Aspects of Automorphic Induction

Home , Cuspidal representation, Tempered representation

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Edward Michael Belfanti, Jr., B.A. Graduate Program in Mathematics

The Ohio State University 2018

Dissertation Committee: James W. Cogdell, Advisor

Sachin Gautam

Roman Holowinsky c Copyright by

Edward Michael Belfanti, Jr.

2018 Abstract

Langlands’ functoriality conjectures predict how automorphic representations of diﬀerent groups are related to one another. Automorphic induction is a basic case of functoriality motivated by Galois theory. Let F be a local ﬁeld of characteristic 0. The local Langlands correspondence for GL(N) states that there is a bijection between N-dimensional, complex

0 representations of the Weil-Deligne group WF and irreducible, admissible representations of GL(N,F ). Given an operation on Weil group representations, one can ask what the corresponding operation is for representations of GL(N,F ). Automorphic induction is the operation on representations of GL(N,F ) corresponding to induction of representations of the

Weil group. Given a cyclic extension E ⊃ F of degree D, automorphic induction is a mapping of representations of GL(M,E) to representations of GL(MD,F ). Once automorphic induction has been established for local ﬁelds, one can ask if the operation applied at each local component of an automorphic representation produces another automorphic representation.

The automorphic induction problem was ﬁrst considered in detail in [Kaz84]. Kazhdan proved the local automorphic induction map exists in the of M = 1. The next major result was [AC89], where it was shown that a global automorphic induction operation exists for prime degree extensions. The local theory was completed in [HH95]; Henniart and Herb showed that the local automorphic induction map exists for arbitrary M and cyclic extensions of arbitrary degree. Moreover, Henniart showed in [Hen01] that the local automorphic induction map constructed in [HH95] is consistent with induction of Weil group representations and the local Langlands correspondence. Finally, Henniart extended the results of [AC89] in [Hen12] to cyclic extensions of any degree and veriﬁed that the resulting

ii mapping of automorphic representations is consistent with the local lifting of [HH95]. All of these results rely on some version of the trace formula and powerful theorems about

L-functions for GL(N).

The goal of this thesis is to give a different proof of local and global automorphic induction when M = 1 and to emphasize some different aspects of the theory. As with the results previously mentioned, the main technical tool is a trace formula, specifically the full global trace formula of Arthur. The proof relies more on the trace formula and less on

L-functions than previous proofs.

Chapter 1 consists of background and preliminaries needed to state the main theorems on local and global automorphic induction. We also discuss various results that are needed to state Arthur’s trace formula.

In Chapter 2, we give a statement of the relevant trace formulas. We also discuss some computations in general rank that could be used in a generalization of our techniques to the case of M > 1.

The main local and global theorems are established in Chapter 3. As with most trace formula arguments, the proof boils down to an application of linear independence of characters.

iii For my family, friends, and all those who helped

along the way

iv Acknowledgments

A dissertation is the work of many people, some of whom deserve special thanks.

Melissa Grasso has been with me through not one, but two ﬁnal semesters during which

I was convinced I would never graduate. Her love, support and companionship over the past seven years, and through some particularly trying months, has helped me more than I can ever repay. Her own successes and diligent hard work provided the motivation I needed to see this project through to the end; I only hope that the completion of this degree justiﬁes in some small way the innumerable sacriﬁces, big and small, she has made for me. With love, thank you.

My parents, Ed and Teresa, have provided boundless love, generosity and advice, not just these past seven years but the past twenty-nine. I feel fortunate every day for their support, and it is an honor to be their son and share any success I may ﬁnd with them.

Thank you always.

My brothers, Nick and Dom, have provided the support and encouragement that their older brother is supposed to provide. Thank you both for never doubting me.

Thanks to Nana, Papa, and Grandma, for the love and support unique to grandparents.

Thanks to my Grandpa, who I know would be proud of me. Thanks to my aunts and uncles

Amy, Paul and Mary, and my cousins William, Charles and Kate, for a fun Christmas before getting back to the grind. I’ve received many kinds words from my aunts Nancy,

Mary, Donna and Julie, which were especially welcome over the last weekend of work on this document.

I would like to thank Linda James, John Grasso, Jennifer Grasso, Erika James and Kevin

Riopelle, for always making me feel welcome in Sudbury.

v Stephen Eigenmann deserves thanks for always helping me keep my priorities straight.

Thanks to Jack Marczak, with whom I started studying math in high school. Thanks also to Larry Marczak, without whose help I may never have mastered logarithms.

I would like to thank the group chat consisting of Matt Flagg, Eric Kim, Rob Monaco and Tomas Castella for providing needed humor over the past seven years and for keeping me apprised of the news and scores I missed while completing this work. Matt deserves special thanks for his commiseration and support throughout more than ten years of school.

John Lynch and I have been friends ever since we realized we were in so many classes together that it was getting weird for us not to talk. Carney forever.

I would like to thank my friends and former roommates Steph Wilker, Scott Jaﬀee and Matt Jaﬀee, who managed to tolerate me through my early years in grad school, and qualifying exams in particular.

Thanks to Catherine Glover and Lyman Gillispie for their friendship, and for getting me out of my oﬃce. A special thanks to Lyman, for always taking seriously my terrible opinions about books that I don’t understand.

I would like to thank my colleague Yilong Wang for interesting conversations, mathematical and otherwise, and for encouragement over the last few weeks of writing.

Dan Moore, PhD, has been a close friend ever since he got over my gum snapping. His technical support kept me sane at a very strenuous time during the completion of this document. Thanks for the many, let’s say bizarre and enlightening, conversations. We’ll always have that Culver’s on the outskirts of middle of nowhere Indiana. Dan could not have supported me without the support he received from Mariel Colman, who deserves special thanks for her patience with us.

Thanks to Solomon Friedberg, whose advice and encouragement when I was at Boston

College inspired me to go to graduate school, and study representation theory in particular.

Jim Cogdell, my advisor, deserves thanks for many hours of conversation about representation theory, as well as for frantically reading this thesis and providing feedback as I pushed every conceivable deadline. It is impossible to believe this document could exist without his wisdom and advice throughout the research and writing process. He deserves

vi special thanks for helping me through a diﬃcult project and situation not of his design.

vii Vita

2007–2011 ...... Boston College B.A. in Mathematics 2011–Present ...... The Ohio State University Graduate Teaching Associate, Graduate Research Associate Fields of Study

Major Field: Mathematics

Studies in Automorphic Representation Theory: James W. Cogdell

viii Table of Contents

Page Abstract...... ii Dedication...... iv Acknowledgments...... v Vita...... viii

Chapters

1 Preliminaries ...... 1 1.1 L-groups...... 1 1.2 Twisted endoscopy for (GN , ω) ...... 11 1.3 Representations of GN ...... 22 1.4 A substitute for global parameters ...... 44 1.5 Langlands-Shelstad-Kottwitz transfer...... 64 1.6 Local intertwining operators...... 79 1.7 Statement of the main theorems ...... 99

2 Trace formulas ...... 108 2.1 The discrete part of the trace formula ...... 108 2.2 Stabilization...... 114 2.3 Contribution of a parameter ψ ...... 121 2.4 A preliminary comparison...... 128 2.5 The stable multiplicity formula...... 135 2.6 An elliptic computation ...... 146 2.7 Globalizing local representations ...... 153 2.8 Orthogonality relations...... 164

3 The case of M = 1...... 170 3.1 Stable multiplicity when M = 1...... 170 3.2 Orthogonality relations when M = 1...... 171 3.3 Spherical characters and weak lifting...... 173 3.4 Main theorems ...... 178

Bibliography ...... 191

ix Appendices

A Odds and ends ...... 201 A.1 Restriction of scalars and endoscopy ...... 201 A.2 The Langlands classiﬁcation...... 205 A.3 Global induced representations ...... 208 A.4 More on the Local Langlands correspondence ...... 219

x Chapter 1 Preliminaries

We collect various results that are needed for the statement of the trace formula and the main theorems.

1.1 L-groups

In this section, we will introduce the L-groups used throughout. We will start with some basic deﬁnitions and then specialize to our case. We follow the discussion from [Art05, §26],

[Bor79, Ch. I] and [KS99, §§1,2].

1.1.1 Let F be either a local or global field of characteristic 0. Let G be a connected, reductive, linear algebraic group defined over F . Assume in addition that G is quasi-split and that the derived group of G is simply connected. These assumptions will simplify the discussion and include many classical situations. Throughout this work, a “general” group will refer to such a G. A Borel subgroup B ⊂ G and maximal torus T ⊂ B will often be referred to as a pair (B,T ). To each pair (B,T ), the theory of algebraic groups (over an algebraically closed field of characteristic 0) assigns a based root datum

∨ ∨ Ψ(B,T ) = (XT , ∆B,T ,XT , ∆B,T ).

See [Spr79, §2] for a more detailed summary. Here XT is the lattice of characters of T ,

∨ ∆B,T is the set of simple roots of T relative to B, X is the lattice of cocharacters of T and

∨ ∆B,T is the set of simple coroots of T relative to B. Any two pairs (B,T ) and (B1,T1) are

1 conjugate to one another. Moreover, any inner automorphism of G which takes (B,T ) to

(B1,T1) induces an isomorphism between Ψ(B,T ) and Ψ(B1,T1). Thus we obtain

Ψ(G) = (X, ∆,X∨, ∆∨), the canonical based root datum of G.

Unless stated otherwise, for now all homomorphisms of algebraic groups are deﬁned over

F . Let Aut(G) denote the set of automorphisms of G. For g ∈ G, let Int(g) denote the automorphism of G given by

Int(g)(x) = gxg−1, x ∈ G.

Then Int(G) ⊂ Aut(G) will denote the image of the map

g 7→ Int(g): G → Aut(G).

The group Out(G) of outer automorphisms is deﬁned to be the quotient

Out(G) = Aut(G)/ Int(G).

The datum Ψ(G) satisﬁes

Aut(Ψ(G)) ' Out(G).

Recall that a splitting of G is a pair (B,T ) together with a set {Xα : α ∈ ∆} of nonzero vectors in the associated root spaces gα ⊂ g, the Lie algebra of G. Suppose a group Γ acts on G by automorphisms. If the Γ-action preserves the splitting (B,T, {Xα}), we call it an L-action relative to (B,T, {Xα}). A Γ-action is called an L-action if it is an L-action relative to some splitting. For any splitting (B,T, {Xα}), there are isomorphisms

Aut(Ψ(G)) ' Out(G)

'{Automorphisms of G which preserve (B,T, {Xα})}.

See [Spr79, Proposition 2.13] for more details.

2 An action of ΓF = Gal(F /F ) on Ψ(G) can be obtained in the following way. Let

ψ∗ : G → G∗ be an isomorphism of G with a split group G∗ deﬁned over F . Such an isomorphism is of course not necessarily deﬁned over F . Then the map

∗−1 ∗ σ 7→ ψ ◦ σ(ψ ):ΓF → Out(G)

deﬁnes an action of ΓF on Ψ(G) denoted by ρG; it does not depend on ψ∗. By ﬁxing a splitting of G, we obtain an L-action of ΓF on G also denoted by ρG. The datum

Ψ(G)∨ = (X∨, ∆∨,X, ∆)

∨ is the dual root datum of G. The group ΓF acts on Ψ(G) via the dual action of ρG which we denote by ρ∨ . An L-group datum for G is a triple (G, ρ , η ) where G b Gb G

1. Gb is a connected reductive group over C,

2. ρ :Γ → Out(G) is an L-action of Γ on G and Gb F b F b

∨ ηG : Ψ(G) → Ψ(Gb)

is a ΓF -equivariant bijection between canonical based root data. In other words, ηG intertwines the actions ρ∨ and ρ . G Gb

For each G, we ﬁx a ΓF -splitting

spl(Gb) = (B,b T,b {Xbα∨ }) of Gb.AΓF -action on Ψ(Gb) then gives an L-action ﬁxing spl(Gb). We deﬁne the semidirect product

L G = Gb o ΓF via the L-action ρ discussed above. Let W be the Weil group of F . See [Tat79] for the Gb F deﬁnition of WF and its basic properties. Via the natural map WF → ΓF , we obtain the 3 Weil form of the L-group

Gb o WF .

Let E be a ﬁnite extension of F over which G is split. Then the action ρ | is trivial. Gb WE Thus we may also consider the ﬁnite form of the L-group,

Gb o ΓE/F , for a ﬁnite extension E of F over which G is split.

We recall [Tat79, Proposition (1.6.1)].

Proposition. Let iv : F → F v be an F -homomorphism. For each ﬁnite extension E of F in F , let Ev = i(E)Fv be the induced completion of E. There exists a continuous homomorphism θv : WFv → WF such that the following diagrams are commutative.

WFv / ΓFv

θv iv WF / ΓF

E× ∼ W ab v / Ev

nv θv × × ∼ ab E \AE / WE

× The map iv is induced by the embedding iv : F → F v. The map nv sends a ∈ Ev to the class of the idele whose v-component is a and whose other components are 1. The map θv is

0 unique up to conjugation by the kernel WF of the map WF → ΓF .

L L Using any form of the L-group, we obtain an embedding Gv ,→ G which is trivial on

Gb. Indeed, the action of ΓFv on Gbv = Gb is the restriction of the action of ΓF , and likewise for WFv and WF . As in the proposition, the embedding is to be regarded as a map up to 0 WF -conjugacy.

1.1.2 Let P(GF ) be the parabolic subgroups of G deﬁned over F and let P(G) be the parabolic subgroups of G deﬁned over F . Let p(GF ) be the conjugacy classes of parabolic

4 subgroups in P(GF ). (Conjugacy relative to G(F ) and G(F ) is the same.) Let p(G) be the conjugacy classes of parabolic subgroups in P(G). Since G is quasi-split, p(GF ) is the same as the set of conjugacy classes of parabolic subgroups (deﬁned over F ) that are stable under

ΓF . There is a canonical bijection between p(G) and subsets of ∆. Since G is quasi-split, p(GF ) corresponds to ΓF -stable subsets of ∆.

Let Pb ⊂ Gb be a parabolic subgroup and consider the normalizer NLG(Pb). A parabolic subgroup of LG is such a normalizer that meets every class in LG modulo Gb. In other words, for NLG(Pb) to be a parabolic subgroup, the map NLG(Pb) → WF must be surjective. Let p(LG) be the set of conjugacy classes of parabolic subgroups of LG. Then the correspondence

∨ ∨ ∆ ↔ ∆ induces a bijection between ΓF -stable subsets of ∆ and ∆ and thus a bijection

L p(GF ) ↔ p( G).

Let P ∈ P(GF ) and let M ⊂ P be a Levi subgroup deﬁned over F . Let Bb be the Borel subgroup determined by the dual root datum Ψ(G)∨ = Ψ(Gb). Let LP ⊃ LB be the standard parabolic subgroup in the class associated to P . Then LM can be identiﬁed with a Levi subgroup of LP , namely the normalizer in LP of a Levi subgroup Mc ⊂ Pb.

1.1.3 We will let GN denote the general linear group GL(N) regarded as a matrix group deﬁned over F . We ﬁx the standard pair (BN ,TN ) consisting of the group BN of upper triangular matrices and the group TN of diagonal matrices in GN . Let

χi : TN → Gm, 1 ≤ i ≤ N, and

λi : Gm → TN , 1 ≤ i ≤ N, denote the standard characters and cocharacters of TN . More precisely,

χi(diag(t1, . . . , tN )) = ti and

λi(t) = diag(1, . . . , t, . . . , 1)

5 with t in the ith place. Then

N N ! M N−1 M N−1 Ψ(BN ,TN ) = Zχi, {χi − χi+1}i=1 , Zλi, {λi − λi+1}i=1 . i=1 i=1

Let GbN = GL(N, C). The pair (BbN , TbN ), the datum Ψ(BbN , TbN ) and the characters χbi and cocharacters λbi are all deﬁned similarly. ∨ Since GN is split, the action of ΓF on Ψ(GN ) is trivial. Thus the dual action on Ψ(GN ) is also trivial. By taking the trivial action on Ψ(GbN ), we see that

η = λ 7→ χ : X∨ → X GN i bi b

is a ΓF -equivariant bijection. The triple (GbN , triv, ηGN ) is an L-group datum for GN .

1.1.4 We recall some notions about induced groups and restriction of scalars. All of this material (and additional details) is from [Bor79, §§I.4,I.5]. Let W be a group and let W 0 ⊂ W be a subgroup of ﬁnite index. Suppose W 0 acts on a group M by automorphisms. Then we let

W W 0 0 0 0 IndW 0 (M) = IW 0 (M) = {f : W → M : f(w w) = w · f(w), w ∈ W , w ∈ W }.

W The group W acts on IW 0 (M) by right translation:

(wf)(w1) = f(w1w).

For the moment we let F ⊂ E be a finite extension in F . Let GE be a connected reductive E-group and set H = RE/F GE. By definition H is the functor from F -algebras to sets defined by the property that if A is an F -algebra then

H(A) = GE(A ⊗F E).

See [PR94, §2.1.2] for more details, in particular for the fact that H is represented by a linear algebraic group deﬁned over F , also denoted by H. We will only use the interpretation

σ of H as a linear algebraic group. Let σ be an automorphism of F . Let GE be the algebraic

6 group deﬁned by applying σ to the polynomials that deﬁne GE. Then

Y H(F ) ' IΓF (G (F )) ' Gσ (F ). ΓE E E σ∈ΓE \ΓF

The group ΓF acts on the right hand side by permuting the factors.

∨ ∨ Fix a pair (BE,TE) in GE and let Ψ(BE,TE) = (XE, ∆E,XE, ∆E) be the associated root datum. Fix a pair (BH ,TH ) in H with associated based root datum Ψ(BH ,TH ) = (X , ∆ ,X∨ , ∆∨ ). Fix in addition an isomorphism ψ : H → Q Gσ . This iso- H H H H σ∈ΓE \ΓF E morphism is not generally deﬁned over F . Then (Q Bσ , Q T σ) is a pair in σ∈ΓE \ΓF E σ∈ΓE \ΓF E Q Gσ . We may choose ψ to take (B ,T ) to this pair. We have σ∈ΓE \ΓF E H H

XH = {χE ◦ σ ◦ ψ : χ ∈ XE, σ ∈ ΓF },

∆H = {αE ◦ σ ◦ ψ : αE ∈ ∆E, σ ∈ ΓF },

∨ −1 ∨ XH = {ψ ◦ σ ◦ λE : λE ∈ XE, σ ∈ ΓF },

∨ −1 ∨ ∨ ∨ ∆H = {ψ ◦ σ ◦ αE : αE ∈ XE, σ ∈ ΓF }.

The group Γ acts on Q Gσ , and composition with σ in the sets above denotes F σ∈ΓE \ΓF E composition with this action. There are isomorphisms

X ' IΓF (X ) ΓE E and

X∨ ' IΓF X∨. ΓE E

Let ρE denote the action of ΓE on GbE, described in 1.1.1, used to form the L-group L GbE. Consider the group IΓF G ΓE bE obtained by inducing ρE. The action of ΓF will be denoted by ρF . It is isomorphic (as a group) to the product of [E : F ] copies of GbE. A choice of a pair (BbE, TbE) with associated

∨ ∨ ΓF root datum (Xb , ∆b , Xb , ∆b ) induces a pair and datum on IΓ GbE by taking TbE BbE ,TbE TbE BbE ,TbE E the same pair and the same datum on each factor. In fact, the character lattice can be

7 identiﬁed with IΓF X as above. Let η denote the Γ -equivariant bijection between X∨ ΓE bE E E E and X . Then η induces a Γ -equivariant bijection between IΓF (X∨) and IΓF (X ). Thus bE E F ΓE E ΓE bE (IΓF G , ρ , η ) is an L-group datum for H. ΓE bE F F

The group GbE embeds in Hb diagonally. This embedding ﬁts into a commutative diagram

L {1} / GbE / GE / ΓE / {1}

L {1} / Hb / H / ΓF / {1}.

(The map ΓE → ΓF is the identity embedding.)

If JE ⊆ ∆E is ΓE-stable then

[ J = {αE ◦ σ : αE ∈ JE}

σ∈ΓE/F is ΓF -stable. The map JE 7→ J is a bijection from the ΓE-stable subsets of ∆E to the

ΓF -stable subsets of ∆. The map PE 7→ RE/F PE is a bijection from the set of parabolic subgroups of GE deﬁned over E to the set of parabolic subgroups of H deﬁned over F ; it induces a bijection between the conjugacy classes of parabolic subgroups. In addition there

L L is a bijection p GE → p H .

Since PE is a Borel subgroup of GE if and only if RE/F PE is a Borel subgroup of H

(indeed, the map JE 7→ J preserves inclusions), the group H is quasi-split over F if and only if GE is quasi-split over E.

1.1.5 The situation of interest is E ⊃ F a cyclic extension of degree D and GE the group

GL(M) viewed as an E-group, denoted by GM,E. Throughout, we will let HM = RE/F GM,E. If A is an F -algebra then

HM (A) = GM (E ⊗F A).

In particular, we have HM (F ) = GM (E). Let N = MD. A choice of F -basis for E gives an embedding GM (E) ,→ GN (F ). The image of this embedding deﬁnes a concrete F -group which represents HM .

The standard pair (BE,TE) in GM,E gives a standard pair (RE/F BE, RE/F TE) in HM .

8 We will denote this standard pair by (BHM ,THM ).

In our basic case, the group HbM can be identiﬁed as

HbM = GL(M, C) × · · · × GL(M, C), with D factors in the product. We regard HbM as a standard Levi subgroup of GbN . Fix a generator σ of ΓE/F . Then σ acts on HbM by a cyclic permutation:

σ · (s1, . . . , sD) = (sD, s1, . . . , sD−1).

Since HM is split over E, the action of ΓE on HbM is trivial. The action of ΓF on HbM is also by cyclic permutations via the map ΓF → ΓE/F . Similar remarks hold for the action of

WF on HbM , namely it factors through the map WF → ΓE/F .

1.1.6 Let us consider the simple example of a quadratic extension E/F with H1 = √ RE/F G1,E and G2. Assume that E = F ( θ). Then H1 is represented by the subgroup of

G2 given by elements of the form   a bθ   m(a, b) =   . b a √ The eigenvalues of m(a, b) are a ± b θ. The character lattice of H1 is spanned by

√ m(a, b) 7→ a ± b θ.

Neither character is deﬁned over F . The nontrivial element σ of the Galois group ΓE/F = hσi × × acts on the character lattice by permuting the two generators. The dual group Hb1 is C ×C . L The Galois group acts by σ · (s1, s2) = (s2, s1). There is an L-homomorphism from H1 to

L G2 given by    k α 1 k     k (α, β) o σ 7→     × σ . β 1

This L-homomorphism will be discussed in more detail in 1.2.6.

9 1.1.7 Assume for the moment that F is global and let v be a place of F . Consider HM,v, the group HM viewed as a group over Fv ⊃ F . Namely, if Av is an Fv-algebra then we have

HM,v(Av) = HM (Av).

We deﬁne an Fv-algebra Ev by setting

Ev = E ⊗F Fv.

Then Y Ev = Ew. w|v

The product is over the places w of E lying over v. As groups over Fv, we have

Y HM,v = REv/Fv GM,Ev = REw/Fv GM,Ew . w|v

Indeed, it suﬃces to look at the points of the groups in any Fv-algebra F ⊂ Fv ⊂ Av. By deﬁnition, HM,v(Av) = RE/F GM (Av) = GM (E ⊗F Av), while REv/Fv GM,Ev (Av) = GM (Ev ⊗Fv Av).

Thus the ﬁrst equality would follow from the fact that E ⊗F Av = Ev ⊗Fv Av. But this is clear, since

Ev ⊗Fv Av = (E ⊗F Fv) ⊗Fv Av = E ⊗F (Fv ⊗Fv Av) = E ⊗F Av.

For the second equality, note that    Y GM (Ev ⊗Fv Av) = GM  Ew ⊗Fv Av w|v Y = GM (Ew ⊗Fv Av) w|v Y = REw/Fv GM (Av). w|v

Recall that a place v of F is inert in E if Ev is a ﬁeld. In other words, there is only one 10 place, w, lying over v and HM,v = REw/Fv GM,Ew . At the other extreme is a split place v which has D places lying over it. In this case, HM,v is isomorphic to a product of copies of

GM,Fv It is thus isomorphic to a Levi subgroup of GN,Fv . A general place is a combination of these two extremes.

1.1.8 Recall that a reductive group over a number field F is said to be unramified at a finite place v if

1. Gv is quasi-split over Fv and

2. Gv is split over an unramiﬁed extension of Fv.

Let v be a place of F which is unramified in E. Then Ew ⊃ Fv is an unramified extension for all w dividing v. The composite over all w lying over v of the fields Ew is an unramified extension of Fv over which HM,v is split. Indeed, the composite splits each factor of Q w|v REw/Fv GM . Consequently HM is unramified at the unramified places of F in E. (Note that HM is quasi-split over F by the discussion in 1.1.4.)

1.2 Twisted endoscopy for (GN , ω)

1.2.1 We continue to let F denote a local or global ﬁeld of characteristic 0. In [KS99, (2.1)], twisted endoscopic data are associated to a triplet (G, θ, a). In their level of generality, G is a connected reductive group over F and θ is an F -automorphism of G. If F is local, then a

1 is an element in H (WF ,Z(Gb)) and determines a character of G(F ). If F is global, then a 1 is taken to be in the quotient of H (WF ,Z(Gb)) modulo the subgroup of everywhere locally trivial elements and determines a character of G(F )\G(A). We will recall the deﬁnition from [KS99, §2.1] restricted to the special case of interest. We will only seriously be concerned with endoscopic data for the triplet (GN , id, aω), where aω determines a character ω satisfying

× × × ker ω = NE/F E in the local case and ker ω = NE/F E \AE in the global case. Such data will be known as (GN , ω)-data.

1 If H is a group with a map p : H → WF and a represents a ∈ H (WF ,Z(Gb)), we also let a denote the pullback to H along p. For our purposes we can and will always assume 11 that θ = id, the identity map on G.

Deﬁnition. Fix a cocycle a that represents a. A tuple (H, H, s, ξ) is an endoscopic datum for (G, id, a) if the following conditions hold.

1. H is a quasi-split group over F .

2. H is a split extension of WF by Hb such that the L-action ρH of WF on Hb determined by this extension coincides with ρ . Hb

3. s is a semisimple element of Gb.

4. ξ : H → LG is an L-homomorphism satisfying the following two conditions.

(a) Int(s) ◦ ξ = a0 · ξ for a 1-cocycle a0 that is equivalent to a if F is local and

everywhere locally equivalent to a if F is global.

0 (b) ξ restricts to an isomorphism of Hb with (Gbs) , the identity component of the

centralizer of s in Gb.

1.2.2 Consider the second condition. According to the deﬁnition, there is an exact sequence

p 1 → Hb → H → WF → 1 and a continuous homomorphism

c : WF → H such that p ◦ c : WF → WF is the identity map. We may deﬁne an action of WF on Hb by

ρH(w) = Int c(w).

We thus obtain a map

ρH : WF → Out(Hb) and the requirement is that this map coincides with the action ρ used to form LH, see Hb 1.1.1.

12 In the fourth condition, that ξ is an L-homomorphism just means that

ξ H / LG

p ! } WF commutes. 0 An endoscopic datum (H, H, s, ξ) is called elliptic if ξ(Z(Hb)ΓF ) ⊂ Z(Gb). 0 0 0 0 An isomorphism from (H, H, s, ξ) to (H , H , s , ξ ) is an element gb ∈ Gb such that

−1 0 0 1. gξb (H)gb = ξ (H ) and

−1 0 2. gsb gb = s modulo Z(Gb).

See [KS99, pp.15-16] for this deﬁnition and a discussion in a more general context. In

−1 particular, if gb ∈ Gb then (H, H, s, ξ) and (H, H, gsb gb , Int(gb) ◦ ξ) are isomorphic. We let

AutG(H) = Aut(G,id,a)(H, H, s, ξ) denote the automorphisms of (H, H, s, ξ) as an endoscopic datum for (G, id, a). Let gb be an isomorphism. We use ξ and ξ0 to identify H and H0 with subgroups of LG. Under this identiﬁcation, we have

−1 0 gbHgb = H .

Let

β : Hb → Hb 0 be the isomorphism obtained by restricting Int(gb) to Hb. We have ﬁxed F -splittings of H and H0 in 1.1.1. Then let α : H → H0 be the unique F -isomorphism dual to β−1 that preserves these splittings. It turns out that α is actually deﬁned over F , see [KS99, (2.1.7)].

According to the previous paragraph, an automorphism gb in Aut(H, H, s, ξ) determines automorphisms β ∈ Aut(Hb) and α ∈ AutF (H). Let Had be the image of H in AutF (H) under h 7→ Int h. We set

OutF (H) = AutF (H)/Had(F ).

13 The map gb 7→ α is a well-deﬁned homomorphism from Aut(H, H, s, ξ) → OutF (H). The image of this homomorphism will be denoted by Out(H, H, s, ξ).

We let E(G, a) denote the isomorphism classes of endoscopic data. As is customary, we will often write H for the datum (H, H, s, ξ). We will see shortly that this convention does not cause any confusion in our primary case of (GN , ω)-data.

1.2.3 The following lemma is useful in particular for studying the endoscopic groups of GN and related groups. An ordinary datum for a group G is a datum for the triplet (G, id, 1).

Lemma. Suppose G1 and G2 are groups representing ordinary endoscopic triplets. Then

E(G1 × G2) = E(G1) × E(G2).

Proof. Suppose for i = 1, 2 that (Hi, Hi, si, ξi) is an endoscopic datum for Gi. Then

(H1 × H2, H1 × H2, s1 × s2, ξ1 × ξ2) is a datum for G1 × G2.

Assume now that (H, H, s, ξ) ∈ E(G1 × G2). Then

Hb = (G\1 × G2)s = (Gb1 × Gb2)s = Gb1,s1 × Gb2,s2 .

−1 Consider ξ (Gbi,si ) ⊂ Hb. It carries an action of WF via ρH. According to [Spr94], there is −1 a group Hi whose L-group is given by ξ (Gbi,si ) with the action of WF coming from ρH. L Then (Hi, Hi, ξ, si) is a datum in E(Gi).

1.2.4 Suppose that F is a local ﬁeld of characteristic 0 and consider the general linear group GN . Then, since the action of WF on GbN is trivial,

1 × × × H (WF ,Z(GbN )) = Hom(WF , C ) ' Hom(F , C ).

The second isomorphism is provided by the canonical class ﬁeld theory map

ab × WF → WF ' F .

We always assume that this map is normalized so that a geometric Frobenius in WF is mapped to a uniformizer of F . See [Tat79] for a better summary. The trivial class in

14 1 H (WF ,Z(GbN )) is just the trivial character. Let

× × ω : F → C

× be a character determining the extension E ⊃ F . That is, ker ω = NE/F E . Then we take a = aω, the pullback of ω to WF . Suppose now that F is global. We need to consider

1 1 H (WF ,Z(Gb))/{a ∈ H (WF ,Z(Gb)) : av is trivial for all v}.

See part 4a) of Deﬁnition 1.2.1 for this condition. Again, we have

1 × × × × H (WF ,Z(GbN )) = Hom(WF , C ) ' Hom(F \AF , C ).

As above, the isomorphism is provided by the canonical class ﬁeld theory map

ab × × WF → WF ' F \AF .

The subspace of everywhere locally trivial elements is just the trivial character of WF into

× C . Again, if × × × ω : F \AF → C

× × is a character with ker ω = NE/F (E \AE), we set a = aω equal to the pullback of ω to WF .

1.2.5 The following material on endoscopic groups is also discussed in [HI12]. Let

  F ×, if F is local of characteristic 0, CF = × ×  F \AF , if F is a number ﬁeld.

Let ω be a character of CF of degree D and let aω be the corresponding character of WF .

ω Let E (N) be the set of equivalence classes of (GN , id, aω)-endoscopic data. We will continue to refer to such data as (GN , ω)-data. We will describe the tuples in this set. The most relevant endoscopic groups for the trace formula are the elliptic ones. As we will see, the set

Eω(N) is empty unless D divides N. Thus we will assume implicitly that this is the case and set N = MD. The main purpose of this section is to establish the following proposition.

15 Proposition. There is a unique class of elliptic (GN , ω)-endoscopic data. The class can be

L represented by the group HM with H = HM and ξHM a homomorphism that restrict to an embedding of HbM as a standard Levi subgroup of GbN .

1.2.6 Consider HM and its L-group

L HM = GL(M, C) × · · · × GL(M, C) o WF .

There is a canonical map

WF → WF /WE ' ΓE/F = hσi.

The subgroup WE acts trivially on HbM and the quotient ΓE/F acts by cyclic permutations, see 1.1.5.

i(w) We write the image of w ∈ WF under the quotient map WF → hσi as σ . The integer i(w) is deﬁned modulo D and satisﬁes i(w1w2) = i(w1) + i(w2). Let ω(σ) = ζ. Then ζ is a primitive Dth root of unity. More generally, we have

i(w) aω(w) = ζ .

Set

D−1 s = diag(zbM , ζzbM , . . . , ζ zbM ) ∈ GbN , zbM ∈ Z(GL(M, C)).

Let   1  M  A = Aσ,M =   . 1M(D−1)

Then conjugation by A gives the action of σ on HbM :

−1 A(gb1,..., gbD)A = (gbD, gb1,..., gbD−1), gb ∈ GL(M, C).

In particular we have AsA−1 = ζ−1s. Deﬁne

L L ξHM : HM → GN

16 by

(bh, w) 7→ (bhAi(w), w).

L Lemma. The tuple (HM , HM , s, ξHM ) is a (GN , ω)-endoscopic datum.

Proof. Only the properties of ξHM need to be checked. The map χHM is over WF . To simplify the notation, we use superscripts to denote the WF -action. To see that ξHM is a homomorphism, note that

w1 ξHM (bh1, w1)(bh2, w2) = ξHM (bh1bh2 , w1w2)

w1 i(w1w2) = (bh1bh2 A , w1w2)

i(w1) −i(w1) i(w1)+i(w2) = (bh1A bh2A A , w1w2)

i(w1) i(w2) = (bh1A bh2A , w1w2)

i(w1) i(w2) = (bh1A , w1)(bh2A , w2)

= ξHM (bh1, w1)ξHM (bh2, w2).

L For the ﬁfth equality, recall that GbN = GbN × WF is a direct product. We will check the last property of Deﬁnition 1.2.1. First,

−1 i(w) −1 sξHM (bh, w)s = (sbhA s , w)

= (sbhζi(w)s−1Ai(w), w)

= (ζi(w)bhAi(w), w)

= aω(w)ξHM (bh, w).

Second, since i(1) = 0 we have ξHM (bh, 1) = (bh, 1) and we have ξHM (HbM ) = GbN,s since aω(1) = 1.

We will refer to the datum of the lemma as the standard datum for (GN , ω).

1.2.7 Since it will play an important role later and is useful for understanding endoscopic

L data, let us describe the automorphisms of our standard datum (HM , HM , s, ξHM ).

L Lemma. The group Out(HM , HM , s, ξHM ) can be identiﬁed with the Galois group ΓE/F . 17 Proof. Let g be an automorphism of (H , LH , s, ξ ). By deﬁnition, b M M HM

gξ (LH )g−1 = ξ (LH ) b HM M b HM M

and

−1 gsb gb = zsb

D−1 for some zb = diag(α, . . . , α) ∈ Z(GbN ). Recall that s = diag(zbM , ζzbM , . . . , ζ zbM ). Suppose zbM = diag(β, . . . , β). We have that s and zsb are conjugate. In particular, they have the D−1 same eigenvalues. The eigenvalues of s are β, ζβ, . . . , ζ β, while the eigenvalues of zsb are αβ,ζαβ,...,ζD−1αβ. Consequently, for some j we have β = αβζj. Thus α itself is a Dth root of unity, say α = ζk. Then

−1 −k k gsb gb = A sA ,

so that

Akg ∈ G = ξ (H ) b bN,s HM bM

and

g ∈ ξ (H )hAi ' H hAi. b HM bM bM o

The map gb 7→ α from Aut(H, H, s, ξ) → Out(H, H, s, ξ) is described in 1.2.2. The image −k of the automorphism gb considered here is σ , viewed as an automorphism of HM . Thus L Out(HM , HM , s, ξHM ) can be identiﬁed with a subgroup of ΓE/F . Since we may take g = Ak, for any k, the group Out(H , LH , s, ξ ) contains σk for any k. b M M HM

L We will write OutN (HM ) for the group Out(HM , HM , s, ξHM ). The notation will be justiﬁed once we prove Proposition 1.2.5.

1.2.8 Let k X LE = GM1,E × · · · × GMk,E, Mi = M, i=1 be a standard diagonal Levi subgroup of GM,E. Then

LHM = RE/F LE = RE/F GM1,E × · · · × RE/F GMk,E

18 is a Levi subgroup of HM . We will refer to such Levi subgroups of HM as standard. Let

HMi = RE/F GMi,E be a factor of the standard Levi subgroup. Associated to HMi is the standard datum (H , LH , s , ξ ) for the pair (G , ω). (Recall that the choice of s Mi Mi i HMi MiD i is not unique. We ﬁx any choice for right now).

We have that

LbHM = HbM1 × · · · × HbMk .

L Recall that ρ is the L-action of WF on HbMi used to form HMi . Then we may form HbMi L LHM via the diagonal action ρ of WF on LbHM : LbHM

ρ (w)(bhM1 ,..., bhMk ) = (ρHM (w)bhM1 , . . . , ρHM (w)bhMk ). LbHM 1 k

The group LGN = GM1D × · · · × GMkD is a Levi subgroup of GN with L-group the direct product GbM1D × · · · × GbMkD × WF .

Let

L L ξL : LGN → GN

be the L-embedding then identiﬁes LbGN with block diagonal matrices in GbN . We may deﬁne an L-homomorphism

ξ : LL → LL LHM HM GN by setting

ξL (hM ,..., hM ) w = (ξH (hM ), . . . , ξH (hM )) × w. HM b 1 b k o M1 b 1 Mk b k

L Then (L , L , s1 × · · · × s , ξ ) is a (L , ω)-datum. Now, we choose the si so that HM HM k LHM GN if s denotes the image of s1 × · · · × sk under ξL, we have Gbn,s = (ξL ◦ ξLH )(LbHM ). GbN GbN M L Then (LH , LH , s , ξL ◦ ξL ) is a (GN , ω)-datum. Such a datum will also be referred M M GbN HM to as standard. We will write ξ in place of ξ ◦ ξ when discussing standard data. LHM L LHM

Lemma. Let (H, H, s, ξ) be an arbitrary (GN , ω)-datum. Then (H, H, s, ξ) is isomorphic to a standard datum. Said diﬀerently, every class of endoscopic data in Eω(N) contains a 19 standard datum.

This shows in particular that Eω(N) is empty unless D divides N.

× Proof. We will ﬁrst analyze s. Recall that aω : WF → C is the pullback to WF of the × × character ω : F → C . We also write aω for the pullback to H of aω via the map p : H → WF . Let hσ ∈ H be an element which satisﬁes aω(hσ) = ζ.

L Write ξ(hσ) ∈ GN as bhσ × wσ. Then

−1 sξ(hσ)s = aω(hσ)ξ(hσ)

implies

−1 sbhσs = ζbhσ

and thus s is conjugate to ζs. Let

k Y ni ps(x) = (x − si) i=1 be the characteristic polynomial of s. We assume that if i 6= j then si 6= sj. Then we have

k k Y ni Y ni ps(x) = (x − si) = (x − ζsi) = pζs(x). i=1 i=1

n Consider the factor (x − s1) 1 . We claim that, for m ≥ 0, the polynomial ps(x) is exactly

m n m−1 n divisible by (x − ζ s1) 1 . Assume ps(x) is exactly divisible by (x − ζ s1) 1 , say sj =

m−1 n1 m n1 ζ s1. Then (x − ζsj) = (x − ζ s1) is an exact divisor of pζs(x). Thus it is an exact divisor of ps(x). Relabeling the si if necessary, we see that the distinct eigenvalues of s are

D−1 D−1 D−1 s1, ζs1, . . . , ζ s1, s2, ζs2, . . . , ζ s2, . . . , sk, ζsk, . . . , ζ sk.

j For each i and j, the eigenvalue ζ si occurs with multiplicity ni. Since (H, H, s, ξ) is

−1 isomorphic to (H, H, gsb gb , Int(gb) ◦ ξ) for any gb, the datum (H, H, s, ξ) is conjugate to a datum with semisimple element

D−1 D−1 diag(s1In1 , . . . , ζ s1In1 , . . . , skInk , . . . , ζ skInk ).

20 From now on we assume s is in this form. In particular, we have

k Y GbN,s = (GL(ni, C) × · · · × GL(ni, C)) = ξ(Hb). i=1

For each i there are D factors of GL(ni, C). Consequently we have n1 + ··· + nk = M.

