Note that, the homomorphism

detρd−ab : Gd−abK → C

× factors through

(5.3) Gd−abK

detρd−ab//

canonicalmap

��;;

;;;;

;;;;

;;;;

;C

×

(Gd−abK )ab

detρd−ab∗

CC��

where(Gd−abK )ab= Gab

K .For the representationVρ of GK and for any integerd satisfying∂K(ρ)≤ d, introduce

IK(Vρ ,ψ,d,c) =∫

(1∇(ϕK )

K

d−ab)0

detρd−ab

({y,K}d−ab

ϕK

)−1ψ(

c−1ArtKd−ab/K

({y,K}d−ab

ϕK

))dµ∇d−ab

K (y)

and

DK(Vρ ,ψ,d,c) = detρd−ab∗

(Art

−1K (c)

),

wherec= c(Vρ ,ψ)∈K× such thatνK(c)= a(Vρ)+n(ψ). Note that, the integral IK(Vρ ,ψ,d,c)exists and is a non-zero complex number.

Proposition 5.1. The expressionDK(Vρ ,ψ,d,c)IK(Vρ ,ψ,d,c)|IK(Vρ ,ψ,d,c)| does not depend on the choice

of d∈ Z satisfying∂K(ρ)≤ d and c∈ K× satisfyingνK(c) = a(Vρ)+n(ψ).

Proof. Let c,c′ ∈ K× satisfyingνK(c) = νK(c′) = a(Vρ)+n(ψ). Let u∈UK so thatc′ =u.c. Then,

DK(Vρ ,ψ,d,c′) = detρd−ab∗

(Art

−1K (c′)

)

= detρd−ab∗

(Art

−1K (u.c)

)

= detρd−ab∗

(Art

−1K (u)

)DK(Vρ ,ψ,d,c),

14 K. I IKEDA

and

IK(Vρ ,ψ,d,c′) =∫

(1∇(ϕK )

K

d−ab)0

detρd−ab

({y,K}d−ab

ϕK

)−1ψ(

c′−1ArtKd−ab/K

({y,K}d−ab

ϕK

))dµ∇d−ab

K (y)

=

∫

(1∇(ϕK )

K

d−ab)0

detρd−ab

({y,K}d−ab

ϕK

)−1ψ(

c−1u−1ArtKd−ab/K

({y,K}d−ab

ϕK

))dµ∇d−ab

K (y)

=∫

(1∇(ϕK )

K

d−ab)0

detρd−ab

({y,K}d−ab

ϕK

)−1ψ(

c−1ArtKd−ab/K

({u−1y,K}d−ab

ϕK

))dµ∇d−ab

K (y)

= detρd−ab

({u,K}d−ab

ϕK

)−1 ∫

(1∇(ϕK )

K

d−ab)0

detρd−ab

({u−1y,K}d−ab

ϕK

)−1ψ(

c−1ArtKd−ab/K

({u−1y,K}d−ab

ϕK

))d

= detρd−ab∗

(Art

−1K (u)

)−1∫

(1∇(ϕK )

K

d−ab)0

detρd−ab

({u−1y,K}d−ab

ϕK

)−1ψ(

c−1ArtKd−ab/K

({u−1y,K}d−ab

ϕK

))dµ

= detρd−ab∗

(Art

−1K (u)

)−1IK(Vρ ,ψ,d,c),

where the last equality follows from the left-invariance ofthe Haar measuredµ∇d−abK . Thus,

DK(Vρ ,ψ,d,c′)IK(Vρ ,ψ,d,c′)

| IK(Vρ ,ψ,d,c′) |= DK(Vρ ,ψ,d,c)

IK(Vρ ,ψ,d,c)

| IK(Vρ ,ψ,d,c) |,

as detρd−ab∗

(Art

−1K (u)

)−1∈ µµµ∞(Kalg), whereµµµ∞(Kalg) denotes the multiplicative group of

all roots of unity in a fixed algebraic closureKalg of K.Next, let d,d′ ∈ Z such that∂K(ρ) ≤ d ≤ d′. Choose anyσ ∈ GK such that, setting

σd′−ab := σ |Kd′−ab, the equalityσd′−ab |Kab= Art−1K (c) holds. Then,

detρd′−ab

(σd′−ab

)= detρd′−ab

∗

(Art

−1K (c)

),

by the commutativity of the diagram (5.3). On the other hand,this σ ∈ GK also satisfiesσd−ab := σ |Kd−ab= σd′−ab |Kd−ab andσd−ab |Kab=Art

−1K (c), and again by the commutative

triangle (5.3),

detρd−ab

(σd−ab

)= detρd−ab

∗

(Art

−1K (c)

).

