L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Luis Alberto LOMELÍ

On automorphicL-functions in positive characteristic Tome 66, n^o5 (2016), p. 1733-1771.

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ON AUTOMORPHIC L-FUNCTIONS IN POSITIVE CHARACTERISTIC

by Luis Alberto LOMELÍ

Abstract. — We give a proof of the existence of Asai, exterior square, and symmetric square local L-functions, γ-factors and root numbers in characteris- ticp – the case ofp= 2 included. Our study is made possible by developing the Langlands-Shahidi method over a global function field in the case of a Siegel Levi subgroup of a split classical group or a quasi-split unitary group. The resulting automorphicL-functions are shown to satisfy a rationality property and a functional equation. A uniqueness result of G. Henniart and the author allows us to show that the definitions provided in this article are in accordance with the local Langlands conjecture for GLn. Furthermore, in order to be self contained, we include a treatise ofL-functions arising from maximal Levi subgroups of general linear groups.

Résumé. — Nous donnons une preuve de l’existence des fonctionsLlocales d’Asai, extérieur et symétrique carré ainsi que des facteursγetεcorrespondants en caractéristiquep– le casp= 2 étant inclus. Notre étude est possible grâce à la méthode de Langlands-Shahidi sur un corps de fonctions global dans le cas d’un sous-groupe de Siegel Levi d’un groupe classique déployé ou d’un groupe unitaire quasi-déployé. Les fonctionsLqui en résultent satisfont une propriété de rationalité et une équation fonctionnelle. Un résultat d’unicité de G. Henniart et de l’auteur permet de montrer que les définitions données dans cet article sont compatibles avec la conjecture de Langlands locale pour GLn. De plus, afin d’être complet, nous décrivons les fonctionsLprovenant d’un sous-groupe maximal de Levi d’un groupe linéaire général.

Introduction

We develop the Langlands-Shahidi method over a global function field in the case of a Siegel Levi subgroup of a split classical group or a quasi- split unitary group. In particular, we complete the study of exterior and

Keywords:AutomorphicL-funcitons, functional equation, Langlands-Shahidi method, local factors.

Math. classification:11F70, 11M38, 22E50, 22E55.

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symmetric squareL-functions of [9, 20] and establish the existence of Asai L-functions, and related local factors, in positive characteristic; these are uniquely characterized in [10]. The cases treated in this article include an induction step, necessary to develop theLSmethod for the classical groups in positive characteristic [21].

A thorough understanding of automorphic L-functions and related local factors for the classical groups also requires Godement-Jacquet L-func- tions [7, 12] as well as Rankin-Selberg products for representations of GLm

and GLn [13]. Indeed, Godement-Jacquet factors appear intrinsically in our study of the symplectic group and Rankin-Selberg products, which are ubiquitous in the Langlands program, are needed in a crucial multiplicativity property for representations obtained via unitary parabolic induction.

In order to be self contained, our treatise includesL-functions arising from maximal Levi subgroups of general linear groups.

Thanks to a characterization of local factors with G. Henniart [9, 10], we can be certain that our definitions are in accordance with the Langlands conjectures for GL_n in positive characteristic [16, 18].

While one purpose of the article is to complete our understanding of the Siegel Levi case for the split classical groups, the cases involving quasi-split unitary groups are new in positive characteristic. Thus, we establish existence of Asaiγ-factors and prove that they satisfy a global functional equation involving partial L-functions; uniqueness of Asai γ-factors is proved in [10]. Furthermore, AsaiL-functions are important in our study of auto- morphicL-functions in the case of a general maximal Levi subgroup of a unitary group [19]. The following theorem summarizes the global results of this article (see Theorem 7.2); its proof begins on §1 and ends on §7.

Theorem. — Letk be a global function field with finite field of con- stantsFq. Let M be a Siegel Levi subgroup of a split classical group or a quasi-split unitary group. Letπbe a cuspidal automorphic representation ofM(Ak). Then, theL-functionsL(s, π, ri)arenice, i.e., for eachi:

(i) (Rationality)L(s, π, ri)has a meromorphic continuation to a rational function onq^−s;

(ii) (Functional equation)L(s, π, r_i) =ε(s, π, r_i)L(1−s,π, r˜ _i).

Remark. — Over a number field, globalL-functions are said to be nice if they have a meromorphic continuation, are bounded on vertical strips, and satisfy a global functional equation. In the function field case, the rationality ofL(s, π, r_i) implies boundedness on vertical strips away from poles.

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The automorphicL-functions of the theorem are indexed by representa- tionsri depending on the classical group, which we now describe. Let F be a non-archimedean local field of characteristic p and let ψ be a non- trivial character of F. Consider M ' GLn as a Siegel Levi subgroup of G= SO2n+1 or SO2n. Let r= Sym² when G= SO2n+1, and let r=∧² whenG= SO2n.

Given a smooth irreducible representation πof GLn(F), let σbe the `- adicn-dimensional Frobenius semisimple representation of the Weil-Deligne group corresponding toπ under the local Langlands correspondence [18].