Thus there is a standard Levi subgroup LHM corresponding to this partition and the standard datum (L , LL , s, ξ ◦ ξ ). Our claim is that the given datum (H, H, s, ξ) HM HM L LHM is isomorphic to this standard datum. By deﬁnition, the semisimple elements are the same.

According to the deﬁnition of 1.2.2, we only need to show that ξ(H) = ξ (LL ). We LHM HM already have that ξ(H) = ξ (L ). We will prove the left side is contained in the right b LHM bHM side. The proof in the other direction is the same.

Recall from Deﬁnition 1.2.1 that there is a homomorphism c : WF → H which splits H.

In particular, H = c(WF )Hb. Let wE ∈ WE and consider c(wE) ∈ H. Then

−1 sξ(c(wE))s = aω(wE)ξ(c(wE)) = ξ(c(wE)),

since WE is the kernel of aω. This implies that ξ(c(wE)) centralizes s, namely

ξ(c(w )) ∈ G × w = ξ (L × w ). E bN,s E LHM bHM E

Let wσ ∈ WF be any element which generates the quotient WF /WE = ΓE/F . Since

−1 sξ(c(wσ))s = ζξ(c(wσ)) we have

−1 ξ(c(wσ)) sξ(c(wσ)) = ζs.

Thus ξ(c(wσ)) acts by cyclic permutation on s in the blocks. In each block, this action is given by a matrix Aσ,ni . Consequently ξ(c(wσ)) ∈ Gbs · diag(Aσ,n1 ,...,Aσ,nk ) as desired.

Now we can prove Proposition 1.2.5.

L Proof of Proposition 1.2.5. Suppose ( H, H, s, ξ) is an elliptic (GN , ω)-datum. According to the previous lemma, we may assume that this is a standard datum. According to the

21 construction of standard data, we have

Z(Hb)ΓF = (F ×)k,

where k is number of factors in LbHM . The elliptic condition of 1.2.2 forces k = 1. Thus H = D−1 HM . Moreover, we may take s to be the semisimple element diag(IM , ζIM , . . . , ζ IM ).

1.2.9 Let LHM be a Levi subgroup of HM as above. The construction of standard data proceeds by ﬁrst realizing LHM as an ω-endoscopic group for a corresponding Levi subgroup

LGN of GN and then embedding LbGN into GbN as a standard Levi subgroup. There is another approach which it will be useful to discuss brieﬂy. According to the lemma, the data constructed according to this process are all conjugate to standard data.

Let λ = λ be the embedding realizing L as a standard diagonal Levi subgroup LHM HM L L L in HM . Dual to λ is an L-embedding λ that realizes LHM as a Levi subgroup of HM . Then the map

L L L ξHM ◦ λ : HM → GN

is an L-embedding that, for an appropriate choice of s, realizes LHM as an ω-endoscopic dataum for GN . It is equivalent to the standard datum represented by LHM .

1.3 Representations of GN

In this section we review, following [Art13, §1.3], the local and global representation theory of GN .

1.3.1 Suppose that F is a local ﬁeld of characteristic 0. We set

  WF , if F is archimedean, LF =  WF × SU(2), if F is non-archimedean.

Let φ be a ﬁnite dimensional, Frobenius-semisimple, continuous representation of LF . (Such a representation is then actually semisimple.) We may associate to φ a local L-function

L(s, φ) and local ε-factor ε(s, φ, ψF ). Here ψF is a nontrivial additive character of F . If F

22 is archimedean, the definitions of L(s, φ) and ε(s, φ, ψF ) can be found in [Tat79, §3]. Now consider the non-archimedean case. Recall that both SU(2) and SL(2, C) have a unique isomorphism class of irreducible representations of each dimension. (An explicit realization of these representations is discussed at 1.3.20, but we don’t need it immediately.) Using this fact, from the representation φ of WF × SU(2), we obtain a corresponding representation of WF × SL(2, C), also denoted by φ. Let | − | be the absolute value on WF obtained by pulling back the absolute value on F × via the class field theory map discussed in 1.2.4. We define a representation of WF by    |w|1/2    µφ(w) = φ w,   , w ∈ WF . |w|−1/2

There is a nilpotent matrix Nφ that satisﬁes    1 1    exp(Nφ) = φ 1,   . 1

The pair Vφ = (µφ,Nφ) is a representation of the Weil-Deligne group in the sense of [Tat79, §4]. Then we set

−s L(s, φ) = Z(Vφ, qF ) and

−s ε(s, φ, ψF ) = ε(Vφ, qF ), where the objects on the right side are deﬁned as in [Tat79, §4].

Suppose G is arbitrary and r is a ﬁnite dimensional representation of LG, see [Bor79,

§2.6]. For each irreducible representation π of G(F ), it is expected that there will be local factors L(s, π, r) and ε(s, π, r, ψF ). One case where these factors are known to exist is when

G = GN1 × GN2 and t r(gb1, gb2): X 7→ gb1Xgb2

is the representation of GbN1 × GbN2 on (N1 × N2)-dimensional complex matrices. These are

23 the local Rankin-Selberg factors

L(s, π1 × πe2) = L(s, π, r) and

ε(s, π1 × πe2, ψF ) = ε(s, π, r, ψF )

for representations

π = π1 ⊗ π2

of GN1 (F ) × GN2 (F ). See [JPS83], [Sha84], [Sha85] and [MW89, Appendice] for these Rankin-Selberg factors. See also [Bor79, §12] for a general discussion and some other examples.

1.3.2 We continue to let G be as in 1.1.1 and let F be local. (Recall that this includes the assumption that G is quasi-split.) Consider L-homomorphisms

L φ : LF → G that are semisimple and continuous. (Here semisimple means that the projection of φ(w) in

Gb is semisimple for each w ∈ WF .) The set of such homomorphisms taken up to Gb-conjugacy is denoted by

Φ(G) = {equivalence classes of φ under Gb-conjugacy}.

The set of equivalence classes of irreducible admissible representations of G(F ) is denoted by

Π(G) = {equivalence classes of irreducible admissible representations of G(F )}.

We have chains of subsets

Φ2,bdd(G) ⊂ Φbdd(G) ⊂ Φ(G) and

Π2,temp(G) ⊂ Πtemp(G) ⊂ Π(G)

24 deﬁned as follows [Art13, p.17].

L Φbdd(G) = {φ whose image in G projects onto a relatively compact subset in Gb}.

L L Φ2(G) = {φ whose image lies in no P ( G}.

Φ2,bdd(G) = Φ2(G) ∩ Φbdd(G).

Πtemp(G) = {tempered representations of G(F )}.

Π2(G) = {essentially square-integrable representations of G(F )}.

Π2,temp(G) = Πtemp(G) ∩ Π2(G).

L Consider the special case of GN . We let Φ(N) = Φ(GN ). Since GN = GL(N, C) × WF , the set Φ(N) can be identiﬁed with the equivalence classes of N-dimensional, semisimple, continuous representations of LF . The subsets deﬁned above simplify in this case:

Φ2(N) = {N-dimensional, irreducible, semisimple, continuous representations of LF } and

Φbdd(N) = {N-dimensional, semisimple, continuous, unitary representations of LF }.

Indeed, a semisimple representation is irreducible if and only if it does not factor through a parabolic subgroup. We will often write Φsim(N) in place of Φ2(N). For the second set, note that any compact subset of GbN is contained in a maximal compact subgroup, which in this case is the complex unitary group. Thus we are justiﬁed in writing Φunit(N) = Φbdd(N). Assume F is non-archimedean. Then there are additional subsets deﬁned as follows.

Πscusp(N) = {supercuspidal representations in Π2(N)}.

Πscusp,temp(N) = {supercuspidal representations in Π2,temp(N)}.

Msim(N) = {members of Φsim(N) that are trivial on SU(2)}.

Msim,bdd(N) = {members of Φsim,bdd(N) that are trivial on SU(2)}.

25 Assume F is archimedean. Then we deﬁne

Πscusp,temp(N) = Π2,temp(N) = Πtemp(N),N = 1.

Πscusp(N) = Π2(N) = Π(N),N = 1.

Πscusp,temp(N) = Πscusp(N) = ∅, N > 1.

Thus we have deﬁned parallel chains

Msim,bdd(N) ⊂ Φsim,bdd(N) ⊂ Φbdd(N) ⊂ Φ(N) (1.3.3) and

Πscusp,temp(N) ⊂ Π2,temp(N) ⊂ Πtemp(N) ⊂ Π(N). (1.3.4)

1.3.5 The following theorem, the local Langlands correspondence for GN , is due to [Lan89] in the archimedean case and [HT01], [Hen00] and [Sch13] in the non-archimedean case.

Consider ﬁrst the special case when N = 1. Then Φ(N) = Φ(1) consists of characters of LF . Such characters are in bijection with characters of F × = Π(1) as in 1.2.4. If χ ∈ Φ(1), we will write χ for the corresponding character of F × also. A character of F × determines a character of GL(N,F ), χ ◦ det, via composition with the determinant.

Theorem (Local Langlands correspondence for GN ). There is a unique family of bijections

φ 7→ πφ : Φ(N) → Π(N) satisfying the following properties.

1. When N = 1, the bijection is given by class ﬁeld theory as described above and in

1.2.4.

2. φ ⊗ χ 7→ πφ ⊗ (χ ◦ det), χ ∈ Φ(1) = Π(1).

3. det ◦φ 7→ ηπφ where ηπφ is the central character of π.

∨ ∨ 4. φ 7→ (πφ) .

5. L(s, π1 × π2) = L(s, φ1 ⊗ φ2). 26 6. ε(s, π1 × π2, ψF ) = ε(s, φ1 ⊗ φ2, ψF ).

In addition, the bijections are compatible with the chains (1.3.3) and (1.3.4) for all N.

In the non-archimedean case, this additional fact is contained in [Zel80], and in particular

§10, where Zelevinsky shows how to deduce the full local Langlands correspondence from the correspondence between supercuspidal representations of GL(N) and N-dimensional, irreducible representations of the Weil group. The archimedean case is part of [Lan89]. See in particular item (v) of Section 3 there.

Note that Πtemp(N) ( Πunit(N) for N ≥ 2. Thus Πunit(N) does not correspond to

Φunit(N). The set Πunit(N) is not in bijection with a naturally deﬁned set of parameters. However, we have

Π2,temp(N) = Π2(N) ∩ Πunit(N) = Π2,unit(N).

1.3.6 We need to consider some subgroups and vector spaces. Let F be a number ﬁeld and let G be a general group.

Let AG be the maximal split torus in the center of G. Let

∗ X (G)F = HomF (G, Gm) be the abelian group of F -rational characters of G. Then we have the associated real vector space

∗ aG = Hom (X (G)F , R) .

We let a∗ be its dual space and let a∗ be its complex dual space. There is a canonical G G,C homomorphism

HG : G(A) → aG

deﬁned by setting

∗ hHG(x), χi = log |χ(x)|, x ∈ G(A), χ ∈ X (G)F .

For future reference, we note that if F0 is a local field and G0 is defined over F0, then the local analogs X∗(G ) , a , a∗ and a∗ can be defined by replacing every instance of F 0 F0 G0 G0 G0,C

27 with F0. In the local case, the map HG0 goes from G0(F0) to aG0 by the same formula as the global map HG.

1 We denote the kernel of HG by G(A) . Let GQ = RF/QG and set

A+ = A ( )0. G,∞ GQ R

This is the central subgroup of G(A) deﬁned by the image of the map

A ( )0 → G(F ⊗ ) ⊂ G( ). GQ R Q R A

+ The restriction of the homomorphism HG to AG,∞ deﬁnes an isomorphism

+ ∼ HG : AG,∞ → aG

and we have

1 + 1 G(A) = G(A) × AG,∞ ' G(A) × aG.

Thus there is an isomorphism

2 + ∼ 2 1 L (G(F )AG,∞\G(A)) → L (G(F )\G(A) ).

1 The important property here is that G(F )\G(AF ) has ﬁnite volume. We will need to use a diﬀerent central subgroup later. The most general situation, as at [Art13, p.122] is to consider a closed subgroup XG ⊂ Z(G(A)), the center of G(A). We require that the product XGZ(G(F )) be closed and co-compact in Z(G(A)). Then the quotient

G(F )XG\G(A) = G(F )\G(A)/XG

+ has ﬁnite volume. In particular, the group AG,∞ satisﬁes these properties. Fix a character χ on the quotient

Z(G(F )) ∩ XG\XG = Z(G(F ))\Z(G(F ))XG.

Then we will consider the space

2 L (G(F )\G(A), χ)

28 of χ-equivariant functions on G(F )\G(A) that are square-integrable modulo XG. Taking + 2 1 XG = AG,∞ and χ the trivial character gives a space isomorphic to L (G(F )\G(A) ).

The central character datum (XG, χ) induces a central character datum for each Levi +,G + subgroup L of G. Write AL,∞ for the kernel in AL,∞ of the composition

+ HL AL,∞ → aL → aG.

+,G Then we can pull the character χ back to XL = (AL,∞)XG to obtain the central character datum (XL, χ) for L.

2 1 1 1.3.7 The space L (G(F )\G(A) ) carries an action R(−) = RG(−) of G(A) by right translation:

2 1 1 (R(g)f)(x) = f(xg), f ∈ L (G(F )\G(A) ), g, x ∈ G(A) .

Recall that

2 1 2 1 Lcusp(G(F )\G(A) ) ⊂ L (G(F )\G(A) )

2 1 consists of the functions f ∈ L (G(F )\G(A) ) with the following property. For every 1 parabolic subgroup P ⊂ G and almost every x ∈ G(A) , the function f satisﬁes Z f(nx) dn = 0. NP (F )\NP (A)

Here NP denotes the unipotent radical of P .

2 1 2 1 The space Ldisc(G(F )\G(A) ) is deﬁned to be the subrepresentation of L (G(F )\G(A) ) which decomposes as a direct sum of irreducible subrepresentations. We obtain a chain of

1 right G(A) -invariant subspaces

2 1 2 1 2 1 Lcusp(G(F )\G(A) ) ⊂ Ldisc(G(F )\G(A) ) ⊂ L (G(F )\G(A) ).

The ﬁrst inclusion is a well-known fact originally due to Gelfand and Piatetski-Shapiro. See

[GP63]. With a character χ ﬁxed as above, there are similarly deﬁned spaces

2 2 2 Lcusp(G(F )\G(A), χ) ⊂ Ldisc(G(F )\G(A), χ) ⊂ L (G(F )\G(A), χ).

29 The chain

Acusp(G) ⊆ A2(G) ⊆ A(G)

1 consists of irreducible unitary representations of G(A) whose restrictions to G(A) are irre- 2 2 1 ducible constituents of the corresponding L -spaces (A2 corresponds to Ldisc(G(F )\G(A) )). If we do not require the representations to be unitary, we obtain the sets

+ Acusp(G) ⊃ Acusp(G)

and

+ A2 (G) ⊃ A2(G).

Such non-unitary representations will only appear incidentally. They cannot be accessed via the trace formula.

1.3.8 We need to say a word about maximal compact subgroups. Fix a minimal Levi subgroup L0 of G. At [Art81, p.9], the notion of a maximal compact subgroup Kv ⊂ G(Fv) being admissible relative to L0 is deﬁned. The notion of a maximal compact subgroup Q K = v Kv ⊂ G(A) being admissible relative to L0 is also deﬁned there. The key properties for us are that, locally or globally, such maximal compact subgroups can be used in an

Iwasawa decomposition relative to any parabolic subgroup P that contains L0. In addition, for almost all v the local group Kv of K is of the form G(OFv ). Given a Levi subgroup L ⊂ G, we let P(L) denote the set of parabolic subgroups with

L as a Levi subgroup. Let P ∈ P(L) be a parabolic subgroup with Levi decomposition

P = LN . Suppose π is a subrepresentation of L2 (L(F )\L( ), χ) and λ ∈ a∗ . We do not P disc A L,C assume that π is irreducible, since later we will want to consider the representation obtained

2 by inducing the full discrete spectrum of a Levi subgroup. Let Vπ ⊂ Ldisc(L(F )\L(A), χ) be the space of functions realizing the representation π.

G Consider the space, denoted by HP (Vπ), consisting of functions φ : G(A) → C satisfying the following properties.

• φ is NP (A)-invariant.

30 • The function

(m, k) 7→ φ(mk): M(A) × K

2 2 belongs to Ldisc(M(F )\M(A), χ) ⊗ L (K).

For y ∈ G(A), there is an action given by

(λ+ρP )(HP (xy)) −(λ+ρP )(HP (x)) (RP (λ, y)φ)(x) = φ(xy)e e , see [Art05, pp.32-33]. Note that the action depends on λ, though the underlying space

G HP (Vπ) does not. This is one realization of the induced representation

G G G IP (πλ) = (RP (λ), HP (Vπ)).

A different realization is discussed in Appendix A.3. Since G is normally fixed, we will usually drop it from all the notation. We use IP (πλ) when we do not need to emphasize the underlying space or the actions. Later, when there are different actions on the same underlying space, we will be more careful.

1.3.9 Consider the case of G = GN . Then

A(N) = A(G) = {IP (π1 ⊗ · · · ⊗ πr): πi ∈ A2(Ni)}. (1.3.10)

Here π1 ⊗ · · · ⊗ πr is a representation of the standard Levi subgroup GN1 × · · · × GNr and P is the corresponding standard parabolic subgroup of upper triangular matrices. (Note that we can take P = GN .) This description follows from the theory of Eisenstein series,

[Lan66], [Lan76], [Art79], [MW95], and the fact (special to GN ) that a (local) representation parabolically induced from a (local) unitary representation is irreducible. See [Ber84] for the non-archimedean case and [Vog86] for the archimedean case. Then we let A+(N) be induced representations of the form

+ IP (π1 ⊗ · · · ⊗ πr), πi ∈ A2 (Ni).

31 This is a full induced representation which is not generally irreducible. We obtain a chain of representations

+ Acusp(N) ⊂ A2(N) ⊂ A(N) ⊂ A (N) (1.3.11) which is roughly parallel to (1.3.4).

1.3.12 Assume that G is unramified at v. In particular, we assume that v is a non- archimedean place. Let Ev ⊃ Fv be a finite unramified extension over which Gv is split.

Then ΓEv/Fv is a ﬁnite cyclic group with canonical generator the (geometric, say) Frobenius automorphism Frobv. Let

L Gv = Gb o ΓEv/Fv = Gb o hFrobvi denote the ﬁnite form of the L-group of Gv. We may choose a pair (Bv,Tv) deﬁned over Fv such that Tv is split over Ev. We have chosen a hyperspecial maximal compact subgroup

Kv of G(Fv) that lies in the apartment of Tv [Tit79] at almost all places, see 1.3.8. Such a maximal compact subgroup is necessary at an unramiﬁed place for the material on conjugacy classes below. Recall that an irreducible representation πv of G(Fv) is spherical (relative to

Kv) if the restriction of πv to Kv contains the trivial representation. In the unramified case we are considering here, such spherical representations are usually called unramified. A local representation which is not unramified is called ramified.

Consider an irreducible admissible representation π of G(A). At almost each place v, it is possible to choose a Kv-ﬁxed vector uv. According to [Fla79], π can be written as a restricted tensor product

O 0 π ' πv. v

The local factors πv are uniquely determined by π and are unramiﬁed for almost all v. By a restricted tensor product, We just mean that the space of π is spanned by pure tensors N w = v wv where wv = uv at almost all v.

32 When Gv is unramiﬁed there is a canonical bijection

πv 7→ c(πv): {unramiﬁed representations of G(Fv)} →

L {semisimple conjugacy classes in Gv that project onto Frobv}.

For a description of this bijection, see [Bor79, §§6-7].

As an example, assume G = GN and note that GN is unramified for all v. The unramified representations of GN (F ) are in bijection with unordered N-tuples of unramified characters of the maximal torus, say (| − |s1 ,..., | − |sN ). See [Car79] for more details. Fv Fv Then the corresponding semisimple conjugacy class is represented by a diagonal matrix diag(|$ |s1 ,..., |$ |sN ) for a uniformizer $ of F . v Fv v Fv v v

L 1.3.13 Let G = Gb o ΓE/F where E ⊃ F is a ﬁnite Galois extension over which G is split.

Let π be an irreducible admissible representation of G(A). Let S be a ﬁnite set of valuations outside of which π and E are unramiﬁed. If v∈ / S, then we have the conjugacy class c(πv)

L L in the L-group Gv and we write cv(π) for the image of c(πv) in G under the canonical

L L conjugacy class of embeddings Gv ,→ G, see 1.1.1. We obtain a mapping

S π 7→ c (π) = {cv(π): v∈ / S} from (irreducible, admissible) representations of G(A) to families of semisimple conjugacy classes in LG. If we identify classes which are the same almost everywhere, then the construction is independent of E and S.

Let P = LNP be a parabolic subgroup of G and let π be a cuspidal representation of L.

The induced representation IP π is generally reducible. An automorphic representation is any representation which can be realized as an irreducible constituent of such an induced representation. This deﬁnition is justiﬁed by [Lan79b]. Let

S S Caut(G) = {c (π): π is automorphic and unramiﬁed outside of S}.

Let S1 be a ﬁnite set of places as above and assume that π1 is unramiﬁed outside of S1, and

S S likewise for S2 and π2. Then c 1 (π1) and c 2 (π2) are equivalent if cv(π1) equals cv(π2) for

33 almost all v. This deﬁnes an equivalence relation on the set

0 [ S Caut(G) = Caut(G). S

The union is over ﬁnite subsets of places that contain S∞. Then we let

0 Caut(G) = {equivalence classes in Caut(G)}.

Set

Πaut(G) = {automorphic representations of G}.

We obtain a function

π 7→ c(π):Πaut(G) → Caut(G).

If G = GN , then Caut(N) consists of equivalence classes of families of conjugacy classes in

GL(N, C). Let H be another connected reductive linear algebraic group. We are still assuming that G is quasi-split, though H need not be for the following discussion. (One can allow for groups G that are not quasi-split, but additional notions must be introduced, namely the notion of relevant parabolic subgroups, see [Bor79, §3.3].) Let

ξ : LH → LG be an L-homomorphism of L-groups. According to [Bor79, §15.1], ξ is required to be continuous and the restriction of ξ to Hb is required to be a complex analytic homomorphism in Gb. Let S be a ﬁnite set of places of F and let c = {cv : v∈ / S} be a family of conjugacy L L classes in H. Then ξ(c) = {ξ(cv): v∈ / s} is a family of conjugacy classes in G. The principle of functoriality says that if πH is an automorphic representation of H then there exists an automorphic representation πG of G such that c(πG) = ξ(c(πH )).

1.3.14 A general automorphic representation of GN is an irreducible constituent of an induced representation

+ ρ = IP (π1 ⊗ · · · ⊗ πr), πi ∈ Acusp(Ni).

34 Such a representation ρ may have many irreducible constituents. However, the representation

ρ does have a canonical constituent. If

O 0 πi = πi,v, 1 ≤ i ≤ r, v let φi,v be the local parameter corresponding to πi,v via Theorem 1.3.5. We obtain a parameter of GN by setting

φv = φ1,v ⊕ · · · ⊕ φr,v.

According to Theorem 1.3.5, the parameter φv corresponds to a representation πv and we obtain an automorphic representation

O 0 π = πv. v Such a π is called isobaric and we write

π = π1 ··· πr, since π is determined by the πi and is independent of their order. We let

r + X Πiso(N) = {π = π1 ··· πr : πi ∈ Acusp(Ni), Ni = N} i=1 denote the set of isobaric representations. The notion of an isobaric representation is related to the Langlands classification. Indeed, fix v and consider the local representation at v of the inducing representation π1 ⊗ · · · ⊗ πr, namely the local representation π1,v ⊗ · · · ⊗ πr,v. If this local representation happens to be in Langlands order then, according to [Zel80, §10], the local factor πv described above is the Langlands quotient of π1,v ⊗ · · · ⊗ πr,v. See [Lan79a] for more details on isobaric representations and Appendix A.2, in particular Theorem A.2.2, for a statement of the Langlands classification.

The following theorem is due to Jacquet-Shalika [JS81].

Theorem (Strong multiplicity one). The map

π 7→ c(π):Πiso(N) → Caut(N)

35 is a bijection.

1.3.15 Any representation which is induced from unitary cuspidal representations is isobaric; otherwise, it isn’t clear which types of representations are isobaric. The following theorem, due to Moeglin-Waldspurger [MW89], answers this question in part and classiﬁes the discrete spectrum of GN in terms of cuspidal automorphic representations.

Theorem (Moeglin-Waldspurger classiﬁcation). The representations π ∈ A2(N) are in bijection with pairs (m, µ) where N = mn and µ is a representations in Acusp(m). Moreover, let Pm = P(m,...,m) be the standard parabolic subgroup with Levi subgroup Lm = LP = QN/m i=1 GL(m) and let σµ be the representation of Lm(A) given by

n−1 n−3 − n−1 x 7→ µ(x1)| det x1| 2 ⊗ µ(x2)| det x2| 2 ⊗ · · · ⊗ µ(xn)| det xn| 2 .

1 Then π is the unique irreducible quotient of IPm (σµ). The restriction of π to GN (A) occurs in the discrete spectrum with multiplicity one.

Note that A2(N) is parametrized by the set

a Acusp(m). m|N

The representations in A2(N) that are not in Acusp(N) are called residual.

The theorem implies that π ∈ A2(N) is isobaric. Indeed, each local factor of π is a Langlands quotient. Furthermore, if π ∈ A(N), it is also isobaric. This follows from the fact that the induced representation π = IP (π1 ⊗ · · · ⊗ πr) is irreducible if the πi are unitary.

1.3.16 As deﬁned at 1.3.7, a cuspidal automorphic representation (of any group G(A)) is by deﬁnition a subrepresentation of the space of cusp forms equipped with the right regular action. In particular, to say a representation is cuspidal implies that the underlying space is a subspace of the space of cuspidal automorphic forms. Theorems 1.3.14 and 1.3.15 allow us to be a bit more cavalier with the notion of cuspidal representations in the case of GN (A).

Let π = (r, Vπ) be an irreducible, admissible representation of GN (A). We will say that π is cuspidal if it is equivalent to an irreducible subrepresentation of the space of cusp forms 36 on GN (A). Theorems 1.3.14 and 1.3.15 imply that if π is cuspidal according to this new deﬁnition, then there is a unique subspace Vπ,cusp of the space of cusp forms such that (r, Vπ)

2 1 is equivalent to (R,Vπ,cusp). Moreover, Vπ,cusp is the only constituent of L (GN (F )\GN (A) ) that is equivalent to (r, Vπ). The same remarks holds with every instance of “cuspidal” replaced by “discrete.” Thus we can, and will, regard Acusp(N) or Adisc(N) as consisting of isomorphism classes of cuspidal or discrete representations.

1.3.17 Let v be a place of F and consider the local Langlands group LFv and the local group GN,v. There is a chain of subsets of continuous, semisimple representations

+ Φsim,bdd,v(N) = Ψcusp,v(N) ⊂ Ψsim,v(N) ⊂ Ψv(N) ⊂ Ψv (N)

deﬁned as follows.

+ Ψv (N) = {N-dimensional representations of LFv × SU(2)}.

Ψv(N) = {N-dimensional unitary representations of LFv × SU(2)}.

Ψsim,v(N) = {irreducible N-dimensional unitary representations of LFv × SU(2)}.

Ψcusp,v(N) = {irreducible N-dimensional unitary representations of LFv × SU(2)

that are trivial on SU(2)}

= Φsim,bdd,v(N).

To make sense of the equality Ψcusp,v(N) = Φsim,bdd,v(N), note that any representation

+ ψ ∈ Ψv (N) determines a parameter φψ in the following way. Recall that both SU(2) and

SL(2, C) have a unique isomorphism class of irreducible representations of each dimension. (An explicit realization of these representations is discussed at 1.3.20, but we don’t need it immediately.) Using this fact, from the representation ψ of LF × SU(2), we obtain a corresponding representation of LF × SL(2, C), also denoted by ψ. Then we may set    |u|1/2 φ (u) = ψ u,   , u ∈ L . (1.3.18) ψ    Fv |u|−1/2

37 By saying Ψcusp,v(N) = Φsim,bdd,v(N), we just mean that the resulting map

+ ψ 7→ φψ :Ψv (N) → Φv(n)

restricts to a bijection from Ψcusp,v(N) to Φsim,bdd,v(N).

1.3.19 The local group LFv has the property that its N-dimensional (continuous, semisimple) representations are in bijection with the irreducible admissible representations of GN,v.

The hypothetical global Langlands group LF , see [Lan79a], [Kot84], [Art02], would play a similar role in the global theory of automorphic representations. In the formulation of

[Kot84], LF would be a locally compact extension

1 → KF → LF → WF → 1

of WF by a compact connected group KF equipped with a conjugacy class of local embeddings

LFv ,→ LF such that the diagram

LFv / WFv / ΓFv

LF / WF / ΓF commutes. Let

Φ(N) = {equivalence classes of N-dimensional, continuous, semisimple

representations of LF }.

The key property of LF would be the following: The set Φ(N) is in bijection with Πiso(N).

For the remainder of this section, we assume that a group LF with this property exists. Then we would expect

Φsim,bdd(N) = Φ2,bdd(N) = {irreducible unitary representations in Φ(N)}

to parametrize unitary cuspidal automorphic representations of GN . However, we still would need to account for the full discrete spectrum A2(N). Indeed, Theorem 1.3.4 states that

38 A2(N) generally contains much more than the unitary cuspidal automorphic representations. For the moment, set

Ψ(N) = {N-dimensional continuous, semisimple, unitary representations of LF × SU(2)}.

Any ψ ∈ Ψ(N) determines a representation φψ ∈ Φ(N) according to equation (1.3.18). Then there is a bijection

∼ ψ 7→ πψ :Ψsim(N) = Ψ2(N) = {irreducible members of Ψ(N)} → A2(N).

Indeed, Ψsim(N) is in bijection with the set of pairs that parametrizes A2(N) in Theorem 1.3.15. Furthermore, we expect a similar bijection

∼ ψ 7→ πψ : Ψ(N) → A(N).

To see this, write a typical parameter ψ as a sum

n X ψ = ψ2,i i=1 of simple parameters in Ψ2(Ni). The ψi need not be distinct in this sum. Let Lψ be the standard Levi subgroup determined by the integers Ni. Then set

π = IGN (π ⊗ · · · ⊗ π ). ψ Lψ ψ1 ψn

The representation πψ lies in A(N), see (1.3.10) and the discussion there.

1.3.20 In this subsection, F needs to be local for the parameter sets to be deﬁned. However, the proof is formal and does not rely on any special properties of the source of the representation, so the proof makes sense for the hypothetical parameter sets we have been discussing. Our goal is to prove the following proposition.

Proposition. The map

ψ 7→ φψ : Ψ(N) → Φ(N) is injective.

39 Assume that ψ is irreducible. Then

ψ ' ψLF ⊗ ΨSU(2).

Here ψLF is an NLF -dimensional unitary representation of LF . The representation ψSU(2) is an NSU(2)-dimensional representation of SU(2). Their dimensions satisfy NLF NSU(2) = N.

Lemma. If ψ is irreducible, then

NSU(2)−1 M NSU(2)−(2k+1) 2 φψ ' ψLF | − | . k=0

Proof. Recall that the irreducible ﬁnite dimensional representations of SU(2) or SL(2, C) are indexed by positive integers and can be realized explicitly on the space Vn of homogeneous polynomials in two variables of degree n. We have dim Vn = n + 1 and Vn−1 realizes the representation of degree n with the action

(sP )(X,Y ) = P ((X,Y )s),P (X,Y ) ∈ C[X,Y ], s ∈ SU(2) or SL(2, C).

A basis of Vn is given by the polynomials

n−i i Pi = X Y , 0 ≤ i ≤ n.   z   Then the action of the matrix Z =   ∈ SL(2, C) is given by z−1

n−2i ZPi = z Pi.

Now, let VLF ⊗ VSU(2) be the space of an irreducible ψ as above. The space is spanned by vectors of the form v ⊗ Pi for 0 ≤ i ≤ NSU(2) − 1. Then w ∈ LF acts by

NSU(2)−1−2i 2 φψ(w): v ⊗ Pi 7→ |w| (ψLF (w)v ⊗ Pi).

Consequently the space spanned by the vectors of the form v ⊗ Pi is an irreducible subspace NSU(2)−1−2i 2 realizing the representation ψLF | − | . This proves the lemma.

40 Lemma. Suppose ψ = ψ1 ⊕ ψ2. Then

φψ = φψ1 ⊕ φψ2 .

Proof. Set   |w|1/2   sw =   . |w|−1/2

For i = 1, 2, let Vi be the space of ψi (and therefore φψi ) and let vi be a vector in Vi. We have

φψ(w)(v1 + v2) = ψ(w, sw)(v1 + v2)

= ψ1(w, sw)v1 + ψ2(w, sw)v2

= φψ1 (w)v1 + φψ2 (w)v2.

Let’s prove the proposition.

Proof of Proposition 1.3.20. Let ψ, λ be elements of Ψ(N). Suppose

k M ψ = miψi i=1 and l M λ = njλj j=1 be the decompositions of ψ and λ into direct sums of distinct irreducible unitary representations. Suppose that φψ ' φλ via an intertwining operator A. Consider the representation

ψ1,L of LF obtained by restricting ψ1 to LF . Let V be the subspace of V = V which F ψ1,LF ψ decomposes as a direct sum of copies of ψ1,L upon restriction to LF . Then AV also F ψ1,LF decomposes as a direct sum of copies of ψ1,LF upon restriction to LF , so we may assume that

V = V and similarly for W . Thus we may assume ψL ' λL and consider parameters ψ1,LF λ F F of the form k M ψ = (ψLF ⊗ νi). i=1 41 and l M λ = (λLF ⊗ µj). j=1

Here νi and µj are irreducible representations of SU(2). We already know that ψLF ' λLF , so it suﬃces to show that the νi and µj are the same. In fact, it suﬃces to show that if νk and µl are the representations of largest dimension, then νk = µl. But this follows from the previous two lemmas by examining the largest power of | − | occuring in φψ and φλ.

1.3.21 We revert back to a global situation. As discussed in 1.3.19, the set Ψ(N) of unitary representations is in bijection with the set A(N), which consists of unitary automorphic representations. The representations in A(N) are called globally tempered. Since φψ can be computed somewhat explicitly, the map ψ 7→ φψ serves to cut out the set of parameters in Φ(N) which correspond to globally tempered representations. One has in addition sets

Ψ+(N) ⊃ Ψ(N) and

+ Ψsim(N) ⊃ Ψsim(N) obtained by removing the condition that the representation ψ of LF × SU(2) be unitary. Then there is a bijection

+ ∼ + ψ 7→ ρψ :Ψ (N) → A (N).

Indeed, a representation ψ ∈ Ψ+(N) can be written as a direct sum of representations

+ ψi ∈ Ψsim(Ni). According to our assumptions about the hypothetical group LF , each ψi + corresponds to an automorphic representations πi = πψi in A2 (Ni). Then we set

+ ρψ = IP (π1 ⊗ · · · ⊗ πr) ∈ A (N).

However, the map

+ ψ 7→ φψ :Ψ (N) → Φ(N) is not injective. Indeed, the set Φ(N) is in bijection with the set of isobaric representations, which are essentially unordered sums of cuspidal representations, while Ψ+(N) consists

42 of unordered sums of square-integrable representations. The representation ρψ will not in general be the representation corresponding to φψ. In particular, ρψ is an induced representation which is not necessarily irreducible.

Let

Ψcusp(N) = Φsim,bdd(N) ⊂ Ψsim(N)

be the subset of representations which are trivial on SU(2). We obtain a chain

+ Ψcusp(N) ⊂ Ψsim(N) ⊂ Ψ(N) ⊂ Ψ (N) (1.3.22)

which is parallel to (1.3.11) in the sense that ψ 7→ ρψ takes (1.3.22) bijectively to (1.3.11).

Thus, if the hypothetical group LF has the expected properties, its parameters would correspond to well-deﬁned sets of automorphic representations. Later, we will use these sets of automorphic representations as a substitute for global parameters.

1.3.23 Let us recall the generalized Ramanujan conjecture [Pia79]. Let U be the unipotent radical of a Borel subgroup B = TU deﬁned over F . Let λ : U → Ga be a generic character. That is, its centralizer in T is the center of G. (See 1.5.7 for such characters in the local case. The global case considered here is similar.) Composing λ : U(A) → A with a nontrivial additive character ψ of F \A gives a generic character of U(F )\U(A). Then we may deﬁne a linear form

2 1 Wλ : L (G(F )\G(A) ) → C

by Z f 7→ f(n)λ(n) dn. U(F )\U(A) 2 1 If π is a subrepresentation of L (G(F )\G(A) ) then we call π globally λ-generic if the restriction of Wλ to the π-isotypic subspace is nontrivial. We call π globally generic if it is λ-generic with respect to some λ.

N0 Conjecture (Generalized Ramanujan conjecture). Let π = v πv be a globally generic cuspidal automorphic representation of a connected reductive group G. Then each πv is tempered.

43 There is a (hypothetical) restriction map

+ + ψ 7→ ψv :Ψ (N) → Ψv (N)

obtained by composing ψ with the conjugacy class of embeddings LFv ,→ LF . The Ramanujan conjecture says that ψ 7→ ψv actually takes Ψ(N) to Ψv(N).

1.3.24 The three left subsets of the chain (1.3.11) consist of isobaric representations. They are thus bijective with the three left subsets of the chain

Csim(N) ⊂ C2(N) ⊂ C(N) ⊂ Caut(N).

We write

Csim(N) = Ccusp(N).

If the global group LF with the expected properties existed, Csim(N) = Ccusp(N) would be in bijection with

Φsim,bdd(N) = Ψcusp(N) = {N-dimensional, unitary, irreducible representations of LF },

while the set Caut(N) would be in bijection with Φ(N).

1.4 A substitute for global parameters

Let F be a number ﬁeld. We will use cuspidal representations of GN = GN,F as a substitute for irreducible N-dimensional representations of the hypothetical global Langlands group LF .

This idea was introduced in [Art13, §1.4]. We will produce a group Lψ and a representation

ω ω ψ of Lψ for each “parameter” ψ. The pair (Lψ, ψ ) can be regarded as an approximation of a representation of LF . While the hypothetical group LF can “see” all the isobaric representations of GN , our approximations will only be able to see the globally tempered automorphic representations in the image of automorphic induction; these are the types of representations that can be accessed via the ω-twisted trace formula, to be introduced in 2.1.

44 1.4.1 Let F be a number ﬁeld and consider GN = GN,F . In 1.3.19, we wrote Ψsim(N) =

Ψ2(N) for the set of equivalence classes of irreducible unitary representations of the hypothetical group LF × SU(2). If a group LF with the expected properties existed, the set of irreducible unitary representations of LF × SU(2) would be in bijection with the set A2(N) of unitary square-integrable automorphic representations of GN . Now, we deﬁne

Ψsim(N) = {ψ = µ ν : µ ∈ Acusp(m),

ν an irreducible representation of SU(2) of dimension N/m}.

The operation is formal. In particular, we do not regard µ ν as a representation of

Gm(A) × SU(2). This deﬁnition of Ψsim(N) does not rely on any groups or constructions other than GN and SU(2). There is a canonical bijection

∼ ψ 7→ πψ :Ψsim(N) → A2(N).

Indeed, the set Ψsim(N) is in bijection with the disjoint union

a Acusp(m). m|N

According to Theorem 1.3.15, this is the same set that parametrizes A2(N). Recall from

1.3.24 that π 7→ c(π) is a bijection from A2(N) to C2(N). Thus we also have a bijection

∼ ψ 7→ c(ψ) = c(πψ):Ψsim(N) → C2(N).

In 1.3.19, we wrote Ψ(N) for the set of unitary representations of LF × SU(2). Now, we set

k X Ψ(N) = {ψ = l1ψ1 ··· lkψk : li ≥ 0, ψi 6= ψj if i 6= j, ψi ∈ Ψsim(Ni),N = liNi}. i=1 (1.4.2)

We regard the members of Ψ(N) as formal unordered sums.

Suppose ψ ∈ Ψ(N) is as in (1.4.2) and write πi for πψi ∈ A2(Ni). Then we set

πψ = IP (π1 ⊗ · · · ⊗ π1 ⊗ · · · ⊗ πk ⊗ · · · ⊗ πk) ∈ A(N).

45 The representation π is repeated li times. Each πi is a unitary automorphic representation of G and the inducing representation is a unitary automorphic representation of Qk Gli . Ni i=1 Ni

The representation πψ is irreducible since the inducing representation is unitary. This fact is special to GN , see [Ber84] and [Vog86]. The bijections described above can then be extended to ∼ ψ 7→ πψ : Ψ(N) → A(N)

and ∼ ψ 7→ c(ψ) = c(πψ) : Ψ(N) → C(N).

Indeed, the ﬁrst bijection follows from the description (1.3.10). The second was discussed in the previous section, see in particular 1.3.24.