Thus, by the commutative diagram (5.1),

detρd′−ab

(σd′−ab

)= detρd−ab

(σd−ab

),

which proves that

DK(Vρ ,ψ,d′,c) = DK(Vρ ,ψ,d,c).

Moreover, changing variables along the continuous (hence measurable) surjection

rd′d : ∇(ϕK)

K

d′−ab։ ∇(ϕK)

K

d−ab


which makes the diagram

(5.4) ∇(ϕK)K

d′−ab∼

{•,K}d′−abϕK

//

rd′d

��

Gd′−abK

ResKd′−ab

Kd−ab

��

∇(ϕK)K

d−ab∼

{•,K}d−abϕK

// Gd−abK

commutative, the following equalities


(1∇(ϕK )

K

d−ab)0

detρd−ab

({y,K}d−ab

ϕK

)−1ψ(

c−1ArtKd−ab/K

({y,K}d−ab

ϕK

))d(rd′

d )∗µ∇d′−abK (y)

=∫

(1∇(ϕK )

K

d′−ab)0

detρd−ab

({rd′

d (y),K}d−abϕK

)−1ψ(

c−1ArtKd−ab/K

({rd′

d (y),K}d−abϕK

))dµ∇d′−ab

K (y)

hold, where the first equality follows from compactness of∇(ϕK)K

d−ab(so the push-forward

measured(rd′d )∗µ∇d′−ab

K on ∇(ϕK)K

d−abunderrd′

d is nothing butdµ∇d′−abK ) and the second

equality is the change of variables formula along the surjection rd′d . Therefore, by commu-

tative diagrams (5.4) and (5.1), and by the fact that

ArtKd−ab/K

(ResK

d′−ab

Kd−ab {y,K}d′−abϕK

)= ArtKd′−ab/K

({y,K}d′−ab

ϕK

)

for everyy∈ ∇(ϕK)K

d′−ab, the equalities


(1∇(ϕK )

K

d′−ab)0

detρd−ab

(ResK

d′−ab


)−1ψ(

c−1ArtKd−ab/K

(ResK

d′−ab


))dµ∇d′−ab

K (y)

=∫

(1∇(ϕK )

K

d′−ab)0

detρd′−ab

({y,K}d′−ab

ϕK

)−1ψ(

c−1ArtKd′−ab/K

({y,K}d′−ab

ϕK

))dµ∇d′−ab

K (y)

= IK(Vρ ,ψ,d′,c),

follow and completes the proof. �

So, by Proposition 5.1, the constant DK(Vρ ,ψ,d,c)IK(Vρ ,ψ,d,c)|IK(Vρ ,ψ,d,c)| dependsonly on the

irreducible representationVρ of GK and on the non-trivial additive characterψ : K+ → C

×

of K.Introduce

eK(V,ψ) := DK(V,ψ,d,c)IK(V,ψ,d,c)| IK(V,ψ,d,c) |

,

whereV is anyirreduciblerepresentation ofGK overC andψ is a fixed non-trivial additivecharacter ofK. Moreover, for anarbitrary continuous representationρ : GK → GL(V) of

16 K. I IKEDA

GK on ann-dimensional vector spaceV overC, with the decompositionVρ =Vρ1 ⊕·· ·⊕Vρm into irreducible subrepresentationsVρ1, · · · ,Vρm of Vρ , define

eK(Vρ ,ψ) :=m

∏j=1

eK(Vρ j ,ψ).

Lemma 5.2. Assume that K⊆ E ⊆ E′ ⊆ E′′ is a tower of finite and separable extensionsof K. Let

χ : GE′′ → GL(C)

be a1-dimensional complex representation of GE′′ . Then

eE′(IndGE′

GE′′χ ,ψ ◦TrE′/K) = eE′′(χ ,ψ ◦TrE′′/K),

whereψ : K+ → C

× is a fixed non-trivial additive character of K.