The Langlands-Shahidi local coefficientC_ψ(s, π, w₀), defined via intertwining operators of induced representations and the uniqueness property of Whittaker models when π is ψ-generic, allows us to give a definition of γ(s, π, r◦ρn, ψ). In [9], the following equality is established

γ(s, π, r◦ρ_n, ψ) =γ(s, r◦σ, ψ),

where we have a Deligne-Langlands γ-factor on the right hand side. It is the case of a quasi-split even unitary group U2n that leads us to the existence of Asaiγ-factors. Namelyγ(s, π, r_A, ψ), whose existence we prove in this article via theLS method in characteristicp. There is an underlying degree-2 extension. If E/F is a quadratic extension of non-archimedean local fields, the local Langlands correspondence for a smooth irreducible representation π of GLn(E) gives a representation σ of W_E⁰. Let ^⊗I(σ) be the representation of W_F⁰ obtained from σ via tensor induction [5].

Compatibility with the local Langlands conjecture via tensor induction is established in [10], where we show the equality

γ(s, π, r_A, ψ) =γ(s,^⊗I(σ), ψ).

In all of these cases, we obtain only oneL-function indexed byr= Sym²,

∧² or r_A depending on the quasi-split classical group. Next, when π is ψ-generic, the cases of Sp_2n and U2n+1 yield

Cψ(s, π, w0) =γ(s, π, r1, ψ)γ(2s, π, r2, ψ),

where γ(s, π, r1, ψ) is a Godement-Jacquet γ-factor and r2 is either an exterior square or a twisted Asai representation (see 6.2 Theorem⁰).

We take the opportunity to build the theory from the ground up, beginning with abelian local factors of Tate’s thesis, which arise as Langlands- Shahidi local coefficients of semisimple rank one classical groups. The theory of principal series is then presented; essential, since an automorphic representation is class one at almost every place. Local coefficients arise in the global theory via Fourier coefficients of appropriately chosen Eisenstein

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series and satisfy a crude functional equation. It is viaγ-factors, defined by means of the local coefficient, that we defineL-functions and root numbers.

We remark that, recording results for future reference along the way, we are able to provide a thorough and linearly ordered proof of Theorem 7.2 beginning in §1. Also, we take [4] to be an appropriate setting for working with quasi-split classical groups over function fields. In particular, unitary groups are defined over a degree-2 finite étale algebraK over a fieldk.

Let us now give a more detailed description of the contents of the article. In §1, the necessary properties ofγ-factors are gathered in the case of GL₁ over a non-archimedean local field F. Already in this setting, lo- cal L-functions and root numbers, or ε-factors, are defined via γ-factors.

In addition, given a degree-2 finite étale algebraE over F, Langlands λ- factors give a connection betweenγ-factors over E and F. The case of a separable quadratic algebraE'F×F arises when considering quasi-split unitary groups over global fields at split places. In §2, we extend the basic definitions to this general setting. Recall that the local coefficient is defined via intertwining operators of parabolically induced representations and the uniqueness property of Whittaker models. Throughout the article, G is either a general linear group or one of the classical groups SO2n+1, Sp_2n, SO2n, U2n or U2n+1. When Gis a classical group, the theory is studied for a standard Siegel parabolic subgroupP = MN; the Siegel Levi sub- group M is isomorphic to GL_n, except for the case of U_2n+1, where it is isomorphic to GLn×U1.

The semisimple rank one local computations of §3 are key to the Langlands-Shahidi method, they give compatibility of the method for the classical groups with local class field theory. The casesG= SL₂ and SU₃ are mentioned in characteristic zero in [15], where it is shown how to extend these results via restriction of scalars. We provide a straight forward and self contained proof whenG= U₃, including the case of residual characteristic two in the setting of a degree-2 finite étale algebraE overF.

In §4, we fix a system of Weyl group element representatives and make the appropriate normalization of Haar measures. This is done via Shahidi’s approach to Langlands’ lemma, which gives an algorithm to decompose Weyl group elements, and the corresponding unipotent radical, leading to- wards a multiplicativity property of the local coefficient. Notice that we can extend the algorithm to include unitary groups over a degree-2 finite étale algebraE over the local field F. When the parabolic subgroup is a

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Borel subgroup, multiplicativity allows one to reduce the theory of principal series to semisimple rank one cases; it can be seen as the formula of Gindikin and Karpelevič in this setting.

The link to the global theory is made via the theory of Eisenstein series, which is available over a global function field [8, 22, 23]. We continue the discussion on the crude functional equation of [20] in §5 in order to include quasi-split unitary groups (see Theorem 5.1). For appropriately chosen Eisenstein series corresponding to globally generic cuspidal automorphic representations, the global intertwining operator appearing in their functional equation decomposes into a product of local intertwining operators discussed in §2. Whittaker models allow one to compute Fourier coefficients, making the connection to the local coefficient possible. The formula of Casselman-Shalika for Whittaker models of unramified principal series and a formula for the intertwining operator in terms ofL-functions, when it is evaluated at the identity, provide the necessary local results required in the proof of the crude functional equation.