1.4.3 Let v be a valuation of F and consider ψ = µ ν ∈ Ψsim(N). Here µ is a cuspidal representation of Gm. The local component µv of µ is an irreducible representation of

Gm,v. According to Theorem 1.3.5, the representation µv corresponds to an m-dimensional

+ representation φv of LFv . The tensor product ψv = φv ⊗ ν is a member of Ψv (N). We + obtain a map Ψsim(N) → Ψv (N) that can be extended to a localization map

+ ψ 7→ ψv : Ψ(N) → Ψv (N).

More precisely, if ψ is as in (1.4.2), then

k M ⊕li ψ 7→ ψv = ψi,v . i=1

The exponent on ψi,v indicates the direct sum of li of ψi,v. Conjecture 1.3.23 can be rephrased as the statement that ψ 7→ ψv maps Ψ(N) to Ψv(N). Indeed, the conjecture states that the representation µv above is tempered and thus ψv = φv ⊗ ν would be a unitary representation of LFv .

1.4.4 Let

× × × ω : F \AF → C

46 be a ﬁnite order character. We will be concerned with representations which are ﬁxed by twisting with ω. Given a representation π of GN (AF ), we may consider the representation

(π ⊗ ω)(g) = ω(det g)π(g).

The representation π ⊗ ω is regarded as acting on the same space as π. To say that π is

ﬁxed by twisting with ω just means that π ⊗ ω ' π. We will usually write ω in place of

ω ◦ det and refer to such representations as ω-ﬁxed or ω-stable. The same notions apply to

× × a local representation πv and ﬁnite order character ωv : Fv → C . We start with a few basic lemmas.

Lemma. Suppose GN (AF ) has ω-ﬁxed representations. Then N is divisible by the order of

ω. Similarly for GN (Fv) and ωv-ﬁxed representations.

Proof. Suppose π is ω-ﬁxed. Let zα = αIN be the central element of GN (AF ) corresponding × N N to α ∈ AF . Let χπ be the central character of π. Then χπ(zα)ω(α) = χπ(zα) and thus ω is the trivial character. The proof is identical in the local case.

Lemma. Suppose µ ∈ Acusp(m). Then µ ⊗ ω ∈ Acusp(m).

Proof. Recall the discussion of 1.3.16. To say that µ⊗ω is cuspidal means that it is equivalent to a subrepresentation of the space of cusp forms. Let χ be the central character of µ. Let f

2 be a function in Lcusp(G(F )\G(A), χ). This space of functions is described in more detail in 1.3.6 and 1.3.7. Let ωf denote the function given by

ωf(g) = ω(g)f(g).

Let P be a parabolic subgroup with unipotent radical NP . Since Z Z (ωf)(nx) dn = ω(nx)f(nx) dn NP (F )\NP (A) NP (F )\NP (A) Z = ω(x) f(nx) dn NP (F )\NP (A) = 0,

47 2 the function ωf lies in Lcusp(G(F )\G(A), χ). Let Iω denote the resulting operator of multiplication by ω:

2 2 Iω : Lcusp(G(F )\G(A), χ) → Lcusp(G(F )\G(A), χ).

By deﬁnition, the representation µ can be realized on a space of functions

2 Vµ ⊂ Lcusp(G(F )\G(A), χ).

2 Recall that R(−) denotes the right regular representation of G(A) on L (G(F )\G(A), χ).

Upon restricting Iω to Vµ, we obtain the following commutative diagram for any g ∈ G(A):

Iω Vµ / IωVµ

(R⊗ω)(g) R(g) Vµ / IωVµ Iω

Thus the representation µ⊗ω, realized on the space Vµ with the action of R⊗ω, is equivalent to a space of cuspidal functions with the action R.

For ψ = µ ν ∈ Ψsim(N), set

ψ ⊗ ω = (µ ⊗ ω) ν.

Then, if ψ = l1ψ1 ··· lkψk ∈ Ψ(N), set

ψ ⊗ ω = l1(ψ1 ⊗ ω) ··· lk(ψk ⊗ ω).

Let

Ψω(N) = {ψ ∈ Ψ(N): ψ ⊗ ω = ψ}.

In addition, we set

ω ω Ψsim(N) = Ψ (N) ∩ Ψsim(N).

ω The set Ψsim(N) is the set of simple ω-stable parameters. Equivalently, it is the set of ω parameters which satisfy k = 1 and l1 = 1 in the notation of (1.4.2). An element ψ ∈ Ψsim(N) is thus an ω-ﬁxed parameter µν. It represents an ω-stable square-integrable representation.

48 We will also write

ω ω ω Φsim(N) = Ψcusp(N) = Ψ (N) ∩ Ψcusp(N) for the subset of parameters that are generic in the sense that ν is trivial. A simple generic parameter is thus an ω-fixed unitary cuspidal automorphic representation. This notion of generic is consistent with that described in 1.3.23. Indeed, a member of the discrete spectrum of GN is generic if and only if it is cuspidal. To see this, first recall that a representation which is globally generic, in the sense described at 1.3.23, is locally generic at each place, in the sense that it has a nonzero Whittaker functional at each place. Whittaker functionals are discussed in more detail at 1.6.2. Such (unitary) representations were classified in [Ber84], and in particular are full induced representations. Theorem 1.3.4 shows however that the local component of a residual representation is not fully induced.

1.4.5 There is also an action of ω on C(N). It suﬃces to describe the action locally. Viewed × Q as a character of AF , the character ω has a factorization ω = v ωv. As before, we may regard ωv as a character of GN (Fv) via composition with the determinant. For almost all v,

ωv is unramiﬁed and thus has an associated conjugacy class c(ωv). The action of ωv is given by c(πv) 7→ c(ωv)c(πv). This is deﬁned for almost all v. Consider the following diagram.

ω C(N) / C(N) O O

ω A(N) / A(N) O O

ω Ψ(N) / Ψ(N)

The vertical maps are the bijections described in 1.4.1. The horizontal maps are the action of ω. They are also bijections. Since

c(πv ⊗ ωv) = c(ωv)c(πv), the upper square commutes. To show the lower square commutes, we just need to prove the

49 following lemma.

Lemma. For any ψ ∈ Ψ(N), we have

πψ⊗ω = (πψ) ⊗ ω.

In particular, ψ ⊗ ω = ψ if and only if πψ ' πψ ⊗ ω.

Proof. Recall from Theorem 1.3.15 that if ψ = µ ν then πψ is the unique irreducible quotient of an induced representation we will denote by I(µ, n) where n is the dimension of

ν. Thus we have a surjection

I(µ ⊗ ω, n) πψ⊗ω.

We also have a surjection

I(µ ⊗ ω, n) 'I(µ, n) ⊗ ω (πψ) ⊗ ω.

Consequently πψ⊗ω ' (πψ) ⊗ ω.

1.4.6 We will assume the following theorem in order to motivate the construction of various parameter sets. The constructions are basically formal and do not rely on the theorem. The proof will come later and is one of the main global results. Recall that the map ξHω was discussed in 1.2.6 and that functoriality was discussed in 1.3.13.

ω Theorem. Suppose φ ∈ Φsim(N) is a simple generic global parameter. In other words, φ is an ω-ﬁxed, unitary, cuspidal automorphic representation of GN . Let Hω be the group deﬁned

× × by ω. That is, if D is the degree of ω and ker ω = NE/F E \AE for an extension E ⊃ F , then Hω = HN/D = RE/F GN/D,E. Then there is a cuspidal representation π ∈ Acusp(Hω) such that c(φ) = ξHω (c(π)).

ω 1.4.7 Let ψ = t1ψ1 ··· tsψr be an element of Ψ (N). Then we have

t1ψ1 ··· tsψs = ψ = ψ ⊗ ω = t1(ψ1 ⊗ ω) ··· ts(ψs ⊗ ω).

50 2 Thus we see that ψ1 ⊗ ω occurs in ψ with multiplicity l1. By considering ψ = ψ ⊗ ω , we see

2 that ψ1 ⊗ ω occurs in ψ with multiplicity l1. Continuing in this way, we see that ψ contains the orbit of ψ1 under hωi, the cyclic group of characters generated by ω, with multiplicity l1. Thus we can rewrite ψ as

k1−1 kr−1 ψ = l1(ψ1 ψ1 ⊗ ω ··· ψ1 ⊗ ω ) ··· lr(ψr ψr ⊗ ω ··· ψr ⊗ ω ).

k Here ki is the smallest integer such that ψi ⊗ ω i = ψi. We will call this the standard form of ψ.

Consider the orbit of a ﬁxed basepoint ψ1 = µ1 ν1 with µ1 a cuspidal automorphic representation of Gm1 and ν1 the representation of SU(2) of dimension n1. The orbit of hωi k −1 on µ1 is {µ1, µ1 ⊗ ω, . . . , µ1 ⊗ ω 1 }. In particular, the stabilizer of µ1 in hωi is generated by ωk1 . We have

k1 #hω i = D/k1 = D1.

Let F ⊂ E1 ⊂ E be the intermediate extension of degree

[E1 : F ] = D/k1 = D1

k1 determined by ω . Then µ1 is a cuspidal automorphic representation of Gm1 ﬁxed by k twisting with ω 1 . According to Theorem 1.4.6, the cuspidal representation µ1 should be obtained by lifting from

H1 = RE1/F Gm1/[E1:F ] = RE1/F Gm1/D1 .

More precisely, we expect there to be a representation π1 ∈ Acusp(H1) such that ξH1 (c(π1)) = c(µ1).

For each 1 ≤ i ≤ r, we deﬁne Hi according to the recipe of the previous paragraph. Let

L Hi → ΓF be the Galois form of the L-group with the associated projection, see 1.1.5. Then

L let Lψ be the ﬁber product over ΓF of the groups Hi:

r Y L Lψ = Hi → ΓF . (1.4.8) i=1

51 Qr L In other words, Lψ is the subset of the direct product i=1 Hi determined by requiring each coordinate to project onto the same element of ΓF .

k 1.4.9 Suppose that ψ1 is ﬁxed by ω 1 , so that

k1−1 ψ = ψ1 (ψ1 ⊗ ω) ··· (ψ1 ⊗ ω )

is in standard form. Then

H1 = RE1/F Gm1/D1

and

L L Lψ = H1 = (RE1/F Gm1/D1 ).

ω If ψ1 = µ1 ν1 as in 1.4.1, then ψ ∈ Ψ (m1n1k1). We would like to produce a map

L L H1 × SL(2, C) → Gm1n1k1 .

It will be a rough approximation of a representation of the hypothetical group LF . The ﬁrst

L L step is to produce a map H1 → Gm1k1 . L L Let ξ1 : H1 → Gm1 be the L-homomorphism, deﬁned as in 1.2.6, realizing H1 as a

(standard) twisted endoscopic group for Gm1 . Denote also by ω the character of ΓF obtained × × × × by inﬂation from ΓF → ΓE/F ' (NE/F E \AE)\(F \AF ). For any two positive integers m and k, we deﬁne an L-embedding

L L L Im,k : Gm = GL(m, C) × ΓF → Gmk = GL(mk, C) × ΓF

by   g      ω(σ)g  g × σ 7→   × σ.  .   ..      ω(σ)k−1g The localizations

L L L Im,k,v : Gm,v → Gmk,v

52 L L are deﬁned similarly. The desired map H1 → Gm1k1 is

ω L L L ξ1 = Im1,k1 ◦ ξ1 : H1 → Gm1k1 .

The L-embedding deﬁned above is dual to an operation on local representations. Let

Lm be the standard Levi subgroup of Gmk corresponding to the partition (m, . . . , m) and let Pm be the upper triangular parabolic subgroup with Levi subgroup Lm. Let Πadm(−) denote the admissible representations of a given group. We deﬁne a function

Im,k,v :Πadm(Gm,v(Fv)) → Πadm(Gmk,v(Fv)) by

k−1 Im,k,v(πv) = IPm (πv ⊗ (πv ⊗ ωv) ⊗ · · · ⊗ (πv ⊗ ωv )).

Let πv ∈ Πadm(Gm(Fv)) be irreducible, unitary and unramified. If v is unramified in E, so that ωv is an unramified character, then Im,k,v(πv) is irreducible, unitary and unramified. The unitary and unramified statements are clear. The irreducibility statement is due to

L Bernstein [Ber84]. Let cv(πv) ∈ Gm be the corresponding semisimple conjugacy class. For such πv we have L Im,k,v(cv(πv)) = cv(Im,k,v(πv)).

L L 1.4.10 In 1.4.11, we will construct a second L-embedding H1 → Gm1k1 . As a ﬁrst step, consider a tower of cyclic ﬁeld extensions F ⊂ E ⊂ L with [E : F ] = d and [L : E] = δ and a positive integer m. We claim that there is a canonical L-embedding

L L RE/F Gm → RL/F Gm.

This is of course just the L-homomorphism corresponding to the base change lifting of automorphic representations of GL(m, AE) to automorphic representations of GL(m, AL) [AC89]. We will construct this map for future reference and completeness.

Let

L (gb0,..., gbd−1) o σ ∈ RE/F Gm = (GL(m, C) × · · · × GL(m, C)) o ΓF .

53 We deﬁne a function

L L β = βL/E : RE/F Gm → RL/F Gm by

(gb0,..., gbd−1) o σ 7→ (gb0,..., gbd−1, gb0,..., gbd−1,..., gb0,..., gbd−1) o σ.

In other words, we put δ copies of (gb0,..., gbd−1) on the diagonal. The map β is a map over

ΓF and we need to show that β is actually a homomorphism.

Fix a generator σ0 of ΓL/F . There are natural surjections

ΓF → ΓL/F → ΓE/F .

The restriction of σ0 to E, which we also denote by σ0, is a generator of ΓE/F . We denote δ(σ) d(σ) the image of an element σ ∈ ΓF in ΓL/F (respectively ΓE/F ) by σ0 (respectively σ0 ). The exponent δ(σ) (respectively d(σ)) is deﬁned modulo dδ (respectively d). Then δ(σ) and d(σ) are equivalent modulo d. The generator σ0 acts on the dual of RL/F Gm by cyclic permutations as in 1.1.5, and similarly for the action of σ0 on the dual of RE/F Gm. On the one hand, we have

β((gb0,..., gbd−1) o σ)β((bh0,..., bhd−1) o τ) = bck o στ where

bck = gbjbhj−δ(σ), 0 ≤ k ≤ δd − 1, k = md + j.

On the other hand, we have

β((gb0,..., gbd−1) o σ · (bh0,..., bhd−1) o τ) = β((gbibhi−d(σ)) o στ) = γbk o στ.

Here γbk = gbjbhj−d(σ) with k as above. But this is equal to bck, since δ(σ) and d(σ) are equivalent modulo d.

1.4.11 We continue with the setup of 1.4.7 and 1.4.9. Recall that H1 = RE1/F Gm1/D1 . We have two L-homomorphisms:

ξ LI ω L m1/D1,E1/F L m1,k1 L ξ1 : H1 −→ Gm1,F −→ Gm1k1,F 54 and β ξ ω L E/E1 L m1/D1,E/F L β1 : H1 −→ RE/F Gm1/D1 −→ Gm1k1,F .

We’ve added some additional subscripts to ξ∗ in order to indicate which ﬁelds are involved.

In all cases, ξ∗ is a map as in 1.2.6 realizing the source as a twisted endoscopic group of the target.

ω ω The maps ξ1 and β1 have two interpretations. One interpretation, discussed at the ω ω L beginning of the section, is that ξ1 and β1 , together with the group H1, are approximations of a representation of the hypothetical global group LF . The second, more concrete

L L interpretation, is that as L-homomorphisms from H1 to Gm1k1,F , these maps correspond to a lifting of automorphic representations according to the principle of functoriality, see

L ω 1.3.13. Note that, since Gm1k1,F = Gbm1k1,F × ΓF is a direct product, we may regard ξ1 ω L and β1 as representations of H1 of degree m1k1.

ω ω L Proposition. The maps ξ1 and β1 are isomorphic representations of H1. In particular, according to the principle of functoriality, they correspond to the same lifting of automorphic representations from H1(AF ) to Gm1k1 (AF ).

ω Proof. It suffices to compute the trace of each representation. We will begin with ξ1 . Let L bh1 o σ denote a typical element of H1 = Hb1 o ΓF . Recall that in 1.2.6 we fixed a matrix A which gave the cyclic permutation action of a generator of the finite Galois group on

the dual group. In our current situation, we ﬁx a generator σE1/F of ΓE1/F . It acts by cyclic permutations on Hb1 as at 1.2.6. Let Aσ ,m /D be the permutation matrix in Gbm1 E1/F 1 1 i (σ) corresponding to the action of σ . Let σ E1/F be the image of σ in Γ . Then E1/F E1/F E1/F

 i (σ)  E1/F bh1Aσ  E1/F,m1/D1  i (σ)  E1/F   ω(σ)bh1Aσ  ξω(h σ) =  E1/F,m1/D1  . 1 b1 o  .   ..     i (σ)  k −1 E1/F ω(σ) 1 h Aσ b1 E1/F,m1/D1

55 Computing the trace in each block, we obtain

k1−1 i (σ) ω X i E1/F tr(ξ (h σ)) = ω(σ) tr h Aσ . 1 b1 o b1 E1/F,m1/D1 i=0

i (σ) E1/F We claim that if σ lies outside Γ then h Aσ = 0. Suppose σ does not lie in Γ , E b1 E1/F,m1/D1 E1 i (σ) E1/F so that i (σ) is nonzero. Then tr h Aσ vanishes and the claim holds in this E1/F b1 E1/F,m1/D1 case. Now, suppose σ lies in ΓE1 but not in ΓE. Consider the maps

ΓF → ΓE/F → ΓE1/F .

Let σE/F be a generator of ΓE/F , so in particular the restriction of σE/F to E1 is equal to

σE1/F . The order of ΓE/F is D, while the order of ΓE1/F is D1. Thus the kernel of the map

D/k1 ΓE/F → ΓE1/F , namely the group ΓE/E1 , is of order D/D1 = k1. It is generated by σE/F .

Dl/k1 The image of σ in ΓE/F is equal to σE/F and is not the identity. Note that ω(σE/F ) = ζD is a primitive Dth root of unity. Then

Dl/k1 Dl/k1 ω(σ) = ω(σE/F ) = ζ .

This is a k1th root of unity that is not equal to 1, and thus

k −1 X1 ω(σ)i = 0. i=0 This proves the claim, and we obtain the formula   k tr h , σ ∈ Γ , ω  1 b1 E tr(ξ1 (bh1 o σ)) =  0, otherwise.

Now we consider the map βω. Let A be deﬁned as above, but now lying in 1 σE/F ,m1/D1

Gbm1k1 and giving the action of σE/F . Then

ω iE/F (σ) β (bh1 σ) = diag(bh1,..., bh1)A . 1 o σE/F ,m1/D1

There are k1 copies of bh1 on the diagonal. Each bh1 lies in GL(m1/D1, C)×· · ·×GL(m1/D1, C), D copies. Since A permutes copies of GL(m /D ) cyclically, the trace vanishes 1 σE/F ,m1/D1 1 1 unless iE/F (σ) = 0. If iE/F (σ) = 0, then the trace is k1 tr bh1, as desired. 56 ω ω 1.4.12 The maps ξi and βi are defined as in the previous subsection. Proposition 1.4.11 says that they are conjugate to one another, but for the moment we continue to differentiate between them. Recall that Lψ was defined by (1.4.8). Then we define

ω L ψξ : Lψ × SL(2, C) → GN = GL(N, C) × ΓF

by r ω M ω ψξ = li(ξi ⊗ νi). i=1 The map

ω L ψβ : Lψ × SL(2, C) → GN

is deﬁned similarly: r ω M ω ψβ = li(βi ⊗ νi). i=1

k1 Recall that if A, B, C and D are matrices then (A ⊗ B)(C ⊗ D) = AC ⊗ BD. Let bh1

denote the image of bh1 in RE/F Gm1k1/D. Then

ω L β1 ⊗ ν1 : H1 × SL(2, C) → Gbm1k1n1

is given by

iE/F (σ) k1 iE/F (σ) (bh1 σ) × s 7→ bh1A ⊗ ν1(s) = (bh ⊗ ν1(s))(A ⊗ In ). o σE/F ,m1k1/D 1 σE/F ,m1k1/D 1

k1 The matrix bh1 ⊗ ν1(s) consists of block matrices of size m1n1k1/D and it is permuted cyclically by A ⊗ I . Thus it lies in the image of LR G in LG . σE/F ,m1k1/D n1 E/F m1n1k1/D m1k1n1 ω ω More generally, the map ψβ associated to ψ ∈ Ψ (N) has its image in a Levi subgroup of L HM , M = N/D, and there is a map

ψω : L × SL(2, ) → LH HM ,β ψ C M

57 such that ψω β L Lψ × SL(2, C) / GN ;

ω ψ ξH HM ,β M & L HM ω ω commutes. Thus it makes sense to also denote Ψ (N) by Ψ (HM ). This is a piece of formal evidence which suggests that each ω-fixed representation in A(N) is in the image of the lifting from HM . Both ψω and ξ are G -conjugacy classes of homomorphisms. Thus ψω is determined β H bN HM ,β up to conjugacy by GbN ⊃ HbM . Recall that at 1.2.7 we showed the outer automorphism group of HM as an endoscopic datum can be identified with the Galois group ΓE/F . This outer automorphism group acts on the space of HbM orbits of L-homomorphisms from L Lψ × SL(2, C) to HM . The size of the orbit is an integer which will play a role later. Suppose now that H ∈ Eω(N) represents a general endoscopic datum. Then we saw in 1.2.6 that H can be represented by a Levi subgroup of HM . Thus we can consider the set Ψω(H) consisting of the maps ψω with the property that the image of ψω lies in β HM ,β L L ω the image of H in HM . An L-homomorphism ψ ∈ Ψ (H) can be identified with a pair

ω ω (ψ, ψH,β) where ψ ∈ Ψ (N) and

ω L ψH,β : Lψ × SL(2, C) → H

satisﬁes

ω ω ξ ◦ ψH,β = ψβ .

ω Here ψH,β is deﬁned as an Hb-orbit up to a group of outer automorphisms coming from the action of GbN which in general properly includes the Galois group.

ω ω ω ω 1.4.13 Let ψ ∈ Ψ (N). We will write ψ for either of the maps ψξ or ψβ . This is justiﬁed, ω ω since according to Proposition 1.4.11, ψξ and ψβ are isomorphic representations. The diﬀerent viewpoints provided by the two realizations of ψω will be useful later. Set

ω Sψ(N) = Cent(im(ψ ), GbN ),

58 ω the centralizer of ψ (Lψ × SL(2, C)) in GbN .

Lemma. The group Sψ(N) is connected.

ω Proof. It suﬃces to show that ξ1 is a direct sum of irreducible representations. Indeed, this ω implies that ψ1,LF ξ is a direct sum of irreducible representations, and by Schur’s lemma the centralizer would be a product of copies of GL(m, C), which is connected. ω The image of ξ1 (bh1 o σ) is computed in the proof of Proposition 1.4.11. Let Vm1k1 be the space on which Gbm1k1 acts by linear transformations. The block matrix form shows that ω ξ1 is a direct sum of k1 representations of dimension m1, all of which are twists of the basic representation ξH1 . Thus it suﬃces to prove that this representation is irreducible.

To simplify the notation, suppose that H = RE/F GN/D and consider the representation L ξH : H → GbN . We just need to prove that this representation is irreducible. We will L identify H with its image under ξH . Let V = VN be the standard N-dimensional space on which GbN acts. We may write

V = V1 ⊕ · · · ⊕ VD

L where each Vi is an N/D-dimensional space preserved by Hb. Then H preserves the decomposition, since ΓF permutes cyclically the Vi. More precisely, if ΓE/F = hσi and wσ 0 L is a preimage of σ in ΓF , then wσVi = Vi+1. Let {0} 6= V ⊂ V be an H-stable subspace

0 with v1 ⊕ · · · ⊕ vD ∈ V a nonzero element. Assume v1 6= 0. Let bh1 ∈ GbN/D be such that bh1v1 6= v1. Then bh1 can be extended to an element of Hb, say bh, that acts by

ξH (bh)(v1 ⊕ · · · ⊕ vD) = hc1v1 ⊕ · · · ⊕ vD.

Then

0 (1 − ξH (bh))(v1 ⊕ · · · ⊕ vD) = v1 − bh1v1 6= 0 ∈ V .

0 0 This implies that V1 ⊂ V . Letting wσ act, we see that V ⊂ V . Consequently, ξH is irreducible.

1.4.14 Suppose now that H ∈ Eω(N) and ψ ∈ Ψω(H). We set

ω Sψ = Sψ(H) = Cent(im ψH,β, Hb). 59 Lemma. The group Sψ(H) is connected.

Proof. Since H is a product of groups of the form RE/F Gm,E, it suﬃces to prove the proposition for HM . We can clarify the situation a bit by considering the following more general set up. Let Gb be a complex, reductive group with an action of ΓF that factors through a cyclic quotient ΓE/F . Let

ψ : G Γ → H Γ Gb b o E/F bM o E/F be an L-homomorphism. It suffices to show that the centralizer of the image of ψ in H Gb bM is connected. Since the image of ψ contains 1 σ, where σ is a fixed generator of Γ , Gb o E/F the centralizer is contained in the fixed points of ΓE/F in HbM . Let gb be an element of Gb. We write the image ψ (g) as Gb b diag(ψ1(gb), . . . , ψD(gb)), with each ψi(gb) in GL(M, C). Thus each ψi is an M-dimensional representation of Gb. The action of σ on HbM is given by conjugation by a matrix Aσ as at 1.2.6. Denote the action of σ −1 σ on gb by gb 7→ gb = (1 o σ)gb(1 o σ ). On the one hand,

ψ (gσ) = diag(ψ (gσ), . . . , ψ (gσ)). Gb b 1 b D b

On the other hand,

ψ (gσ) = A diag(ψ (g), . . . , ψ (g))A−1 = (ψ (g), ψ (g), . . . , ψ (g)). Gb b σ 1 b D b σ D b 1 b D−1 b

σ Thus, all the representations ψi are σ conjugates of one another. Since ψi and ψi have the same centralizer in GL(M, ) and the centralizer of ψ lies in the ﬁxed points of A , C Gb σ the centralizer of ψ is isomorphic to the centralizer of ψ in GL(M, ) for any i. Such a Gb i C centralizer is connected. This proves the lemma.

1.4.15 If HM = RE/F GN/D, we write

ω ω Ψell(N) = Ψ2 (H)

60 ω L for the set of parameters such that the associated map ψH does not have its image in L for any proper Levi subgroup L ( HM . At 2.5.9, we will discuss a diﬀerent characterization.

Lemma. We have

ω ω Ψsim(N) ⊂ Ψell(N).

ω Proof. Such a parameter ψ ∈ Ψsim(N) is of the form µ1 ν1 where µ1 is a cuspidal automorphic representation of Gm1 ﬁxed by tensoring with ω. This implies that D divides L m1 and m1n1 = N where n1 = dim ν1. In this case H1 = RE/F Gm1/D and Lψ = H1. The map

ω L L ψH : Lψ × SL(2, C) → H = RE/F GN/D is given by

bh1 o σ × s 7→ bh1 ⊗ ν1(s) o σ.

But bh1 × s 7→ bh1 ⊗ ν1(s) is an irreducible representation of Gm1/D × SL(2, C) in GL(N/D, C).

Thus it does not factor through a Levi subgroup of GL(N/D, C).

1.4.16 Endoscopy exhibits a general phenomenon, described informally and in some generality as follows. For the moment, assume F is a local or global ﬁeld of characteristic 0 and that the parameter sets and endoscopic data are deﬁned appropriately.

For simplicity, consider a triple (G, id, 1) as in 1.2.1. Let E(G) be the corresponding set of equivalence classes of endoscopic data. For any G0 ∈ E(G), we assume a set of parameters

0 Ψ(G ) has been defined. If F is local the set was defined in 1.3.17 for GN and can be defined in a similar way for arbitrary G. If F is global, one must use representations of Lψ, or some other replacement, in the absence of a global Langlands group. Then there is a bijection

0 0 0 0 0 {(G , ψ ): G ∈ E(G), ψ ∈ Ψ(G )} ←→ {(ψ, s): ψ ∈ Ψ(G), s ∈ Sψ}.

For a proof and more thorough discussion of this bijection, see [Art13, pp. 36-37].

Since our situation is rather simple, this bijection, which is important in the constructions throughout [Art13], is straightforward and will not play an explicit role. First, consider HM regarded as a representative of the unique elliptic (GN , ω)-datum. According to Lemma 61 A.1, the set E(HM ) of equivalence classes of endoscopic data for HM is in bijection with

E(GM,E), the set of equivalence classes of ordinary endoscopic data for GM,E. This latter set can be described by applying the results of 1.2.6 when ω is the trivial character (so that D = 1). The endoscopic data are essentially a Levi subgroup of L ⊂ GM,E and a semisimple element in GbM,E whose centralizer is Lb. It turns out that, since Sψ is connected, the semisimple element s does not play a particularly important role. Given a pair (G0, ψ0)

0 0 0 with G ∈ E(GM,E) and ψ ∈ Ψ(G ), the parameter ψ ∈ Ψ(G) is determined as in 1.4.12 by

0 0 the composition ξG0 ◦ψ and we may take s = ξG0 (s ). Conversely, given (ψ, s), the centralizer of s is essentially the smallest Levi subgroup through which ψ factors. Parameters which factor through a proper Levi subgroup can be handled by induction.

1.4.17 Let W be a group with a normal subgroup W 0 ⊂ W with W 0\W a cyclic group of order D, say W 0\W = hσi. Write also

D−1 a W = W 0σk. k=0

× 0 Let ω : W → C a character such that W ⊂ ker ω and ω(σ) 6= 1.

Lemma. Let ψ be an irreducible representation ψ : W → GL(N, C) acting on a space V . Suppose that ψ ⊗ ω ' ψ. Then ψ is induced from a representation of W 0.

Proof. Fix an operator A on V such that

Aψ(w)A−1 = ω(w)ψ(w) for all w ∈ W . Since A is an operator on a ﬁnite dimensional complex vector space, it has an eigenvector. We may scale A if necessary so that V A, the space of A-ﬁxed vectors, is nonzero.

Suppose that AvA = vA and consider ψ(w0)vA for w0 ∈ W 0. Then, since W 0 ⊂ ker ω,

Aψ(w0)vA = ψ(w0)AvA = ψ(w0)vA.

That is,

ψ(W 0)V A ⊂ V A.

62 Let V 0 ⊂ V A be an irreducible subrepresentation of W 0. We claim that

W 0 V ' IW 0 V .

This is equivalent to showing that

D−1 M V = ψ(σk)V 0. k=0 Since V is irreducible and D−1 X ψ(σk)V 0 k=0 is W -stable, we certainly have D−1 X V = ψ(σk)V 0. k=0 Pm k 0 We must show that the sum is direct. We will prove that k=0 ψ(σ )V is direct by induction on m. The case of m = 0 is clear.

Suppose m+1 X k 0 ψ(σ )vk = 0. k=0 Then m m+1 0 X k 0 ψ(σ )vm+1 = − ψ(σ )vk. k=0 Note that

Aψ(σk) = ω(σk)ψ(σk)A.

Let ω(σ) = ζ, a Dth root of unity. Then

m m m+1 m+1 0 m+1 0 X k 0 X k k 0 ζ ψ(σ )Avm+1 = Aψ(σ )vm+1 = − Aψ(σ )vk = − ζ ψ(σ )Avk. k=0 k=0

0 0 Note that vm+1 and vk lie in the space of A-ﬁxed vectors. This implies m m m+1 m+1 0 X k k 0 X m+1 k 0 ζ ψ(σ )vm+1 = − ζ ψ(σ )vk = − ζ ψ(σ )vk. k=0 k=0 Then m X m+1 k k 0 (ζ − ζ )ψ(σ )vk = 0, k=0

63 0 so that vk = 0.

1.5 Langlands-Shelstad-Kottwitz transfer

In this section we let F be local. The discussion follows [Art13, §2.1], [KS99, Chapters 3,5] and [HI12, §§1-2].

The general theory of [KS99] applies to a triplet (G, θ, a). Such triplets are described at the beginning of 1.2.1. The cocycle a determines a character ω of G(F ). We will restrict ourselves to cocycles that determine a character of ﬁnite order. We will always assume that

θ = id is the identity automorphism, so we will often write (G, ω) instead of the full triplet.

Recall that we are assuming that G is quasi-split and that Gder is simply connected. This latter condition implies that there is no diﬀerence between “regular” and “strongly regular” elements. We will refer only to “regular” elements even though some of the discussion is only true for “strongly regular” elements in general. We have tried to point out where the

“strongly regular” assumption is most essential.

1.5.1 Let γ ∈ G(F ). Then there is the associated (geometric) conjugacy class

−1 Ogeom(γ) = {g γg : g ∈ G(F )}.

We will let

Ogeom(G) = {Ogeom(γ): γ ∈ G(F )}. denote the set of all such geometric conjugacy classes. The set Ogeom,ss(G) will denote the subset of Ogeom(G) consisting of semisimple conjugacy classes:

Ogeom,ss = {γ ∈ Ogeom(G): γ is semisimple}.

Let

W = W (G, T ) be the Weyl group of T in G. Let O ∈ Ogeom,ss be a semisimple conjugacy class. Every such class meets T . Let t(O) lie in the intersection O ∩ T . Then we deﬁne the abstract norm map

64 NG by setting NG(O) = t(O):

NG : Ogeom,ss(G) → T/W.

To see that the map is well-deﬁned, note that if t1 and t2 both lie in O ∩T , then t1 and t2 are conjugate by W . The map is surjective, since every element of T is semisimple. Moreover, the map is injective since the conjugacy classes are disjoint. Thus NG is a bijection. See [KS99, (3.2)] for more details.

Consider an endoscopic datum (H, H, s, ξ) ∈ Eω(G), deﬁned at Deﬁnition 1.2.1. The following is Theorem 3.3.A of [KS99].

Theorem. There is a canonical map

AH/G : Ogeom,ss(H) → Ogeom,ss(G) from the set of semisimple conjugacy classes in H(F ) to the set of semisimple conjugacy classes in G(F ). The map AH/G is deﬁned over F , in the sense that it respects the action of ΓF .

Proof. We sketch the proof by constructing the map AH/G. Suppose (BH ,TH ) is a pair in

H. Recall that we have ﬁxed spl(Gb) and spl(Hb) in 1.1.1. Here we switch notation slightly from 1.1.1, so that the pair in spl(Hb) is (BH , TH ) and the pair in spl(Gb) is (B, T ). We assume s ∈ T , ξ(TH ) ⊂ T and ξ(BH ) ⊂ B. Associated to the pairs (BH ,TH ) and (BH , TH ) is an isomorphism ∼ TbH → TH .

Similarly there is an isomorphism ∼ Tb → T .

Then

TbH 'TH 'T' Tb and thus there is an isomorphism

i : TH ' T.

65 Then WH = W (H,TH ) can be identiﬁed with a subgroup of W and there is a map

iW : TH /WH → T/W.

Note that iW need no longer be an isomorphism. The map AH/G is the composition

−1 NH iW NG AH/G : Ogeom,ss(H) −→ TH /WH −→ T/W −→ Ogeom,ss(G).

It is independent of any choices.

That AH/G is defined over F follows from the next fact, Lemma 3.3.B of [KS99]. If TH is defined over F there exists a Borel subgroup BH ⊃ TH and a pair (B,T ) with T defined over F such that the isomorphism i is defined over F . Such an isomorphism is referred to as an admissible embedding. Note that B need not be defined over F .

1.5.2 For γ ∈ G, let Gγ be the centralizer

−1 Gγ = {x ∈ G : x γx = γ}.

0 A semisimple element γ ∈ G is regular (respectively strongly regular) if (Gγ) (respectively

Gγ) is a torus. Since we are assuming that Gder is simply connected, every regular element is automatically strongly regular. See [Har11, Remark 4.12] for more discussion. From now on, we will only use the term regular, though in general most of the following results are only true for strongly regular semisimple elements. In particular, we only discuss orbital integrals at semisimple elements. A semisimple element γ ∈ H is called G-regular if AH/G(γ) consists of regular elements. Being G-regular is stronger than being H-regular.

Lemma. A G-regular semisimple element of H is H-regular.

See [KS99, Lemma 3.3.C]. In general, an H-regular element of H does not need to be

G-regular. Indeed, consider the example described in 1.1.6. Then m(a, 0) is an H-regular element of H which is not G-regular; a semisimple element of GN is regular if and only if it has distinct eigenvalues.

H Let γ ∈ H(F ) be a G-regular semisimple element. In particular, Hγ is a is an abelian group deﬁned over F . Under our assumptions, we can deﬁne the stable conjugacy class 66 H H Ost(γ ) of γ to be the intersection

H H Ost(γ ) = Ogeom(γ ) ∩ H(F ).

The stable conjugacy class of a regular semisimple γ ∈ G(F ) is deﬁned similarly. Suppose

γH ∈ H(F ) is semisimple and G-regular. We say that γH is a norm of γ ∈ G(F ) if γ lies

H in the image of the conjugacy class of γ under AH/G. If this image contains no points of G(F ) then we say γH is not a norm.

1.5.3 The next lemma gives some indication of the relationship between GN -conjugacy and

× HM -conjugacy in our main case of interest. As usual, ω is a character of F that satisﬁes

× ker ω = NE/F E . We view ω also as a character of GN (F ) via the determinant. Regarding

HM (F ) as a subgroup of GN (F ) via a choice of basis as in 1.1.5, the restriction of ω to

HM (F ) is trivial. Indeed, HM (F ) ' GM,E(E) and we have

detGN (F )h = NE/F detGM,E (E)h ∈ ker ω, h ∈ HM (F ).

(Choosing a basis will not aﬀect the results, since what follows only depends on conjugacy classes.) We have the following lemma from [Kaz84] and [HH95].

Lemma. Suppose γ ∈ GN (F ) is regular. Then GN,γ(F ) is contained in the kernel of ω if and only if γ is conjugate to an element of HM (F ).

Proof. Note that rank GN = rank HM . If γ ∈ HM ∩ GN,reg, then GN,γ = HM,γ. If in addition γ ∈ GN (F ), then GN,γ(F ) = HM,γ(F ) ⊂ ker ω.

Assume then that γ ∈ GN,reg and GN,γ(F ) ⊂ ker ω. Since γ is regular, we have Qk × GN,γ(F ) ' i=1 Fi , where Fi ⊃ F is a ﬁeld extension. Consider the element α = × (1,..., 1, αi, 1,..., 1) ∈ GN,γ(F ), where αi ∈ Fi . Then

× detGN (F )α = NFi/F αi ∈ ker ω = NE/F E .

× × Thus NFi/F Fi ⊆ NE/F E for all i. Local class ﬁeld theory says that Fi ⊇ E. We have that

k X [Fi : F ] = N = MD. i=1 67 Since [E : F ] = D, k X [Fi : E] = M. i=1

Thus GN,γ is conjugate to a torus in GL(M)E which we may identify as a torus in HM .

1.5.4 Let K be a suitable maximal compact subgroup of G(F ). The group K should be chosen to be admissible relative to some ﬁxed minimal Levi subgroup M0 as in 1.3.8. See also [Art89, p.25] for more on this choice. Then H(G) will denote the local Hecke algebra of smooth, compactly supported, left and right K-ﬁnite functions on G(F ).

Let g (respectively gγ) denote the Lie algebra of G (respectively Gγ). We set

D(γ) = det (1 − Ad(γ)) . g/gγ

If γ ∈ G(F ) satisﬁes Gγ(F ) ⊂ ker ω and f ∈ H(G), we set Z ω ω 1/2 −1 IG(γ, f) = fG(γ) = |D(γ)| f(x γx)ω(x) dx. Gγ (F )\G(F )

The integral is well deﬁned for such γ. If γ does not satisfy Gγ(F ) ⊂ ker ω, then we set

ω IG(γ, f) = 0. In particular, if the center of G(F ) is not contained in the kernel of ω, then ω ω every number IG(γ, f) is equal to 0 by deﬁnition. The number IG(γ, f) is the (ω-twisted) orbital integral of f at γ. If ω ≡ 1 is the trivial character, we will generally drop ω from the notation for the orbital integral and from all notation that follows.

Let us say a word about the measures used in the integrals above in our main case of

HM and GN . We adopt the conventions of [HH95, pp.140-141]. We ﬁx the Haar measures on GN (F ) and HM (F ) that give the maximal compact subgroups measure one. For any

GN -regular γ in HM (F ), the centralizers GN,γ(F ) and HM,γ(F ) are isomorphic tori. We choose the measure on the torus that gives the maximal compact subgroup measure one.

This deﬁnes the quotient measure used in the integrals above. In particular, isomorphic centralizers are given the same measure. We may also give all conjugate tori the same

0 measure; if γ and γ are conjugate to one another, then GN,γ(F ) and GN,γ0 (F ) are given the same measure.