Assume thatK ⊆ E ⊆ E′ is a tower of finite and separable extensions ofK. Let

r ′1, r′2 : GE′ → GL(V)

be two continuous representations ofGE′ on ann-dimensional spaceV overC. By BrauerInduction Theorem, there areZ-linear combinations

r ′1 =s

∑i=1

ni IndGE′

GE′i

χi

and

r ′2 =s

∑i=1

mi IndGE′

GE′i

τi ,

whereni ,mi ∈ Z (note that 0 is allowed in this setting),GE′i

are certain finite index sub-groups ofGE′ , andχi ,τi : GE′

i→GL(C) are certain 1-dimensional complex representations

of GE′i

for i = 1, · · · ,s. Then,

IndGEGE′

r ′1 = IndGEGE′

(s

∑i=1

ni IndGE′

GE′i

χi) =s

∑i=1

ni IndGEGE′i

χi

and

IndGEGE′

r ′2 = IndGEGE′

(s

∑i=1

mi IndGE′

GE′i

τi) =s

∑i=1

mi IndGEGE′i

τi .

Hence, the identities

eE(IndGEGE′

r ′1,ψ ◦TrE/K)

eE(IndGEGE′

r ′2,ψ ◦TrE/K)=

eE(∑si=1ni IndGE

GE′i

χi ,ψ ◦TrE/K)

eE(∑si=1mi IndGE

GE′i

τi ,ψ ◦TrE/K)

=∏s

i=1eE(IndGEGE′i

χi ,ψ ◦TrE/K)ni

∏si=1eE(IndGE

GE′i

τi ,ψ ◦TrE/K)mi

=∏s

i=1eE′i(χi ,ψ ◦TrE′

i /K)ni

∏si=1eE′

i(τi ,ψ ◦TrE′

i /K)mi


and

eE′(r ′1,ψ ◦TrE′/K)

eE′(r ′2,ψ ◦TrE′/K)=

eE′(∑si=1ni Ind

GE′

GE′i

χi ,ψ ◦TrE′/K)

eE′(∑si=1mi Ind

GE′

GE′i

τi ,ψ ◦TrE′/K)

=∏s

i=1eE′(IndGE′

GE′i

χi ,ψ ◦TrE′/K)ni

∏si=1eE′(Ind

GE′

GE′i

τi ,ψ ◦TrE′/K)mi

=∏s

i=1eE′i(χi ,ψ ◦TrE′

i /K)ni

∏si=1eE′

i(τi ,ψ ◦TrE′

i /K)mi

follow from Lemma 5.2 at once. Therefore,

eE(IndGEGE′

r ′1,ψ ◦TrE/K)

eE(IndGEGE′



eE′(r ′2,ψ ◦TrE′/K),

which proves the following theorem:

Theorem 5.3. Assume that K⊆ E ⊆ E′ is a tower of finite and separable extensions of K.Fix a non-trivial additive characterψ : K+ → C

× of K. Let

r ′1, r′2 : GE′ → GL(V)

be two continuous representations of GE′ on an n-dimensional space V overC. Then,

eE(IndGEGE′

r ′1,ψ ◦TrE/K)

eE(IndGEGE′



eE′(r ′2,ψ ◦TrE′/K).

Note that, the assignment(V,ψ) eK(V,ψ)

which associates with each pair(V,ψ), consisting of a continuous representation ofGK onann-dimensional vector spaceV overC and a non-trivial additive characterψ : K+ → C

×

of K, a non-zero number eK(V,ψ) ∈ C

× has the following properties:

(i) If dim(V) = 1 andχ : GK → C

× is the corresponding quasi-character, then

eK(V,ψ) = εTateK (χ ,ψ),

wheredµ∇abK corresponds to the additive Haar measuredµψ onK+;

(ii) For any finite separable extensionE/K, the mapping

(V,ψ ◦TrE/K) eE(V,ψ ◦TrE/K)

which associates with each pair(V,ψ ◦TrE/K), consisting of a continuous rep-resentation ofGE on ann-dimensional vector spaceV overC and a non-trivialadditive characterψ : K+ → C

× of K, a non-zero number eE(V,ψ ◦TrE/K) ∈ C

×

is inductive in degree 0:(a) Additivity–For any short exact sequence

0→V ′ →V →V ′′ → 0

of continuous representations ofGE,

eE(V,ψ ◦TrE/K) = eE(V′,ψ ◦TrE/K)eE(V

′′,ψ ◦TrE/K);

18 K. I IKEDA

(b) For any towerK ⊆ E ⊆ E′ of finite separable extensions ofK,

eE(IndGEGE′

r ′1,ψ ◦TrE/K)

eE(IndGEGE′



eE′(r ′2,ψ ◦TrE′/K),

wherer ′1, r

′2 : GE′ → GL(V)

are any two continuous representations ofGE′ on ann-dimensional spaceVoverC.