A refinement of the crude functional equation leads to a definition ofγ- factors, given in §6. WhenG= GLm+n, SO2n+1, SO2nand U2n, the crude functional equation satisfied by the local coefficient involves only one partial L-function of a cuspidal automorphic representation ofM(Ak). These cases give the main induction of the Langlands-Shahidi method for the classical groups. WhenG= Sp_2n,n >1, or U_2n+1, the crude functional equation gives rise to two individual functional equations in terms of γ-factors. In

§6.3, we provide all of the properties necessary to uniquely characterize γ-factors (see [9, 10]). In [11], we give the characterization for the case of a general linear group.

We define local L-functions and root numbers, orε-factors, in §7. First, for tempered representations. Then, in general via Langlands classification and analytic continuation. The global functional equation takes its final form in terms of completedL-functions and global root numbers in §7.5.

Letπbe a cuspidal automorphic representation ofM(Ak), then L(s, π, r_i) =ε(s, π, r_i)L(1−s,π, r˜ _i),

where ˜π denotes the contragredient of π and the representations r_i are described in §5.1. This includes exterior square, symmetric square, Asai and Rankin-SelbergL-functions. Notice that the functional equation of Tate’s thesis over a global function field is obtained by studying a Größencharakter viewed as an automorphic representation of the maximal torusM'GL₁ ofG= SL2.

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The approach taken in this article does not make use of Lie algebras, other than to check that our definition ofL-functions agrees with the definition usingL-groups via the adjoint action of^LM on^Lnand the Satake parametrization for unramified principal series. For a general smooth irreducible representation, the results of [9, 10] and the appendix, show that ourγ-factors and related local factors agree with those of the corresponding representation of the Weil-Deligne group via the local Langlands correspondence.

I would like to thank F. Shahidi for his guidance and support, in addition to many interesting mathematical discussions held over the course of several years. I also wanted to take the opportunity to thank G. Henniart for many insightful remarks; joint work on the characterization of γ-factors helped solidify the induction step of the Langlands-Shahidi method for the classical groups. Mathematical conversations with L. Lafforgue and J.-K. Yu were very helpful while writing this paper; I am thankful to them for taking interest in this project. The author also thanks the anonymous referee for making very good observations. Work was begun at Purdue University and continued at the Institut des Hautes Études Scientifiques during the academic year 2010 – 2011, where a first version of the article was produced.

A final version was written while the author was at the Max-Planck Institut für Mathematik and the Instituto de Matemáticas de la PUCV. I am very grateful to these institutions for their hospitality and support.

1. Abelian γ-factors

LetOF be the ring of integers of a non-archimedean local fieldF. LetpF

be the maximal ideal ofOFand letq_F denote the cardinality ofOF/p_F. For every continuous non-trivial character ψ : F → C^× of level l, fix a Haar measure µψ of F such that µψ(OF) = q^l/2_F . Let C_c^∞(F) be the Bruhat- Schwartz space of complex valued locally constant functions on F with compact support. The Fourier transformFψ:C_c^∞(F)→C_c^∞(F) is defined by

Fψ(f)(x) = Z

F

f(y)ψ(xy)dµψ(y).

Given a functionf with domainF anda∈F^×, letf^a denote the function onF defined by the rulef^a(x) =f(ax). Then

(1.1) Fψ^a◦Fψ^b(f) = ab⁻¹

1 2

Ff^−ab⁻¹,

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for everyf ∈C_c^∞(F),a, b∈F^×. Notice that the level ofψ^a isl−ordF(a) andµ_ψa =|a|^1/2_F µ_ψ. Also, every continuous non-trivial additive character ofF is of the formψ^a, for somea∈F^×.

Fix Haar measuresµ^×_ψ onF^× so that dµ^×_ψ(x) = qF

qF−1 dµψ(x)

|x|_F . Thenµ^×_ψ(O_F^×) =µ_ψ(OF).

Given a continuous characterχ:F^×→C^×,f ∈C_c^∞(F), ands∈C, the integral

ζψ(s, χ, f) = Z

F^×

f(x)χ(x)|x|^s_Fdµ^×_ψ(x),

defines a Laurent power series on q_F^−s. These ζ-functions converge and define a rational function onq^−s_F in some right half plane (<(s)>0 ifχ is unitary). Given a continuous multiplicative characterχ, a continuous non- trivial additive characterψ, and a ∈F^×, the abelian γ-factorγ(s, χ, ψ^a) is the unique function on the variables satisfying Tate’s local functional equation

(1.2) ζψ(1−s, χ⁻¹,Fψ^a(f)) =γ(s, χ, ψ^a)ζψ(s, χ, f),

for every f ∈C_c^∞(F). This γ-factor is a rational function on q^−s_F . Equa- tion (1.2), applied twice in combination with (1.1), gives

(1.3) γ(s, χ, ψ)γ(1−s, χ⁻¹, ψ) = 1.

The relationship betweenµψ^a andµψ, together with (1.2), yields (1.4) γ(s, χ, ψ^a) =χ(a)|a|^s−_F ¹²γ(s, χ, ψ).