68 1.5.5 The two different notations for the orbital integral of f at γ are designed to emphasize two different aspects of the definition. By fixing γ, we obtain a linear form

ω IG(γ, −): H(G) → C.

By ﬁxing f, we obtain a function

ω fG(−): G(F ) → C.

ω In particular, we may view fG as a function on the set of regular elements in G(F ). We write

ω ω I (G) = {fG : f ∈ H(G)}

for the space of such functions. By deﬁnition there is a surjective mapping

ω ω f 7→ fG : H(G) → I (G).

Members of Iω(G) will sometimes be called invariant functions.

1.5.6 The functions in Iω(G) have a spectral interpretation. One considers a representation

(π, Vπ) of G(F ) and a map ω π : G(F ) → Aut(Vπ) such that

ω ω π (g1gg2) = π(g1)π (g)π(g2)ω(g2).

This simplification of the general definitions, see [Art13, pp.52-53], simplifies further in our case. Suppose (π, Vπ) is a representation of G(F ) and π ' π ⊗ ω. (Recall that π ⊗ ω is the representation on Vπ defined by (π ⊗ ω(g)) = ω(g)π(g).) Then there is an operator A = Aπ

on V = Vπ such that π(g)A = ω(g)Aπ(g).

Thus A is an intertwining operator from (π ⊗ ω, V ) to (π, V ). Then, we may deﬁne a map

g 7→ π(g)A = πA : G(F ) → Aut(V ).

69 The map πA satisﬁes

A π (g1gg2) = π(g1gg2)A = π(g1)π(g)π(g2)A

A = π(g1)π(g)Aπ(g2)ω(g2) = π(g1)π (g)π(g2)ω(g2).

Fix such an A. Then

Z πA(f) = f(x)πA(x) dx = π(f)A G(F )

is an operator on V and we can consider tr(π(f)A). Then set

ω A fG(π) = tr(π (f)) = tr(π(f)A).

The notation is slightly misleading, as the trace depends on the choice of A and not just on ω and π. Later, we will describe canonical choices for the operator A. As before, the diﬀerent notations are meant to indicate that the trace deﬁnes both a function on any set of

(admissible) representations of G(F ) and a linear form on H(G).

Recall that Πtemp(G) denotes the set of tempered representations of G(F ). In general, one expects a rough duality between regular conjugacy classes and tempered representa-

ω tions. As evidence of this duality, note that either of the functions {fG(γ)}γ∈Greg(F ) or ω {fG(π)}π∈Πtemp(G) determines the other, in the following sense. The Weyl integration formula ω ω implies that if fG(γ) vanishes for all γ then fG(π) also vanishes. See [HH95, p. 145] for more ω ω details. Since the regular set is dense in G(F ), this shows that {fG(γ)} determines {fG(π)}. The other direction is more diﬃcult, and it is a theorem of Moeglin and Waldspurger [MW18], proved in certain cases by Kazhdan [Kaz86] and Henniart-Lemaire [HL13]. They showed

ω ω that if fG(π) vanishes for all tempered representations then all regular fG(γ) vanish too. The spectral interpretation gives a diﬀerent description via the trace Paley-Wiener

Theorem. See [BDK86], [Rog88], [CD90], [DM08] and [HL13]. Roughly speaking, the trace

Paley-Wiener Theorem says that if a function λ on Π(G) shares enough properties with a

ω ω function of the form fG, then λ must be of the form fG.

70 1.5.7 The transfer factor ∆0 is deﬁned in §5.3 of [KS99] for triplets (G, θ, a) where G is quasi-split and θ preserves an F -splitting of G. See also [LS87, §3.7]. As usual, we will focus on the case of (G, ω) with G satisfying the conditions of 1.1.1 and ω of ﬁnite order.

We will need the notion of a Whittaker datum (B, χ) for G. Here B is a Borel subgroup of G deﬁned over F with unipotent radical U and χ is a generic character of U(F ). Recall that a character of U(F ) is generic if its centralizer in T (F ) is equal to Z(F ). Let ψF be a non-trivial additive character of F and let (B,T, {Xα}) be an F -splitting of G, see 1.1.1.

The choice of root vectors {Xα} leads to a surjection

X Y U Ga α∈∆(T,b) deﬁned over F . The set ∆(T, b) consists of the simple roots of T in the Lie algebra of B.

Composing X with the natural map

Y X Y xα 7→ xα : Ga → Ga α gives a map U → Ga deﬁned over F , since the splitting was chosen to be over F . We obtain a map U(F ) → F which can be composed with ψF to yield a generic character of U(F ). Every generic character can be obtained this way. The pair (B, χ) is called a Whittaker datum.

H 1.5.8 Let H represent a (G, ω)-datum. The factor ∆0(γ , γ) is a complex valued function on

{G − regular semisimple elements in H(F )} × {regular semisimple elements in G(F )}.

H H The factor vanishes unless γ is a norm of γ. There is not a unique choice of ∆0(γ , γ). In [KS99, (5.3)], the construction of the factor depends on a choice of splitting for G. However, the factor can be normalized via ε-factors. Assume that T (respectively TH ) is a torus in a splitting for G (respectively H) with Borel subgroup B. The virtual ΓF -representation

∗ ∗ V = X (T ) ⊗ C − X (TH ) ⊗ C

71 is of degree 0 and has an associated local factor εL(V, ψ)[Tat79, (3.6)]. In [KS99, §5.3], the normalized transfer factor ∆G,H is deﬁned by

∆G,H = εL(V, ψ)∆0

The normalized factor ∆G,H depends only on the Whittaker datum (B, χ).

The transfer factor ∆G,H satisﬁes two important properties under conjugation [KS99, Lemma 5.1.B, Theorem 5.1.D].

H H Proposition. 1. If γ1 , γ2 ∈ H(F ) are stably conjugate, we have

H H ∆G,H (γ1 , γ) = ∆G,H (γ2 , γ).

2. If g ∈ G(F ), we have

H −1 H ∆G,H (γ , g γg) = ω(g)∆G,H (γ , γ).

1.5.9 For the pair (GN , ω) and its twisted endoscopic groups, rational conjugacy and stable conjugacy are the same. Indeed, for the group GN , the existence of the rational canonical form implies that for γ ∈ GN (F ) the set Ogeom(γ)∩GN (F ) is equal to the rational conjugacy class Orat(γ):

Ogeom(γ) ∩ GN (F ) = Orat(γ).

However, we can also deduce this fact, as well as the corresponding statement for HM , by using the formalism introduced in [Lan79c]. For additional expositions, see [Art05, §27] or

[Har11].

Let γ ∈ G(F ) be a regular element. (In general, the following statements are only true for strongly regular elements.) Let T = Gγ be the centralizer of γ in G. Then T is a maximal torus deﬁned over F . Suppose γ0 ∈ G(F ) is stably conjugate to γ. That is, there is

0 −1 a g ∈ G(F ) such that γ = g γg. Let σ ∈ ΓF . Then

γ = σ(γ) = σ(gγ0g−1) = σ(g)σ(γ0)σ(g)−1 = σ(g)γ0σ(g)−1 = σ(g)g−1γgσ(g)−1.

72 The element gσ(g)−1 lies in T (F ) and

−1 σ 7→ gσ(g) :ΓF → T (F )

0 1 0 is a 1-cocycle of ΓF in T (F ). If G is a group deﬁned over F , we will write H (F,G )

1 0 −1 1 for H (ΓF ,G (F )). Then we regard σ 7→ gσ(g) as an element of H (F,T ). The stable conjugacy class of γ consists of ﬁnitely many G(F )-conjugacy classes. One checks that

g−1γg = γ0 7→ (σ 7→ gσ(g)−1) induces a well deﬁned bijection between the set of G(F )-conjugacy classes in the stable conjugacy class of γ and

ker(H1(F,T ) → H1(F,G)).

Thus, to show that stable conjugacy and rational conjugacy coincide (for regular elements)

1 in some group G, it suﬃces to show that H (F,T ) is trivial for each γ. (Recall that T = Gγ.)

This is a very rare property, and the only cases that come to mind are GN and its inner forms [AC89, Ch. 2].

Let γ ∈ HM (F ) be a GN -regular element. Then HM,γ is isomorphic to a direct product

0 of groups of the form RE0/F G1,E0 for E ⊃ E. Thus, it suﬃces to prove the following lemma.

Lemma. We have

1 H (F, RE/F G1,E) = {1}.

Proof. There are at least two ways to see this. The ﬁrst way is to note that there is a

ΓF -equivariant bijection (R G )(F ) ' IΓF G (F ), E/F 1,E ΓE 1,E see 1.1.4 which follows [Bor79, §5]. The lemma follows by applying Shapiro’s Lemma followed by Hilbert’s Theorem 90.

A second option is to apply Tate-Nakayama duality [Kot84, §3], which in this setting is an isomorphism ∗ 1 ∼ ΓF H (F,T ) → π0 Tb .

73 × The ﬁxed point set of ΓF in the dual torus is isomorphic to C , which is connected.

1.5.10 Having discussed the transfer of conjugacy classes and transfer factors, we may introduce the notion of transfer of functions. We continue to work with a general pair (G, ω).

The special case of (GN , ω) will be discussed in more detail shortly. Let f ∈ H(G) and let H represent a (G, ω)-datum. Fix any transfer factor ∆(−, −) for H and G. Suppose γH is a G-regular stable conjugacy class in H(F ). Then we deﬁne a number f H (γH ) by

H H X H ω f (γ ) = ∆(γ , γ)fG(γ). γ∈Oreg(G)

The sum is over the regular (rational) conjugacy classes in G(F ). For a fixed γH , the sum is finite. Note that each term of the sum is well defined thanks to Proposition 1.5.8. The number f H (γH ) depends on the choice of transfer factor.

Consider the special case when H = G represents the trivial endoscopic datum for the triplet (G, id, 1). Then ∆(−, −) can be taken to be 1 whenever it is nonzero. More precisely,

∆(γ0, γ) = 1 if γ and γ0 are stably conjugate and vanishes otherwise. Then we obtain

G G X f (γ ) = fG(γ). G γ∈Ostab(γ )

Thus f G(γG) is the sum of the orbital integrals of representatives of the G(F )-conjugacy classes in the stable class of γG. The number f G(γG) is the stable orbital integral of the function f at the stable conjugacy class γG. We write S(G) for the space of functions on the set of stable regular conjugacy classes obtained this way. That is,

S(G) = {f G : f ∈ H(G)}.

A linear form S on H(G) is called stable if its value at f depends only on f G.

The following two theorems were long standing obstacles in the subject. Both theorems hold for the general triplet (G, θ, a) discussed brieﬂy at the beginning of 1.2.1. Since we only gave the deﬁnitions of endoscopic datum in the case when θ = id, we state the theorems only in this case. Fix the pair (G, ω).

74 Theorem 1.5.11 (Langlands-Shelstad-Kottwitz conjecture). For any endoscopic datum

H ∈ Eω(G), transfer factor ∆(−, −) and f ∈ H(G), the function f H lies in S(H).

Theorem 1.5.12 (Fundamental lemma). Suppose in addition that F is non-archimedean and

G is unramiﬁed. If f is the characteristic function of a hyperspecial maximal compact subgroup of G(F ), then f H is the image in S(H) of the characteristic function of a hyperspecial maximal compact subgroup of H(F ).

For the archimedean case of Theorem 1.5.11, some of the main references are [She82],

[She08a], [She10], [She08b] and [She12]. For a diﬀerent exposition, see [Ren11]. For Theorem

1.5.11 in the non-archimedean case and Theorem 1.5.12, the main references are [Wal97],

[Wal06], [Wal08], [Wal09] and [Ngˆo10]. See [Art13, p.54] for a more complete discussion of the references and the history surrounding these conjectures.

1.5.13 Let H be a (GN , ω)-datum. We have seen that for H and GN that there is no diﬀerence between rational conjugacy and stable conjugacy. Any F -basis of E induces an embedding of H(F ) in GN (F ). All such embeddings are conjugate to one another. Let

H γ ∈ H(F ) be a GN -regular class. Then the corresponding class in GN (F ) is the class

H H generated by the image of γ in GN (F ) under any embedding. Consequently ∆(γ , γ)

H H vanishes unless γ is GN (F )-conjugate to γ . Since we can always take γ itself as a representative of the associated class γ, we will write ∆(γH ) = ∆(γH , γH ). The matching condition becomes the following statement.

H Theorem. Let f ∈ H(GN ). Then there exists f ∈ H(H) such that for each GN -regular γH ∈ H(F ) we have

I (γH , f H ) = ∆(γH )Iω (γH , f). H GN

As discussed above, technically speaking f H is not a function, but rather a member of

S(H) = I(H). See 1.5.5 for I(H). However, it will occasionally be useful to choose any function f H representing the conjugacy class function. Given f ∈ H(N) and φ ∈ H(H), we will say that f and φ have matching orbital integrals if their images in I(H) are equal, and in particular if the statement of the theorem holds for f and φ. When H = H1 = RE/F GL1,E, 75 certain cases of Theorem 1.5.13 were proved by Kazhdan [Kaz84]. With some restrictions on f, Herb proved Theorem 1.5.13 in [Her95]. There she also proved Theorem 1.5.13 for any f when E is an unramiﬁed extension of F . The cases of the theorem proved in [Her95] were suﬃcient for [HH95]. The special case of Theorem 1.5.12 for automorphic induction is also due to Waldspurger [Wal91].

ω 1.5.14 If H ∈ E (N), we have the normalized transfer factor ∆G,H (−, −) discussed in

1.5.8. It depends only on the choice of a Whittaker datum (BN , χ) for GN . Since GN has a standard upper triangular Borel subgroup and splitting, we can regard the normalized transfer factor as depending only on a choice of additive character. Then there is a canonical transfer map

H f 7→ f : H(GN ) → I(H).

We showed in Lemma 1.2.7 that the automorphism group of HM as an endoscopic datum can be identiﬁed with the Galois group ΓE/F . The factor ∆G,HM is invariant under the action of

HM such automorphisms on the ﬁrst variable. Thus, f lies in the subspace of ΓE/F -invariant ω functions in I(HM ). We let I (HM ) denote the set of such Galois invariant functions and

ω ω let H (HM ) be the preimage of I (HM ) under the map f 7→ fHM . (We use the superscript ω to indicate the natural space of functions when discussing ω-twisted endoscopy. It may have been preferable to use a superscript ΓE/F to indicate Galois-invariant functions, but as we shall see shortly the corresponding space of functions for Levi subgroups of HM is smaller than the set of Galois-invariant functions.)

The non-elliptic (GN , ω)-endoscopic data are represented by Levi subgroups of HM . Let

ω L = LHM ∈ E (N) be a Levi subgroup of HM . At 1.2.8, we introduced the standard datum associated to L, which realized L as a (GN , ω)-datum by ﬁrst realizing L as a (LN , ω)-datum for a corresponding Levi subgroup of GN and then using the standard diagonal embedding for

Levi subgroups of GN . At 1.2.9, we showed that L can also be realized as a (GN , ω)-datum by first using the standard Levi embedding in HM followed by the standard endoscopic map ξHM realizing HM as a (GN , ω)-datum. The different realizations are conjugate to one another, and correspond to two different ways of factoring the transfer map f 7→ f L.

76 Both factorizations rely on the Harish-Chandra descent map, which we shall recall. Both realizations also show that, in general, the automorphism group of L as an endoscopic datum for (GN , ω) will contain automorphisms induced by the Weyl group of a Levi subgroup and the Galois group.

1.5.15 We now recall the Harish-Chandra descent formulas. Let G be a general group and

1 let ω be a character of G(F ) determined by an element of H (WF ,Z(Gb)) as in 1.2.1. Let L be a Levi subgroup and let P ∈ P(L) be a parabolic subgroup. Let P = LNP be the Levi decomposition of P . We set

Z Z ω 1/2 −1 −1 fL (m) = δP (m) f(k mnk)ω(k) dn dk, m ∈ L(F ). K NP

The notation is slightly misleading, since the function fL does not depend only on L.

However, the orbital integrals of fL at G-regular points in L(F ) depend only on L. For such descent formulas in great generality, see [Art88a, §8]. Consequently, we obtain a well deﬁned

ω function fL ∈ I(L) that satisﬁes the following descent formula for G-regular elements of L(F ):

ω ω IG(γ, f) = IL (γ, fL). (1.5.16)

(We should note that the notation IL(−, −) is reserved for a diﬀerent linear form in most of

Arthur’s work. Since those forms do not appear here, there is no harm in using IL(−, −) to

L denote an orbital integral on L. The “proper” notation would be IL (−, −).)

Similar formulas hold on the spectral side. More speciﬁcally, let L ⊂ GN be a Levi subgroup whose center lies in the kernel of ω and let πL be an ω-ﬁxed representation of

L(F ). Let AπL be a ﬁxed operator on VπL satisfying πL(m)A = ω(m)AπL(m). The induced

GN representation π = IL (πL) can be realized on a space of functions with values in V πL. If φ is such a function, the map Aπ : φ 7→ AπL ◦ φ satisﬁes π(g)Aπ = ω(g)Aππ(g). Then we have

ω tr (π(f) ◦ Aπ) = tr (πL(fL ) ◦ AπL ) . (1.5.17)

Induced representations and intertwining operators are discussed in more detail in the next section. However, we have mentioned them here because the Harish-Chandra descent map

77 ω f 7→ fL is the simplest form of an endoscopic transfer of functions. Indeed, for any group G its quasi-split Levi subgroups represent endoscopic data. Then formula (1.5.16) says

ω that f 7→ fL is the Langlands-Shelstad-Kottwitz transfer for the pair (G, L). Dual to the transfer mapping f 7→ fL, one expects a mapping of representations πL → πG that satisﬁes a character relation. Formula 1.5.17 says that the mapping πL 7→ πG is given by parabolic induction, the most basic case of a transfer of representations.

1.5.18 Now we may discuss the transfer map in general. Let L = LHM be a Levi subgroup of HM . As discussed at 1.2.8, L represents a (GN , ω)-datum. In particular, according to Theorem 1.5.11, there is a local transfer map

f 7→ f L : H(N) → I(L).

The transfer factors can be chosen consistently in Levi subgroups, as least for GN -regular elements. See [HI12, §4]. When the transfer factors are chosen consistently, the following diagram commutes:

H f7→f M ω H(N) / I (HM )

f7→fL HM HM GN f 7→(f )L I(L ) I(L) GN L / fL 7→(fL ) GN GN

Either of the composite maps give the transfer f 7→ f L. They correspond to the two diﬀerent methods for realizing L as a (GN , ω)-datum mentioned at the end of 1.5.14.

1.5.19 Let HM represent the unique class of elliptic (GN , ω)-data. The following proposition is the analog of [Art13, Corollary 2.1.2]. As discussed in the proof of [Art13, Proposition

2.1.1], this is an important proposition used in the proof of the stabilization of the trace formula, which will be discussed in §2.2 and is discussed more thoroughly in [Art13, §3.2].

The work of Moeglin and Waldspurger has established the twisted trace formula in full generality. Thus we regard the proposition as known. For an indication of the proof, see

[Art13, Proposition 2.1.1] and [Art13, Corollary 2.1.2].

78 Proposition. The twisted transfer mapping

HM ω f 7→ f : H(N) → I (HM ) is surjective.

ω ω Let I (L) ⊂ I(L) be the image of the Harish-Chandra descent map from I (HM ) to I(L). According to the previous proposition, the space Iω(L) is the image of the transfer map f 7→ f L from H(N) to I(L). In order to give a more concrete description of Iω(L), we need to understand the image of the Harish-Chandra transfer map. This is provided by

[Wal14, Lemme 4.3]. As in the previous proposition, the image is “as large as possible.”

ω Lemma. Let G be a group and let L ⊂ G be a Levi subgroup. The image of the map f 7→ fL consists of all functions φ ∈ I(L) whose orbital integrals satisfy

ω −1 ω IL (gmg , φ) = ω(g)IL (m, φ).

for every G-regular element m ∈ L(F ) and g ∈ G(F ) such that gmg−1 ∈ L(F ).

1.6 Local intertwining operators

There are two types of local intertwining operators we will need to consider. We will start with the intertwining operators for representations that are ﬁxed by ω. Then we will recall the basic properties of the local intertwining operators between induced representations. We follow closely the treatment of [Art13, §§2.2-2.5]. Much of the discussion will have to do with normalizing intertwining operators following Shahidi as well as the special normalizations available when working on GN . The global operator that occurs in the discrete part of the trace formula is a certain self-intertwining operator. It is not normalized, but it is explicit. Later, we will compare the local factors of global intertwining operators with the explicit self-intertwining operators that we construct and normalize here.

79 1.6.1 Let F be a local ﬁeld of characteristic 0. Let E be a cyclic extension of F of degree

D and let ω = ωE/F be a character determined by the extension. In other words, ω is a × × character of F with the property that ker ω = NE/F E . The existence of such a character

ω is guaranteed by local class ﬁeld theory. Let GN = GN,F be the group GL(N) regarded as an F -group.

Let π be an admissible representations of GN (F ). (From now on, all local representations are assumed to be admissible.) We write ω also for the character of GN (F ) obtained by composing ω with the determinant. The twisted representation π ⊗ ω acts on the same space as π according to the formula

(π ⊗ ω)(g) = ω(g)π(g).

We will primarily be concerned with representations that are ω-ﬁxed. In other words, representations (π, Vπ) that are isomorphic to their twist (π ⊗ ω, Vπ). Given an ω-ﬁxed representation (π, Vπ), there is an automorphism Aπ of Vπ satisfying

Aπ ◦ π(g) = ω(g)π(g) ◦ Aπ, g ∈ GN (F ).

Such an operator Aπ is just an intertwining operator from (π, Vπ) to (π ⊗ ω, Vπ). If GN (F ) has an ω-ﬁxed representation, then D divides N, see 1.4.4, so we assume that this is the case.

1.6.2 Let π be an irreducible ω-ﬁxed representation. According to Schur’s lemma, the intertwining operator Aπ is unique up to a scalar. We will normalize this operator using Whittaker models.

Whittaker data were discussed in 1.5.7. We have ﬁxed the standard upper triangular

Borel subgroup TN UN = BN ⊂ GN . Since GN also has a standard splitting, see 1.1.3, the

Whittaker datum (BN , χ) can be regarded as canonical up to the choice of a nontrivial additive character ψF of F , which we now regard as ﬁxed. Suppose ﬁrst that π is an

ω-ﬁxed tempered representation of GN (F ). It is a fact that any tempered representation of

GN (F ) is χ-generic for any χ. See [HH95, §§2.1-2.3] for a short summary of the relevant

80 results, which are based on the classiﬁcation theorems of [Zel80] and [Ber84]. Let λπ be a

(BN , χ)-Whittaker functional. Thus λπ is a nonzero linear form on Vπ,∞ ⊂ Vπ, the space of smooth vectors for π, such that

λπ(π(n)v) = χ(n)λπ(v), n ∈ UN (F ), v ∈ V∞.

The form λπ is unique up to a scalar multiple [Sha74].

Fix an operator Aπ intertwining (π, Vπ) and (π ⊗ ω, Vπ) as above. Then

(λπ ◦ Aπ)(π(n)v) = λπ(Aππ(n)v)

= λπ(ω(n)π(n)Aπv)

= λπ(π(n)Aπv)

= χ(n)λπ(Aπv)

= χ(n)(λπ ◦ Aπ)(v).

That is, (λπ ◦ Aπ) is a χ-Whittaker functional on Vπ. By the uniqueness of the Whittaker functional, λπ ◦ Aπ = c · λπ for some c 6= 0. From now on, we take Aπ to be the unique intertwining operator between (π, Vπ) and (π ⊗ ω, Vπ) satisfying λπ = λπ ◦ Aπ, that is c = 1.

Note that, since they only diﬀer by a scalar, a diﬀerent choice of λπ leads to the same Aπ.

Thus Aπ can be regarded as canonical up to the choice of ψF used to deﬁne χ.

1.6.3 We now discuss a particular realization of the induced representation that is well- suited for discussing twisting by ω. A diﬀerent realization, which is better for discussing intertwining operators, is introduced below. In this subsection, we allow G to be a general group over F , since it does not complicate the discussion.

Let L ⊂ G be a Levi subgroup of G deﬁned over F . Let P ∈ P(L) be a parabolic

subgroup of G with unipotent radical NP . Let (πL,VπL ) be an admissible representation of L(F ). Deﬁne a space of functions as follows:

G 1/2 HP (πL) = {f : G(F ) → VπL : f(mng) = δP (m)πL(m)f(g), m ∈ L(F ), n ∈ NP (F )}.

81 G The group G(F ) acts on HP (πL) by right translation:

0 0 0 R(g)f(g ) = f(g g), g, g ∈ G(F ).

G G The resulting representation of G(F ), IP (πL) = (R, HP (πL)) is one realization of the induced representation. When G has been ﬁxed we drop it from the notation.

Later, we will be interested in the family of induced representations obtained by twisting

π . For λ ∈ a∗ , we consider the twist L L,C

λ(HL(m)) πL,λ(m) = πL(m)e .

Then πL,λ is a representation of L(F ) acting on the same space as πL. The vector space a∗ and the map H were discussed in 1.3.6. As λ varies, we obtain a family of induced L,C L G representations IP (πL,λ).

Suppose f ∈ HP (πL). Consider the function (ωf) given by (ωf)(g) = ω(g)f(g). Then

(ωf)(mng) = ω(mng)f(mng)

1/2 = ω(m)ω(g)δP (m)πL(m)f(g) 1/2 = δP (m)(πL ⊗ ω)(m)(ωf)(g).

Thus there is a linear isomorphism

f 7→ ωf : HP (πL) → HP (πL ⊗ ω).

Denote this map by Iω. Then there is a commutative diagram for any g:

Iω HP (πL) / HP (πL ⊗ ω)

R(g)⊗ω R(g) HP (πL) / HP (πL ⊗ ω) Iω

In other words, Iω intertwines the representations (R ⊗ ω, HP (πL)) and (R, HP (πL ⊗ ω)).

1.6.4 Let L be a block diagonal Levi subgroup in GN . The standard splitting of GN induces a splitting of L. In particular, if (BN , χ) is a Whittaker datum for GN and we set

82 BL = BN ∩ L, then (BL, χ) is a Whittaker datum for L. In this way, a choice of splitting and Whittaker datum for GN induces splittings and Whittaker data for all block diagonal Levi subgroups.

Assume πL ' πL ⊗ ω and let AπL be any intertwining operator between πL and πL ⊗ ω.

GN Consider the induced representation π = IP πL. We deﬁne

Aπ : HP → HP by

(Aπf)(g) = AπL (f(g)).

The operator Aπ intertwines π and π ⊗ ω.

Suppose now that the inducing representation πL has a Whittaker functional λπL .

Associated to λπL is a nontrivial Whittaker functional λπ for the induced representation, see [Sha81, Proposition 3.1]. Even if the induced representation is reducible, the Whittaker

functional is unique up to scalars [Rod73]. Now let AπL be the intertwining operator normalized using Whittaker functionals as in 1.6.2 and consider the associated intertwining operator Aπ for the induced representation π, where we assume the parabolic subgroup P is standard. We have the following lemma [HI12, Lemma 4.4].

Lemma. Let λπ be a Whittaker functional for the induced representation π. Then λπ ◦Aπ =

λπ.

Thus the canonical intertwining operator obtained from parabolic induction coincides with the intertwining operator obtained from normalizing via Whittaker models.

When working with twisted representations, inducing from ω-ﬁxed data provides a

ω ω functor from Πadm(L) → Πadm(GN ). However, there are induced representations of GN (F ) that are ω-fixed but are not induced from ω-fixed representations. These representations are said to be ω-discrete and in the theory of ω-fixed representations they behave more like discrete series representations than induced representations. The basic example is the

83 representation πω,D of GD(F ) induced from

1 ⊗ ω ⊗ · · · ⊗ ωD−1,

a unitary character of the maximal torus of GD(F ). It is not immediately clear what operator intertwines πω,D with πω,D ⊗ ω. This is part of the discussion of intertwining operators to come.

1.6.5 Fix a group G and a unitary character ω of G(F ). The Langlands classiﬁcation is stated in Appendix A.2. For now, we just recall that every irreducible, admissible representation can be obtained in an essentially unique way as the unique irreducible quotient of a standard representation, namely an induced representation ρ = (R, HP (τλ)) where τ is a tempered representation of L(F ), P ∈ P(L) is a parabolic subgroup and λ is

∗ ∗ a positive character in aL. See 1.3.6 for aL. The notion of a character being positive is discussed in A.2. Standard representations are generally not irreducible. However, since every tempered representation for GN (F ) is generic, every standard representation has an essentially unique nontrivial Whittaker functional. Thus one expects to be able to normalize intertwining operators between standard representations by using Whittaker functionals.

We expect an analog of the Langlands classification for ω-fixed representations. This type of twisted Langlands classification is discussed in [HL13], see §2.12 there. In particular, we have the following useful fact.

G Lemma. Let (R, HP (τλ)) be a standard representation with Langlands quotient (π, Vπ). Then τ is ω-ﬁxed if and only if π is ω-ﬁxed.

Proof. Note that, since ω is unitary, if τ is tempered then τ ⊗ ω is tempered. As we saw at 1.6.3, the representations (R ⊗ ω, HP (τλ)) and (R, HP ((τ ⊗ ω)λ)) are isomorphic. Now, suppose that τ is ω-ﬁxed. Then (R, HP ((τ ⊗ ω)λ)) and (R, HP (τλ)) are isomorphic standard representations. We have maps

(R, HP ((τ ⊗ ω)λ)) ' (R ⊗ ω, HP (τλ)) (π ⊗ ω, Vπ).

84 and

(R, HP ((τ ⊗ ω)λ)) ' (R, HP (τλ)) (π, Vπ).

Since (π ⊗ ω, Vπ) and (π, Vπ) are irreducible quotients of the same standard representation, they must be isomorphic.

Conversely, assume that (π, Vπ) ' (π ⊗ ω, Vπ). According to the ﬁrst map above, the representation (π ⊗ ω, Vπ) is the Langlands quotient of (R, HP ((τ ⊗ ω)λ)). We also have

(R, HP (τλ)) (π, Vπ) ' (π ⊗ ω, Vπ).

Since (R, HP ((τ ⊗ ω)λ)) and (R, HP (τλ)) are standard representations with the same Lang- lands quotient, we must have τ ⊗ ω ' τ.

Thus the bijection provided by the Langlands classiﬁcation between Σ(N), the set of standard representations, and Π(N) restricts to a bijection Σω(N) ←→ Πω(N). The set

Σω(N) consists of ω-ﬁxed standard representations. According to the lemma, this is also the set of standard representations obtained from ω-ﬁxed inducing data.

Consider a standard representation (R, HP (τλ)). Suppose A : HP (τλ) → HP (τλ) inter- 0 0 twines R and R ⊗ ω. Let H ⊂ HP (τλ) be the subspace such that HP (τλ)/H is the unique 0 irreducible quotient. Then A preserves H . Indeed,

0 −1 0 (R, HP (τλ)/AH ) ' (R ⊗ ω , HP (τλ)/H ).

The representation on the right is irreducible. Thus the representation on the left is an irreducible quotient of (R, HP (τλ)). The uniqueness in the Langlands classiﬁcation implies 0 0 that AH = H . Consequently, the operator A gives a well-deﬁned intertwining operator 0 0 between the quotients (R, HP (τλ)/H ) and (R ⊗ ω, HP (τλ)/H ).

Now let π be any ω-ﬁxed representation of GN (F ). Let ρ = (R, HP (τλ)) be the ω-ﬁxed standard representation with π as its Langlands quotient. Since ρ has a nonzero Whittaker functional, there is an operator Aρ intertwining ρ and ρ⊗ω normalized as in 1.6.2. According to the previous paragraph, Aρ induces an operator Aπ intertwining π and π ⊗ ω. This is the normalized intertwining operator for π.

85 1.6.6 We mention a general situation which serves as a transition to the discussion of the other type of intertwining operators we will consider. Consider an induced representation

π = (R, HP (πL)) as in 1.6.3. We introduced there the linear map Iω which intertwines

(R ⊗ ω, HP (πL)) and (R, HP (πL ⊗ ω)). If (R, HP (πL)) happened to be ω-ﬁxed, one would hope that Iω would provide the normalized intertwining operator Aπ. The problem is that

Iω does not send HP (πL) to itself. In other words, the spaces of functions HP (πL) and

HP (πL ⊗ ω) are not equal, even if the induced representation is ω-ﬁxed. In an attempt to solve this problem, we will have to introduce additional intertwining operators.

1.6.7 Now we switch gears and discuss the intertwining operators that exist between induced representations. We will also be concerned with the normalization of these operators.

We continue to let F denote a local field of characteristic 0. For now, we let G be a general group defined over F and let L0 be a fixed minimal Levi subgroup defined over F . Recall that a maximal compact subgroup K ⊂ G(F ) satisfying certain properties was fixed at 1.5.4.

Fix a Levi subgroup L containing L0.

Let P ∈ P(L) be a parabolic subgroup of G with Levi decomposition P = LNP . Let

KL = K ∩ L(F ). Consider the following space of functions on K:

HP (π) = {φ : K → Vπ : φ is K-ﬁnite and φ(nmk) = π(m)φ(k),

n ∈ NP (F ) ∩ K, m ∈ KL, k ∈ K}.

For λ ∈ a∗ we have the family of twisted representations π , see 1.6.3. Then H (π) = L,C λ P

HP (πλ). Since the space HP (π) does not depend on λ, we will put the λ-twist in the action

RP (λ) of G(F ) on HP (π).

We need some more notation before deﬁning the action RP (λ). For g ∈ G(F ), we write the Iwasawa decomposition relative to P as

g = NP (g)L(g)K(g),NP (g) ∈ NP (F ),L(g) ∈ L(F ),K(g) ∈ K.

86 We deﬁne a homomorphism HP : G(F ) → aL by

HP (g) = HL(L(g)).

We let ρP denote the usual half sum of the roots of AL, the maximal split torus in the center of L, in NP . The action RP (λ) of G(F ) on HP (π) is given by

(λ+ρP )(HP (kg)) (RP (λ, g)φ)(k) = π(L(kg))φ(K(kg))e , g ∈ G(F ), k ∈ K.

The representation IP (πλ) = (RP (λ), HP (π)) is isomorphic to IP (πλ) = (R, HP (πλ)).

Indeed, suppose φ ∈ HP (π). We may deﬁne a function fφ ∈ HP (πλ) by

1/2 fφ(nmk) = δP (m) πλ(m)φ(k), n ∈ NP (F ), m ∈ L(F ), k ∈ K.

The map φ 7→ fφ depends on λ. It provides an intertwining isomorphism from (RP (λ), HP (π)) to (R, HP (πλ)). Let P,P 0 ∈ P(L) be parabolic subgroups of G. There is an operator

JP 0|P (πλ): HP (π) → HP 0 (π)

that intertwines the induced representations IP (πλ) and IP 0 (πλ). For the real part of λ in a

∗ certain aﬃne chamber in aL, the operator is given by the absolutely convergent integral

Z 0 0 0 (λ+ρP )(HP (n )) 0 JP 0|P (πλ)φ (k) = π(L(n ))φ(K(n ))e dn . NP 0 (F )∩NP (F )\NP 0 (F ) As above, L(n0) and K(n0) are deﬁned using the Iwasawa decomposition relative to P . The integral can be analytically continued to a meromorphic function of λ ∈ a∗ . Some of the L,C main references for the basic properties of these operators are [Har75], [KS80] and [Sha81].

We have a ﬁxed additive character ψF . A splitting of G together with the measure on F that is self-dual with respect to ψF determine a measure on the quotient NP 0 (F ) ∩

NP (F )\NP 0 (F ). See [Art13, pp.82-83] for more details and discussion. Any of our groups

H = RE/F GM,E have a standard splitting. Thus we can regard the measure on NP 0 ∩

NP (F )\NP 0 (F ) as depending only on ψF . For w ∈ K, let l(w) denote the linear map from

−1 HP (π) to HwP w−1 (wπ) deﬁned by l(w)φ(k) = φ(w k). 87 1.6.8 Our goal is to normalize the local intertwining operators JP 0|P (πλ). Before discussing the normalization more precisely, let us state the following theorem, [Art89, Theorem 2.1].

Let L ⊂ L0 be a Levi subgroup of L0. Recall that for any Q ∈ P(L0) and R ∈ PL0 (L) there

0 is a unique element Q(R) ∈ P(L) such that Q(R) ⊂ Q and Q(R) ∩ L = R. If φ ∈ VQ(R)(π), then for k ∈ K the functions φk on KL0 deﬁned by φk(kL0 ) = φ(kL0 k) lies in VR(π).

Theorem. There exist meromorphic, scalar valued functions

0 rP 0|P (πλ),P,P ∈ P(L), π ∈ Π(L(F )) such that the normalized intertwining operators

−1 RP 0|P (πλ) = rP 0|P (πλ) JP 0|P (πλ) have analytic continuation as meromorphic functions of λ ∈ a∗ and such that the following L,C properties hold.

• RP 0|P (πλ)RP (λ, g) = RP 0 (λ, g)RP 0|P (πλ).

0 00 • RP 00|P (πλ) = RP 00|P 0 (πλ)RP 0|P (πλ), for any P , P , P in P(L).

0 L0 • (RQ(R0)|Q(R)(πλ)φ)k = RR0|R(πλ)φk, φ ∈ HQ(R)(π), k ∈ K, for R, R in P (L) and Q ∈ P(L0).

• If π is unitary, then

∗ RP 0|P (πλ) = RP |P 0 (π−λ).

−1 • l(w)RP 0|P (πλ)l(w) = RwP 0w−1|wP w−1 ((wπ)wλ), for any w ∈ K.

∨ • If F is archimedean, RP 0|P (πλ) is a rational function of {λ(α ): α ∈ ∆P }. If F is a

non-archimedean ﬁeld with residue ﬁeld of order q, RP 0|P (πλ) is a rational function of −λ(α∨) {q : α ∈ ∆P }.

• If π is tempered, rP 0|P (πλ) has neither zeros nor poles with the real part of λ in the positive chamber attached to P .

88 • Suppose that F is non-archimedean, that G and π are unramiﬁed and that K is

hyperspecial. Then if φ ∈ HP (π) is ﬁxed by K, the function RP 0|P (πλ)φ is independent of λ.

Arthur shows in [Art89, §2] that this theorem can be reduced to the special case that

L is maximal in G (that is, that dim(AL/AG) = 1) and π is square-integrable modulo the center of AL. More precisely, if the functions rP 0|P (πλ) are deﬁned in this case and the properties above hold, there is a canonical way to obtain the normalizing factors for all representations and all Levi subgroups. Arthur provides a construction of the normalizing factors in this special case, and thus in general. In the archimedean case, the normalization can be done for any group using ratios of L-functions following a suggestion of Langlands.

The construction described in [Art89, §4] for the p-adic case (originally due to Langlands) is not in terms of ratios of L-functions.

1.6.9 When G is quasi-split and π is generic, Shahidi proved Langlands’ conjecture on normalizing factors, see [Sha90]. We will recall the results more precisely in a moment. To study the full discrete spectrum in arbitrary rank, it turns out that one has to normalize the intertwining operators in situations where the inducing data is unitary but not necessarily generic. This is a signiﬁcant problem in Arthur’s work on classical groups, see in particular

[Art13, §§2.3-2.5]. However, GN is unique in that every tempered representation is generic. Consequently, we can normalize intertwining operators for arbitrary inducing data using

Shahidi’s normalization in the generic case combined with the Langlands classification, recalled in A.2. The normalization in [Art13] is carried out in three steps. We will discuss the first step here, which is the most difficult and due mostly to Shahidi, and the latter two steps, as well as the modifications needed for the ω-twisting, in the remainder of the section.

We brieﬂy recount the general setup before giving the simpler construction for GN . Fix parabolic subgroups P and P 0 in P(L). Fix a generic representation π of L(F ) and consider

∗ the twists π for λ ∈ a . Let n (respectively n 0 ) be the Lie algebra of the unipotent λ L,C bP bP radical of Pb (respectively Pb0). The group LL acts via the adjoint action on the quotient

89 bnP 0 /bnP 0 ∩ bnP : L ρP 0|P : L → Aut(bnP 0 /bnP 0 ∩ bnP ).

In this situation, Shahidi constructed the local L-function

∨ L(s, πλ, ρP 0|P )

and local ε-factor

∨ ε(s, πλ, ρP 0|P , ψF ).

Moreover, he proved that the quotient

∨ ∨ −1 ∨ −1 rP 0|P (πλ, ψF ) = L(0, πλ, ρP 0|P )ε(0, πλ, ρP 0|P , ψF ) L(1, πλ, ρP 0|P ) satisﬁes the conditions of Theorem 1.6.8. The original reference for general G is [Sha90].

For GN , there is the earlier paper [Sha84]. For a full account, see [Sha10]. Shahidi’s L-functions, and thus the normalizing factors, are built from the rank one case.