Thus, combining with Deligne, Dwork, and Langlands’ Theorem, next theorem followsimmediately:

Theorem 5.4. Langlands’ localε-factor εLanglandsK (V,ψ) for the pair(V,ψ) consisting of

a continuous representation of GK on an n-dimensional vector space V overC and a non-trivial additive characterψ : K+ → C

× of K is given by

εLanglandsK (V,ψ) = eK(V,ψ).

Recall that, the local root numberWK(V,ψ) of a continuousn-dimensional representa-tionV of GK overC with respect to the non-trivial additive characterψ : K+ → C

× of K isdefined by

WK(V,ψ) =εLanglands

K (V,ψ)

| εLanglandsK (V,ψ) |

.

Thus, the following corollary directly follows from Theorem 5.4.

Corollary 5.5. The local root number WK(V,ψ) of a continuous n-dimensional represen-tation V of GK overC with respect to a non-trivial additive characterψ : K+ → C

× of Kis given by

WK(V,ψ) =DK(V,ψ,d,c)IK(V,ψ,d,c)| DK(V,ψ,d,c)IK(V,ψ,d,c) |

,

where d∈ Z satisfying∂K(V)≤ d and c∈ K× satisfyingνK(c) = a(V)+n(ψ).

6. AN EXPLICIT DESCRIPTION OFε -FACTORS OFWEIL-DELIGNE REPRESENTATIONS

OVERC

Fix a non-trivial additive characterψ : K+ → C

× of K.First, letρ : WK → GL(V) be a continuous semisimple representation ofWK on ann-

dimensional spaceV overC. Let

V =V1⊕·· ·⊕Vk

be the decomposition of the representationV of WK into irreducible representations ofWK .Then, by the additivity ofεLanglands

K (V,ψ) as a function ofV, and by (3.5), the followingidentities

εLanglandsK (V,ψ) = εLanglands

K (V1,ψ) · · ·εLanglandsK (Vk,ψ)

= εLanglandsK (VGal

1 ⊗ωs1,ψ) · · ·εLanglandsK (VGal

k ⊗ωsk,ψ)


1 ,ψ)q−s1[a(V

Gal1 )+n(ψ)dim(VGal

1 )]K · · ·εLanglands

K (VGalk ,ψ)q

−sk[a(VGalk )+n(ψ)dim(VGal

k )]K


1 ,ψ) · · ·εLanglandsK (VGal

k ,ψ)q−∑k

j=1 sj [a(VGalj )+n(ψ)dim(VGal

j )]

K ,

follow immediately. Therefore, combining with Theorem 5.4, we have the following theo-rem:


Theorem 6.1. Langlands’ localε-factor εLanglandsK (V,ψ) for the pair(V,ψ) consisting of

a continuous semisimple representation of WK on an n-dimensional vector space V overC

and a non-trivial additive characterψ : K+ → C

× of K is given by

εLanglandsK (V,ψ) = eK(V

Gal1 ,ψ) · · ·eK(V

Galk ,ψ)q

−∑kj=1 sj [a(VGal

j )+n(ψ)dim(VGalj )]

K ,

where V=V1⊕·· ·⊕Vk is the decomposition of the representation V of WK into irreduciblerepresentations Vj of WK , with Vj = VGal

j ⊗ωsj for some Galois type representation VGalj

of WK and sj ∈ C, for j = 1, · · · ,k.

Recall that, the local root numberWK(V,ψ) of a continuous semisimplen-dimensionalrepresentationV of WK overC with respect to the non-trivial additive characterψ : K+ →C

× of K is defined by

WK(V,ψ) =εLanglands

K (V,ψ)

| εLanglandsK (V,ψ) |

.

Thus, the following corollary directly follows from Theorem 6.1.

Corollary 6.2. The local root number WK(V,ψ) of a continuous semisimple n-dimensionalrepresentation V of WK overC with respect to a non-trivial additive characterψ : K+ →C

× of K is given by

WK(V,ψ) =WK(VGal1 ,ψ) · · ·WK(V

Galk ,ψ)

q−∑k


j )]

K

| q−∑k


j )]

K |




Now, let (V,N) be a Weil-Deligne representation ofWDK over C. Without loss ofgenerality, we can assume that(V,N) is aΦ-semisimple representation ofWDK overC as

εLanglandsK ((V,N),ψ) = εLanglands

K ((V,N)ss,ψ).