Root numbers, orε-factors, are defined via the equation (1.5) ε(s, χ, ψ) =γ(s, χ, ψ) L(s, χ)

L(1−s, χ⁻¹),

whereL(s, χ) = 1 ifχ is ramified, andL(s, χ) = (1−q^−s_F χ($_F))⁻¹ifχ is unramified. They satisfy

ε(s, χ, ψ)ε(1−s, χ⁻¹, ψ) = 1, (1.6a)

ε(s, χ, ψ^a) =χ(a)|a|^s−_F ¹²ε(s, χ, ψ).

(1.6b)

Additionally, ifχis unramified andl= 0 thenε(s, χ, ψ) = 1, i.e., (1.7) γ(s, χ, ψ) =L(1−s, χ⁻¹)

L(s, χ) .

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Extend every f ∈C_c^∞(F^×) to a function inC_c^∞(F) by settingf(0) = 0.

ThenC_c^∞(F) = span(1OF, C_c^∞(F^×)), where1^A denotes the characteristic function of a setA⊂F. TakingN withχ_1+pN

F = 1 and settingf =11+p^N_F

in (1.2) yields the formula γ(s, χ, ψ) =

Z

p^l−N_F

χ⁻¹(x)|x|^−s_F ψ(x)dµψ(x).

The integral is zero on small couronnes around 0, and converges as a principal value integral. Therefore, the formula is valid for alls∈C, and it is also possible to write

γ(s, χ, ψ) = Z

F

χ⁻¹(x)|x|^−s_F ψ(x)dµψ(x),

which converges as a principal value integral. Combining this with (3) gives γ(s, χ, ψ)⁻¹=

Z

F

χ(x)|x|^s−1_F ψ(x)dµψ(x)

= (1−q_F⁻¹) Z

F^×

χ(x)|x|^s_Fψ(x)dµ^×_ψ(x).

Given a separable quadratic extension E/F of locally compact non- archimedean fields, let η_E/F : F^× → C^× be the character given by the rule: η_E/F(x) = 1 if x ∈ N_E/F(E^×); η_E/F(x) = −1 if x /∈ N_E/F(E^×).

Define

λ(E/F, ψ) = R

F^×η_E/F(x)ψ(x)dµ^×_ψ(x)

R

F^×ηE/F(x)ψ(x)dµ^×_ψ(x) .

It satisfies

λ(E/F, ψ)λ(E/F, ψ) = 1.

To avoid confusion, given a continuous character χ⁰ : E^× → C^× and a continuous non-trivial characterψ⁰ : E → C^×, write γE(s, χ⁰, ψ⁰) for the correspondingγ-factor. Then

(1.8) λ(E/F, ψ)γE(s, χ◦N_E/F, ψ◦Tr_E/F)

=γ(s, χ|F^×, ψ)γ(s, η_E/Fχ|F^×, ψ).

To simplify notation, it is convenient to writeψ_E=ψ◦TrE/F.

Now, letE'F×F. We can assumeEis the separable quadratic algebra F×F by fixing a basis. For every x= (x₁, x₂)∈ E, let |x|_E =|x₁x₂|_F,

¯

x= (x2, x1), N_E/F(x) =x¯x, and Tr_E/F(x) =x+ ¯x. A smooth character χ : E^× → C^× is of the form χ = χ₁⊗χ₂, with χ₁ and χ₂ continuous characters ofF^×. Complex valued locally constant functionsf ∈C^∞(E^×)

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are extended to complex valued locally constant functionsC^∞(E) by setting f(x) = 0 for x ∈ E −E^×. Embed F diagonally in E, in order to interpretχ|_F× andψ_E =ψ◦Tr_E/F correctly. Let µ_ψ_E =µ_ψ×µ_ψ. Then, equation (1.8) also holds in this case withη_E/F = 1 and λ(E/F, ψ) = 1, i.e.:

γE(s, χ◦NE/F, ψ◦TrE/F) =γ(s, χ1χ2, ψ)².

2. On the local coefficient and the classical groups Given an algebraic groupGdefined over a fieldk, or an algebraA over k,Gwill denote the group of rational points, i.e.,G=G(k) orG(A). Let F be a non-archimedean local field.

2.1. The general linear group

Let Gbe the general linear group GLn1+n2. Let E be either the field F or a degree-2 finite étale algebra over F. Let B = TU be the Borel subgroup ofGwith maximal torusTand unipotent radicalU, whose group of rational pointsB overEconsists of upper triangular matrices. The non- trivial characterψ:F →C^× extends to a character ofU, also denoted by ψ, by settingψ(u) =ψE(u1,2+· · ·+un₁+n₂−1,n₁+n₂), foru= (ui,j)∈U. Here,ψE =ψ◦Tr_E/F ifE =F×F, andψE=ψifE=F. Consider the maximal Levi subgroupM'GLn1×GLn2, whose group of rational points overEis

M =

g1 0 0 g₂

|g₁∈GL_n₁(E), g₂∈GL_n₂(E)

.

LetP=MNdenote the standard parabolic subgroup ofGwith unipotent radicalN.

Given smooth irreducible representations π1 of GLn1(E) and π2 of GLn₂(E), letπbe the representationπ1⊗˜π2with spaceV. Here, ˜π2denotes the contragredient representation ofπ₂. Consider the unitarily induced representation

I(s, π) = ind^G_P(|det(·)|_E^s² π₁⊗ |det(·)|⁻_E^s²˜π₂).