For GN , the combinatorics are more straightforward than in general. We follow now the discussion from [AC89, Chapter 2, §2] for the simpler description of the normalizing factors in this case. In what follows, we work with GN ; this is the only group for which we will need precise results about the normalizing factors.

In general, the factor rP 0|P (πλ) takes the form

Y ∨ ∗ 0 a rP |P (πλ) = rα(π, λ(α )), λ ∈ L,C. α∈ΣQ∩ΣP

Here ΣP denotes the roots of AL, the maximal split torus in the center of L, in P and

rα(π, s)

is a meromorphic function of one complex variable. The function rα(π, s), or more properly

∨ rα(π, λ(α )) is the normalizing factor in the rank one case, and the formula above shows how to obtain the general normalizing factor from the rank one case.

90 The ﬁxed Levi subgroup L has an isomorphism

∼ L → Gn1 × · · · × Gnr ⊂ GN , for a partition (n1, . . . , nr) of N. Let α be a root of AL in GN . Let αe be the corresponding root of Gn1 × · · · × Gnr under the isomorphism above. The image of AL is just the center of Gn1 × · · · × Gnr . A typical central element is of the form (a1In1 , . . . , arInr ). Then for distinct integers 1 ≤ p, q ≤ r, we have

α(a I , . . . , a I ) = a /a . e 1 n1 r nr i j

Thus every root α corresponds to an ordered pair of distinct integers (p, q) with 1 ≤ p, q ≤ r.

According to the isomorphism above, the ﬁxed representation π can be written as

π1 × · · · × πr, πi ∈ Π(Gni (F )).

Suppose ﬁrst that π is tempered. Then we set

−1 −1 rα(π, s) = L(s, πp × πeq)ε(s, πp × πeq, ψF ) L(s + 1, πp × πeq) .

The function rα(π, s) is meromorphic in s and satisﬁes

∨ ∗ rα(πλ, s) = rα(π, λ(α ) + s), λ ∈ iaM .

Thus the deﬁnition can be extended by the formula above to all essentially tempered representations πλ of L(F ). Now, we extend from the essentially tempered case to a general representation via the

Langlands classiﬁcation, see A.2 for a statement, following [Art89, p.30]. Let π be an arbitrary representation in Π(L(F )). The representation π is the Langlands quotient of a

L representation IR(σµ), where LR is an admissible Levi subgroup of L, σ is a representation in Π (L (F )), and µ is a point in the positive chamber of a∗ /a∗ attached to R. The temp R LR L function r (σ ), Λ ∈ a∗ , has no pole or zero when Λ = µ. Then we deﬁne R|R Λ LR,C

rP 0|P (πλ) = rP 0(R)|P (R)(σµ+λ).

91 The normalizing factor rP 0|P (πλ) is deﬁned for any representation π ∈ Π(L(F )). We ﬁx these normalizing factors for the remainder of this work. The normalizing factor is a ratio of L-functions and ε-factors for any π ∈ Π(L(F )). That such a normalization is possible is a property special to GN , since every tempered representation is generic. We obtain normalized intertwining operators

−1 RP 0|P (πλ) = rP 0|P (πλ) JP 0|P (πλ):(RP (λ), HP (π)) → (RP 0 (λ), HP 0 (π)).

This completes the ﬁrst step of the normalization.

1.6.10 The intertwining map we are actually interested in is a certain self-intertwining operator from HP (π) back to itself that only exists in certain cases. Under certain conditions to be discussed, will see that the self-intertwining map is an operator that intertwines

IP (π) with its ω-twist. Moreover, this self-intertwining operator will be normalized. We continue the normalization process following the steps in [Art13, §§2.3-2.5]. Shahidi’s general constructions apply to the case of a generic inducing representation for any group G, and

[Art13, Corollary 2.5.2] discusses the relationship between Shahidi’s normalization and

Arthur’s normalization in this case. For our groups, every tempered representation is generic, so using the Langlands classiﬁcation as above we expect to be able to normalize intertwining operators for arbitrary inducing data. While we follow Arthur’s general pattern, the situation is much simpler, since we have access to special Weyl group representatives which do not generally exist.

Suppose for the moment that G is a general group. Fix P ∈ P(L), π ∈ Π(L) and w ∈ W (L). Recall that W (L) is deﬁned as the quotient NG(AL)/L. We may choose a representative w˜ of w in the normalizer NG(L) of L in G. We obtain another representation

(wπ)(m) = π(w ˜−1mw˜), m ∈ L(F )

of L(F ) on the underlying space Vπ of π. The notation wπ is justiﬁed, since the equivalence class of the representation is independent of the choice of w˜ ∈ NG(L). Indeed, if we1 and we2

92 are two representatives of w, we may write we2 = we1m1 for m1 ∈ L. In this case,

−1 −1 (we2π)(m)π(m1) = π(m1) (we1π)(m).

However, as part of the normalization, we will be ﬁxing representatives of Weyl group elements and consequently we prefer a slightly more precise notation. Given a representative we, we will write (wπ,e Vπ) for the resulting representation. Associated to (wπ,e Vπ) is the induced representation IP (wπe ) which acts on a space of functions on K with values in V = V . Note that the space of the induced representation depends on the choice of wπe π representative of w. Let

P 0 =w ˜−1P w.˜

Since we are now interested only in induced representations with the twisting parameter λ equal to 0, we write RP rather than RP (0). There is an intertwining isomorphism

l(w, ˜ π):(RP 0 , HP 0 (π)) → (RP , HP (wπe )) obtained by left translation:

0 0 −1 0 (l(w, ˜ π)φ )(x) = φ (w ˜ x), φ ∈ HP 0 (π), x ∈ G(F ).

1.6.11 We’ve seen that the isomorphism class of wπ is well-deﬁned independently of the representative of w. In particular, the induced representations IP (wπe ) are all isomorphic for any choice of representative we. Thus we can speak of the induced representation IP (wπ). However, the space of the induced representation depends on the choice of representative, and consequently the intertwining operator l(w,e π) does as well. For the applications we have in mind, we need to ﬁx representations and intertwining operators, and this involves choosing representatives of Weyl elements. For any w, there is a representative weS ∈ NG(L) that depends only on a choice of splitting S for G. See [Art13, pp. 89-90] for this construction.

However, the resulting mapping

w 7→ weS : W (L) → NG(L)

93 is not multiplicative. The resulting family of intertwining operators l(w,e π) is consequently not multiplicative. The problem of being multiplicative is solved by [Art13, Lemma 2.3.4] in his situation by adding certain signs related to ε-factors and λ-factors that arise also in the work of Shahidi. The present situation is much simpler.

Consider ﬁrst GN and a Levi subgroup L ⊂ GN . Consider the associated Weyl group

W (L). Every w in W (L) can be represented by a permutation matrix weper. The resulting map

w 7→ weper : W (L) → NG(L) is multiplicative in w. We ﬁx these representatives and will write we instead of weper. They diﬀer from the representative weS by a sign, which must somehow be related to the signs discussed at [Art13, p.90]. Now consider two Weyl elements w1 and w2 with associated representatives we1 and we2. Then w]1w2 = we1we2. This implies that

l(we2, we1π) ◦ l(we1, π) = l(w]2w1, π),

0 −1 compare with [Art13, Lemma 2.3.4]. Continuing to write P = we P we, combining with the results from the previous subsection, we have operators

RP (w,e π) = l(w,e π) ◦ RP 0|P (π):(RP , HP (π)) → (RP 0 , HP 0 (π)) → (RP , HP (wπe )).

At 1.6.2, we normalized the operator Aπ by requiring it to preserve the Whittaker functional. The operator RP (w,e π) is normalized in a similar sense. Indeed, suppose that π is generic. A ﬁxed (BL, χ)-Whittaker functional λπ induces a canonical (BN , χ)-Whittaker functional on the induced representation IP (π). See [Art13, (2.5.1)] for the integral formula for the (BN , χN )-functional and more discussion. Denote this induced Whittaker functional on IP (π) by Λπ.

Proposition. The operator RP (w,e π) satisﬁes

Λ = Λ ◦ R (w, π). π wπe P e

The proposition justiﬁes calling RP (w,e π) “normalized.” See [Art13, pp.112-114] for a

94 more general statement and discussion of this proposition, which is implicit in [Sha90].

1.6.12 The third step of Arthur’s normalization is to normalize an intertwining operator when π ' wπ and thus obtain a normalized self-intertwining operator for (RP , HP (π)) by composition with l(w,e π) ◦ RP 0|P (π). Note that in Arthur’s situation, this normalized intertwining operator need not be a scalar, since the induced representation is not generally irreducible. Such local representations that are ﬁxed by w occur as factors of global representations that occur in the global trace formula, and are implicit in what we do later, but are not the main focus. In the case of GN , when the inducing data is unitary, the induced representations are all irreducible and the normalized self-intertwining operator is the identity, see [Art13, Corollary 2.5.2]. Now we consider a slightly more general situation that takes into consideration the twisting by ω on representations of GN (F ).

Consider a representation π of L(F ) acting on the space Vπ. Associated to π are two

−1 −1 representations, wπe and π ⊗ ω , acting on Vπ. Suppose wπe is isomorphic to π ⊗ ω . −1 For the moment, let T (π, w,e ω ): Vπ → Vπ be any intertwining operator from (wπ,e Vπ) −1 −1 to (π ⊗ ω ,Vπ). Then T (π, w,e ω ) induces an intertwining operator, also denoted by −1 −1 T (π, w,e ω ), from (RP , HP (wπe )) to (R, HP (π ⊗ ω )) by its pointwise action:

−1 −1 (T (π, w,e ω )f)(k) = T (π, w,e ω )f(k), f ∈ HP (wπe ), k ∈ K.

Recall that Iω was multiplication by ω in the model (R, HP (π)) for the induced representation.

Let φ 7→ fφ be the map from 1.6.7 that intertwines (RP , HP (π)) with (R, HP (π)). Let f 7→ φf be its inverse. Then we may consider the composition

−1 (f 7→ φf ) ◦ Iω ◦ (φ 7→ fφ):(RP , HP (π)) → (R, HP (π)) → (R ⊗ ω , HP (π ⊗ ω))

−1 → (RP ⊗ ω , HP (π ⊗ ω)).

We will denote this map also by Iω. Note that if ω is unramiﬁed then the spaces HP (π ⊗ ω) and HP (π) are equal. However, the action RP depends on the inducing data, so that

(RP , HP (π)) and (RP , HP (π ⊗ ω)) need not be isomorphic even if ω is unramiﬁed. This notation inconsistency will never cause a problem, as the action RP will never be referred

95 to without an underlying space.

Consider the composition of all the operators we have deﬁned:

−1 Iω◦T (π, w,e ω ) ◦ l(w,e π) ◦ RP 0|P (π):(RP , HP (π)) → (RP 0 , HP 0 (π)) → −1 −1 (RP , HP (wπe )) → (RP , HP (π ⊗ ω )) → (RP ⊗ ω , HP (π)). (1.6.13)

−1 Thus, under the assumption that wπe is isomorphic to π ⊗ ω , we have constructed a self- intertwining operator. It shows explicitly that (RP , HP (π)) is an ω-stable representation. Of

−1 the four operators in the composition, only one of them, T (π, w,e ω ) is either unnormalized −1 or non-canonical. We deal next with the normalization of the operator T (π, w,e ω ).

−1 −1 1.6.14 We consider T (π, w,e ω ) as a map from (wπ,e Vπ) to (π ⊗ ω ,Vπ). Thus it satisﬁes

−1 −1 −1 T (π, w,e ω )(wπe )(m) = (π ⊗ ω )(m)T (π, w,e ω ).

Suppose ﬁrst that π is generic. Recall that we have ﬁxed the standard splitting of GN and that it induces a splitting on a block diagonal Levi subgroup L. Moreover, a Whittaker datum (BN , χ) for GN restricts to a Whittaker datum (BL, χ) for L. Fix a (BL, χ)-Whittaker functional λπ for π. For n ∈ NL(F ) and v ∈ Vπ,∞, we have

−1 −1 λπ((π ⊗ ω )(n)v) = λπ(ω (n)π(n)v)

= λπ(π(n)v)

= χ(n)λπ(v).

−1 Thus λπ is a (BL, χ)-Whittaker functional for π ⊗ ω . Next, consider

−1 λπ((wπe )(n)v) = λπ(π(we nwe)v).

The representative we was chosen to be a permutation matrix in the normalizer of L. That is, we acts on L by permuting the blocks of L and thus we preserves the splitting of L induced −1 by the standard splitting of GN . In particular, we nwe lies in the unipotent radical of BL and we have

−1 −1 λπ(π(we nwe)v) = χ(we nwe)λπ(v). 96 Again, since we preserves the splitting, it preserves χ and thus λπ is also a (BL, χ)-Whittaker functional for wπe . We regard λπ as the ﬁxed (BL, χ)-Whittaker functional for π, wπe and −1 −1 π ⊗ ω . Now, consider λπ ◦ T (π, w,e ω ). Then

−1 −1 (λπ ◦ T (π, w,e ω ))((wπe )(n)v) = λπ(T (π, w,e ω )(wπe )(n)v) −1 −1 = λπ((π ⊗ ω )(n)T (π, w,e ω )v) −1 = χ(n)λπ(T (π, w,e ω )v) −1 = χ(n)(λπ ◦ T (π, w,e ω ))(v).

By the uniqueness of the Whittaker functional (more speciﬁcally, regarding λπ as a Whittaker

−1 functional for wπe ), we have that λπ ◦ T (π, w,e ω ) = c · λπ for some nonzero number c. As we did in 1.6.2 for the intertwining operator Aπ, we normalize the intertwining operator

−1 T (π, w,e ω ) so that c = 1. This normalization is independent of the choice of Whittaker functional λπ.

1.6.15 Now we consider the case of non-generic representations. In other words, let π be any irreducible representation of L(F ) and consider the same two representations wπe and −1 π ⊗ ω that we are assuming are isomorphic. We have the ﬁxed Borel subgroup BL of L.

A Langlands triple (P1, π1, λ1), discussed in more detail in A.2, will be called standard if

P1 ⊃ BL. According to the Langlands classiﬁcation, there is a standard triple (P1, π1, λ1) associated to π. In other words, L1 is the Levi component of P1 that contains TN , the diagonal maximal torus of GN and L, π1 is a tempered representation of L1(F ) and π is the Langlands quotient of the induced representation IL (π ). P1 1,λ1 −1 The triple (P1, π1 ⊗ ω , λ1) is also a standard triple. Indeed, since ω is unitary the

−1 representation π1 ⊗ ω is again tempered, which is the only entry in the triple that has changed.

−1 Consider the parabolic subgroup wPe 1we of L. The representative we was chosen to −1 −1 normalize BL. Thus wPe 1we ⊃ BL is a standard parabolic subgroup of L. It has wLe 1we

97 −1 as a Levi subgroup. We may view wπe 1 as a representation of wLe 1we :

−1 (wπe 1)(wme 1we ) = π1(m1).

−1 −1 In addition, we can deﬁne wλe 1 as a character for wLe 1we ; the triple (wPe 1we , wπe 1, wλe 1) is standard.

−1 We claim next that the standard representations associated to the data (P1, π1 ⊗ ω , λ1)

−1 and (wPe 1we , wπe 1, wλe 1) are isomorphic. According to the uniqueness statement of the Langlands classiﬁcation, it suﬃces to show that the respective Langlands quotients are isomorphic.

First, note that

−1 −1 (R, HP1 ((π1 ⊗ ω )λ1 )) = (R, HP1 ((π1,λ1 ) ⊗ ω ))

−1 ' (R ⊗ ω , HP1 (π1,λ1 )).

−1 This latter representation has (π ⊗ ω ,Vπ) as its unique irreducible quotient.

−1 There is a similar type of computation for the datum (wPe 1we , wπe 1, wλe 1). First, note that

( , −1 (wπ )) = ( , −1 (w(π ))). R HwPe 1we e wλe 1 R HwPe 1we e 1,λ1

Consider a function f in the latter space. By deﬁnition it transforms according to the rule

−1 −1 1/2 −1 −1 f (wm1w )(wn1w )m = δ −1 (wm1w )(w(π1,λ ))(wm1w )f(m). e e e e wPe 1we e e e 1 e e

−1 Consider the function we f. It satisﬁes

−1 −1 (we f)(m1n1m) = f(wme 1n1mwe ) −1 −1 −1 = f((wme 1we )(wne 1we )(wme we )) 1/2 −1 −1 −1 = δ −1 (wm1w )(w(π1,λ ))(wm1w )f(wmw ) wPe 1we e e e 1 e e e e = δ1/2(m )(π )(m )(w−1f)(m). P1 1 1,λ1 1 e

98 −1 The map f 7→ we f provides an intertwining isomorphism

( , −1 (w(π ))) → (w , (π )). R HwPe 1we e 1,λ1 eR HP1 1,λ1

The representation (w , (π )) has (wπ, V ) as its unique irreducible quotient. eR HP1 1,λ1 e π −1 We are assuming that the representations (wπ,e Vπ) and (π⊗ω ,Vπ) are isomorphic. The −1 −1 previous computations show that (P1, π1 ⊗ ω , λ1) and (wPe 1we , wπe 1, wλe 1) are standard triples with isomorphic Langlands quotients. The uniqueness aspect of the Langlands classiﬁcation implies that the corresponding standard representations IL ((π ⊗ ω−1) ) and P1 1 λ1 L I −1 ((wπ1)wλ ) are isomorphic to one another. wPe 1we e e 1

Let λπ1 be a ﬁxed Whittaker functional for π1. Then λπ1 is also a Whittaker functional for (π ⊗ ω−1) and (wπ ) . As discussed in 1.6.11, the choice of λ induces 1 λ1 e 1 wλe 1 π1 a Whittaker functional on the corresponding induced representations IL ((π ⊗ ω−1) ) P1 1 λ1 L and I −1 ((wπ1)wλ ). These induced representations are isomorphic, so we may ﬁx a wPe 1we e e 1 −1 L L −1 unique intertwining operator T (π1, w, ω ) from I −1 ((wπ1)wλ ) to I ((π1 ⊗ ω )λ ) e wPe 1we e e 1 P1 1 that preserves the Whittaker functional. As discussed in 1.6.5, an intertwining operator between standard representations induces an operator on the Langlands quotients. Thus

−1 −1 the operator T (π1, w,e ω ) induces an operator from (wπ,e Vπ) to (π ⊗ ω ,Vπ). This is our −1 choice for the operator T (π, w,e ω ). We have now normalized all of the operators that occur in the composition at (1.6.13).

The resulting operator on HP (π) is the desired operator intertwining the representations

−1 (RP , HP (π)) and (RP ⊗ ω , HP (π)).

1.7 Statement of the main theorems

We will recall the local results of [HH95] and state a version that applies to a slightly diﬀerent class of representations. In the end, we will not prove these more general results, but our study of the case of M = 1 and the setup of this and other sections will hopefully provide some of the background material needed to establish the more general results.

99 1.7.1 We continue with the local notation of the previous section. Thus F is a local ﬁeld of characteristic 0 and E is a ﬁnite cyclic extension of degree D determined by a character

ω = ωE/F . The local transfer factors, characters and orbital integrals were introduced in Section 1.5. Before we recall the main theorem, we need to introduce the relevant notion of local lifting.

H A local transfer factor ∆G (−) is deﬁned in [HH95, §3] when F is non-archimedean and at [Hen10, §3.7] when F is archimedean. It is shown in [HS12, Appendix A] that the transfer

H factor ∆G (−) is a multiple of the Langlands-Shelstad-Kottwitz transfer factor discussed in Section 1.5. As we shall see, such a constant is built into the deﬁnition of local transfer of representations, so one may regard the transfer factors in what follows as a special case of the Langlands-Shelstad-Kottwitz factors or as those used in [HH95]. We will discuss some further results related to constants and normalization after stating the main theorems.

Let N = MD. Let GN,reg(F ) denote the set of regular elements in GN (F ). By ﬁxing an

F -basis of E, we obtain as usual an embedding HM (F ) ' GM,E(E) ,→ GN (F ). Fix such an embedding. We will regard HM (F ) as a subgroup of GN (F ) via this ﬁxed embedding. Thus we may consider the set HM (F ) ∩ GN,reg(F ) consisting of GN -regular elements of HM (F ).

The set HM (F ) ∩ GN,reg(F ) is dense in HM (F ).

Let f ∈ H(N) and let φ ∈ H(HM ) be smooth, compactly supported functions. The notion of f and φ having matching orbital integrals has been discussed previously, see in particular Theorem 1.5.13. We continue to write f 7→ f HM for this relationship of having matching orbital integrals, even though f HM is not uniquely determined by the condition of having matching orbital integrals with f.

Now, we recall [HH95, Deﬁnition 3.7]. Let π ∈ Π(HM (F )) be an irreducible, admissible

ω representation of HM (F ). Let πN ∈ Π (GN (F )) be an irreducible, admissible, ω-ﬁxed representation of GN (F ). Let AπN be any ﬁxed operator intertwining πN and πN ⊗ ω.

We say that πN is an ω-lift of π if the following condition holds: There is a nonzero

HM number c = c(π, πN ,AπN ) such that for any functions f ∈ H(GN,reg(F )) and f ∈

100 H(HM (F ) ∩ GN,reg(F )) with matching orbital integrals, we have

HM tr (πN (f) ◦ AπN ) = c tr π(f ).

Note that the deﬁnition only applies to functions which are supported on the GN -regular set. Indeed, in [HH95] the main local theorems are proved only using such functions.

Now we may state the main theorem of [HH95]. Assume E and F are non-archimedean.

Theorem (Local automorphic induction). Let π be a class of smooth irreducible tempered representations of HM (F ) ' GM,E(E). Then π has an ω-lift, πN , which is ω-stable, tempered and does not depend on ω. Every class of smooth irreducible ω-stable tempered representations of GN (F ) is obtained. Moreover, π is a discrete series representation if and only if πN is ω-discrete. In the case that π is a discrete series representation, the other classes of smooth irreducible tempered representations of H(F ) lifting to πN are the conjugates of π under

ΓE/F .

This result is proved when M = 1 in [Kaz84]. In fact, the theorem applies to a broader class of representations that includes some generic representations which are non-tempered, see [HH95, Theorem 2.4]. Given π, we will often write AIE/F π for the representation πN satisfying the conditions of the theorem. The proofs in [Kaz84] and [HH95] rely on the simple trace formula and L-functions. We will reprove this theorem when M = 1 using the full trace formula, which is introduced in the next chapter.

Let us explain the statement that πN “does not depend on ω.” Suppose it is known that

0 × 0 × π has an ω-lift πN . Let ω be another character of F with ker ω = NE/F E . Then we may 0 0 0 deﬁne exactly as above the notion of an ω -lift of π. Suppose πN is the ω -lift of π. Then to 0 say that the lifting does not depend on ω just means that πN and πN are isomorphic.

1.7.2 The statement of Theorem 1.7.1 when F is archimedean is the same. We will not reprove Theorem 1.7.1 in the archimedean case. The archimedean analog of Theorem 1.7.1 is contained, as a very special case, in Shelstad’s work on endoscopy for real groups. Some of the main references are [She82], [She08a], [She10] and [She08b], but see [Art13, p.43, p.307] for a more complete discussion. Her results include the statement that the archimedean 101 L-packets of [Lan89] satisfy endoscopic character relations. The deﬁnition of ω-lift given here is a special case of such endoscopic character relations.

The archimedean case of Theorem 1.7.1 is also proved in [Hen10]. Moreover, Henniart showed that the ω-lifting in the archimedean case is consistent with the archimedean Theorem

1.3.5. We will use the archimedean version of Theorem 1.7.1 to prove Theorem 1.7.1 for non-archimedean F .

1.7.3 Henniart has shown in [Hen01, §7] that ω-lifting is consistent with the local Langlands correspondence, Theorem 1.3.5, in the following sense. Let π be a representation of HM (F ) regarded as a representation of GM,E(E). According to Theorem 1.3.5, associated to π is an M-dimensional representation φ of the group LE. Let πN be the ω-lift of π as in

Theorem 1.7.1 and let φN be the N-dimensional representation of LF associated to πN . Then Henniart showed that φ is equal to ILF φ. N LE

1.7.4 No particular choice of intertwining operator or transfer factor is chosen in [HH95], aside from satisfying various global identities and internal compatibility conditions. In 1.5.8, we chose a speciﬁc normalizing factor that depends only on a choice of Whittaker datum

(BN , χ). Since GN has a standard Borel subgroup and splitting, we can regard the transfer factor as depending only on a choice of additive character ψF for F . In addition, we have normalized the intertwining operator Aπ depending on the choice of Whittaker datum, see

1.6.2. We may also regard Aπ as depending only on ψF . When these choices are made for the transfer factor and intertwining operator are made, Henniart-Lemaire and Hiraga-Ichino have proved the following theorem.

Theorem. The number c in Theorem 1.7.1 is equal to 1.

The fact that c is independent of the representation π is proved in [HL10] in the non-archimedean case. The archimedean case is in [Hen10]. Using the independence of representation, it is shown in [HI12] that c = 1 via explicit computation after reducing to the case of M = 1 and π the trivial character.

102 1.7.5 We continue to assume that F is a non-archimedean local field of characteristic 0. The set Ψ(N) was defined at 1.3.17. Since F is non-archimedean, it consists of N-dimensional unitary representations of WF × SU(2) × SU(2). Let ψe ∈ Ψ(N) be such a representation which is self-dual. Recall that stable linear forms were defined at 1.5.10. One of the main local theorems of [Art13] is the fact that a natural linear form on a classical group associated to ψ is stable. Moreover, this stable form can be decomposed into a weighted sum of irreducible characters of a classical group. The representations which contribute to the sum constitute the L-packet, if ψ is trivial on the second copy of SU(2), or the A-packet, if ψ is non-trivial on the second copy of SU(2), associated to ψ˜. Our current situation is vastly simpler, though we will still rephrase some of the ideas of the previous section in terms of linear forms and state them in slightly greater generality. At the very least, the discussion here will put the orthogonality relations introduced later in a better context.

ω Let ψ ∈ Ψ (N) be an ω-ﬁxed, N-dimensional, unitary representation of WF × SU(2) ×

SU(2). It determines an ω-ﬁxed Langlands parameter φψ according to (1.3.18) and thus a representation πψ via Theorem 1.3.5. The representation πψ is ω-ﬁxed according to item

2 of Theorem 1.3.5. Since πψ is ω-stable, there is an associated normalized intertwining operator Aπψ as in 1.6.2 and 1.6.5. Then we may consider the normalized linear form

f 7→ tr πψ(f) ◦ Aπψ , f ∈ H(N).

The representation πψ is unitary but not necessarily generic, so the normalization of the intertwining operator Aπψ requires the Langlands classiﬁcation. The representations πψ are the types of representations which occur as local components of ω-ﬁxed globally tempered automorphic representations.

The transfer map

HM ω f 7→ f : H(N) → I (HM ) was discussed at 1.5.14. According to Proposition 1.5.19, the map is surjective. Thus,

H ω associated to an invariant function f M ∈ I (HM ) is a number

HM f (ψ) = tr πψ(f) ◦ Aπψ . 103 ω H In this expression, we have chosen any function f whose image in H (HM ) is equal to f M .

Since the map f 7→ f HM has a kernel, we must verify that f HM (ψ) is well-deﬁned. Suppose

HM HM then that f1 and f2 lie in H(N) and that f1 = f2 . Then f1 and f2 have the same ω-twisted regular orbital integrals. It follows from the Weyl integration formula, see [HH95,

§3.11] for the formula in this context, that their traces at any ω-ﬁxed π are the same, and thus the linear form is well deﬁned.

H H ω The linear form f M 7→ f M (ψ) is deﬁned on the set of invariant functions I (HM ).

At 1.5.5, we discussed the map from H(HM ) → I(HM ). Thus we may pull the form

H H ω f M 7→ f M (ψ) back to H (HM ). Denote the resulting linear form by φ 7→ φ(ψ) for

ω ω φ ∈ H (HM ). By deﬁnition, φ 7→ φ(ψ) depends only on the image of φ in I (HM ). Since

I(HM ) = S(HM ), the form φ 7→ φ(ψ) is stable by deﬁnition, see 1.5.10. We can try to express this linear form in terms of irreducible characters.

ω Theorem. Let ψ ∈ Ψ (N). There is a nonzero number c and a representation π of HM (F ) such that if f 7→ f HM we have

HM HM f (ψ) = tr πψ(f) ◦ Aπψ = c tr π(f ).

In the formulation of [Art13, §2.2], this is essentially a way of deﬁning packets for the group HM . Since HM (F ) ' GL(M,E), it is not surprising the resulting packets only contain one representation. However, we will only be able to separate representations of HM (F ) up to the action of ΓE/F . This is consistent with the results of [Art13], where representations for even orthogonal groups are separated up to the action by an involution. Part of the theorems of automorphic induction is the fact that the ﬁber of automorphic induction essentially consists of Galois orbits, see Theorem 1.7.1. Thus we can formulate the local theorems for any function, regardless of its support, and we expect that the theorems can be proved using the full trace formula for the class of representations cut out by the parameter set Ψω(N), a set of irreducible unitary representations which properly contains the set of tempered representations and includes some non-generic representations. We will only prove the theorems when M = 1, and thus will not deal with any non-tempered representations.

104 We should mention that the analogous results for base change, namely the existence of local base change satisfying a character relation, see [AC89, Ch.1], have been extended to all unitary representations in [BH16]. We have not attempted to apply their methods to automorphic induction, though it is reasonable to believe that the methods could be applied to extend the character relations of local automorphic induction to unitary and elliptic representations.

1.7.6 Global situations will require us to understand slightly more general local liftings. See [HL10] for a better discussion.

Suppose for the moment that E ⊃ F is a cyclic extension of number ﬁelds of degree D.

Then we may consider the F -group HM = RE/F GM,E. Let v be a place of F . Consider the

Fv-algebra Y Ev = E ⊗F Fv ' Ew. w|v The product is over the places w of E lying over v. We saw before at 1.1.7 that

Y HM,v ' REw/Fv GM,Ew . w|v

× × Let ω = ωE/F be a character of F \AF that determines the extension E ⊃ F . When viewed × as a character of AF , the character ω factors as

Y ω = ωv. v

× The local character ωv is a character of Fv that determines the extension Ew ⊃ Fv.

The group HM,v represents a (GN,v, ωv)-datum. Indeed, HM,v is a Levi subgroup of

REw/Fv GN/[Ew:Fv],Ew , which represents the unique class of elliptic (GN,v, ωv)-data. The group HM,v can be viewed as an elliptic endoscopic group for a corresponding Levi subgroup of GN,v. This is the larger class of groups that we need to consider. We expect to be able to break up the lifting at a general place into an automorphic induction step, as described above, and a parabolic induction step. Since HM,v is a (GN,v, ωv)-datum, there is an associated transfer map f 7→ f HM,v .

105 We revert back to the local notation introduced at the beginning of this section. According to the previous paragraph, we need to understand lifting representations of L ⊂ HM to

GN . The transfer of functions from GN to Levi subgroups of HM (equivalently from GN to non-elliptic (GN , ω)-data) was discussed at 1.5.14-1.5.19. Dual to the diagram at 1.5.18 is the general diagram for automorphic induction:

AI ω E/F Π (N)o Π(HM ) O O G N HM IL I GN L ω Π (LGN ) o Π(L) AIE/F

1.7.7 The main global theorems for automorphic induction in the case of cyclic extensions of degree p were proved in [AC89]. For a diﬀerent treatment of the global theorems, as well as a discussion of their consistency with the local lifting of [HH95] and [Hen10], see

[Hen12]. For some other papers discussing and proving additional global results related to base change and automorphic induction, see [LR98] and [Raj02].

We let E ⊃ F be a cyclic extension of number ﬁelds of degree D determined by a

× × character ω = ωE/F of F \AF . Then HM = RE/F GM,E is the global restriction of scalars, viewed as the representative of the unique class of (GN , ω)-data. We write AIEv/Fv for the local automorphic induction map based on character relations, which we recall generally involves parabolic induction. N0 Let π = v πv be an irreducible, admissible representation of HM (AF ). Suppose that local automorphic induction is known for each πv. In other words, the ωv-stable representation AIEv/Fv πv satisfying the conditions of Theorem 1.7.1 exists. Then we set

O 0 AIE/F π = AIEv/Fv πv. v

The representation AIE/F π is an ω-stable, irreducible, admissible representation of GN (AF ).

Theorem 1.7.8. • Suppose π is an automorphic representation of HM (AF ) in the

discrete spectrum. Then AIE/F π is automorphic.

0 • Suppose π is another automorphic representation of HM (AF ) in the discrete spectrum 106 0 σ 0 with AIE/F π = AIE/F π . Then there is an element σ ∈ ΓE/F such that π = π .

• Every ω-elliptic automorphic representation of GN (AF ) is of the form AIE/F π for some discrete spectrum automorphic representation π.

• Suppose π is cuspidal. Then AIE/F π is cuspidal if and only the conjugates of π under

ΓE/F are distinct.

We will only prove this theorem when M = 1. We hope that the setup and methods used in the case M = 1 can be extended to prove the theorem for any M. Using the description

(1.3.10) of the set A(N), the main theorem can be used to extend the lifting to globally tempered automorphic representations. The methods used here do not give any information about automorphic representations that are not isobaric.

A key step in the proofs of automorphic induction in [Kaz84], [AC89] (where it is proved in the context of base change) and [HH95] is the following fact which we state as a proposition.

The proof in the latter two cases uses results from the theory of L-functions, while the former requires only the Chebotarev density theorem. We will establish this proposition as part of the proof of the previous theorem just using the global trace formula, at least in the case M = 1 that we consider.

0 Proposition 1.7.9. Consider two discrete spectrum representations π and π of HM . Sup-

0 σ 0 pose AIEv/Fv πv = AIEv/Fv πv for almost all v. Then there exists σ ∈ ΓE/F such that π = π .

The proposition is weaker than the main theorem, but is interesting on its own as a type of “strong multiplicity one” for automorphic induction.

107 Chapter 2 Trace formulas

In this chapter, we will collect the results on trace formulas that apply to GN and its

(GN , ω)-endoscopic groups for any N. Later, we will apply these results when N = [E : F ] to establish the theorems stated in the previous section in the base case.

2.1 The discrete part of the trace formula

In this section, we will let F be a global ﬁeld and introduce the global trace formula, following

[Art13, §3.1]. The discrete part of the trace formula is an explicitly deﬁned linear form that contains the trace of the right regular representation on the discrete spectrum as well as some additional terms that contain a contribution from global intertwining operators.

2.1.1 For now, we let G be a connected, reductive, linear algebraic group deﬁned over F Q and set A = AF . We ﬁx a suitable maximal compact subgroup K = v Kv in G(A) and minimal Levi subgroup L0 which are in good position. See 1.3.8 for more discussion of the requirements on K and L0. At 1.3.6 we introduced the objects needed to make sense of the spaces

2 2 Ldisc(G(F )\G(A), χ) ⊂ L (G(F )\G(A), χ), as well as the corresponding spaces for Levi subgroups of G. Global induced representations were introduced at 1.3.8. (We had been using χ to denote the generic character in a

Whittaker datum. See 1.5.7. Since Whittaker data do not appear explicitly in what follows,

108 we now reserve χ and related notation for central characters as in 1.3.6.) We will write

G 2 HP (χ) = HP Ldisc(L(F )\L(A), χ) for the space of the induced representation obtained from the entire discrete spectrum of

L(A). We will use the notation IP (χ, λ) = (RP (χ, λ), HP (χ)) to denote the associated induced representation. The action R (χ, λ) for λ ∈ a∗ was also discussed at 1.3.8.When P L,C

λ = 0, we will write IP (χ) = IP (χ, 0) and RP (χ) = RP (χ, 0). These induced representation, as well as the intertwining operators deﬁned below, are discussed in more detail in A.3. In

[Art13], Arthur generally uses IP (χ) also to denote the action of the induced representation.

We will generally try to use RP (χ), since at times we will need to keep close track of the underlying vector spaces.

M¨ullerhas shown [M¨ul89] that the restriction of

Z RP (χ, f) = f(x)RP (χ, x) dx, f ∈ H(G), G(A) to the discrete spectrum is a trace class operator. In addition, the absolute convergence of the spectral side of the trace formula is now known [FLM11], [FL11]. Following Arthur, we will work with subrepresentations of IP (χ) determined by their archimedean inﬁnitesimal characters rather than the results on absolute convergence.

Let π be an irreducible, admissible representation of G(A). Then one can, roughly speaking, consider the norm of the imaginary part of the inﬁnitesimal character of the archimedean component of π, a positive real number we will denote by t(π). See [Art82, §3] or [Art13, p.123] for a more complete discussion of the deﬁnition of t(π) and what follows.

Then we may consider IP,t(χ), the subrepresentation of IP (χ) consisting of the irreducible constituents π of IP (χ) with t(π) = t. Then there is a direct sum decomposition

M IP (χ) = IP,t(χ). t≥0

The space of the induced representation IP,t(χ) will be denoted by HP,t(χ) and the restriction of the action RP (χ) will be denoted by RP,t(χ). The role of the parameter t, as far as convergence of the discrete part of the trace formula is concerned, will be discussed at the 109 end of the section.

−1 Let H(G, χ) be the χ -equivariant Hecke algebra of smooth, K-ﬁnite functions on G(A) −1 satisfying f(zx) = χ(z) f(x) for z ∈ XG and compactly supported modulo XG. The central subgroup XG was introduced at 1.3.6. For f ∈ H(G, χ), we obtain a convolution operator on HP,t(χ): Z RP,t(χ, f) = f(x)RP,t(χ, x) dx. G(A)/XG

2.1.2 Let L = L(L0) be the set of Levi subgroups of G that contain L0. The set L is ﬁnite. We deﬁne the relative Weyl group

G W (L) = W (L) = NG(AL)/L.

L L G In addition, for L ∈ L(L0), set W0 = W (L0). Let aL be the complement of aG in aL deﬁned to be the kernel of the restriction surjection aL aG. See 1.3.6 for the deﬁnition of the spaces aL and aG. The set of regular elements in the Weyl group W (L) is the set

W (L)reg = {w ∈ W (L) : det(w − 1) G 6= 0}. aL

Equivalently, they are the Weyl group elements whose ﬁxed point set is as small as possible, namely aG.

The homomorphism HL(−) was deﬁned at 1.3.6. We deﬁne the homomorphism HP :

G(A) → aL by using the Iwasawa decomposition G(A) = NP (A)L(A)K:

HP (nmk) = HL(m), n ∈ NP (A), m ∈ L(A), k ∈ K.

The linear form ρP on aL is deﬁned as usual as half the sum of the positive roots (with multiplicity) of AL in NP . The local versions of some of these objects were already used at 1.6.7.

∗ + ∗ Let (aP ) be the positive cone in aL associated to P . If φ ∈ HP (χ) is a smooth function and λ ∈ (aG)∗ has real part in the cone ρ + (a∗ )+ in (aG)∗, then for x ∈ G( ) and L C P P L A

110 P,P 0 ∈ P(L) the integral

Z −(λ+ρ 0 )(H 0 (x)) (λ+ρP )(HP (nx)) MP 0|P (χ, λ)φ (x) = e P P φ(nx)e dn NP 0 (A)∩NP (A)\NP 0 (A) converges. For such λ we obtain an operator

MP 0|P (χ, λ): HP (χ) → HP 0 (χ).

The notation here diﬀers slightly from [Art13, p.124].

Let we be a representative of w in G(F ). Now, we assume that P ∈ P(L) and that 0 −1 −1 P = w P = we P we ∈ P(L). Then l(w) will be the mapping from HP 0 (χ) to HP (χ) −1 deﬁned by left translation by we :

l(w): HP 0 (χ) → HP (χ) where

0 0 −1 (l(w)φ )(g) = φ (we g), g ∈ G(A).

For λ ∈ (aG)∗ such that the intertwining integral converges, we set L C

MP (w, χ, λ) = l(w) ◦ MP 0|P (χ, λ).

This operator intertwines the actions RP,t(χ, λ) and RP,t(χ, wλ) on HP,t(χ). The operator has analytic continuation to a meromorphic function of λ ∈ (aG)∗ . The values at λ ∈ i(aG)∗ L C L are analytic and unitary. Thus the value at λ = 0 gives an operator MP,t(w, χ) on HP,t(χ) which intertwines the action RP,t(χ) with itself. This global intertwining operator originally arose in Langlands’ Eisenstein series theory [Lan76]. See [Art79] and [MW95] for other expositions.

G The discrete part of the trace formula is the linear form Idisc,t = Idisc,t, t ≥ 0, on H(G, χ) deﬁned by

X L G −1 X −1 Idisc,t(f) = |W ||W | | det(w − 1) G | tr (MP,t(w, χ)RP,t(χ, f)) . (2.1.3) 0 0 aL L∈L w∈W (L)reg

We will discuss the sense in which this linear form converges at the end of the section.