Thus, assume that(V,N) is a Φ-semisimple representation ofWDK overC. Then, asVis a semisimple representation ofWK , combining Theorem 6.1 with the definition of theLanglands’ localε-factor εLanglands

K ((V,N),ψ) of the Weil-Deligne representation(V,N)with respect toψ:


K (V,ψ).det(−Φ |V IK /V

IKN),

we obtain the following theorem on explicit description of Langlands’ε-factors of Weil-Deligne representations ofWDK overC with respect toψ:

Theorem 6.3. Langlands’ localε-factorεLanglandsK ((V,N),ψ) for the pair((V,N),ψ) con-

sisting of aΦ-semisimple Weil-Deligne representation(V,N) of WDK overC and a non-trivial additive characterψ : K+ → C

× of K is given by

εLanglandsK ((V,N),ψ)= eK(V

Gal1 ,ψ) · · ·eK(V

Galk ,ψ)q



K .det(−Φ |V IK /V

IKN),




20 K. I IKEDA

Recall that, every indecomposable Weil-Deligne representation (V,N) of WDK overChas the form

(V,N) = (Vo,No)⊗Sp(d),

where(Vo,No) is an irreducible representation ofWDK overC, which means thatVo is anirreducible representation ofWK overC andNo = 0, and Sp(d) is the special representationof WDK overC for some 1≤ d ∈ Z. Then the underlying Weil group representation of(Vo,No)⊗Sp(d) overC is Vo⊕Vo(1)⊕ ·· ·⊕Vo(d−1), whereVo( j) denotesVo⊗ω j forj = 1, · · · ,d−1.

Theorem 6.4. Let (V,N) be an indecomposable representation of WDK overC with thedecomposition


where(Vo,No) is an irreducible representation of WDK overC and Sp(d) is the specialrepresentation of WDK over C for some1 ≤ d ∈ Z. Then, Langlands’ localε-factorεLanglands

K ((V,N),ψ) for the pair ((V,N),ψ), whereψ : K+ → C

× denotes a non-trivialadditive character of K is given by

εLanglandsK ((V,N),ψ)=

(eK(V

Galo ,ψ)q−so[a(VGal

o )+n(ψ)dim(VGalo )]

K

)d.q

−(d−1)d

2 (n(ψ)dim(Vo)+a(Vo))K .det(−Φ |

V IK /VIKN),

where Vo =VGalo ⊗ωso for some Galois type representation VGal

o of WK and for some so ∈C.

Proof. Let (V,N) be an indecomposable representation ofWDK overC with the decompo-sition


where(Vo,No) is an irreducible representation ofWDK overC and Sp(d) is the special rep-resentation ofWDK overC for some 1≤ d ∈ Z. The underlying Weil group representationof (Vo,No)⊗Sp(d) overC isVo⊕Vo(1)⊕·· ·⊕Vo(d−1), whereVo( j) denotesVo⊗ω j forj = 1, · · · ,d−1. Therefore,


K ((Vo,No)⊗Sp(d),ψ)

= εLanglandsK (Vo,ψ)εLanglands

K (Vo(1),ψ) · · ·εLanglandsK (Vo(d−1),ψ).det(−Φ |

V IK /VIKN)

= εLanglandsK (Vo,ψ)εLanglands

K (Vo⊗ω1,ψ) · · ·εLanglandsK (Vo⊗ωd−1,ψ).det(−Φ |

V IK /VIKN)

= εLanglandsK (Vo,ψ)d.q

−(d−1)d


V IK /VIKN),

by part (ii) in Section 2.2. Now, by Theorem 6.1,


K (Vo,ψ)d.q−

(d−1)d2 (n(ψ)dim(Vo)+a(Vo))


IKN)

=(

eK(VGalo ,ψ)q−so[a(VGal


K

)d.q

−(d−1)d


V IK /VIKN),

whereVo =VGalo ⊗ωso for some Galois type representationVGal

o of WK and for someso ∈C. �

Given a Weil-Deligne representation(V,N) of WDK overC. Following [14], define thelocal root numberWK((V,N),ψ) of (V,N) with respect to the additive characterψ : K× →C

× by

WK((V,N),ψ) =εLanglands

K ((V,N),ψ)

| εLanglandsK ((V,N),ψ) |

.


Then, we have the following corollary of Theorem 6.3.