More precisely, let V(s, π) be the space of functions f : G → V satisfying the following properties: there exists a compact open subgroup K of G such that f(gk) = f(g) for every g ∈ G, k ∈ K; and, f(mng) =

|det(m1)|_E^s² π1(m1) ⊗ |det(m2)|⁻_E^s² ˜π2(m2)δ

1 2

P(m)f(g), for every m =

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(m₁, m₂) ∈ M, n ∈ N, g ∈ G, where δ_P denotes the modulus character ofP. Then, I(s, π) is the right regular representation ofGon the space of functions V(s, π).

Elements of the Weyl group W = W(T,G) will be identified with a fixed representative in H(F), where H is the normalizer of T in G and F is embedded diagonally if E =F ×F. This is done globally in such a way that the theory matches the semisimple rank one local theory of §3, and similarly for Haar measures. Letw₀=w_lw_l,Mthe Weyl group element which is the product of the longest Weyl group elementwland the longest Weyl group elementw_l,M with respect to M. We have the character ψ_M ofM, defined in a way which isw0 compatible withψ:

ψM(u) =ψ(w0uw₀⁻¹), u∈U∩M.

LetM⁰ =w0Mw⁻¹₀ , and let P⁰ be the corresponding standard parabolic subgroup ofGwith unipotent radicalN⁰. Given a representationπofM, let w0(π) denote the representation ofM⁰given byπ(w₀⁻¹m⁰w0),m⁰ ∈M⁰. Ifπ isψ_M-generic, thenw₀(π) isψ_M⁰-generic, whereψ_M⁰(u⁰) =ψ_M(w₀⁻¹u⁰w₀), u⁰∈M⁰∩U.

The intertwining operator A(s, π, w₀) : V(s, π) → V(−s, w₀(π)), is defined by

A(s, π, w₀)f(g) = Z

N⁰

f(w₀⁻¹n⁰g)dn⁰,

where the integral converges in a right half plane. In fact, it is a rational operator in the sense of Waldspurger, §IV of [29].

Let C_c^∞(G, V) be the space of locally constant functions with values in the space V of π. For every s ∈ C, there is a surjective map Ps : C_c^∞(G, V)→V(s, π), given by

Psϕ(g) = Z

P

|det(mp,1)|⁻_E^s²|det(mp,2)|_E^s² δ

1 2

P(mp)π⁻¹(mp)ϕ(pg)dp.

Here, everyp∈P can be written asp=mpnp for somemp∈M,np∈N; furthermore,mp is identified with (mp,1, mp,2)∈GLn₁(E)×GLn₂(E) by means of the embedding ofM inside G. We note that the Haar measure dponP is right invariant.

LetKbe a special maximal compact open subgroup ofG. For eachs∈C, consider the functionf_s⁰=Ps(1P w0B∩K). Notice thatψrestricted toM∩U defines a character, which we denote byψM. Then, ifπisψM-generic with Whittaker functionalλψ_M, the function

λψ(s, π)f_s⁰= Z

N⁰

λψ_M(f_s⁰(w₀⁻¹n))ψ(n)dn

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defines a non-zero polynomial function on

q^−s_F , q_F^s (Theorem 1.4 of [20]).

It is possible to use the previous integral to define a Whittaker functional for I(s, π). The local coefficient is defined using the uniqueness property of Whittaker models and is given by

Cψ(s, π, w0) = λψ(s, π)f_s⁰

λψ(−s, w0(π))A(s, π, w0)f_s⁰. It is a non-zero rational function onq_F^−s [20, 24].

2.2. Siegel Levi subgroups of classical groups

Let G = Gn be either a split classical group SO2n+1, Sp_2n, SO2n, or a quasi-split unitary group U2n, U2n+1. A definition of the split classical groups over a non-archimedean local field or a global function field kcan be found in §3 of [20], including the case chark = 2. A definition of the quasi-split unitary groups, suitable for computing with local coefficients, is provided below.

LetK be a degree-2 finite étale algebra over a fieldk. Letθ denote the non-trivial involution of K over k. Write ¯x = θ(x), for x ∈ K. Extend conjugation to elementsg= (gi,j)∈GLm(K), by setting ¯g= (¯gi,j). Given a positive integern, consider

h2n(x) =

n

X

i=1

¯

xix_2n+1−i−

n

X

i=1

¯

x_2n+1−ixi, x∈K²ⁿ,

h_2n+1(x) =

2n

X

i=1

¯

x_ix_2n+2−i−¯x_n+1x_n+1, x∈K²ⁿ⁺¹.

The unitary groupG= U_m, m= 2n orm= 2n+ 1, is defined so that its group of rational points is given by

G={g∈GLm(K)|hm(gx) =hm(x) for everyx∈K^m}.

Remark. — Throughout the article, we assume that split classical groups are defined overkand non-split quasi-split classical groups are obtained via a degree-2 extensionK/k.