111 2.1.4 Consider now a pair (G, ω) where G is as in 2.1.1 and ω is a character of G(A) which is trivial on G(F ). The twisted representation theory of (G, ω) is trivial unless ω is trivial

+ on AG,∞, so we may as well assume that this is the case. In addition we assume that ω is ω G,ω trivial on XG. We deﬁne the operator RP (χ, y) = RP (χ, y) on HP (χ) by

ω (RP (χ, y)φ)(x) = ω(xy)φ(xy), φ ∈ HP (χ), x, y ∈ G(A).

ω There is also the t-analog RP,t(χ, y) acting on HP,t(χ). Let

Iω : HP (χ) → HP (χ)

ω be the operator of multiplication by ω. Then RP,t(χ, y) = RP,t(χ, y) ◦ Iω. The operator ω G,ω RP,t(χ, f) = RP,t (χ, f), f ∈ H(G, χ), is deﬁned by Z ω ω RP,t(χ, f) = f(y)RP,t(χ, y) dy. G(A)/XG

ω G,ω The linear form Idisc,t = Idisc,t is deﬁned as in 2.1.3. More precisely,

ω X L G −1 X −1 ω I (f) = |W ||W | | det(w − 1) G | tr MP,t(w, χ)R (χ, f) . (2.1.5) disc,t 0 0 aL P,t L∈L w∈W (L)reg

ω For the rest of this section, we will write Idisc,t(−) in all cases rather than refer explicitly to the untwisted version, which one of course obtains by taking ω to be the trivial character.

ω The linear form Idisc,t(−) depends on a choice of G(A)-invariant measure on G(A)/XG. We will not be explicit about the choice of measure.

ω 2.1.6 Now we will discuss the convergence properties of Idisc,t(−) that we will use. Let

Πunit,t(G(A), χ) denote the set of irreducible, admissible, unitary representations of G(A) with parameter t which transform under XG according to χ. For any π ∈ Πunit,t(G(A), χ), the operator π(f) ◦ Iω is deﬁned by the formula Z π(f) ◦ Iω = f(y)π(y)Iω dy. G(A)/XG

112 ω The linear form Idisc,t(−) is a sum of characters tr(π(f) ◦ Iω) for π ∈ Πunit,t(G(A), χ). The ω multiplicity adisc(−) is deﬁned by the equality

ω X ω Idisc,t(f) = adisc(π) tr (π(f) ◦ Iω) , f ∈ H(G, χ). π∈Πunit,t(G(A),χ)

ω G,ω The function adisc(−) = adisc(−) is in general complex-valued. It is the important piece of global information on the spectral side of the trace formula. See [Art88b, §4] for more details and for a construction of a natural set properly contained in Πunit,t(G(A), χ) that ω still contains the support of adisc(−).

Fix a function f ∈ H(G, χ). The set of π ∈ Πunit,t(G(A), χ) for which tr (π(f) ◦ Iω) is nonzero is ﬁnite. Denote this ﬁnite set of representations by Πt(f). It turns out that Πt(f) depends only on the Hecke type of f, which we recall.

Let S∞ denote the ﬁnite set of archimedean valuations of F . We let

Y A∞ = {a ∈ A : av = 0 if v∈ / S∞} = Fv = F∞ v∈S∞ and

∞ A = {a ∈ A : av = 0 if v ∈ S∞} be the subrings of archimedean and ﬁnite adeles in A. In addition, if S is any set of valuations of F that contains S∞, then we set

S A = {a ∈ A : av = 0 if v ∈ S} and

∞ ∞ AS = {a ∈ A : av = 0 if v∈ / S}.

∞ ∞ ∞ A Hecke type for G is a pair (τ∞, κ ) where κ is a compact open subgroup of G(A ) and τ∞ is a ﬁnite set of irreducible representations of a maximal compact subgroup K∞ of

∞ ∞ S ∞ G(A∞). There is a decomposition of κ as a product κS K , where S ⊃ S∞ is ﬁnite, κS is ∞ S a compact open subgroup of G(AS ) and K is a product over v∈ / S of hyperspecial maximal ∞ compact subgroups Kv ⊂ G(Fv). If κ = K∞κ is contained in the maximal compact

∞ subgroup K, we call (τ∞, κ ) a Hecke type for f if f is bi-invariant under translation by

113 ∞ κ and f transforms under left and right translation by K∞ according to representations in the set τ∞. Every f ∈ H(G, χ) is assumed to be K-ﬁnite and thus has a Hecke type. Then, the ﬁnite set Πt(f) depends only on a choice of Hecke type for f. See for example [Art88b,

2 Lemma 4.1], which relies on the decomposition of L (G(F )\G(A), χ) of [Lan76].

2.2 Stabilization

We will limit ourselves to a discussion of stabilization in the primary case of interest of the triplet (GN , id, ω). We deal with a global ﬁeld F . The discussion follows [Art13, §3.2]. As in the local case discussed earlier, HM = RE/F GM represents the unique elliptic member of Eω(N). A general datum in Eω(N) localizes to an element of Eωv (N) that need not be elliptic.

ω 2.2.1 For H ∈ E (N), there is a global transfer factor ∆H,G(−, −) = ∆(−, −), see [KS99, (7.3)]. It is a function of adelic variables γH and γ that lie in the adelized varieties of

GN -regular classes in H and GN . The global transfer factor allows us to deﬁne a global transfer mapping which takes a function f ∈ H(GN ) = H(GN (A)) to a function

H H X H f (γ ) = ∆(γ , γ)fG(γ). γ

The global transfer factor can be written as a product

H Y H ∆(γ , γ) = ∆v(γv , γv) v of local transfer factors. The product is not canonical, because the local transfer factors are not uniquely determined. The local transfer factors can be taken to be the normalized transfer factors discussed in 1.5.8. See [Art13, p.137] for a thorough discussion. The transfer Q mapping therefore takes a decomposable function f = v fv in H(GN ) to the decomposable function

H H Y H H f (γ ) = fv (γv ). v

114 Theorems 1.5.11 and 1.5.12 together imply that

H O 0 f ∈ I(H) = I(Hv). v

Hv We discussed at 1.5.14 and the subsections that followed that the local transfer map fv 7→ fv

ωv has its image in a subalgebra I (Hv) determined by the automorphism group of Hv as a

(GN,v, ωv)-datum. Thus the global transfer map sends a decomposable function into the subalgebra

ω O 0 ωv I (H) = I (Hv). v For the global trace formula, we will need to use equivariant functions. That is, functions which transform on a central subgroup according to a ﬁxed character. We will convert the local transfer map into a map of equivariant function spaces, following [HH95, §7.6].

The requirements for the central character datum for any group were discussed at

1.3.6. Let ZN denote the center of GN . For XGN , we can take the full center ZN (AF ) = × Z(GN (AF )) ' AF . Then χ is an automorphic character of

× × Z(GN (F ))\Z(GN (AF )) ' F \AF

which we will always assume is unitary. Let ZHM denote the center of HM . Then ZHM (AF ) ' × AE. We let AHM denote the maximal F -split torus in the center of HM . Any choice of

F -basis for E determines an embedding HM ,→ GN , and every such embedding restricts to an isomorphism of AHM with ZN . Then our group XHM will be AHM (AF ). Often we will just write ZN for AHM because the identiﬁcation does not depend on the embedding of × × × HM ,→ GN . The group E AF \AE is compact.

Consider the space H(GN , χ). Since ZN (AF ) is a restricted product of local groups

ZN (Fv) relative to ZN (Fv) ∩ Kv, we have

O 0 H(GN , χ) = H(GN,v, χv). v

Here χv is the local component at v of the character χ and H(GN,v, χv) is the algebra of

−1 smooth, χv -equivariant functions that are compactly supported modulo ZN (Fv). Let χHM

115 be an automorphic character of ZN (AF ) ⊂ HM (AF ). There is an analogous decomposition

O 0 H(HM , χHM ) = H(HM,v, χHM ,v). v

We need to make a choice of χHM , depending on χ, such that there is a global transfer map that goes from H(GN , χ) to H(HM , χHM ). Since both H(GN , χ) and H(HM,v, χHM ,v) decompose into restricted products, it suﬃces to deﬁne the transfer map for equivariant functions locally. As usual, the transfer is most properly viewed as a map from H(GN , χ) to the space of invariant functions

O 0 I(HM , χHM ) = I(HM,v, χHM ,v). v Before discussing the modified transfer map, let us briefly discuss measures. As mentioned in the previous section, we do not want to be overly explicit about measures. However, we will discus just a few internal compatibilities. Haar measures on GN (Fv) and HM (Fv) were used already to define orbital integrals, see 1.5.4. The product of these measures gives a

Haar measure on GN (AF ) and HM (AF ). As mentioned at 1.5.4, we take the same measure on isomorphic tori. We also need measures on the center ZN (AF ). For each place v of F , we choose the Haar measure on ZN (Fv) which gives volume one to the maximal compact subgroup. The product of the measures gives a Haar measure on ZN (AF ). For all maximal compact subgroups then, at almost all places the measure is assigning a hyperspecial maximal compact subgroup volume one.

Now we switch to local notation. Orbital integrals were deﬁned at 1.5.4 for compactly supported functions. The deﬁnition is identical for equivariant functions. Following [HH95,

§7.6], we will use Theorem 1.5.13 to easily deduce the analogous result for equivariant functions. That is, functions transforming according to χv or χH,v. The general situation is discussed in [KS99, (5.5)]. Let ∆HM (−) be the transfer factor deﬁned in [HH95, §3]. It is GN shown in [HS12, Appendix A] that ∆ (−) is a scalar multiple of ∆HM (−). Lemma 4.2 GN ,HM GN of [HH95] is the following. Let γ ∈ HM (F ) and let zα ∈ ZN (F ) be the central element αIN . Then

∆HM (z γ) = ωMD(D−1)/2(α)∆HM (γ). GN α GN 116 Since ∆ (−) is a multiple of ∆HM (−), the same formula holds for ∆ (−). GN ,HM GN GN ,HM

Let f ∈ H(GN ) be a function. Fix the character χ of ZN (F ). Associated to f is the equivariant function fe deﬁned by Z fe(g) = χ(z)f(zg) dz. ZN (F )

The resulting map

f 7→ fe : H(GN ) → H(GN , χ)

is surjective. Let χHM be the character of ZN (F ) deﬁned by

−MD(D−1)/2 χHM (zα) = ω (α)χ(zα).

Then there is a similar map from H(HM ) to H(HM , χHM ). Let fe be a function in H(GN , χ), H assumed to be the image of f ∈ H(GN ). Let f M be a function in H(HM ) with the same

HM regular orbital integrals as f. Then consider the function fe ∈ H(HM , χHM ). We claim that fe and feHM have matching orbital integrals. Since f is compactly supported, changing the order of integration in the following computation is justiﬁed. Note that D(zγ) = D(γ) and ω(zγ) = ω(γ). We have

Z Iω (γ, f) = |D(γ)|1/2 f(g−1γg)ω(g) dg GN e e GN,γ (Fv)\GN (Fv) Z Z = |D(γ)|1/2 χ(z)f(zg−1γg)ω(g) dz dg GN,γ (F )\GN (F ) ZN (F ) Z Z = |D(γ)|1/2 χ(z) f(zg−1γg)ω(g) dg dz ZN (F ) GN,γ (F )\GN (F ) Z = χ(z)Iω (zγ, f) dz. GN ZN (F )

The exact same computation gives

Z HM HM IHM (γ, fe ) = χHM (z)IHM (zγ, f ). ZN (F )

117 Note that ∆GN ,HM (γ)χ(z) = ∆GN ,HM (zγ)χHM (z). We obtain Z ∆ (γ)Iω (γ, f) = ∆ (γ)χ(z)Iω (zγ, f) dz GN ,HM GN e GN ,HM GN ZN (F ) Z = χ (z)∆ (zγ)Iω (zγ, f) dz HM GN ,HM GN ZN (F ) Z HM = χHM (z)IHM (zγ, f ) dz ZN (F )

HM = IHM (γ, fe ),

as desired.

Switching back to global notation, the pair (ZN (AF ), χHM ) will be the central character datum on HM , where

−MD(D−1)/2 Y χHM (h) = ω (h)χ(h) = χHM ,v(hv). v

The central character data of GN and HM induce central character data on all Levi subgroups and thus all the groups and endoscopic groups we will discuss.

When using equivariant functions, the deﬁnition of the convolution operator π(f) at

1.5.6 changes slightly. If for the moment F is local and XG ⊂ G(F ) is the central subgroup in the central character datum, we set

Z π(f) = f(x)π(x) dx. XG\G(F )

2.2.2 Next, we deﬁne and compute a global coeﬃcient that occurs in the trace formula. For H ∈ Eω(N), consider the group

Γ Γ Γ Γ Γ Z(Hb) = Z(Hb) /(Z(Hb) ∩ Z(GbN ) ) = Z(Hb) /Z(GbN ).

This group is defined and discussed in general at [Art13, (3.2.2)]. It is finite only when H is elliptic. Indeed, the finiteness of this group in general is an alternative definition of an elliptic datum. If H = HM represents the unique elliptic (GN , ω)-datum, then it is trivial.

118 Let ker1(F, −) denote the locally trivial classes in H1(F, −). Then set

1 −1 1 κ(GN ,HM ) = | ker (F,Z(GbN ))| | ker (F,Z(HbM ))| and

Γ −1 −1 i(GN ,HM ) = κ(GN ,HM )|Z(HbM ) | |OutN (HM )|

−1 = κ(GN ,HM )|OutN (HM )| .

The number i(GN ,HM ) occurs in the stabilization of the global trace formula for the pair

(GN , ω). For a discussion of these numbers in general, see [Art13, p.136]. Note that the representation of Γ on Z(H ) is isomorphic to IΓF ( , 1). By Shapiro’s lemma, to prove F bM ΓE C that κ(G, H) equals 1 it suﬃces to know that if λ is a character of ΓF that is locally trivial × then λ itself is trivial. But this is clear, since λ induces a locally trivial character on AF via the maps

× ab λ × AF −→ ΓF −→ C which then must be trivial. Since OutN (HM ) ' ΓE/F , see 1.2.7, we see that i(GN ,HM ) = D−1.

2.2.3 Recall that Theorem 1.7.1 holds for archimedean local ﬁelds, see the discussion at

1.7.2. Let π be a representation of HM (AF ) and let πN be a representation of GN (AF ).

Suppose moreover that for every archimedean place v, we have πN,v = AIEv/Fv πv. The numbers t(π) and t(πN ) depend on a choice of norm on the space of linear forms in which the archimedean inﬁnitesimal characters lie. If complementary norms on the respective spaces for HM and GN are chosen, then t(π) = t(πN ). See [Art13, pp.132-133] for more discussion.

2.2.4 For the moment, let G denote a general group and consider the pair (G, ω). Suppose G0 represents a datum (G0, G0, ξ0, s0) in Eω(G). In general, one must introduce auxiliary data (Ge0, ξe0) where Ge0 is a suitable central extension of G0 and ξe0 : G0 → LGe0 is an L- embedding. This issue is handled in [KS99] and is discussed at [Art13, pp.132-133]. Such central extensions and auxiliary data are not needed in our main case of G = GN , and in the

119 brief discussion about stabilization that follows we assume these additional constructions are not needed either.

Stabilizing (the discrete part of) the trace formula is the process of establishing a decomposition

ω X 0 G0 G0 Idisc,t(f) = i(G, G )Sbdisc,t(f ), f ∈ H(G, χ), (2.2.5) 0 ω G ∈Eell(G)

0 G0 for stable linear forms Sdisc,t = Sdisc,t attached to the elliptic endoscopic groups. (See 1.5.10 for the deﬁnition of a stable form in the local case. A linear form on H(G, χ) is stable if its

N G0 value at f depends only on the image of f in v S(Gv, χv).) The linear forms Sbdisc,t(−) are stable linear forms on H(G0, χ0). If G is quasi-split, the claim is that

ω X 0 G0 G0 Idisc,t(f) − i(G, G )Sbdisc,t(f ), f ∈ H(G, χ) 0 ω 0 G ∈Eell(G),G 6=G is a stable linear form. In particular, there is no stabilization problem for G = HM and

ω the trivial character, since HM has no ordinary elliptic endoscopic groups other than itself. Indeed, note that GN has only itself as an ordinary elliptic datum. The reader who is bothered by the restriction of scalars operation can glance at A.1.

We have been using, and will continue to use, the formulation of the stabilization of the discrete part of the trace formula of [Art13, §3.2]. The stabilization of the ordinary trace formula is due to Arthur. At the time of publication of [Art13], the stabilization of the twisted trace formula was ongoing work of Moeglin and Waldspurger. Our understanding is that the stabilization of the twisted trace formula is now complete. See [Art13, p.136] for a discussion of issues surrounding the stabilization of the trace formula.

We state the specialization of the general stabilization (2.2.5) as a theorem. Note that

HM ω f lies in H (HM , χHM ).

Theorem. There is an equality

GN ,ω −1 HM HM Idisc,t (f) = D Idisc,t(f ), f ∈ H(GN , χ). (2.2.6)

120 2.3 Contribution of a parameter ψ

The main theorems will follow from (2.2.6). In this section, we will show how the parameters

ψ intervene in (2.2.6). This section follows [Art13, §3.3]. Though we only discussed stabilization brieﬂy at 2.2.4, the notions introduced there are enough to make sense of what follows for G a general (still quasi-split) group and ω a unitary automorphic character of

G(A). However, much of what we discuss in this section is concerned with decompositions of invariant trace formulas. Since our primary concern is the stabilization of (2.2.6), which only involves invariant trace formulas, the situation is much simpler than [Art13, §3.3].

2.3.1 Recall from 1.3.13 that we have the set Caut(G) of equivalence classes of families c

L of semisimple conjugacy classes in G. We will need a subset Caut(G, χ) ⊂ Caut(G) that is compatible with the central character datum (XG, χ). Any class c ∈ Caut(G) determines a character ζv on Z(G(Fv)) for almost all v via the central character of the unramiﬁed representation of G(Fv) attached to cv. Then

Caut(G, χ) ={c ∈ Caut(G): χ extends to a character ζ on Z(G(F ))\Z(G(A))

which restricts to ζv on Z(G(Fv)) for almost all v}.

Consider the operator RP,t(χ, f) occurring in the the discrete part of the ordinary trace formula (2.1.3). The representation IP,t(χ) = (RP,t(χ), HP,t(χ)) is isomorphic to a direct sum of induced representations:

M IP,t(χ) ' IP (πL). 2 {πL⊂Ldisc(L(F )\L(A),χ): t(IP (πL))=t}

Note that we are abusing notation slightly and using πL to denote a particular realization of a discrete spectrum representation, since in this generality the group L might not satisfy multiplicity one in the discrete spectrum. Fix a summand π = IP (πL). Even though π may not be irreducible, we may still associate to π a class c(π) ∈ Caut(G, χ). Indeed, any irreducible constituent of π is an automorphic representation with an associated class in

Caut(G, χ) and all such constituents have the same class. This common class will be denoted

121 by c(π) even when π is not irreducible. If c is a class in Caut(G, χ), we write

M IP,t,c(χ) ' IP (πL)

{πL : c(IP (πL))=c, t(IP (πL))=t} for the subrepresentation of IP,t(χ) consisting of the summands whose class is equal to c.

We also write RP,t,c(χ, f) for the action of a Hecke function on this subspace. The global intertwining operator MP,t(w, χ), introduced at 2.1.2, preserves the space of the induced representation IP,t,c(χ), which we denote by

M HP,t,c(χ) = HP (πL).

{πL : c(IP (πL))=c, t(IP (πL))=t}

Let MP,t,c(w, χ) denote the restriction of MP,t(w, χ) to HP,t,c(χ). Then

X tr (MP,t(w, χ)RP,t(χ, f)) = tr (MP,t,c(w, χ)RP,t,c(χ, f)) .

c∈Caut(G,χ)

There is a c-variant of Idisc,t,

X M G −1 X −1 Idisc,t,c(f) = |W ||W | | det(w − 1) G | tr (MP,t,c(w, χ)RP,t,c(χ, f)) , 0 0 aM M∈L w∈W (M)reg

and thus a decomposition

X Idisc,t(f) = Idisc,t,c(f). (2.3.2)

c∈Caut(G,χ)

2.3.3 Similar remarks hold in the twisted case. First, there is a subset of Caut(G, χ)

ω associated to a pair (G, ω). We let Caut(G, χ) be the subset of Caut(G, χ) such that for almost all v the class cv satisﬁes

cv = cv(ω)cv.

(Note that this is an equality of conjugacy classes.)

ω For c ∈ Caut(G, χ), the operator

ω MP,t,c(w, χ)RP,t,c(χ, f) = MP,t,c(w, χ)RP,t,c(χ, f)Iω

ω preserves the subspace HP,t,c(χ). If c lies in the complement of Caut(G, χ), the corresponding

122 subspace is not preserved and the operator does not contribute to the trace. Thus, if we set

ω X L G −1 X −1 ω I (f) = |W ||W | | det(w − 1) G | tr MP,t,c(w, χ)R (χ, f) , disc,t,c 0 0 aL P,t,c L∈L w∈W (L)reg (2.3.4)

there is a decomposition

ω X ω Idisc,t(f) = Idisc,t,c(f). (2.3.5) ω c∈Caut(G,χ)

−1 2.3.6 Recall that H(G, χ) is the χ -equivariant Hecke algebra of G(A) relative to a suitably chosen maximal compact subgroup

Y K = Kv ⊂ G(A). v

For the moment, let S ⊃ S∞ be a ﬁnite set of valuations outside of which G is unramiﬁed. We let

S Y K = Kv. v∈ /S

By assumption, each Kv for v∈ / S is a hyperspecial maximal compact subgroup. Then H(G, KS, χ) will denote the space of functions in H(G, χ) that are bi-invariant under KS.

The algebra H(G, χ) is a direct limit

H(G, χ) = lim H(G, KS, χ). −→ S

S At 1.3.13, we deﬁned the sets Caut(G). The χ-variant is deﬁned as above. Then

C (G, χ) = lim CS (G, χ). aut −→ aut S These two limits are related through the unramiﬁed Hecke algebra

S S ∞ S S S Hun = Hun(G) = Cc (K \G(A )/K )

S on G(A ).

When v is unramiﬁed and Kv is hyperspecial, there is a natural bijection between

L the semisimple classes in Gv that project onto the Frobenius and homomorphisms from

123 S S H(G(Fv),Kv) to C. See for example [Bor79, §7]. Thus an element c ∈ Caut(G, χ) determines a complex-valued character

S S h 7→ bh(c ): Hun → C.

S S There is an action of Hun on H(G, K , χ),

S S f 7→ fh, f ∈ H(G, K , χ), h ∈ Hun, characterized by multipliers; namely, if π is an irreducible unitary representation of G(A) which is unramiﬁed outside of S, then

S tr(π(fh)) = bh(c (π)) tr(π(f)).

S 2.3.7 We can describe the decomposition (2.3.2) in terms of eigenvalues of Hun. Any f ∈ H(G, χ) belongs to H(G, KS, χ) for some S. Fix S and f and consider the linear form

S h 7→ Idisc,t(fh): Hun → C.

The expansion (2.1.3) for Idisc,t(−) implies that the linear form h 7→ Idisc,t(fh) is a ﬁnite sum

S of eigenforms. More precisely, there are linear forms Idisc,t,cS (−) on H(G, K , χ) satisfying

S Idisc,t,cS (fh) = bh(c )Idisc,t,cS (f). (2.3.8) such that X Idisc,t(fh) = Idisc,t,cS (fh) cS In particular, X Idisc,t(f) = Idisc,t,cS (f). cS S The sum is over a ﬁnite subset of Caut(G, χ), which depends on f only through a choice of

Hecke type. If c ∈ Caut(G, χ), we can then write

X Idisc,t,c(f) = Idisc,t,cS (f), (2.3.9) cS 7→c

124 S where the sum is over the preimage of c in Caut(G, χ). This is a diﬀerent expression for the summand of c on the right hand side of (2.3.2). For a given f it is independent of the choice of S.

Similar remarks hold for the ω-twisted trace formula:

ω X ω Idisc,t,c(f) = Idisc,t,cS (f). (2.3.10) cS 7→c

ω S S,ω Now, we take c ∈ Caut(G, χ) and c 7→ c indicates the preimage of c in Caut (G, χ).

2.3.11 Let us write Cω(G, χ) = lim Cω,S(G, χ) A −→ A S for the set of equivalence classes of families cS of semisimple conjugacy classes in LG that are compatible with χ and ω and such that cv projects to a Frobenius class in WFv for all v. ω Suppose H ∈ Eell(G) represents an elliptic endoscopic datum for the pair (G, ω). Then we may consider the analogous set of equivalence classes

C (H, χ ) = lim CS(H, χ ). A H −→ A H S Then there are canonical maps

H H ω c 7→ c : CA(H, χ ) → CA(G, χ).

We can take Cω(G, χ) to be the domain of summation in (2.3.5) if we set the summand to 0 A for c ∈ Cω(G, χ) Cω (G, χ), and similarly in the non-twisted case. A r aut The following lemma, Lemma 3.3.1 in [Art13], converts the decomposition (2.2.5) into

ω a parallel decomposition for Idisc,t,c(−). We will soon apply the lemma to our simple case, namely the equality (2.2.6).

We will not copy out the proof, but just the deﬁnitions

H H H H X 0 X H0 H0 Sdisc,t,cH,S (f ) = Idisc,t,cH,S (f ) − ι(H,H ) Sdisc,t,cH0,S (f ) 0 0 H6=H ∈Eell(H) cH ,S 7→cH,S

125 and

H H X H Sdisc,t,cH (f ) = Sdisc,t,cH,S (f ). cH,S 7→cH Lemma. 1. Suppose H is quasi-split. Then there is a decomposition

H H X H H H Sdisc,t(f ) = Sdisc,t,c(f ), f ∈ H(H, χH ), H c ∈CA(H,χH )

H The linear form Sdisc,t,cH (−) is stable and satisﬁes the analogs of (2.3.8) and (2.3.9).

It vanishes for all c outside a ﬁnite subset of CA(H, χH ) that depends on f only through a choice of Hecke type.

2. For all c ∈ Cω(G, χ), there exists a decomposition A

ω X X H H Idisc,t,c(f) = i(G, H) Sbdisc,t,cH (f ), f ∈ H(G, χ). ω H H∈Eell(G) c 7→c

The sum is over the classes in CA(H, χH ) that map to c.

2.3.12 For completeness, we include two more pieces of general notation before specializing

ω ω to our main situation. Fix a class c ∈ C (G, χ) and let H ∈ Eell(G). We write

H H X H H Idisc,t,c(f ) = Idisc,t,cH (f ) cH 7→c and

H H X H H Sdisc,t,c(f ) = Sdisc,t,cH (f ). cH 7→c H These are sums over classes c ∈ CA(H, χH ) that map to c.

Now we will see what happens for our pair (GN , ω). We ﬁx the character χ and recall

ω that Eell(N) is represented by HM . Theorems 1.3.14 and 1.3.15 imply

ψ 7→ c(ψ):Ψω(N, χ) →∼ Cω(N, χ) is a bijection, see also 1.4.1. They also show cS 7→ c is injective on the preimage of Cω(N, χ) in Cω(N, χ). Indeed, Cω(N, χ) corresponds to a set of isobaric representations, see the A

126 S S discussion at 1.3.24, so if c1 and c2 correspond to the same isobaric representation, they must be equal. The sum over cS in (2.3.10) is therefore not needed for any S. Similar

H ,S H remarks on sums over c M 7→ c M hold, since HM (AF ) ' GM (AE). Associated to ψ is the real parameter t(ψ) = t(πψ). Write

ω ω Idisc,ψ(f) = Idisc,t(ψ),c(ψ)(f), f ∈ H(GN , χ). (2.3.13)

We set

HM HM HM HM ω HM Idisc,ψ(f ) = Idisc,t(ψ),c(ψ)(f ), ψ ∈ Ψ (N, χ), f ∈ H(HM , χHM ), (2.3.14) following the general notation above. We obtain the following specialization as a corollary of Lemma 2.3.11.

Corollary. Let ψ ∈ Ψω(N, χ) be a parameter representing an automorphic representation of GN (A). We have

ω −1 HM HM HM Idisc,ψ(f) = D Ibdisc,ψ(f ), f ∈ H(GN , χ), f 7→ f .

Proof. The computation is straightforward using the general deﬁnitions we have introduced.

We carry it out just to see how things simplify here. Assume that f and f HM are matching

−1 functions and start with any t and c. We computed the coeﬃcient i(GN ,HM ) = D at 2.2.2. The relevant deﬁnitions are in 2.3.11, where in particular we see that

SHM (−) = IHM (−). disc,t,cHM ,S disc,t,cHM ,S

Moreover, as discussed in the previous paragraph, the maps cS 7→ c and cHM ,S 7→ cHM are injective when cS represents an automorphic representation and thus the corresponding term

127 Idisc,t,cHM ,S (−), or its stable analog, may be nonzero. Starting with Lemma 2.3.11, we have

X Iω (f) = D−1 SHM (f HM ) disc,t,c disc,t,cHM cHM 7→c(ψ) X X = D−1 SHM (f HM ) disc,t,cHM ,S cHM 7→c cHM ,S 7→cHM X = D−1 SHM (f HM ) disc,t,cHM ,S cHM 7→c X = D−1 IHM (f HM ) disc,t,cHM ,S cHM 7→c X = D−1 IHM (f HM ) disc,t,cHM cHM 7→c

−1 HM HM = D Idisc,t,c(f ).

In particular, we have

ω ω −1 HM HM −1 HM HM Idisc,ψ(f) = Idisc,t(ψ),c(ψ)(f) = D Idisc,t(ψ),c(ψ)(f ) = D Idisc,ψ(f ).

2.4 A preliminary comparison

We follow [Art13, §3.4]. The goal is to ﬁnd the initial consequences of the existence and

ω HM description of the forms Idisc,ψ(−) and Idisc,ψ(−).

2.4.1 Consider ﬁrst the pair (GN , ω). Theorems 1.3.14 and 1.3.15 characterize the automorphic spectrum in terms of families c. We have seen that

ω ∼ ω ω ψ 7→ c(ψ):Ψ (N, χ) → C (N, χ) ⊂ CA(N, χ) is a bijection. Said diﬀerently, the set Cω(N, χ) represents the set of ω-stable representations in the automorphic χ-spectrum of GN . Consider the expansion (2.1.5) for GN . Any representation that intervenes in this formula is a member of Aω(N, χ), see the description (1.3.10), and is thus associated to a parameter in Ψω(N, χ). Consequently all the representations occurring in (2.1.5) are represented by the set Cω(N, χ). The summand of c in (2.3.5)

128 therefore vanishes in this case unless c = c(ψ) for some ψ ∈ Ψω(N, χ). Using the deﬁnition

(2.3.13), the decomposition (2.3.5) reduces to

ω X ω Idisc,t(f) = Idisc,t,c(f) ω c∈Caut(N,χ)

X ω = Idisc,t(ψ),c(ψ)(f) {ψ∈Ψω(N,χ): t(ψ)=t}

X ω = Idisc,ψ(f) {ψ∈Ψω(N,χ): t(ψ)=t}

for any f ∈ H(GN , χ).

2.4.2 We will establish a similar formula for H . We write Cω(H , χ ) for the set of M A M HM ﬁbers of the mapping

C (H , χ ) → Cω(N, χ ). A M HM A HM

According to the L-homomorphism ξH deﬁned at 1.2.6. Thus we obtain an injection of Cω(H , χ ) into Cω(N, χ ). We will often use this injection to regard Cω(H , χ ) as A M HM A HM A M HM a subset of Cω(N, χ ). Consider two classes cHM and cHM in C (H , χ ) that are in A HM 1 2 A M HM

HM the same ﬁber. This is a compact way of saying that, for almost all v, the classes c1,v and

HM c2,v are conjugate to one another by AutN (HM,v), the local automorphism group of HM,v as a datum for GN,v. At this stage, we cannot conclude that the associated representations are locally conjugate to one another at all places, and in particular we cannot conclude that they are globally conjugate to one another. A priori, this ﬁber could be inﬁnite even if it represents an automorphic representation.

The set Cω(H , χ ) is designed for the subalgebra Hω(H , χ ) ⊂ H(H , χ ), A M HM M HM M HM in the sense that distinct equivalence classes in Cω(H , χ ) can be separated by the A M HM ω subalgebra H (HM , χHM ). Note that, by the remarks in the previous paragraph, even if f ω lies in H (HM , χHM ), we cannot say that the function

HM Idisc,t,−(f): CA(HM , χHM ) → C

129 is constant on the ﬁbers. However, the function

IHM (f): Cω(H , χ ) → , disc,t,− A M HM C is a sum over the fibers of the map C (H , χ ) → Cω(H , χ ). Again, a priori this A M HM A M HM fiber could be infinite, but the sum is finite for fixed f.

Proposition. Suppose f HM ∈ Hω(H , χ ), t ≥ 0 and c ∈ Cω(H , χ ). If M HM A M HM

(t, c) 6= (t(ψ), c(ψ)) for some ψ ∈ Ψω(N, χ), then

HM HM Idisc,t,c(f ) = 0.

Proof. In Corollary 2.3.12, we showed that

ω −1 HM HM Idisc,t,c(f) = D Idisc,t,c(f ).

HM HM ω Suppose Idisc,t,c(f ) is nonzero. Then Idisc,t,c(f) is nonzero. The remarks at 2.4.1 imply there is some ψ such that (t, c) = (t(ψ), c(ψ)).

HM ω Corollary. For any t ≥ 0 and f ∈ H (HM , χHM ), we have

HM HM X HM HM Idisc,t(f ) = Idisc,ψ(f ). {ψ∈Ψω(N,χ): t(ψ)=t}

Proof. This follows from (2.3.2), the previous proposition and deﬁnition (2.3.14):

HM HM X HM HM Idisc,t(f ) = Idisc,t,c(f )

c∈Caut(HM ,χHM )

X HM HM = Idisc,t,c(f ) ω c∈Caut(HM ,χHM )

X HM HM = Idisc,t(ψ),c(ψ)(f ) {ψ∈Ψω(N,χ) t(ψ)=t}

X HM HM = Idisc,ψ(f ). {ψ∈Ψω(N,χ): t(ψ)=t}

130 2.4.3 Associated to t ≥ 0 and c ∈ Cω(H , χ ) is a subspace A M HM

2 2 Ldisc,t,c(HM (F )\HM (A), χHM ) ⊂ Ldisc(HM (F )\HM (A), χHM ).

It is deﬁned as the direct sum

2 M Ldisc,t,c (HM (F )\HM (A), χHM ) = m(π)π {π : t(π)=t, c(π)=c} of the irreducible representations π of HM (A) attached to t and c that occur in the space 2 Ldisc (HM (F )\HM (A), χHM ) with positive multiplicity m(π). Applying Theorem 1.3.15, the

Moeglin-Waldspurger classiﬁcation of the discrete spectrum, to GM,E(AE) = H(AF ), we see that m(π) is either 0 or 1. Thus we may write

2 M Ldisc,t,c (HM (F )\HM (A), χHM ) = π.

{π∈Πdisc(HM ,χHM ): t(π)=t, c(π)=c} As discussed in 2.4.2, at this stage we cannot conclude that if π is a constituent of the space

2 Ldisc,t,c(HM (F )\HM (A), χHM ) then the other representations occurring in the sum are the global Galois conjugates of π. Then

2 M 2 Ldisc (HM (F )\HM (A), χHM ) = Ldisc,t,c (HM (F )\HM (A), χHM ) . t,c

If ψ ∈ Ψω(N, χ), we set

2 2 Ldisc,ψ (HM (F )\HM (A), χHM ) = Ldisc,t(ψ),c(ψ) (HM (F )\HM (A), χHM ) .

Corollary. Suppose t ≥ 0 and c ∈ Cω(H , χ ). If A M HM

(t, c) 6= (t(ψ), c(ψ))

for some ψ ∈ Ψω(N, χ), then

2 Ldisc,t,c(HM (F )\HM (A), χHM ) = 0.

131 In particular, we have a decomposition

2 M 2 Ldisc(HM (F )\HM (A), χHM ) = Ldisc,ψ(HM (F )\HM (A), χHM ). ψ∈Ψω(N,χ)

Proof. Assume that (t, c) is not of the form (t(ψ), c(ψ)). Proposition 2.4.2 says that the

HM ω HM HM linear form Idisc,t,c(−) vanishes on H (HM , χHM ). We have the expansion of Idisc,t,c(f ) given by (2.3.4):

X L HM −1 X −1 HM |W0 ||W0 | | det(w − 1) HM | tr MP,t,c(w, χHM )RP,t,c(χHM , f ) . aL L∈LHM w∈W (L)reg

We are assuming it is equal to 0. If L is proper in HM , it is a product of groups of the form RE/F GMi,E for Mi < M. The argument is by induction on M, so we assume that the corollary holds for the group HMi with Mi < M. Then the corollary holds for the group L.

If M = 1, then HM = RE/F G1,E, and we are in the base case of the induction; we are not assuming anything in that case.

HM The operator RP,t,c(χHM , f ) in the summand of L is induced from the (t, c)-component of the automorphic discrete spectrum of L. More precisely, it is induced from the component of the discrete spectrum of L given by some partition of (t, c) among the factors of L. This operator then vanishes, by our assumption on (t, c) and the induction hypothesis, and so therefore does the summand of L.

HM HM The only term in the formula for Idisc,t,c(f ) that we have not discussed is

tr RHM (χ , f HM ) , HM ,t,c HM

HM HM the summand corresponding to L = HM . It also vanishes, since Idisc,t,c(f ) vanishes by the assumptions on (t, c) and all the summands corresponding to L ( HM vanish by the discussion H of the previous paragraph. Note that by deﬁnition the operator R M (χ , f HM ) is the HM ,t,c HM

HM 2 operator of right convolution by f on the space Ldisc,t,c(HM (F )\HM (A), χHM ). Its trace is equal to the sum X tr π(f HM ) , t(π) = t, c(π) = c, (2.4.4) π over the irreducible subrepresentations of this space.

132 HM ω The function f is restricted to lie in the subspace H (HM , χHM ). However, this is still enough to imply that the sum (2.4.4) is empty. Indeed, recall that if

H ∗ f M (h) = f HM (h−1)

then

HM HM ∗ HM tr(π(f )π(f )) = kπ(f )kHS.

To prove that the sum (2.4.4) is empty, it suﬃces to show that for any π there is a function

HM ω HM f ∈ H (HM , χHM ) such that π(f ) is nonzero. Indeed, suppose π0 occurs in the sum

HM ω HM (2.4.4) and let f0 be a function in H (HM , χHM ) such that π0(f0 ) is nonzero. Then

X HM HM ∗ HM HM ∗ X HM HM ∗ tr π(f0 ∗ f0 ) = tr π0(f0 )π0(f0 ) + tr π(f0 )π(f0 ) π π6=π0

HM X HM = kπ0(f0 )kHS + kπ(f0 )kHS π6=π0

HM > kπ0(f0 )kHS

> 0, contradicting the fact that the sum vanishes. Thus, if we can prove that for each π there is a function such that π(f HM ) 6= 0, then the fact that the sum (2.4.4) vanishes for every f HM will force the sum to be empty.

N0 HM N0 HM Let x = v xv be a nonzero vector in the space of π. If f = v fv is a decomposable function in H(HM , χHM ), then

HM O 0 HM O HM O π(f )x = πv(fv )xv = πv(fv )xv ⊗ xv, v v∈S v∈ /S

HM where S is a ﬁnite set of places outside of which fv is the characteristic function of the maximal compact subgroup and xv is a vector ﬁxed by the maximal compact subgroup.

HM Thus, it suﬃces to show that at a ﬁxed place v there is a symmetric function fv such that

HM πv(fv ) is a nonzero operator.

The problem is now local, so ﬁx a local representation π. We write H instead of HM,v.

ω There is the local Hecke algebra H (H, χH ) consisting of functions which are invariant

133 under the local automorphism group AutGN (H) consisting of automorphisms of H as a

(GN , ω)-datum. Call such functions symmetric. We would like to show that π(f) is nonzero

ω for some f ∈ H (H, χH ). Suppose this is not the case. Thus, if f is symmetric then π(f) = 0.

Let W denote the local automorphism group of H. Let W0 be the subgroup of W that ﬁxes

π. Let w1, . . . , wk be coset representatives so that

k a W = W0wi. i=1

Note that (wπ)(f) = π(w−1f). For any function f, the function

k sym X X X f = wf = (w0wi)f

w∈W i=1 w0∈W0 is symmetric. We are supposing that if g is any symmetric function then π(g) = 0, and thus

0 = π(f sym)

 k  X X = π  (w0wi)f i=1 w0∈W0 k X X −1 −1 = (wi w0 π)(f) i=1 w0∈W0 k X X −1 = (wi π)(f) i=1 w0∈W0 k X −1 = #W0 (wi π)(f). i=1

−1 Since f is arbitrary and the representations wi π, 1 ≤ i ≤ k are distinct, this is impossible. The hypothesis was that π(−) vanished on the subalgebra of symmetric functions, so this

ω must be false. Consequently, π(−) does not vanish identically on H (H, χH ). This was the statement we needed to prove at the beginning of this ﬁnal paragraph.