Corollary 6.5. The local root number WK((V,N),ψ) of a Φ-semisimple Weil-Delignerepresentation(V,N) of WDK over C with respect to a non-trivial additive characterψ : K+ → C

× of K is given by

WK((V,N),ψ)=WK(VGal1 ,ψ) · · ·WK(V

Galk ,ψ)

q−∑k


j )]


IKN)

| q−∑k


j )]


IKN) |

,




7. A FORMULA FOR ε -FACTORS OF ADMISSIBLE REPRESENTATIONS OFWAK OVERC

There is an obvious analogue of Theorem 6.3 for admissible representations of the Weil-Arthur groupWAK of K overC.

Theorem 7.1. Langlands’ localε-factor εLanglandsK (ϕ,ψ) for the pair(ϕ,ψ) consisting of

an admissible representationϕ :WAK →GL(V) of WAK on an n-dimensional vector spaceV overC and a non-trivial additive characterψ : K+ → C

× of K is given by

εLanglandsK (ϕ,ψ)= eK(V

Gal1 ,ψ) · · ·eK(V

Galk ,ψ)q



K .det(−Φ |(Vρϕ )IK /(Vρϕ )

IKNϕ),

where(ρϕ ,Nϕ) is theΦ-semisimple Weil-Deligne representation of WDK overC corre-sponding toϕ, and Vρϕ = V1⊕ ·· ·⊕Vk is the decomposition of the representation Vρϕ ofWK into irreducible representations Vj of WK , with Vj = VGal

j ⊗ωsj for some Galois type

representation VGalj of WK and sj ∈ C, for j = 1, · · · ,k.

Let ϕ : WAK → GL(V) be an admissible representation ofWAK on ann-dimensionalvector spaceV overC. Assume that

Vϕ =Vo⊗V(d),

whereVo is an irreducible representation ofWK overC andV(d) is the uniqued-dimensionalrepresentation of SL(2,C) overC. Then, the correspondingΦ-semisimple Weil-Delignerepresentation(ρϕ ,Nϕ) of WDK overC is indecomposable and has the form

(ρϕ ,Nϕ) = (Vo,0)⊗Sp(d),

where(Vo,0) is an irreducible representation ofWDK over C and Sp(d) is the specialrepresentation ofWDK overC for this particular 1≤ d ∈ Z. Then the underlying Weilgroup representationVρϕ of (Vo,0)⊗Sp(d) overC is Vo⊕Vo(1)⊕·· ·⊕Vo(d−1), whereVo( j) denotesVo⊗ω j for j = 1, · · · ,d−1. The analogue of Theorem 6.4 for admissiblerepresentations of the Weil-Arthur groupWAK of K is the following result:

Theorem 7.2. Let ϕ : WAK → Gl(V) be an admissible representation of WAK on an n-dimensional space V overC with the decomposition

Vϕ =Vo⊗V(d)

where Vo is an irreducible representation of WK overC and V(d) is the unique d-dimensionalirreducible representation ofSL(2,C) overC for some1≤ d ∈ Z. Then, Langlands’ local

22 K. I IKEDA

ε-factor εLanglandsK (ϕ,ψ) for the pair (ϕ,ψ), whereψ : K+ → C

× denotes a non-trivialadditive character of K is given by

εLanglandsK (ϕ,ψ)=

(eK(V

Galo ,ψ)q−so[a(VGal


K

)d.q

−(d−1)d


(Vρϕ )IK /(Vρϕ )IKNϕ),

where Vo =VGalo ⊗ωso for some Galois type representation VGal

o of WK and for some so ∈C,and(ρϕ ,Nϕ) is theΦ-semisimple Weil-Deligne representation of WDK overC correspond-ing to ϕ.

Given an admissible representationϕ : WAK → GL(V) of WAK on ann-dimensionalvector spaceV overC. By the definition ofεLanglands

K (ϕ,ψ) given in (4.1), define the localroot numberWK(ϕ,ψ) of ϕ with respect to the additive characterψ : K× → C

× by

WK(ϕ,ψ) =WK((ρϕ ,Nϕ),ψ).

Then, we have the following corollary of Theorem 7.1.