For every positive integer n, let Φ_n be then×nmatrix with ij-entries (δi,n−j+1). Fix the Siegel Levi subgroupMofGso that its group of rational

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pointsM is embedded inGas follows:

M =









 g

1

Φntg⁻¹Φn



|g∈GL_n(k)







ifG= SO_2n+1;

M = g

Φntg⁻¹Φn

|g∈GL_n(k)

ifG= Sp_2n or SO_2n;

M =









 g

z

Φntg¯⁻¹Φn



|g∈GLn(K), z∈ker(N_K/k)







ifG= U2n+1;

M = g

Φntg¯⁻¹Φn

|g∈GLn(K)

ifG= U2n.

LetT be the maximal torus whose group of rational points T consists of diagonal matrices, and letB=TUbe the Borel group ofGwith unipotent radical U, whose group of rational points B consists of upper triangular matrices. LetP=MNbe the standard parabolic subgroup ofGwith Levi Mand unipotent radicalN.

If Gis a split classical group, let E be the non-archimedean local field F. If G is a quasi-split unitary group, let E be a degree-2 finite étale algebra overF. HenceM 'GLn(E), unlessG= U2n+1. In this latter case, M ' GLn(E)×E¹, where E¹ = ker(NE/F). Given a smooth irreducible representationπofM, consider the unitarily induced representation (2.1) I(s, π) =

(ind^G_P(|det(·)|^s_Eπ) ifG= SO_2n,Sp_2n or U_2n+1 ind^G_P(|det(·)|_E^s² π) ifG= SO2n+1 or U2n

. In the case of U2n+1, notice that a smooth irreducible representationπofM is of the formπ=π⁰⊗ν, whereπ⁰ is a smooth irreducible representation of GLn(E) andν is a smooth character ofE¹. Also in this case, the character

|det(·)|_E of GL_n(E) is extended to a character of M, trivial onE¹. The space of I(s, π) is denoted by V(s, π). The Weyl group element w0 = wlwl,M will be identified with a fixed representative in G. Notice thatP is self-associate. The intertwining operator A(s, π, w₀) : V(s, π)→ V(−s, w0(π)) is then defined by

A(s, π, w0)f(g) = Z

N

f(w₀⁻¹ng)dn,

There is a surjective mapPs:C_c^∞(G, V)→V(s, π),s∈C, defined by Psϕ(g) =

Z

P

|det(mp)|^−is_E δ

1 2

P(mp)π⁻¹(mp)ϕ(pg)dp,

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where i = 1 or 1/2 depending on (2.1), and each p ∈ P has p = m_pn_p, mp∈M,np∈N. LetK be a maximal compact open subgroup ofGsuch thatG=PK, and consider the functionf_s⁰=Ps(1P w0B∩K).

The non-trivial character ψ :F → C^× extends to a character of ψE of E. As before, it then gives a character of the unipotent upper triangular matrices in GLn(E). This defines a non-degenerate characterψM ofM∩U. LetG1be a classical group of semisimple rank one, and assumeG=Gn is a classical group of the same type asG1 withn >1. Given a non-negative integer m, let Im denote the m×m identity matrix. Then, the group of rational pointsG₁ is identified with a subgroup ofG:

(2.2) G₁'









 I_n−1

g I_n−1



|g∈G₁







⊂G.

The characterψ_M is then extended to a character ψof U so thatψ|G1∩U

agrees with the semisimple rank one definitions of the next section. Then, if πis aψ_M-generic irreducible representation ofM with Wittaker functional λψ_M, the function

λψ(s, π)f_s⁰= Z

N

λψ_M(f_s⁰(w₀⁻¹n))ψ(n)dn,

is an non-zero Laurent polynomial function onq^−s_F (Theorem 1.4 of [20]).

The local coefficient is then given by

C_ψ(s, π, w₀) = λ_ψ(s, π)f_s⁰

λψ(−s, w0(π))A(s, π, w0)f_s⁰. It is a non-zero rational function onq_F^−s [20, 24].

3. The local coefficient for semisimple rank one classical groups

3.1. Definitions

First, letG= U₂or U₃, letEbe either the algebraF×F or letE/F be a separable quadratic extension of non-archimedean local fields. LetE→E, x7→x, denote conjugation on¯ E. IfE=F×F, letβ= (0,1). Assume that E/F is an extension of local fields, then: if char(F) = 2, there is aβ∈ O_E^× with galois conjugate ¯β ∈ O^×_E such that Tr_E/F(β)∈ O_F^×, N_E/F(β)∈ O^×_F, andE =F(β); if char(F) 6= 2, then there is aβ ∈ E such that ¯β =−β andE=F(β). Fix such aβ.

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If G= U3, the elements of the maximal torus T =M are of the form t= diag(a, b,¯a⁻¹),a∈E,b ∈E¹ = ker(N_E/F). The elements ofU are of the form

n(x, z(x, y)) =





1 x z(x, y) 1 x¯

1



, x∈E, y∈F,

wherez(x, y)∈E satisfies the equation TrE/F(z(x, y)) = NE/F(x). In the case of a field extensionE/F, this implies that

z(x, y) =

(y+β(TrE/F(β))⁻¹NE/F(x) if char(F) = 2

1

2N_E/F(x) +βy if char(F)6= 2. IfE =F×F,x= (x1, x2), then

z(x, y) = (y,−y) +βx1x2.