We’ve shown that the sum (2.4.4) is empty. Consequently, we know that the space

2 Ldisc,t,c(HM (F )\HM (A), χHM ) is equal to 0. Finally, note that we already have that

2 M 2 Ldisc (HM (F )\HM (A), χHM ) = Ldisc,t,c (HM (F )\HM (A), χHM ) . t,c

134 The summand corresponding to (t, c) vanishes unless (t, c) = (t(ψ), c(ψ)) for some ψ. This is the ﬁnal statement of the lemma.

2.4.5 The corollary has its own corollary, a weak lifting result, which we will state as a proposition; this is the way the results of this section will actually be applied later. Let π be an automorphic representation of HM (A). The L-homomorphism ξHM was deﬁned at

1.2.6. Let AIwπ be an isobaric automorphic representation of GN (A), if it exists, such that

ξHM (c(π)) = c(AIwπ). In other words, the local conjugacy classes agree at almost all places. This is what we mean by a “weak lift.” Strong multiplicity one, Theorem 1.3.14, implies that, if such a representation exists, it must be unique. Then the previous corollary gives the following.

Proposition. Let π be a discrete spectrum automorphic representation of HM (A). Then

AIwπ exists.

Proof. According to the previous corollary, there is some ψ ∈ Ψω(N, χ) such that

2 π ∈ Ldisc,ψ(HM (F )\HM (A), χHM ).

0 This latter space was deﬁned to be the subspace of automorphic representations π of HM (A) 0 0 whose class c(π ) satisﬁes ξH (c(π )) = c(ψ). Then πψ, the automorphic representation of

GN (A) corresponding to ψ, is the desired representation AIwπ.

2.5 The stable multiplicity formula

We will start by stating the stable multiplicity formula as in [Art13, §4.1]. We will go on to describe the simpliﬁcation of the terms.

2.5.1 Adjusting only for some slight changes in notation, we recall [Art13, Theorem 4.1.2].

ω Theorem (Stable multiplicity formula). Given ψ ∈ Ψ (N, χ) and HM representing the unique elliptic endoscopic datum in Eω(N), we have

HM HM −1 0 HM HM Sdisc,ψ(f ) = mψ|Sψ| σ(Sψ)ε (ψ)f (ψ) (2.5.2) 135 HM ω for any f ∈ H (HM , χHM ).

In this section, we will recall, or introduce when necessary, the terms in the stable multiplicity formula and explain how they simplify. In general, certain cases of this theorem must be proved by induction as part of what is called the “standard model” in [Art13], see

Chapter 4 there and in particular Proposition 4.5.1. In our special case of HM and GN when M = 1, there is no induction step. This is the only case of the theorem we will prove.

HM On the left hand side, we have already seen that Sdisc,ψ(f), the stable part of the discrete

HM part of the trace formula for HM , is just Idisc,ψ(f), the invariant trace formula for HM . See ω ω Section 2.3. At 1.4.12, we discussed the identiﬁcation Ψ (N, χ) = Ψ (HM , χHM ).

ω 2.5.3 Consider ﬁrst the term mψ. Recall that ψ ∈ Ψ (HM , χ) can be identiﬁed with

AutN (HM )-orbits of L-homomorphisms

ψω : L × SL(2, ) → LH . HM ψ C M

See 1.4.12. Let Ψ(HM , χHM ) be the corresponding set of orbits under the subgroup HbM of

AutN (HM ). Then there is a surjective mapping

ω ψHM 7→ ψ : Ψ(HM , χ) → Ψ (HM , χ).

The integer mψ is defined to be the size of the fiber of ψ under this mapping. It is finite, since we computed in 1.2.7 that the outer automorphism group of HM in GN is isomorphic to the Galois group ΓE/F .

2.5.4 The group Sψ was deﬁned to be the centralizer in HbM of a representative of ψ. See

1.4.14. Then Sψ is the set of connected components of the group

Γ Sψ = Sψ/Z(HbM ) E/F .

We proved in 1.4.14 that Sψ is connected. Consequently |Sψ| = 1.

136 2.5.5 Next we consider the number σ(−). For more details on the construction, see [Art13, §4.1]. Following the notation there, we let S be a union of connected components in a complex reductive group. Let S0 be the identity component and set

Z(S) = Cent(S, S0),

the centralizer of S in S0. Fix a maximal torus T ⊂ S0. Then we may form the Weyl set

W (S) = Norm(T,S)/T consisting of automorphisms of T induced by S. The Weyl set W (S) contains the subset of regular elements

Wreg(S) = {w ∈ W (S) : the ﬁxed points of w acting on T is a ﬁnite set}.

Let par(w) denote the parity of the number of positive roots of (S0,T ) mapped by w to negative roots. Then set

s0(w) = (−1)par(w).

The number i(S) is deﬁned by the formula

X i(S) = |W (S)|−1 s0(w)| det(w − 1)|−1.

w∈Wreg(S)

Let Sss denote the set of semisimple elements of S. For s ∈ Sss, we set

0 Ss = Cent(s, S ).

If Σ ⊂ S is invariant under conjugation by S0, let E(Σ) be the equivalence classes in

0 0 −1 Σss = Σ ∩ Sss under the following equivalence relation: s is equivalent to s zs(s ) for

0 0 0 0 s ∈ S and z ∈ Z(Ss ) . Then we set

0 Sell = {s ∈ Sss : |Z(Ss )| < ∞} and

Eell(S) = E(Sell).

137 In this case the equivalence relation is just S0-conjugacy.

Proposition. There exist unique numbers σ(S1) deﬁned when S1 is a connected complex reductive group such that

X −1 0 i(S) = |π0(Ss)| σ(Ss )

s∈Eell(S) and such that

−1 σ(S1) = σ(S1/Z1)|Z1|

for any subgroup Z1 ⊆ Z(S1).

Note that σ(S1) = 0 if the center of S1 is inﬁnite. We will discuss the σ(−) more in our special case in 2.5.8.

2.5.6 In [Art13], the character εG(ψ), deﬁned for a classical group G, is the special value of a character deﬁned on Sψ. Since this latter group is trivial in our situation, the associated character is trivial as well.

2.5.7 The ﬁnal term to discuss is f HM (ψ). Since the left hand side of (2.5.2) is a linear

HM N0 HM form, we may assume that f = v fv is a tensor product of local functions. In this case, the linear form f HM (ψ) is deﬁned by

HM Y HM,v f (ψ) = fv (ψv). v

H HM,v The form is extended by linearity to all f . Let us recall the local term fv (ψv).

Consider first the case of an inert place v. Then Ev is a cyclic field extension of Fv of degree D determined by the local character ωv. The local parameter ψv satisfies ψv ⊗ωv = ψv. Then this is the linear form introduced at 1.7.5. It is well-defined, as discussed there.

Next suppose that v is split, namely that

, Ev = E ⊗F Fv ' Fv × · · · × Fv with D factors in the product. In this case,

HM,v(Fv) ' GM,v(Fv) × · · · × GM,v(Fv), 138 again with D factors in the product. In particular, the group HM,v is isomorphic to a

Levi subgroup of GN,v. The local factor ωv is the trivial character, and thus the condition

ψv ⊗ ωv ' ψv does not impose any conditions on ψv. Any Levi subgroup Lv ⊂ GN,v has an associated Harish-Chandra constant term

fv 7→ fv,Lv : H(GN,v(Fv)) → H(Lv(Fv)),

discussed at 1.5.15. As mentioned there, the function fv,Lv is properly viewed as a member of I(Lv(Hv)). Now, we would like to deﬁne

HM,v fv (ψv) = tr πψv (fv)

HM,v as above, where fv is a function that maps to fv under fv 7→ fv,Lv . However, in this case the linear form is not well-deﬁned unless we know that πψv is induced from Lv. Until we know this, the form f HM (ψ) cannot be deﬁned. The same issue occurs at any non-inert place.

We will only be able to consider the stable multiplicity formula when we have proved that

πψv is of the right form at all places. Eventually, we will prove this for all the automorphic representations in the image of automorphic induction. Now, when the form is deﬁned, we have by deﬁnition

HM f (ψ) = tr πψ(f) ◦ Aπψ .

We will use the right hand side for the time being, since it is always deﬁned. The intertwining operator is the self-intertwining operator deﬁned as the product of the local intertwining operators A normalized via local Whittaker functionals in 1.6. πψv

2.5.8 When σ(Sψ) = 0, the right hand side of (2.5.2) vanishes. Thus establishing the formula in this case is equivalent to a vanishing statement for the left hand side of (2.5.2).

The set of parameters where both sides vanish is related to a chain of subsets

Ψsim(HM ) ⊆ Ψ2(HM ) ⊆ Ψell(HM ) ⊂ Ψdisc(HM ) ⊂ Ψ(HM )

139 deﬁned as follows:

Ψsim(HM ) = {ψ ∈ Ψ(HM ): |Sψ| = 1}.

Ψ2(HM ) = {ψ ∈ Ψ(HM ): Sψ is ﬁnite}.

Ψell(HM ) = {ψ ∈ Ψ(HM ): Sψ,s is ﬁnite for some s ∈ Sψ,ss}.

Ψdisc(HM ) = {ψ ∈ Ψ(HM ): Z(Sψ) is ﬁnite}.

These sets are deﬁned for classical groups at [Art13, p.172]. The heuristics described there suggest the following lemma.

Lemma. If M = 1, then

Ψ(HM ) = Ψsim(HM ) and all the sets are the same. If M > 1, then

Ψsim(HM ) = Ψ2(HM ) = Ψell(HM ) ( Ψdisc(HM ) ( Ψ(HM ).

Proof. According to the proof of Lemma 1.4.14, the centralizer Sψ is contained in the fixed points of ΓF and is isomorphic to a product GL(n1, C) × · · · × GL(nk, C). Suppose first that Γ M = 1. Then Sψ is Z(HbM ) E/F and the first claim holds.

Suppose now that M ≥ 2 and that Sψ,s is finite for some s. We must have k = 1, since otherwise Sψ has a nontrivial center. So, assume Sψ ' GL(m, C). Then we must have × m = 1, since otherwise Sψ has a nontrivial maximal torus. Thus Sψ ' C . But then Sψ is trivial. Thus we have shown that Ψell(HM ) ⊂ Ψsim(HM ), which proves the first three equalities. Moreover, a parameter ψ with Sψ = GL(M, C) satisfies Z(Sψ) = 1 even though

Sψ is nontrivial. Such a ψ lies in Ψdisc(HM ) but does not lie in Ψsim(HM ), which proves for

M ≥ 2 that Ψsim(HM ) is properly contained in Ψdisc(HM ). For the second strict equivalence, any parameter whose centralizer does not consist of a single copy of GL(n, C) cannot be discrete.

Having described these sets of formal parameters, let us return to the number σ(−) from

2.5.5. As mentioned at the end of 2.5.5, we have that σ(S1) = 0 if the center of S1 is inﬁnite.

140 0 In our case, Sψ = Sψ, so that, unless ψ lies in Ψdisc(HM ), the right hand side of (2.5.2) vanishes immediately. If ψ lies in Ψ2(HM ), then the equality with Ψsim(HM ) implies that

σ(Sψ) = 1.

2.5.9 Earlier, at 1.4.15, we gave an alternate deﬁnition of the set Ψ2(HM ). Denote the set

0 deﬁned by this earlier deﬁnition as Ψ2(HM ).

Lemma. We have

0 Ψ2(HM ) = Ψ2(HM ).

Moreover, both sets are equal to the set of parameters of the form

k−1 ψ = ψ1 (ψ1 ⊗ ω) ··· (ψ1 ⊗ ω )

k where ψ1 satisﬁes ψ1 ⊗ ω = ψ1 and k is minimal for this property.

Proof. It suﬃces to prove the second statement.

ω Consider an arbitrary parameter ψ and the associated map ψβ as in 1.4.12. Each ω ω L summand βi ⊗ νi of ψβ factors through a group of the form Hm for some m < M. Thus, the parameter does not factor through a Levi subgroup if and only if there is only one

0 summand. This proves that Ψ (HM ) consists of single ω-orbits, as desired.

ω Consider again a typical parameter ψ. The centralizer of ψβ is isomorphic to GL(l1) ×

· · · × GL(lr). The centralizer Sψ is ﬁnite if and only if r = 1 and l1 = 1, which proves that

Ψ2(HM ) also consists of single ω-orbits.

ω ω ω We continue to write Ψ2 (HM ) = Ψell(GN ) as at 1.4.15. We regard the set Ψell(GN ) as being deﬁned by this equality. Such parameters are elliptic for (GN , ω), in the sense that they are not induced from ω-stable representations. This gives a diﬀerent interpretation

ω of the set Ψell(GN ) that is more intrinsic. The set of elliptic representations contains the ω-ﬁxed discrete spectrum. This is a general phenomenon. The notion of elliptic depends on the particular lifting or endoscopic situation, while the discrete spectrum is intrinsic and independent of endoscopy. Moreover, one expects that the discrete spectrum will be contained in the set of elliptic representations and that the discrete spectrum of any elliptic 141 endoscopic group will lift into the elliptic representations. These statements don’t say much in the case M = 1 that we eventually consider, but they are less trivial when M > 1.

As we shall discuss later, speciﬁcally at 3.1.1, when M = 1 every ω-ﬁxed parameter is elliptic. When M > 1, we expect that handling the elliptic parameters will be similar to the discussion in Chapter3. There is an additional subtlety when M > 1 that we do not have to handle. If M > 1, then the sets Ψ2(HM ) and Ψdisc(HM ) are not equal. Proposition 4.5.1 of [Art13] suggests that the entire set Ψdisc(HM ) can be handled by induction.

HM HM HM HM 2.5.10 Consider the left hand side Sdisc,ψ(f ) = Idisc,ψ(f ) of (2.5.2). According to Corollary 2.3.12, we have

HM HM ω Idisc,ψ(f ) = DIdisc,ψ(f) whenever f 7→ f HM . We will now consider the spectral expansion

ω X L GN −1 X −1 ω Idisc,ψ(f) = |W0 ||W0 | | det(w − 1) GN | tr MP,ψ(w)RP,ψ(f) aL L w

ω of Idisc,ψ(f). The sums are the same as in (2.1.5) and are deﬁned at 2.1.2. Our goal is to compute the terms in the expansion more explicitly, so we can relate them to the (currently, not generally deﬁned) linear form appearing on the right hand side of (2.5.2). As we mentioned above, we will actually relate the spectral expansion to the linear form on GN that is equal to f HM 7→ f HM (ψ) when the latter form exists.

ω The term in the expansion we really need to discuss is tr MP,ψ(w)RP,ψ(f) ; the other terms are numbers that are straightforward to compute. Similar computations are carried out in the base change context in [AC89, Ch. 3, §4] and [BH16, §4]. In some of the discussion that follows, we will need to be more explicit than usual about the space on which representations are acting. We will continue to use the convention (R,V ) to denote a vector space V with an action R by some group. Note that the terms in the trace formula associated to conjugate Levi subgroups are equal. Denote the conjugacy class of a Levi

GN L −1 −1 subgroup L by {L}. There are |W0 ||W0 | |W (L)| groups per conjugacy class. Then

142 we may rewrite the spectral expansion as

ω X −1 X −1 ω Idisc,ψ(f) = |W (L)| | det(w − 1) GN | tr MP,ψ(w)RP,ψ(f) . aL {L} w

2.5.11 Fix the parameter ψ. At 2.1.2 we had to ﬁx a minimal Levi subgroup which we may ﬁx to be the diagonal torus in GN . For any Levi subgroup L containing L0 and regular Weyl group element w ∈ W (L) there is an associated linear operator

ω MP,ψ(w)RP,ψ(f) = MP,ψ(w)RP,ψ(f)Iω.

Fix a representative L of a conjugacy class, which we may as well assume is a standard diagonal representative, and w as well. Though the operator depends on a choice of P in

P(L), the trace does not. Thus there is no harm in ﬁxing P either. Recall that we have the induced representation IP (χ) = IP (χ, 0) of GN (A), see 1.3.8 and 2.1.1. It is induced from 2 the full discrete spectrum RL,Ldisc(L(F )\L(A), χ) . We will continue to use the notation

(RP (χ), HP (χ)) = (RP (χ, 0), HP (χ)) to denote the action and underlying space.

There is a unique conjugacy class {L} of Levi subgroups of GN such that (πψ,Vψ), the representation associated to ψ, is equivalent to an irreducible subrepresentation of

(RP (χ), HP (χ)) for P ∈ P(L). Since the term associated to this unique conjugacy class is the only nonzero operator in the ψ-part of the discrete part of the trace formula, we assume that L is a standard diagonal representative of this class and P is the associated upper triangular parabolic subgroup in P(L). Note that we can take P = GN . If P = GN , there is no induction, the global intertwining operator is trivial and ψ is a discrete parameter. In other words, πψ is an ω-stable discrete spectrum representation of GN (A).

Let (RP,ψ, HP,ψ) be the ψ-part of the induced representation (RP (χ), HP (χ)). Let us be a bit more precise about this space. Consider a discrete series representation πL of L(A), real- 2 ized as (RL,VπL ) ⊂ (RL,Ldisc(L(F )\L(A), χ)). Then we have the associated representation

(RP , HP (VπL )). The parameter ψ determines a set of representations (RL,VπL ):

−1 I (ψ) = {(RL,VπL ):(RP , HP (VπL )) ' (πψ,Vψ)}.

143 The notation is meant to indicate the “preimage” of ψ, or more accurately πψ, under parabolic induction. This diﬀers from the notation used in [Art13], where he would use something like Ψ2(L, ψ), see for example page 174 there. Then

M (RP,ψ, HP,ψ) = (RP , HP (VπL )) ⊂ (RP (χ), HP (χ)). I−1(ψ)

If πψ happens to be discrete, then according to Theorem 1.3.15 there is only one summand. Otherwise, as we shall see momentarily, there is more than one summand.

ω 2.5.12 In even considering the linear form tr MP,ψ(w)RP,ψ(−) , we have basically assumed the following lemma.

Lemma. The subspace HP,ψ is stable under each of the operators Iω, RP,ψ(f) and MP,ψ(w).

ω In particular, the linear form tr MP,ψ(w)IP,ψ(−) is well-deﬁned.

Proof. Let us discuss the terms in order. Fix a discrete spectrum representation πL =

−1 (RL,VπL ) of L(A) in I (ψ) and consider the space HP (VπL ) ⊂ HP,ψ. Recall that H(VπL ) is the space of left NP (A)-invariant functions φ on GN (A) such that the function (m, k) 7→ 2 2 2 φ(mk) lies in VπL ⊗ L (K) ⊂ Ldisc(L(F )\L(A), χ) ⊗ L (K). Consider Iωφ. Then,

(Iωφ)(ng) = ω(ng)φ(ng) = ω(g)φ(g) = (Iωφ)(g).

Thus Iωφ is NP (A)-invariant. Suppose now that

k X 2 φ(mk) = fi(m) ⊗ gi(k), fi ∈ VπL , gi ∈ L (K). i=1 Then k X (Iωφ)(mk) = (Iωfi)(m) ⊗ (Iωgi)(k). i=1 2 2 Note that Iωgi lies in L (K), since ω is unitary. Thus Iωφ lies in IωVπL ⊗ L (K). Thus, we

144 need to prove that (RP , HP (IωVπL )) ' (πψ,Vψ). Note that we have a diagram

Iω HP (VπL ) / HP (IωVπL )

(RP ⊗ω)(g) RP (g) HP (VπL ) / HP (IωVπL ) Iω

Recall Lemma 1.4.5, which implies that πψ ⊗ ω ' πψ. Now, we have

(RP , HP (IωVπL )) ' (RP ⊗ ω, HP (VπL )) ' (πψ ⊗ ω, Vψ) ' (πψ,Vψ),

as desired.

The next term is RP,ψ(f). This is just the restriction of the operator RP (χ) on HP (χ) to the subspace HP,ψ. The operator RP (χ) preserves all of the irreducible subspaces of

HP (χ). In particular, it preserves HP,ψ.

Lastly, we discuss the global intertwining operator MP,ψ(w). This operator was ﬁrst introduced at 2.1.2. It is discussed in more detail in A.3. For the moment, we simply recall that

MP (w, χ):(RP (χ), HP (χ)) → (RP (χ), HP (χ)) is a unitary operator the intertwines the standard induced representation on HP (χ). As is explained in more detail in A.3, if πL = (RL,VπL ) is a discrete series representation, then in fact

MP (w, χ):(RP , HP (Vπ)) → (RP , HP (wVπ)).

Here wVπ indicates the space of functions

{wf : f ∈ Vπ} deﬁned by setting

wf(m) = f(w−1mw).

From this, we can conclude that MP (w, χ) preserves the space HP,ψ. Indeed, suppose now that (RP , HP (VπL )) ' (πψ,Vψ). Then we need to show that (RP , HP (wVπL )) ' (πψ,Vψ) as well. Since MP (w, χ) is a nonzero operator between the irreducible representations

145 (RP , HP (Vπ)) and (RP , HP (wVπ)), it is an isomorphism. Thus

(RP , HP (wVπL )) ' (RP , HP (VπL )) ' (πψ,Vψ).

The restriction of MP (w, χ) to HP,ψ is the operator we have been calling MP,ψ(w).

2.6 An elliptic computation

ω When ψ is elliptic, we can compute Idisc,ψ(−) in more detail. When M = 1, this is the only case. As mentioned at 2.5.8, we have σ(Sψ) = 1 when ψ is elliptic.

2.6.1 If ψ is elliptic, then we can ﬁx a base point ψ1 = µ1 ν1 and write

k1−1 ψ = ψ1 (ψ1 ⊗ ω) ··· (ψ1 ⊗ ω ).

k Implicitly, the parameter ψ1, and thus the cuspidal representation µ1, are ﬁxed by ω 1 and this is the minimal k1 with this property.

The parameter µ1 ν1 represents a discrete spectrum representation of Gm1n1 (A). Then

Lψ = Gm1n1 × · · · × Gm1n1 represents the unique conjugacy class of Levi subgroups with a −1 discrete spectrum representation that lies in I (ψ). We ﬁx Lψ and consider the term in

ω Idisc,ψ(−) associated to Lψ.

−1 2.6.2 We claim that associated to w ∈ W (Lψ)reg are k1 representation of Lψ in I (ψ) that contributes to tr (MP,ψ(w)RP,ψ(f)Iω). Since we are only interested in the trace, we may instead consider tr (IωMP,ψ(w)RP,ψ(f)). Since RP,ψ(f) preserves irreducible subspaces, we are interested in subspaces of HP,ψ that are preserved by IωMP,ψ(w).

−1 So, ﬁx Vπ ∈ I (ψ) and consider the associated subspace HP (Vπ ) ⊂ H . We may Lψ Lψ P,ψ restrict the operator IωM (w) to HP (Vπ ) to obtain the map P,ψ Lψ

IωMP,ψ(w)|HP (Vπ ) : HP (VπL ) → HP (wVπL ) → HP (IωwVπL ). Lψ ψ ψ ψ

2 Thus we are interested in subspaces Vπ ⊂ L (L (F )\L ( ), χ) satisfying the condition Lψ disc ψ ψ A

Vπ = IωwVπ . Lψ Lψ

146 Assume then that Vπ satisﬁes the condition Vπ = IωwVπ . The representation Lψ Lψ Lψ π is isomorphic to a tensor product π ⊗ · · · ⊗ π with each π a discrete spectrum Lψ L1 Lk1 Li representation of G ( ) realized on the space of functions V = V . The space of m1n1 A i πLi functions Vπ is generated by functions of the form f1 ⊗ · · · ⊗ f with fi ∈ Vi. Assume f Lψ k1 is such a function. Then

wf = w(f1 ⊗ · · · ⊗ fk1 ) = fw(1) ⊗ · · · ⊗ fw(k1).

Then the condition Vπ = IωwVπ says that Lψ Lψ

(ωf ) ⊗ · · · ⊗ (ωf ) ∈ Vπ w(1) w(k1) Lψ and thus ωfw(i) ∈ Vi. Said diﬀerently, we have

Vw(i) = Iω−1 Vi.

The representation πLψ is thus determined by any single representation πLi . For 0 ≤ j ≤ j k1 − 1, we may set π1 = ω πψ1 and use the condition to determine a representation of Lψ in −1 I (ψ). The elliptic condition implies that these are k1 distinct representations, otherwise

πψ1 would be ﬁxed by a power of ω less than k1.

−1 2.6.3 We continue with a ﬁxed w. Consider now one of the k1 spaces VπL ∈ I (ψ) that satisfy VπL = IωwVπL . Then we have

IωMP,ψ(w): HP (VπL ) → HP (VπL ).

Of course, these spaces carry representations. Written a bit more precisely, we have

IωMP,ψ(w):(RP , HP (VπL )) →(RP , HP (wVπL )) →

−1 −1 (RP ⊗ ω , HP (IωwVπL )) = (RP ⊗ ω , HP (VπL )).

−1 In other words, IωMP,ψ intertwines (RP , HP (VπL )) with (RP ⊗ ω , HP (VπL )). We have another operator which does this. Indeed, consider the local representations πψ,v. At 1.6.2, we deﬁned a normalized intertwining operator Aπψ,v when πψ,v is generic by requiring the

147 operator to preserve a ﬁxed Whittaker functional. The non-generic case was discussed at

1.6.5. Moreover, we gave a concrete description of this local intertwining operator at (1.6.13).

Then the global operator Aπψ is the tensor product of the local self-intertwining operators.

When πψ is generic, the operator Aπψ preserves any ﬁxed Whittaker functional. Since the induced representations are irreducible, the operators IωMP,ψ(w) and Aπψ diﬀer by a scalar that we can compute.

2.6.4 Suppose ﬁrst that the inducing representation πLψ is cuspidal, so that it has local and global Whittaker functionals. In addition, suppose ψ is discrete so that πψ = πLψ and we are trying to compare Iω with Aπψ . From the global integral for the Whittaker functional, see 1.3.23, we see that Iω also preserves the Whittaker functional. Consequently we have

Iω = Aπψ in this case.

2.6.5 Now we assume that Lψ is a proper Levi subgroup. We continue to assume that πLψ is cuspidal, so that Aπψ is normalized using Whittaker functionals.

Recall from 2.1.2 that MP (w, χ) is the composition of operators l(w) ◦ MP 0|P (χ). At a place v, the localization of the global operator M 0 (χ), acting on (RP , HP (Vπ )), P |P Lψ

0 is the the unnormalized operator JP |P (πLψ,v) from 1.6.7. Since we can always choose global representatives of Weyl group elements to be permutation matrices, the localization of the operator l(w) is the operator l(w, π ) from 1.6.10. The global operator e Lψ,v −1 Iω :(RP , HP (wVπ )) → (RP ⊗ ω , HP (Vπ )) preserves the Whittaker functional. Its Lψ Lψ localization is the operator I ◦ T (π , w, ω−1) from 1.6.14, since this local operator was ωv Lψ,v e v designed to also preserve Whittaker functionals. Let rP (w, πLψ ) be the global normalizing factor. It is the product over all v of the corresponding local normalizing factors

−1 rw P |P (πLψ,v) from 1.6.9. It is an inﬁnite product that converges in a half-plane and has analytic continuation. Then

−1 rP (w, πLψ ) IωMP,ψ(w, χ) = Aπψ , since both sides preserve a ﬁxed Whittaker functional. See in particular Proposition 1.6.11.

148 Lemma. Suppose Vπ = IωwVπ . Then Lψ Lψ

rP (w, πLψ ) = 1.

Proof. Denote the factors of our Levi subgroup Lψ by Li, 0 ≤ i ≤ k1 −1. Fix a representation for L0, say π0, acting on the space V0. The condition Vπ = IωwVπ says that V = Lψ Lψ w(i) −1 Iω−1 Vi. The right action on this space is isomorphic to πi ⊗ ω . So, we have πwk(0) = −k π0 ⊗ ω . A root α in this setting is an ordered pair of distinct integers (i, j) with 0 ≤ i, j, ≤ k − 1.

For the rest of this proof, roots will be denoted as such ordered pairs. Since P is ﬁxed, we may call (i, j) positive if it lies in ΣP and negative if it does not, and thus lies in ΣP . The factor associated to the root α is the value at s = 0 of

−1 −1 L(s, πi × πej)ε(s, πi × πej) L(s + 1, πi × πej) .

Taking the value at s = 0 and applying the global functional equation, we obtain

L(1, π × π ) ei j . L(1, πi × πej)

−m We may write πi = π0 ⊗ ω i for nonnegative integers mi. Then this factor looks like

mi −mj mi−mj L(1, πe0 ⊗ ω × π0 ⊗ ω ) L(1, πe0 ⊗ ω × π0) −m m = m −m . L(1, π0 ⊗ ω i × πf0 ⊗ ω j ) L(1, πe0 ⊗ ω j i × π0)

By deﬁnition, the factor rw−P |P (πL) is a product

m −m Y L(1, π0 ⊗ ω i j × π0) −1 e rw P |P (πLψ ) = m −m L(1, πe0 ⊗ ω j i × π0) (i,j)∈Σw−1P |P

The exponents on ω are to be read modulo k1. Let (i, j) be a root. We have that

mw(i) − mw(j) = (mi + 1) − (mj + 1) = mi − mj.

Consequently, all the roots (i, j), w−1(i, j), ··· , w−k1+1(i, j) in the w−1-orbit of (i, j) that

−1 lie in Σw−1P |P contribute the same ratio to the factor rw P |P (πLψ ). Consider now the root

149 (j, i). The ratio associated to (j, i) is the inverse of the ratio associated to (i, j), and all the roots in the w−1-orbit of (j, i) contribute the same ratio.

Fix (i, j). Suppose ﬁrst that the w−1-orbit of (i, j) is equal to the w−1-orbit of (j, i).

Thus we have w−r(i, j) = (j, i) for some r. Then

mi − mj = mw−1(i) − mw−1(j) = mj − mi.

Consequently, the ratio associated to (i, j) is equal to 1. Thus any root in the orbit of (i, j)

−1 contributes a ratio equal to 1 in the product for rw P |P (πLψ ). We consider now a ﬁxed root (i, j) and its negative (j, i) that have distinct w−1-orbits.

We can assume that (i, j) is negative. It suﬃces to show that the orbits of the roots (i, j)

−1 and (j, i) have the same number of roots in Σw−1P |P . Consider the w -orbit

(i, j), w−1(i, j), . . . , w−k1+1(i, j), w−k1 (i, j) = (i, j).

By a sign change, we just mean that a root (r, s) is positive and w−1(r, s) is negative, or vice versa. The number of sign changes in the above orbit must be even. Hence there are the same number of sign changes from negative to positive as from positive to negative. A root in the orbit that lies in Σw−1P |P corresponds to a sign change from positive to negative. Now, consider the orbit

(j, i), w−1(j, i), . . . , w−k1+1(j, i), w−k1 (j, i) = (i, j).

A sign change from positive to negative in the orbit of (i, j), say from w−r+1(i, j) to w−r(i, j), gives a sign change from negative to positive, w−r+1(j, i) to w−r(j, i), in the orbit of (j, i).

The number of sign changes from positive to negative in the orbit of (i, j) is equal to the number of sign changes in the orbit of (j, i) from positive to negative, which is equal to the number of roots in the orbit of (j, i) that lie in Σw−1P |P . This proves that the orbits of (i, j) and (j, i) have the same number of roots in Σw−1P |P , which completes the proof.

150 2.6.6 When πL is cuspidal and satisﬁes VL = IωwVπ , we have now shown that ψ ψ Lψ

IωMP,ψ(w, χ) restricted to VLψ is equal to Aπψ . When πLψ is not cuspidal, so the local components are not necessarily generic, we still have a normalized operator Aπ obtained Lψ by using the Langlands classiﬁcation, see 1.6.5. We can again ask if IωMP,ψ(w, χ) is equal to Aπ . The process of extending such an equality of operators from the cuspidal case Lψ to the discrete case by taking “Langlands quotients of operators” is discussed at [Art13,

Lemma 4.2.3]. In our main case of M = 1 that we consider later, we only have to deal with cuspidal inducing data, so we will not worry about extending the arguments.

−1 2.6.7 The last term for us to compute is the number | det(w − 1) G | . The vector space a N Lψ aLψ is a k1 dimensional vector space. Let e1, . . . , ek1 be a basis for aLψ . The Weyl element w is assumed to be regular. By relabeling the basis vectors if needed, the w action can be realized as the action of the k1 × k1 permutation matrix corresponding to the cycle

(1 2 . . . k1). Since w ﬁxes the subspace spanned by e1 + ... + ek1 , which is just the subspace a ⊂ a , it also acts on aGN as a (k − 1) × (k − 1) matrix. The (k − 1) × (k − 1) GN Lψ Lψ 1 1 1 1 matrix of the action of w − 1 is   −1 −1    .   1 −1 .     . .   1 .. .       ..   . −1 −1   1 −2

k −1 We claim that the determinant of this matrix is (−1) 1 k1. We do expansion along the first row, which produces two nonzero terms. The deleting the first row and the first column gives a matrix of the same form of size (k1 − 2) × (k1 − 2). By induction, the

k −2 determinant of such a matrix is (−1) 1 (k1 − 1). The contribution from the upper left

k −2 corner is (−1)(−1) 1 (k1 − 1). Now we consider the contribution from the upper right corner. Deleting the ﬁrst row and the last column gives an upper triangular matrix with every diagonal entry equal to 1. Its determinant then is 1. The contribution (−1)k1−2(−1)(1).

151 Thus the determinant is

k1−2 k1−2 k1−1 k1−1 (−1)(−1) (k1 − 1) + (−1) (−1)(1) = (−1) (k1 − 1) + (−1) (1)

k1−1 = (−1) k1

−1 −1 as desired. Consequently, for any w we have | det(w − 1) G | = k . a N 1 Lψ

ω 2.6.8 Now we can ﬁnish the computation of Idisc,ψ(−) for elliptic ψ. Associated to each −1 regular Weyl element w are k1 representations πL ∈ I (ψ) that satisfy IωwVπ . We ψ Lψ denote this set by I−1(ψ, w). Note that there are (k − 1)! regular elements in the Weyl group. Then

ω X −1 X −1 Idisc,ψ(f) = |W (L)| | det(w − 1) GN | tr (IωMP,ψ(w)RP,ψ(f)) aL {L} w∈W (L)reg

−1 X −1 = |W (L )| | det(w − 1) G | tr (I M (w)R (f)) ψ a N ω P,ψ P,ψ Lψ w∈W (Lψ)reg 1 1 X X = tr IωMP,ψ(w)RP,ψ(f)|HP (Vπ ) k1! k1 Lψ w∈W (L ) π ∈I−1(ψ,w) ψ reg Lψ 1 1 X X = rP (w, πLψ ) tr (Aψπψ(f)) k1! k1 w∈W (L ) π ∈I−1(ψ,w) ψ reg Lψ 1 1 X X = tr (Aψπψ(f)) k1! k1 w∈W (L ) π ∈I−1(ψ,w) ψ reg Lψ 1 1 = (k1 − 1)!k1 tr (πψ(f)Aψ) k1! k1 1 = tr (πψ(f)Aψ) k1

In particular, D HM HM ω Idisc,ψ(f ) = DIdisc,ψ(f) = tr (πψ(f)Aψ) . k1

Thus, to prove the stable multiplicity formula (2.5.2) whenever the term f HM (ψ) is deﬁned, it suﬃces to prove that mψ is equal to D/k1.

2.6.9 Now we will calculate mψ when ψ is elliptic. We will be using the notation from

1.4.7. Then Lψ is the group H1 = RE1/F Gm1/[E1:F ] where E1 is a subextension E ⊃ E1 ⊃ F 152 ω of degree D1 = D/k1. Consider the map β1 from 1.4.11. The map βE/E1 maps into the L ΓE/E1 -ﬁxed points of RE/F Gm1/D1 . The automorphisms of the image that are not induced L by the dual group of RE/F Gm1/D1 are induced by ΓF . The action of ΓF on RE/F Gm1/D1 factors through the action of ΓE/F . Since the image of βE/E1 lies in the ΓE/E1 -ﬁxed points, the distinct automorphisms are given by the quotient ΓE1/F of ΓE/F , which has order D/k1. ω The same remarks hold when we consider the map β1 ⊗ ν1 : Lψ → Gbm1k1n1 in 1.4.12.

2.7 Globalizing local representations

If we are to prove local theorems using the global trace formula, we need to be able to take local objects and show that they occur as the localization of a global object at some place. More precisely, we need to show that a local (square-integrable) representation can be obtained as the local factor of an automorphic representation whose other local factors have nice properties. Since the functions we deal with in this section are all functions on HM ,

H we temporarily abandon our convention of using f M to indicate functions on HM and opt instead for f or an appropriate variant. For the properties that inspire the representation constructed below, see [Art13, §6.2]. The argument we give here uses ideas and constructions from [HH95, §8].

2.7.1 Let F0 be a non-archimedean local ﬁeld of characteristic 0. For our local theorems, there is a ﬁxed cyclic extension E0 ⊃ F0. We start with the globalization of such an extension.

Lemma. Let E0 ⊃ F0 be a finite cyclic extension of non-archimedean local fields. Then there exists an extension of number fields E ⊃ F with the following properties.

• There is an inert place v0 of F such that if w0 is the place of E lying over v0 then the

extension Ew0 = Ev0 ⊃ Fv0 is isomorphic to E0 ⊃ F0. Moreover, ΓE/F ' ΓE0/F0 .

• All the archimedean places of F split in E.

This particular globalization is stated at [Hen10, §8.2], where it is not assumed that E0

153 and F0 have characteristic 0. See [Hen84, Lemme 3.6] and [Art13, Lemma 6.2.1] for some examples of such globalizations.

2.7.2 A simple version of the trace formula is used in [Kaz86] and [HH95] to deduce local results for all tempered representations. Along the way to the local theorems, they necessarily deduce global results for automorphic representations with prescribed behavior at two places. A key step in the proofs is to show that a given local representation occurs as the local component of an automorphic representation with certain properties at additional local places. We recall the simple trace formula (as used in [HH95], see speciﬁcally §§7.1-7.2 there) and use it to globalize square-integrable representations.

Let E ⊃ F be a finite cyclic extension of number fields. We will apply the simple trace formula to HM = RE/F GM,E. Recall that central character data for HM were discussed at 2.2.1. We fix a unitary character χ of ZN (AF ) that is trivial on ZN (F ) and let χv denote the local component of χ at a place v of F . Then we have the local algebra of

−1 χv -equivariant functions H(HM (Fv), χv). Let ZHM denote the center of HM and note that

ZHM = RE/F ZM,E. × The center of HM (Fv), namely (RE/F ZM,E)(Fv), is isomorphic to Ev . The central × character χv is deﬁned on ZN (Fv) ' Fv , a group we can recognize as the Fv-points of the split part of the center of HM . If v is inert, in other words, if Ev is a ﬁeld, then

× × ZN (Fv)\ZHM (Fv) ' Fv \Ev is compact.

Fix two ﬁnite places v1, v2 of F . Let f be a smooth function on HM (AF ) with the following properties.

N ∞ 1. f = v fv is a tensor product of local functions where fv ∈ Cc (HM (Fv), χv) for each place v of F .

154 0 2. For almost all places v, fv = fv :   χ (z)−1, if x = zk, z ∈ Z (F ), k ∈ H (O ), 0  v N v M Fv fv (x) =  0, otherwise.

3. fv1 is a linear combination of matrix coeﬃcients of supercuspidal representations of

HM (Fv1 ).

4. fv2 is supported on the set of regular elliptic elements of HM (Fv2 ) whose image in

HM,ad(Fv2 ) is strongly regular.

Note that properties 1 and 2 together imply that f lies in the global χ−1-equivariant

Hecke algebra H(HM (AF ), χ).

For the ﬁnal point, note that HM,ad = HM /ZHM satisﬁes

HM,ad(Fv) = HM (Fv)/ZHM (Fv),

since as discussed earlier, see 1.5.9, the relevant cohomology groups are trivial.

We have the right regular representation RHM (−) of HM (AF ) on the space

2 L (HM (F )ZN (AF )\HM (AF ), χ), see 1.3.7. A function f ∈ H(HM (AF ), χ) acts via the associated convolution operator

RHM (f): Z

RHM (f) = f(h)RHM (h) dh. ZN (AF )\HM (AF ) The measure in the integral is discussed at 2.2.1. N Suppose now that γ is a rational element in HM (F ) ⊂ HM (AF ). Suppose f = v fv is a decomposable function in H(HM (A), χ). Then we deﬁne the global orbital integral by setting Z −1 IHM (γ, f) = f(h γh) dh. HM,γ (AF )\HM (AF ) Q 1/2 Local orbital integrals were deﬁned at 1.5.4. Since the factor v |D(γ)|v introduced there

155 satisﬁes the product formula, we have

Y IHM (γ, f) = IHM,v (γ, fv). v

In particular, if f is a decomposable functions then IHM (γ, f) is nonzero if and only if

IHM,v (γ, fv) is nonzero for each v.