Corollary 7.3. The local root number WK(ϕ,ψ) of an admissible representationϕ :WAK →GL(V) of WAK on an n-dimensional vector space V overC with respect to a non-trivialadditive characterψ : K+ → C

× of K is given by

WK(ϕ,ψ)=WK(VGal1 ,ψ) · · ·WK(V

Galk ,ψ)

q−∑k


j )]


IKNϕ)

| q−∑k


j )]


IKNϕ) |

,

where(ρϕ ,Nϕ) is theΦ-semisimple Weil-Deligne representation of WDK overC corre-sponding toϕ, and Vρϕ = V1⊕ ·· ·⊕Vk is the decomposition of the representation Vρϕ ofWK into irreducible representations Vj of WK , with Vj = VGal

j ⊗ωsj for some Galois type

representation VGalj of WK and sj ∈ C, for j = 1, · · · ,k.

8. ε -FACTORS OFLANGLANDS PARAMETERS

The main reference for this part is [3].

8.1. ε-factors of Weil-Deligne homomorphisms ofWDK in G . Following [3], a triple(ρ ,G ,N) consisting of a complex Lie groupG with reductive identity componentG o, ahomomorphism

ρ : WK → G ,

which is continuous on the inertia subgroupIK of WK andρ(ΦK) semisimple inG , andNa nilpotent element in the Lie algebrag of G , such that

Adρ(w)N =|| w ||K N,

for everyw∈WK is called aWeil-Deligne homomorphism of WDK in G . Two Weil-Delignehomomorphisms(ρ ,G ,N) and(ρ ′,G ,N′) of WDK in G are said to be equivalent, if thereexistsg∈ G o such thatρ ′ = gρg−1 andN′ = Ad(g)N.

Fix an additive characterψ : K+ → C

× of K. For any finite-dimensional representation

r : G → GL(V)

of the Lie groupG on ann-dimensional vector spaceV overC, which is furthermore as-sumed to be algebraic onG o, define Langlands’ local constantεLanglands

K ((ρ ,G ,N), r,ψ) of


the Weil-Deligne homomorphism(ρ ,G ,N) of WDK in G with respect to the representationr of G onV overC and with respect toψ by

εLanglandsK ((ρ ,G ,N), r,ψ) = εLanglands

K

((Vr◦ρ ,N),ψ

),

where (Vr◦ρ ,dr(N)) is an n-dimensionalΦ-semisimple Weil-Deligne representation ofWDK onV overC, becauser ◦ρ(ΦK) is semisimple in GL(V), and

Ad(r ◦ρ)(w)dr(N) =|| w ||K dr(N),

for everyw∈WK . Thus, by equations (3.1), (3.2), and (3.3), define

(8.1) εLanglandsK ((ρ ,G ,N), r,ψ) = εLanglands

K

(Vr◦ρ ,ψ

).det(−Φ |

VIKr◦ρ/(Vr◦ρ )dr(N)IK

),

(8.2) εLanglandsK ((ρ ,G ,N), r,ψ; t) = εLanglands

K ((ρ ,G ,N), r,ψ) ta((Vr◦ρ ,dr(N))),

and

(8.3) εLanglandsK (s,(ρ ,G ,N), r,ψ) = εLanglands

K ((ρ ,G ,N), r,ψ; t) |t=q−s

respectively.Reformulating Theorem 6.3, we get the following theorem:

Theorem 8.1. Langlands’ localε-factorεLanglandsK ((ρ ,G ,N), r,ψ) of the Weil-Deligne ho-

momorphism of WDK in G with respect to the representation r ofG on V overC and withrespect toψ is given by

εLanglandsK ((ρ ,G ,N), r,ψ)= eK(V

Gal1 ,ψ) · · ·eK(V

Galk ,ψ)q



K .det(−Φ |V

IKr◦ρ/(Vr◦ρ )dr(N)IK

),

where Vr◦ρ =V1⊕·· ·⊕Vk is the decomposition of the representation Vr◦ρ of WK into irre-ducible representations Vj of WK , with Vj =VGal

j ⊗ωsj for some Galois type representation

VGalj of WK and sj ∈ C, for j = 1, · · · ,k.

8.2. Admissible homomorphisms ofWAK into G and their ε-factors. Let G be a com-plex Lie group with reductive identity componentG o. A homomorphism

ϕ : WAK → G

is called an admissible homomorphism of the Weil-Arthur group WAK into G , if (i) ϕ istrivial on an open subgroup of the inertia groupIK of K; (ii) ϕ(ΦK) is semisimple; and (iii)ϕ |SL(2,C) is an algebraic homomorphism. Two admissible homomorphisms ϕ1 andϕ2 ofWAK in G are equivalent ifϕ1 andϕ2 areG o-conjugate.