To abbreviate, writez=z(x, y). The non-trivial characterψof F defines a character ofU, also denoted byψ, via the relationship

ψ(n(x, z)) =ψ◦Tr_E/F(x).

Let N_α = {n(x, z(x,0))|x∈E} and N_2α = {n(0, z(0, y))|y∈F}. Then N =NαN2α,Nα∩N2α ={I3}. Notice that, as a topological group,E = F⊕βF. Letµbe the Haar measure onE given by

Z

E

f(x1+βx2)dµ(x1+βx2) = Z

F

Z

F

f(x1+βx2)dµψ(x1)dµψ(x2).

Uniqueness of Haar measures gives

(3.1) µψ_E=cµ.

The measure µψ_E of E defines a Haar measure on Nα, and the measure µψ ofF defines a Haar measure on N2α. Then, fix the Haar measure on N =NαN2α to satisfy dn=c dµ dµψ. IfE =F×F, c= 1. Otherwise,c depends on the level ofψ_E with respect to ψ. If char(F) = 2, the level of ψE equals l and c= 1. It is an exercise to compute the constant c in the case of char(F)6= 2, note that care must be taken for dyadic fields.

If G= SO3, the elements of the maximal torusT =M are of the form t= diag(a,1, a⁻¹). The elements ofU are of the form

n(x) =





1 −2x −x²

1 x

1



, x∈F.

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This includes the case char(F) = 2. The self-dual Haar measure µ_ψ of F defines a Haar measure onN ={n(x)|x∈F}. The characterψofF is then extended to a characterψofU satisfyingψ(n(x)) =ψ(x).

IfG= GL2, SL2, then the elements ofU are of the form n(x) =

1 x 1

, x∈E.

And, the character ψ of F is extended to a character of U by setting ψ(n(x)) =ψE(x).

IfG= U₂, then

n(x) =

1 x 1

, x∈F.

Recall that xis identified with (x, x) ∈ E in the case E = F ×F. The characterψofF is extended to a character ofUby settingψ(n(x)) =ψ(x).

Fix representatives of the Weyl group element w0appearing in the definition of the local coefficient:

(3.2) w0=

0 1

−1 0

, ifG= GL2, SL2, or U2, and

(3.3) w₀=





0 0 1

0 −1 0

1 0 0



, ifG= SO₃, or U₃.

3.2. Compatibility with class field theory

Recall that we take E =F in the caseGis a split classical group, and we letE be a degree-2 étale algebra over F if Gis a quasi-split unitary group. If Gis a general linear group, E can be as in either of these two cases.

Proposition 3.1. — Let χ, χ1, and χ2 be continuous characters of GL₁(E).

• If G = GL2 and π is the smooth representation of T given by π(diag(t₁, t₂)) =χ₁(t₁)χ⁻¹₂ (t₂), then

Cψ(s, π, w0) =γ(s, χ1χ2, ψ).

• If G = SO3 and π is the smooth representation of T given by π(diag(t,1, t⁻¹)) =χ(t), then

Cψ(s, π, w0) =γ(s, χ², ψ).

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• If G = SL2 and π is the smooth representation of T given by π(diag(t, t⁻¹)) =χ(t), then

Cψ(s, π, w0) =γ(s, χ, ψ).

• If G = U₂ and π is the smooth representation of T given by π(diag(t,¯t⁻¹)) =χ(t), then

Cψ(s, π, w0) =γ(s, χ|_F^×, ψ).

• If G = U3 and ν is a continuous character of E¹, extend ν to a character ofE^× via Hilbert’s Theorem 90 (i.e., by setting ν(z) = ν|E¹(¯zz⁻¹) for z ∈ E^×). If π is the smooth representation on T given byπ(diag(t, z,¯t⁻¹)) =χ(t)ν(z), then

C_ψ(s, π, w₀) =λ(E/F, ψ)γ_E(s, χν, ψ_E)γ(2s, η_E/Fχ|_F×, ψ).

Remark. — When Gis a quasi-split unitary group and E = F ×F, embedF diagonally inE. The characterχis then of the formχ₁⊗χ₂, and χ|_F^× is given byχ|_F^×((x, x)) =χ1(x)χ2(x).

Proof. — The case of U3 is presented, with all details given in the case whenE=F×F or E/F is an extension of local fields with char(F) = 2.

With the above notation:

λψ(−s, w0(π))(A(s, π, w0)f_s⁰)

= Z

N

A(s, π, w0)f_s⁰(w₀⁻¹n(u, v))ψE(u)dn(u, v)

= Z

N

Z

N

f_s⁰(w⁻¹₀ n(x, z)w₀⁻¹n(u, v))ψE(u)dn(x, z)dn(u, v).