Following the same lines of reasoning of Section 4.1 (look at2.1 of [3] for details),there exists a bijective correspondence

(ρ ,G ,N) ϕ

between the equivalence classes of Weil-Deligne homomorphisms(ρ ,G ,N) of WDK in acomplex Lie groupG and theG o-conjugacy classes of admissible homomorphismsϕ ofthe Weil-Arthur groupWAK of K in G .

Fix an additive characterψ : K+ → C

× of K. For any finite-dimensional representation

r : G → GL(V)

of the Lie groupG on ann-dimensional vector spaceV over C, which is furthermoreassumed to be algebraic onG o, define Langlands’ local constantεLanglands

K (ϕ, r,ψ) of the

24 K. I IKEDA

admissible homomorphismϕ of WAK in G with respect to the representationr of G onVoverC and with respect toψ by

(8.4) εLanglandsK (ϕ, r,ψ) = εLanglands

K

((ρϕ ,G ,Nϕ), r,ψ

),

where(ρϕ ,G ,Nϕ) is the Weil-Deligne homomorphism ofWDK in G corresponding to theadmissible homomorphismϕ of WAK in G , and further set

(8.5) εLanglandsK (ϕ, r,ψ; t) = εLanglands

K

((ρϕ ,G ,Nϕ), r,ψ; t

),

and

(8.6) εLanglandsK (s,ϕ, r,ψ) = εLanglands

K

(s,(ρϕ ,G ,Nϕ), r,ψ

)

respectively.Reformulating Theorem 8.1, we get the following theorem:

Theorem 8.2. Langlands’ localε-factor εLanglandsK (ϕ, r,ψ) of the admissible homomor-

phismϕ of WAK in G with respect to the representation r ofG on V overC and withrespect toψ is given by

εLanglandsK (ϕ, r,ψ)= eK(V

Gal1 ,ψ) · · ·eK(V

Galk ,ψ)q



K .det(−Φ |V

IKr◦ρϕ /(Vr◦ρϕ )

dr(Nϕ )IK),

where Vr◦ρϕ =V1⊕·· ·⊕Vk is the decomposition of the representation Vr◦ρϕ of WK into irre-ducible representations Vj of WK , with Vj =VGal



8.3. ε-factors of local Langlands parameters ofG. The following setting is central inLanglands reciprocity and functoriality principles. Assume that G is a connected reductivealgebraic group over the local fieldK. TheL-groupLG of G is a complex Lie group with areductive connected componentLGo = G overC.

Recall that, an admissible homomorphism

ϕ : WAK → LG

of the Weil-Arthur groupWAK in theL-groupLG of G is called a local Langlands parameterof G. From Section 8.2, there exists a unique Weil-Deligne homorphism(ρϕ ,

LG,Nϕ) ofWDK in LG corresponding to the local parameterϕ.

Now, fix an additive characterψ : K+ → C

× of K. For any finite-dimensional represen-tation

r : LG→ GL(V)

of theL-groupLG of G on ann-dimensional vector spaceV overC, which is furthermoreassumed to be algebraic onLGo, Langlands’ local constantεLanglands

K (ϕ, r,ψ) of a localLanglands parameterϕ : WAK → LG of G with respect to the representationr of LG onV overC and with respect toψ has an explicit expression, which is the content of thefollowing corollary of Theorem 8.2:

Corollary 8.3. Langlands’ localε-factorεLanglandsK (ϕ, r,ψ) of the local Langlands param-

eterϕ : WAK → LG with respect to the representation r ofLG on V overC and with respectto ψ is given by

εLanglandsK (ϕ, r,ψ)= eK(V

Gal1 ,ψ) · · ·eK(V

Galk ,ψ)q



K .det(−Φ |V

IKr◦ρϕ /(Vr◦ρϕ )

dr(Nϕ )IK),


where Vr◦ρϕ =V1⊕·· ·⊕Vk is the decomposition of the representation Vr◦ρϕ of WK into irre-ducible representations Vj of WK , with Vj =VGal



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YEDITEPE UNIVERSITY, DEPARTMENT OF MATHEMATICS, 26 AGUSTOS YERLESIMI , INONU MAH .,KAYISDAGI CAD ., 34755, ATASEHIR, ISTANBUL, TURKEY

E-mail address: [email protected]

ε-FACTORS OF WEIL-DELIGNE REPRESENTATIONS

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