Using the Bruhat decomposition withz6= 0 gives this equal to Z

N

Z

N−{I3}

f_s⁰(diag(¯z⁻¹,zz¯ ⁻¹, z)n(−x¯zz⁻¹,z)w¯ 0n(−xz⁻¹, z⁻¹)n(u, v))

×ψE(u)dn(x, z)dn(u, v), and noticing that n(−xz⁻¹, z⁻¹)n(u, v) = n(u, v)n(−xz⁻¹, z⁰), for some z⁰∈E, makes this equal to

Z

N

|¯z|^−(s+1)_E χ⁻¹(¯z)ν(z)ψ_E(−xz⁻¹)dn(x, z) λ_ψ(s, π)f_s⁰ .

Therefore

Cψ(s, π, w0)⁻¹= Z

N

|¯z|^−(s+1)_E χ⁻¹(¯z)ν(z)ψE(xz⁻¹)dn(x, z).

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Consider the casesE =F×F or char(F) = 2 (it is an exercise to adapt the following discussion to the remaining case when E/F is an extension of local fields with char(F)6= 2). Notice that z(x,(β+ ¯β)⁻¹x¯xy) = (β+ β)¯ ⁻¹x¯x(y+β). Temporary notation: ifE =F×F and y ∈F, identify y with (y,−y)∈E. Then, the integral

Z

E

Z

F

|¯z(x, y)|^−(s+1)_E χ⁻¹(¯z(x, y))ν(z(x, y))ψE(xz(x, y)⁻¹)dµψ(y)dµψ_E(x) is equal to

Z

F

Z

E

y+β

−s−1

E |x|^−2s−1_E χ⁻¹((β+ ¯β)⁻¹x¯x(y+β))ν(y+β)

×ψE((β+ ¯β)(y+β)⁻¹x¯⁻¹)dµψ(x)dµψE(y).

Hence, after takingx7→(β+ ¯β)(y+β)⁻¹x, Cψ(s, π, w0)⁻¹= c χ⁻¹(β+ ¯β)

Z

F

|y+β|^s−1_E χ(y+β)ν(y+β)dµψ(y)

× Z

E

|x|^−2s−1_E χ⁻¹(N_E/F(x))ψE(x⁻¹)dµψ_E(x), wherec is the constant of equation (3.1). This is introduced by going from the product Haar measure onN to that ofE. Now, notice that

χ⁻¹(β+ ¯β)γ(2s, χ|_F^×, ψ)⁻¹ Z

F

|y+β|^s−1_E χ(y+β)ν(y+β)dµψ(y)

= Z

F

|x|_F Z

F

|yx+βx|^s−1_E χ(yx+βx)ν(yx+βx)ψ((β+ ¯β)x)dµ_ψ(y)dµ_ψ(x)

= Z

F

Z

F

|y+βx|^s−1_E χ(y+βx)ν(y+βx)ψ◦TrE/F(y+βx)dµ_ψ(y)dµ_ψ(x)

=c⁻¹ Z

E

|z|^s−1_E χ(z)ν(z)ψE(z)dµψ_E(z)

=γE(s, χν, ψE)⁻¹.

Also, takingx7→x⁻¹ followed byx7→x, gives¯ Z

E

|x|^−2s−1_E χ⁻¹(N_E/F(x))ψE(¯x⁻¹)dµψ_E(x)

=γE(2s, χ◦N_E/F, ψE)⁻¹

=λ(E/F, ψ)γ(2s, χ|F^×, ψ)⁻¹γ(2s, ηE/Fχ|F^×, ψ)⁻¹, where the Langlandsλ-factor comes from equation (1.8). Thus

Cψ(s, π, w0) =λ(E/F, ψ)γE(s, χν, ψE)γ(2s, ηE/Fχ|_F^×, ψ).

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4. Principal series 4.1. Weyl group elements

For every simple root α, fix a Weyl group element representative wα

in such a way that is in accordance with the semisimple rank one theory.

Specifically, given a classical groupG=Gn (n >2), for each i, 1 6i6 n−1, letα_i be the simple roote_i−e_i+1 in the Bourbaki notation. Embed Gα_i= GL2 inGso that

Gα_i =















 I_i−1

g

I_2n−2i−2

Φ2tg¯⁻¹Φ2

I_i−1







|g∈GL2(E)











⊂G.

Then, the representativeswαi are fixed for 16i6n−1. The wα’s corresponding to the remaining simple roots are given by the embedding of (2.2), unlessG= SO₄. In this latter case, let w_α₁ correspond to the simple root α1=e1−e2as before, and fix

wα₂ =







−1 1 1

−1







for the simple rootα₂=e₁+e₂.

Given an element wof the Weyl group, it has a reduced decomposition into a product of simple roots. The representative ofwgiven by this product is independent of the reduced decomposition.

4.2. Multiplicativity and Haar measures

For future reference, Langlands’ lemma is recalled below for a quasi-split connected reductive groupGdefined over a fieldk. Shahidi’s proof (see §2.1 of [24], originally written for non-archimedean local fields of characteristic zero), provides an algorithm to properly decompose the unipotent radical Nof a parabolic subgroupPofGwith respect to the decomposition of the corresponding Weyl group elements. If Gis viewed as a reductive group scheme, the algorithm further extends to include quasi-split unitary groups with respect to a degree-2 finite étale algebra overk(see 4.4.5 of [4]).