Lemma. If f satisﬁes the four properties above, then the image of the operator RHM (f) on the space of L2-automorphic forms is contained in the space of cuspidal automorphic forms:

2 2 RHM (f): L (HM (F )ZN (AF )\HM (AF ), χ) → Lcusp(HM (F )ZN (AF )\HM (AF ), χ)

2 Let RHM ,cusp(f) denote the restriction of the operator to the space of cuspidal L -automorphic forms. Its trace satisﬁes

X tr RHM ,cusp(f) = vol(HM,γ(E)Z(AF )\HM,γ(AE))IHM (γ, f). {γ}

The sum ranges over the set of regular elliptic conjugacy classes in ZN (F )\HM (F ).

This lemma is originally due to Deligne-Kazhdan. See for example [DKV84, A.1]. The version here is stated at [HH95, Lemma 7.2], where they discuss how it follows from the

original theorem of Deligne and Kazhdan which is a sum over ZHM (F )\HM (F ). Henniart and Herb state this formula as a sum over HM (F ), which is not consistent with [DKV84] or their proof of [HH95, Lemma 7.3].

2.7.3 Later, we will establish the local theorems for spherical characters, namely those

× characters of E0 that are trivial on OE0 , using the full trace formula, but without having to use any delicate globalization of local characters. The local theorems for spherical characters imply the local theorems for generic spherical representations, which should be a useful observation if one wishes to extend the methods in this work to arbitrary rank. We will need to be able to globalize a typical character in a speciﬁc way in order to deduce the local theorems from the spherical case. More precisely, we will want to globalize a local character so that the local theorems are known for all the other factors in the resulting

156 global character.

Let F0 be a non-archimedean local field of characteristic 0. Let E0 ⊃ F0 be a fixed cyclic extension of degree D. Let E ⊃ F be a cyclic extension of number fields degree D globalizing the local extension E0 ⊃ F0 as in Lemma 2.7.1. As in 2.7.1, we continue to write v0 for the place of F such that Ev0 ⊃ Fv0 is isomorphic to E0 ⊃ F0. We will be considering the global restriction of scalars HM = RE/F GM,E.

Let L ⊆ GN be a Levi subgroup and let πL be an irreducible square-integrable represen-

GN tation of L(F0). The representation π = IL πL is an irreducible tempered representation, and every irreducible tempered representation of GN (F0) is such an induced π. The same is then true for HM (F0) ' GM,E0 (E0). Moreover, representations corresponding to irreducible parameters in the local parameter set Ψω(N) are quotients of representations induced from an essentially discrete series representation. Thus the basic case of Theorem 1.7.1 is that of a discrete series representation π of HM (F0).

Recall [Kaz86, Theorem K]. It says that if F1 is a non-archimedean local ﬁeld of characteristic 0, G is a reductive group over F1 and G(F1) has compact center, then every square-integrable representation π of G(F1) has a pseudocoeﬃcient. Namely, there is a function fπ whose trace at a tempered representation τ of G(F1) is given by   1, if τ ' π, tr τ(fπ) =  0, otherwise.

No claim is made about the value of tr τ(fπ) for representations τ that are not tempered. We will use the following version of [Kaz86, Theorem K], stated also at [HH95, §8.3], that allows for a central character. More precisely, if G is a reductive group over F1 and

G(F1) has arbitrary center, choose a central subgroup Z ⊂ G(F1) that is co-compact in

−1 Z(G(F1)) and a unitary character χ on Z. Let H(G(F1), χ) be the corresponding χ - equivariant Hecke algebra and let π be a square-integrable representation of G(F1) whose central character restricted to Z is equal to χ. Then π has a pseudo-coeﬃcient, namely a function fπ ∈ H(G(F1), χ) such that, if τ is a tempered representation of G(F1) whose

157 central character restricted to Z is equal to χ, then   1, if τ ' π, tr τ(fπ) =  0, otherwise.

Moreover, [Kaz86, Theorem K] states that the function fπ can be chosen so that its regular non-elliptic orbital integrals all vanish, and we assume in what follows that this has been done. Suppose that all of the regular elliptic orbital integrals of fπ vanish. In other words, all of the regular orbital integrals of fπ vanish. Then all of the numbers tr τ(fπ) would be equal to 0, which is impossible. Consequently fπ has some regular elliptic orbital integral which does not vanish.

Proposition. Let π0 be a square-integrable representation of HM (F0). There is an automorphic representation π of HM (AF ) with the following properties:

• π is cuspidal.

• πv0 ' π0.

• At any place v of F which does not split completely in E, the local component πv is spherical.

N Proof. We need to construct a function f = v fv that satisﬁes the conditions of 2.7.2 and such that the geometric side of the trace formula evaluated at f is nonzero. By choosing the local components of f appropriately, the non-vanishing of the spectral side of the trace formula will provide a cuspidal representation of HM (AF ) with the desired local properties.

Denote the restriction of the central character of π0 to ZN (F ) by χ0. Then we let χ be any extension of χ0 to an automorphic character on ZN (AF ). We ﬁx this global character; the trace formula in this proof will be taken relative to χ. All local functions will need to be taken in the local equivariant Hecke algebras. We noted in 2.7.2 that when v is inert the quotient ZN (Fv)\Z(HM (Fv)) is compact. In particular, this condition holds at v0.

According to [Kaz86, Theorem K], we may take fv0 to be a pseudocoeﬃcient of π0.

Following [HH95, §§8.2-8.3], we will choose a special global element γ ∈ HM (F ). The element γ will be required to have speciﬁc local properties at a ﬁnite set of places. 158 There is a regular elliptic element γv0 ∈ HM (Fv0 ) where the orbital integral of fv0 does not vanish. Fix some such γv0 .

Let v1 and v2 be two distinct places of F that split completely in E. Choose any function fv1 (respectively fv2 ) on HM (Fv1 ) (respectively HM (Fv2 )) that satisﬁes property 3 (respectively property 4) of 2.7.2 whose orbital integrals don’t vanish on the regular set.

Let γv1 (respectively γv2 ) be a regular element of HM (Fv1 ) (respectively HM (Fv2 )) where the orbital integral of fv1 (respectively fv2 ) is nonzero. That the matrix coeﬃcient of a supercuspidal representation has nonvanishing orbital integrals follows from the expression of the orbital integral in terms of the character, see [Art87], though there must be an easier way to see this. To ﬁnd a function supported on the regular elliptic set with a nonvanishing regular orbital integral, one can take the characteristic function of the intersection of a compact open subgroup with the regular elliptic set.

Fix in addition a third split place v3 diﬀerent from v1 and v2. Choose γv3 to be a

GN -regular element of HM (Fv3 ) such that its image in GN,ad(Fv3 ) is strongly regular.

v0,v1,v2,v3 Note that χv is trivial on ZN (OFv ) for all but ﬁnitely many v. Let Sram be the union of the following sets, with v0, v1, v2 and v3 removed if necessary:

• The set of v where χv is nontrivial on ZN (OFv ).

• The set of archimedean v.

• The set of v where KHM (Fv), the maximal compact subgroup of HM (Fv), is not equal

to HM (OFv ).

v0,v1,v2,v3 v0,v1,v2,v3 Then Sram is a ﬁnite set of places. For each place v ∈ Sram , let fv be any spherical function whose orbital integral does not vanish at some regular γv. The orbital integral of a smooth function deﬁnes a smooth function on the regular set.

Thus we may choose a global element γ ∈ HM (F ) such that the orbital integral of fv at

v0,v1,v2,v3 γ is nonzero for each fv, v ∈ Sram or v = v0, v1, v2. Moreover, if γ is taken to be suﬃciently close to γv0 it will be a regular elliptic element in HM (Fv0 ) and thus a regular elliptic element in HM (F ). In addition, we choose γ to be suﬃciently close to γv3 so that its

159 image in GN,ad(Fv3 ) is strongly regular. That we can choose a global element γ with these properties is justiﬁed by weak approximation. See for example [PR94, §7.1].

v0,v1,v2,v3 It remains to choose functions for v not in Sram and not equal to v0, v1 or v2. Let us deal with the places other than v3 ﬁrst. Let Kv = KHM (Fv) be the maximal compact

v0,v1,v2,v3 subgroup. By assumption on the set Sram , we have Kv = HM (OFv ) ' GM,E(OEv ). If

γ, viewed as an element of HM (Fv), lies in the maximal compact subgroup, we may take

0 the local function fv to be fv . Thus we must deal with the case when γ does not lie in the maximal compact subgroup.

Let 1KvγKv be the characteristic function of KvγKv, which is spherical by deﬁnition. −1 Since we actually need to work with χv -equivariant functions, we must consider the corresponding equivariant characteristic function:

Z e1KvγKv (x) = χv(zv)1KvγKv (zvxv) dzv. ZN (Fv)

0 For v 6= v3, we take fv to be e1KvγKv . Note that, if γ lies in Kv, this is just fv . We need to show that its orbital integral at γ is nonzero. If hv is regular, let

0 −1/2 IH (hv, fv) = |D(hv)| IH (hv, fv).

We have that Z

IHM (γ, e1KvγKv ) = χv(zv)IHM (zvγ, 1KvγKv ). ZN (Fv)

See 2.2.1. Since D(zvhv) = D(hv) when zv is central, we get that Z I (γ, 1 ) = |D(γ)|1/2 χ (z )I0 (z γ, 1 ) dz . HM eKvγKv v v HM v KvγKv v ZN (Fv)

The discriminant factor is nonzero, so it suﬃces to show that the integral is nonzero.

Suppose z is such that I0 (z γ, 1 ) 6= 0. Then in particular, there is some v HM v KvγKv −1 hv ∈ HM (Fv) such that hv zvγhv ∈ KvγKv. Thus for some k1,v, k2,v ∈ Kv we have −1 hv zvγhv = k1,vγk2,v. This is an equality of elements in HM (Fv), which we may view as a subgroup of GN (Fv). Since the embeddings of HM (Fv) into GN (Fv) are conjugate to one

160 another, the determinant of an element of HM (Fv) is well deﬁned. We have

det(zv) det(γ) = det(k1,v) det(γ) det(k2,v) and thus

det(zv) = det(k1,v) det(k2,v).

By assumption, both det(k ) and det(k ) lie in O× . Let α be the element of F × such 1,v 2,v Fv v v that zv = αvIN . Then,

αN = det(z ) = det(k ) det(k ) ∈ O× . v v 1,v 2,v Fv

Since |α |N = 1, we also have |α | = 1. That is, α lies in O× . Since z then lies in Z (O ), v v v v v Fv v N Fv our assumptions on v imply that χ (z ) = 1 for any z that satisﬁes I0 (z γ, 1 ) 6= 0. v v v HM v KvγKv We have then

Z I (γ, 1 ) = |D(γ)|1/2 I0 (z γ, 1 ) dz . HM eKvγKv HM v KvγKv v ZN (OFv ) It suﬃces to show that

Z I0 (z γ, 1 ) = 1 (h−1z γh ) dh HM v KvγKv KvγKv v v v v HM,γ (Fv)\HM (Fv) is positive for each zv ∈ ZN (Ov). Since the characteristic function is nonnegative and

HM (Fv) ⊃ HM,γ(Fv)Kv, we have Z Z −1 −1 1KvγKv (hv zvγhv) dhv ≥ 1KvγKv (hv zvγhv) dhv HM,γ (Fv)\HM (Fv) HM,γ (Fv)\(HM,γ (Fv)Kv)

−1 For hv ∈ HM,γ(Fv)Kv and zv ∈ ZN (Ov), we have 1KvγKv (hv zvγhv) = 1. Since the measure of H (F )\(H (F )K ) is positive, the orbital integral I0 (z γ, 1 ) is positive for M,γ v M,γ v v HM v KvγKv each zv ∈ ZN (OFv ). This completes the proof that the orbital integral of e1KvγKv does not vanish at γ.

The last place to deal with is v3. Recall that γ was chosen so that its image in GN,ad(Fv3 ) is strongly regular. As at [HH95, p.175], there is a compact open neighborhood Uv3 of γ in 0 −1 0 × −1 0 0 HM,γ(Fv3 )∩GN,reg(Fv3 ) such that if γv3 ∈ Uv3 and gv3 γv3 gv3 ∈ Fv3 Uv3 then gv3 γv3 gv3 = γv3 .

161 Indeed, the centralizer of γ in the adjoint group is a maximal torus by assumption. Thus we may choose a small compact neighborhood of γ in the GN -regular set that is disjoint from its conjugates under the Weyl group of HM,γ. Since elements of a maximal torus which are conjugate to one another are conjugate by the Weyl group, we can ﬁnd such a

0 set Uv3 . We shrink Uv3 further if necessary so that if zv3 uv3 = uv3 for zv3 ∈ ZN (Fv3 ) and 0 uv3 , uv3 ∈ Uv3 then χv3 (zv3 ) = 1. Let Γv3 be a small compact open neighborhood of the identity in HM (Fv3 ). Then we follow [HH95] and deﬁne   −1 −1  χv (zv3 ), if xv3 = zv3 hv3 gv3 hv , zv3 ∈ ZN (Fv3 ), hv3 ∈ Γv3 , gv3 ∈ Uv3 , f (x ) = 3 3 v3,Uv3 v3  0, otherwise.

It is a well deﬁned function in H(HM (Fv3 ), χv3 ) which depends on Uv3 , which we may freely shrink around γ.

We need to know that the orbital integral of f is nonzero at γ. So, we consider v3,Uv3 Z f (h−1γh ) dh . v3,Uv3 v3 v3 v3 HM,γ (Fv)\HM (Fv)

−1 Suppose f(hv3 γhv3 ) is nonzero. Then

−1 0 0−1 0 hv3 γhv3 = zv3 hv3 gv3 hv3 , zv3 ∈ ZN (Fv3 ), hv3 ∈ Γv3 , gv3 ∈ Uv3 .

Then,

0 0−1 −1 γ = zv3 hv3 hv3 gv3 hv3 hv3 .

Since γ lies in Uv3 , the right side does too. Since gv3 lies in Uv3 , the ﬁrst condition on Uv3 implies that

0 0−1 −1 hv3 hv3 gv3 hv3 hv3 = gv3

and thus

γ = zv3 gv3 .

The second condition on Uv3 implies that χv3 (zv3 ) = 1. Consequently,

0 f (h−1γh ) = f(z h0 g h −1) = χ−1(z ) = 1. v3,Uv3 v3 v3 v3 v3 v3 v3 v3 v3

162 That is, h 7→ f (h−1γh ) is a characteristic function on H (F ). Suppose h h0 lies v3 v3,Uv3 v3 v3 M v v3 v3 in HM,γ(Fv)Γv3 . Then

0 0 f (h −1h−1γh h0 ) = f (h −1γh0 ) = 1. v3,Uv3 v3 v3 v3 v3 v3,Uv3 v3 v3

Since the measure of HM,γ(Fv3 )\HM,γ(Fv3 )Γv3 is positive, we have Z Z f (h−1γh ) dh ≥ f (h−1γh ) dh > 0. v3,Uv3 v3 v3 v3 v3,Uv3 v3 v3 v3 HM,γ (Fv)\HM (Fv) HM,γ (Fv)\HM,γ (Fv)Γv3 This completes the proof that the orbital integral of f at γ is nonzero. v3,Uv3 We have chosen a function f at each place of F . We set f = f ⊗ N f . The v Uv3 v3,Uv3 v6=v3 v orbital integral of f at γ is nonzero by construction. We claim that U can be chosen so Uv3 v3 that the geometric side of the trace formula at f is nonzero. We will use the following Uv3 lemma. See [Fli87, §I.3] for a proof.

Q Lemma. Let G be a be a connected, reductive group deﬁned over F . Let C = v Cv be a compact subset of G(A) with Cv = G(OFv ) for almost all v. Then there are only ﬁnitely many regular conjugacy classes in G(A) with a representative in G(F ) which intersect C.

0 We will apply the lemma to the group HM = ZN \HM . Note that, for any ﬁeld F ⊃ F , we have H (F 0) = Z (F 0)\H (F 0). Let C ⊂ H ( ) be a compact subset such that M N M Uv3 M AF Z ( )C contains the support of f . We can and will assume that the v component N AF Uv3 Uv3 of C is equal to H (O ) for almost all v. Set C = Z ( )\Z ( )C ⊂ H ( ). Uv3 M Fv Uv3 N AF N AF M AF Then C is a compact subgroup to which we may apply the lemma. Uv3

Let γ, γ1,..., γn be the ﬁnitely many rational elements in HM (F ) whose conjugacy class in H ( ) meets C . We may shrink the support of U so that γ is the only conjugacy M A Uv3 v3 class in H ( ) with a rational representative that meets C . Indeed, for 1 ≤ i ≤ n, M A Uv3 the conjugacy class in H (F ) of γ meet Z (F )U . Since we are only interested in M v3 i,v3 N v3 v3 regular conjugacy classes, the union of the classes γ1,v3 ,..., γn,v3 is closed in HM (Fv3 ). We shrink U if needed so that Z (F )U avoids these orbits. Fix this choice of U and the v3 N v3 v3 v3 associated function f with its support Z ( )C. Then the only conjugacy class in H ( ) Uv3 N A M A with a rational representative that meets C is the conjugacy class of γ. Now let γ0 be a

163 conjugacy class that contributes a nonzero term to the geometric side of the trace formula.

0 0 Then γ is a rational element in ZN (F )\HM (F ) whose conjugacy class meets C. Then γ

0 and γ are in the same HM (A)-conjugacy class. This implies that γ is in the same rational class as γ, since elements in HM (F ) ' GM (E) that are conjugate over any extension are conjugate over E.

Since the geometric side of the trace formula is nonzero at f , Lemma 2.7.2 implies Uv3 there is a cuspidal representation π on the spectral side whose trace at f is nonzero. Uv3 The function was chosen to be spherical at all non-split place, so the only question is what happens at v0. If the local representation πv0 is tempered we are done, since fv0 was chosen to be a pseudo-coeﬃcient of π0. Since the simple trace formula only involves cuspidal representations, we know that πv0 is unitary generic. It is thus a full induced representation.

But the pseudo-coeﬃcient vanishes on any induced representation, so that πv0 cannot be induced. Thus it must be discrete, in particular tempered, and consequently πv0 = π0.

2.8 Orthogonality relations

Let G be a general group over a local ﬁeld F and consider the Hecke algebra H(G) = H(G(F )).

There are two natural families of linear forms on H(G). One is the family f 7→ fG(γ) =

IG(γ, f) of regular orbital integrals. The other is the family f 7→ fG(π) = tr π(f) of characters of tempered representations. These objects were originally discussed at 1.5.4-1.5.6, as well as the ω-variants. Given any linear form Λ on H(G), one can ask to what extent Λ can be described as a linear combination of forms coming from one of these two families. It turns out that, if one restricts to a subspace of H(G), the span of the tempered characters contains those linear forms satisfying some natural finiteness properties. We will discuss the expansion of a particular linear form in this section, as well as the orthogonality relations satisfied by the coefficients in its expansion by tempered characters.

We follow the relevant parts of the discussion in [Art13, §6.5]. We assume here that F is a non-archimedean local ﬁeld of characteristic 0. As usual, E ⊃ F will be a cyclic extension of degree D.

164 2.8.1 Recall that for any group G with Levi subgroup L ⊂ G we have a map

f 7→ fL : H(G) → I(L).

If L = G, it is the map f 7→ fG. If L ( G, it is the Harish-Chandra descent map, see 1.5.15. Recall from 1.5.5 that members of I(L) are sometimes called invariant functions. A function f is called cuspidal if for every proper Levi subgroup L ( G the function fL, viewed as a function on Πtemp(L), vanishes. Similar deﬁnitions hold for equivariant functions and tempered representations with a ﬁxed central character: If Z ⊂ G(F ) is a central subgroup and χ is a character on Z, a function f ∈ H(G, χ) will be called cuspidal if fL vanishes on

Πtemp(G, χ) for every proper Levi subgroup L. Since the map f 7→ fL vanishes on the kernel of the map f 7→ fG, it also makes sense to say that fG ∈ I(G) (or I(G, χ)) is cuspidal.

The orthogonality relations in [Art13, §6.5] are described using the set Tell(G). The set

Tell(G) consists of certain triples that parametrize the set of elliptic tempered representations, see [Art13, §3.5] for the deﬁnitions and [Art93] for the full story. The triplets consist of, roughly speaking, a Levi subgroup L ⊂ G, a square-integrable representations σ of L(F ) and an element of the R-group of σ. The fact that a representation induced from a unitary representation of a Levi subgroup of GN (F ) is irreducible implies that the R-group for GN (F ) is trivial. For HM (F ) ' GM (E) then, a tempered representation whose character does not vanish on the regular elliptic set must be square-integrable. The analogous fact is true for

ω-ﬁxed representations of GN (F ). Namely, every ω-ﬁxed, tempered, elliptic representation of GN (F ) is ω-discrete. See [HH95, §5.3]. To avoid introducing more unnecessary objects, we will describe the results below only in our special case, but see [Art13, §6.5] for the general discussion.

Let fHM ∈ Icusp(HM ) be a cuspidal invariant function. To fHM , one associates the complex-valued function

τ 7→ fHM (τ):Π2(HM ) → C.

The function τ 7→ fHM (τ) has ﬁnite support on Π2(HM ). We obtain a linear map

fHM 7→ (τ 7→ fHM (τ)) 165 from Icusp(HM ) to the space of ﬁnitely supported complex-valued functions on Π2(HM ). According to Arthur [Art13, p.348], either [Kaz86, Theorem A] or the orthogonality relations of [Art93, §6] implies that this linear map is an isomorphism.

The main tool in [Art93] is the ordinary local trace formula of [Art91]. The twisted local trace formula has been established in [MW18]. At [MW18, §6.7], a variant with central characters is discussed. The discussion there combined with the observation at [Art13, p.348] that we may use functions ﬁxed by the automorphism group of the endoscopic group, which

ω in our case is ΓE/F , allows for the following minor extension. Let Π2 (HM , χ) denote the set of equivalence classes under ΓE/F with central character χ. Here, and later, we use {π} to denote the ΓE/F -conjugates of π. Then the linear map

HM HM (f )HM 7→ ({π} 7→ (f )HM ({π}))

ω ω from Icusp(HM , χ) to the space of ﬁnitely supported complex-valued functions on Π2 (HM , χ) is an isomorphism.

ω ω Let φ be an a parameter in Φ2 (HM ), or equivalently Φell(GN ). Assume moreover it has central character χ. As at [Art13, (6.5.4)], we write

HM X HM HM ω f (φ) = cφ({π})(f )HM ({π}), f ∈ Hcusp(HM , χHM ). (2.8.2) ω {π}∈Π2 (HM ,χ)

The complex numbers cφ({π}) are uniquely determined. Theorem 1.7.5 states that there is a unique representation π with cφ(π) nonzero. (Or more precisely, a unique ΓE/F -class {π} of representations.)

2.8.3 This subsection will introduce various inner products as at [Art13, §6.5] as well as discuss some results that rely on the local trace formula. As mentioned, the local trace formula has been generalized to the twisted case in [MW18]. Inner products in particular are discussed at [MW18, Ch.1 §7]. We follow the exposition of [Art13, §6.5], appropriately generalized to our twisted case. Since we have not introduced the set Tell(G) in general, we will continue working only with our special case.

Let Γell(HM ) denote the set of elliptic conjugacy classes in HM (F ). There is a canonical

166 measure dγ that is supported on the set of regular classes in Γell(HM ), see [Art93, §1]. Then

HM HM ω for f , g ∈ Hcusp(HM , χHM ), we set Z H H H H I M (f M , g M ) = f M (γ)(gHM ) (γ) dγ. HM HM Γell(HM )

By deﬁnition, the number IHM (f HM , gHM ) depends only on f HM and gHM as invariant functions, so we will also write

H H H H H Ib M f M , g M = I(f M , g M ). HM HM

ω ω The expression deﬁnes a Hermitian inner product on Hcusp(HM , χHM ) or Icusp(HM , χHM ). The local trace formula implies there is a spectral expansion

X HM HM HM HM HM I (f , g ) = m(τ)(f )HM (τ)(g )HM (τ). (2.8.4)

τ∈Π2(HM ,χ)

Here m(τ) is the number of distinct ΓE/F -conjugates of τ. See [Art93, Corollary 3.2].

We have been using the fact that for HM (F ) ' GLM,E(E) every elliptic, tempered representation is square-integrable. We mentioned at 2.8.1 that the analogous result is true for ω-stable representations of GN (F ); any tempered representation of GN (F ) that is ω-elliptic is ω-discrete. Such a result would presumably also follow from a twisted version of

[Art93]. We will denote the set of ω-discrete representations by Πω−disc(N). For f ∈ H(N, χ) = H(G , χ), we set f ω = f ω . See 1.5.4-1.5.6 for this notation. The N N GN twisted pairing on H(GN , χ) is deﬁned similarly to the ordinary case on HM : Z N,ω N,ω ω ω ω ω I (f, g) = Ib (fN , gN ) = fN (γ)gN (γ) dγ, f, g ∈ H(N, χ). Γell(N)

The set Πω−disc(N) is in bijection with the set of parameters Φω−ell(N) consisting of single

ω-orbits of irreducible representations of WF × SU(2).

ω ω We write fN (φ) = fN (πφ). Instead of writing the twisted spectral expansion as a sum over Πω−disc(N), we may write it as a sum over Φω−ell(N):

N,ω X −1 ω ω I (f, g) = |d(φ)| fN (φ)gN (φ). (2.8.5) φ∈Φω−ell(N)

167 We need to describe the number d(φ). The parameter φ is of the form

k1−1 φ = φ1 ⊕ (φ1 ⊗ ω) ⊕ · · · ⊕ (φ1 ⊗ ω ).

The number d(φ) is in general the determinant of an element of the R-group or Weyl group.

In the present context, the relevant determinant was computed already at 2.6.7 and we have

|d(φ)| = k1.

Now consider HM as an elliptic endoscopic group for (GN , ω). In general, one can deﬁne stable analogs of the inner products as at [Art13, p.351]. As in the case of global trace formulas, see in particular 2.2.4, the stable form associated to HM is just the invariant form

H I M (−, −) above. There is a local analog of the global coefficient i(GN ,HM ) defined at 2.2.2. The local coefficient is also equal to D−1. There is an equality

N,ω −1 HM HM HM I (f, g) = D Ib (f , g ), f, g ∈ Hcusp(N). (2.8.6)

Compare this formula with (2.2.6). As mentioned at [Art13, pp.351-352], the stabilization of pairings in the case of ordinary endoscopy can be found in [Art96, §2] and the twisted analogs given here are expected to follow similarly.

2.8.7 Let m(φ) be the number of distinct ΓE/F -conjugates of φ ∈ Φ(HM ). Equivalently, ω it is the order of the preimage of ξHM ◦ φ under the map φ 7→ ξHM ◦ φ : Φ(HM ) → Φ (N). See 1.2.6. The analog of [Art13, Proposition 6.5.1] is straightforward in our case.

HM HM ω Lemma. Suppose f and g lie in Hcusp(HM , χHM ). Then

H H X H I(f M , g M ) = m(φ)f M (φ)gHM (φ). (2.8.8) ω φ∈Φ2 (HM ,χ)

ω Proof. Let f be any function in Hcusp(N, χ) whose image in I (HM , χHM ) is equal to the

HM ω HM image of f in I (HM , χHM ). In other words, the functions f and f have the same regular orbital integrals. The existence of f is guaranteed by Proposition 1.5.19. Deﬁne g with respect to gHM similarly. Consider the equality (2.8.6):

IN,ω(f, g) = D−1IbHM (f HM , gHM ).

168 The twisted spectral expansion (2.8.5) implies that

X −1 ω ω −1 HM HM HM |d(φ)| fN (φ)gN (φ) = D Ib (f , g ). φ∈Φω−ell(N,χ)

ω HM ω The form fN (φ) is by deﬁnition the form f (φ), see 1.7.5 and 2.5.7. Since Φ2 (HM , χHM ) =

Φω−ell(N, χ), we can rewrite the left side to obtain

X −1 H −1 H H H |d(φ)| f M (φ)gHM (φ) = D Ib M (f M , g M ). ω φ∈Φ2 (HM ,χ) Now, we just observe that

D|d(φ)|−1 = m(φ).

This follows by identifying φ as an N-dimensional representation of LE and the description of the representation ILF φ. See A.4 for details. LE

2.8.9 We can convert the expansions (2.8.4) and (2.8.8) into orthogonality relations for the coeﬃcients cφ(π). This is the form in which the expansions will be used later.

Proposition. We have   m(φ )−1, if φ = φ , X −1  1 1 2 m(π) cφ1 (π)cφ2 (π) = ω  π∈Π2 (HM ,χ)  0, otherwise.

Proof. This follows from the equality of the expansions (2.8.4) and (2.8.8) exactly as in

[Art13, Corollary 6.5.2] by rewriting (2.8.8) according to formula (2.8.2).

169 Chapter 3 The case of M = 1

The goal of this chapter is to prove the main local and global theorems in the case M = 1 and N = D. We believe that is represents the base case of an inductive argument that proves the local and global theorems using the background and methods presented here.

The case of M = 1 is self-contained.

This chapter involves both local and global arguments. If E ⊃ F is either a local or global extension, we will generally let H = H1 = RE/F G1,E and G = GD = GD,F . As usual, × × × ω will be a character of either F or F \AF that determines the local or global extension. In either case, ω is a character of degree D. In both the local and global case, we regard H as the standard representative of the unique class of (G, ω)-elliptic endoscopic data.

3.1 Stable multiplicity when M = 1

× × In this section, the extension E ⊃ F is global and ω is a character of F \AF .

3.1.1 The stable multiplicity formula (2.5.2) was discussed in some generality in 2.5. When ψ is not elliptic, the formula amounts to a vanishing statement, see 2.5.8. For M > 1, this has to be proved by induction. As we shall see in a moment, when M = 1 there are no non-elliptic parameters. As discussed at 2.6.8 and 2.6.9, when ψ is elliptic, (2.5.2) holds whenever the form f H 7→ f H (ψ) is deﬁned.

Lemma. Suppose ψ ∈ Ψω(D). Then there is a cuspidal automorphic representation µ of

k Gm(AF ), m | D, and a positive integer k with the property that ω is the smallest power of

170 ω ﬁxing µ such that

k−1 ψ = µ (µ ⊗ ω) ··· (µ ⊗ ω )

In particular, every ω-ﬁxed parameter is elliptic.

Proof. Let ψ ∈ Ψω(N) be a parameter with standard form

k1−1 kr−1 ψ = l1(ψ1 ψ1 ⊗ ω ··· ψ1 ⊗ ω ) ··· lr(ψr ψr ⊗ ω ··· ψr ⊗ ω ).

k as in 1.4.7. In particular, ψi is ﬁxed by ω i and ki is minimal for this property. Consider a ﬁxed orbit

0 k1−1 ψ = ψ1 (ψ1 ⊗ ω) ··· (ψ1 ⊗ ω ).

0 Writing ψ1 = µ1 ν1, let m1 be the degree of µ1 and let n1 be the degree of ν1. Then ψ corresponds to a representation of Gm1n1k1 that is ﬁxed by twisting by ω. According to the

ﬁrst lemma in 1.4.4, we must have that D | m1n1k1. Thus, D ≤ m1n1k1 ≤ D. Consequently,

k1−1 ψ = ψ1 (ψ1 ⊗ ω) ··· (ψ1 ⊗ ω ).

Moreover, the parameter

k1−1 λ = µ1 (µ1 ⊗ ω) ··· (µ1 ⊗ ω )

ω is an ω-ﬁxed parameter in Ψ (m1k1). Consequently m1k1 = D and thus λ = ψ.

The exact same proof shows that a local parameter φ ∈ Φω(D) must also consist of a single ω-orbit and thus correspond to an ω-discrete representation of G(F ).

3.2 Orthogonality relations when M = 1

We discuss some reﬁnements to the orthogonality relations of 2.8 when M = 1. The local and global theorems will be proved in several stages. As we establish the local theorems in more cases, the orthogonality relations simplify and give additional results. We will prove one such simple result in this section.

The orthogonality relations will be applied at inert places of global extensions. Thus we

171 may assume, as we did in 2.8, that E ⊃ F is a cyclic extension of degree D of non-archimedean local ﬁelds of characteristic 0. We let H = RE/F G1,E and let G = GD.

3.2.1 The expansion (2.8.2) of the linear form f H 7→ f H (φ) applied to functions f H ∈

ω Hcusp(H, χH ). Since M = 1, the group H has no proper Levi subgroups and consequently ω ω Hcusp(H, χH ) = H (H, χH ). The square-integrable representations of H are just the unitary characters, a set we denote by Πunit(H) or Πunit(H, χH ) when the central character is ﬁxed. The expansion of the linear form f H (φ) simpliﬁes to

H X ω f (φ) = cφ({π})fH ({π}), f ∈ H (H, χH ). (3.2.2) ω {π}∈Πunit(H,χH )

3.2.3 Suppose that the main local theorem, Theorem 1.7.1, holds for π. In other words, there is a representation πG = AIE/F π of G(F ) and a nonzero constant c such that

tr (πG(f) ◦ AπG ) = c tr π(f), f ∈ H(G, χ).

Let φG be the parameter of G corresponding to πG. Then φG is necessarily elliptic and

H H 0 ω-fixed. Thus there is an associated linear form f 7→ f (φG) and coefficients cφG ({π }) 0 for {π } ∈ Π(H, χH ). By the uniqueness of the coefficients, we have   c, {π0}'{π}, 0  cφ({π }) =  0, otherwise.

This observation implies the following useful lemma.

Lemma. Suppose Theorem 1.7.1 holds for π. Let πG = AIE/F π be the corresponding representation of G(F ) with parameter φG. Let φ be another ω-ﬁxed parameter of G(F ) and suppose cφ({π}) 6= 0. Then φG = φ.

172 Proof. We have

X 0 −1 0 0 m({π }) cφG ({π })cφ({π }) 0 {π }∈Πunit(H,χH )

−1 = m(π) cφG ({π})cφ({π})

X 0 −1 0 0 + m({π }) cφG ({π })cφ({π }) 0 {π}6={π }∈Πunit(H,χH )

−1 = m({π}) cφG ({π})cφ({π})

6= 0.

By Proposition 2.8.9, we must have φG = φ.

3.3 Spherical characters and weak lifting

Let F0 be a non-archimedean local field of characteristic 0. As usual, let E0 ⊃ F0 be a cyclic × × extension of degree D and let ω0 be a character of F0 satisfying ker ω0 = NE0/F0 E0 . In addition we fix a cyclic extension of number fields E ⊃ F globalizing E0 ⊃ F0 as in Lemma

2.7.1. We let v0 be the place of F such that Ev0 ⊃ Fv0 ' E0 ⊃ F0. Let H = RE/F G1,E and × let G = GD,F . Any basis of E over F induces an isomorphism H(F ) ' E . For any place v

× × × of F , the basis also induces an isomorphism H(Fv) ' Ev . Let ω be a character of F \AF × × × satisfying ker ω = NE/F E \AE. Viewing ω as a character of AF , we have a factorization Q ω = v ωv. In particular, we have ωv0 = ω0. In this section, we will construct the weak lift of a certain class of automorphic characters

× of H(AF ). The computation is related to something we have seen before, and is essentially formal. However, when combined with the global trace formula, the existence and form of the weak lift of the trivial character will imply Theorem 1.7.1 is true for spherical characters,

× namely those characters of E0 ' H(F0) that are trivial on the maximal compact subgroup. Note that, in general, the maximal compact subgroup of H(F ) must be isomorphic to O× v Ev

but need not be equal to H(OFv ). Indeed, this latter group is not even always deﬁned.

173 × 3.3.1 Let χF0 be a character of F0 . Then

× × χE0 = χF0 ◦ NE0/F0 : E0 → C deﬁnes a character of E× that satisﬁes χσ0 = χ for any σ ∈ Γ . The map 0 E0 E0 0 E0/F0

× × χF0 7→ χF0 ◦ NE0/F0 : Π(F0 ) → Π(E0 )

× is just the GL(1) case of local base change. Any character χE0 of E0 that is fixed by ΓE0/F0 × can be written in the form χE0 = χF0 ◦ NE0/F0 for some character χF0 of F0 . × A character χE0 of E0 defines a character of H(Fv0 ) by composition with any isomorphism × E0 ' H(F0) determined by a basis of E0 over F0. If χE0 is fixed by ΓE0/F0 , then the resulting character of H(Fv0 ) is independent of the choice of isomorphism. In this case, we will also write χE0 for the resulting character of H(Fv0 ).

In particular, let | − |E0 be the absolute value on E0 that extends | − |F0 , the absolute value on F . Then | − | is Γ -ﬁxed and consequently any spherical character | − |z , 0 E0 E0/F0 E0 z ∈ C, is ΓE0/F0 -ﬁxed. The following proposition appears as [HH95, Proposition 4.5]. Recall that the automorphic induction lifting was discussed at 1.7.1. We state the lemma for arbitrary M and

N = MD, though we will only apply it when M = 1.

Lemma. Let π0 be a representation of GM (E0) and suppose π0,N is the representation × automorphically induced from π0. Let χF0 be a character of F0 viewed also as a character of GN (F0). Similarly, regard χF0 ◦ NE0/F0 as a character of GM (E0). Then χF0 ⊗ πF0 is the representation automorphically induced from (χF0 ◦ NE0/F0 ) ⊗ πE0 .

Applying the lemma when M = 1 reduces the study of automorphic induction of ΓE0/F0 - × ﬁxed characters of E0 to understanding automorphic induction for the trivial character.

× 3.3.2 Fix an F -basis of E and thus an isomorphism H(AF ) ' AE. Let πH be a character of × H(AF ) and let πE be the corresponding character of AE according to the ﬁxed isomorphism.

Let σ be a generator of ΓE/F . The characters πH and πE have the same stabilizer in ΓE/F .

174 Suppose that the stabilizer of πE in ΓE/F is nontrivial. In other words, there is a subextension F ⊂ F1 ( E such that ΓE/F1 ⊂ ΓE/F ﬁxes πE. We’ve discussed the local GL(1) base change map given by the local norm. The same remarks hold globally. More precisely, since π is ﬁxed by Γ there is an automorphic character π of × such that E E/F1 F1 AF1

πE = πF1 ◦ NE/F1 . × The character ω satisﬁes ker ω = NE/F AE. Suppose the extension F1 ⊃ F is determined by ωk. That is, ker ωk = N × . Since [F : F ] is equal to the order of ωk, we have F1/F AF1 1 D/k [F1 : F ] = D/k. The Galois group ΓE/F1 is cyclic of order k and we may take σ as a

F1/F generator. Let AIw denote the weak automorphic lifting from F1 to F discussed at 2.4.5. The lifting takes a character of × to an ωk-ﬁxed automorphic representation of G ( ). AF1 D/k AF

F1/F Consider the automorphic representation πF = AIw πF1 . The representation

k−1 πF ⊗ (πF ⊗ ω) ⊗ · · · ⊗ (πF ⊗ ω )

is an automorphic representation of a standard Levi subgroup LD/k ⊂ G. The induced representation

k−1 πF (πE) = ILD/k πF ⊗ (πF ⊗ ω) ⊗ · · · ⊗ (πF ⊗ ω ) is an automorphic representation of G(AF ). The corresponding formal parameter is elliptic and πF (πE) contributes a nonzero term to the discrete part of the ω-twisted trace formula for G, as discussed in 2.6. The representation πF (πE) does not depend on the choice of F -basis of E.

Suppose now that πG is an ω-ﬁxed representation of G(AF ) that is not cuspidal. It is then an induced representation

k−1 ILD/k µF ⊗ (µF ⊗ ω) ⊗ · · · ⊗ (µ ⊗ ω ).

k k Here µ is a cuspidal representation of GD/k(AF ) ﬁxed by ω . As before, ω determines an

F1/F extension F1 and a character πF1 that satisﬁes µF = AIw πF1 . By composing πF1 with × the norm NE/F1 , we obtain a character of AE, πE, which satisﬁes πF (πE) = πG.

175 Proposition. We have

E/F πF (πH ) = AIw πH .

Proof. As in 1.3.13, we let c(−) denote the family of conjugacy classes associated to an automorphic representation. Let ξH denote the map realizing H as a twisted endoscopic group for G as in 1.2.6. Then the class of the right side is ξH (c(πH )). (This last piece of × notation is also explained at 1.3.13.) Let πE be the character of AE corresponding to πH . We have the character π of × satisfying N π . Then π deﬁnes a character π F1 AF1 E/F1 F1 F1 H,1 L L of H1 = RF1/F G1,F1 . There is the L-homomorphism from H1 to H for the base change lifting, discussed at 1.4.10. In the notation of 1.4.10, this L-homomrphism is βE/F1 . We have that βE/F1 (c(πH,1)) = c(πH ). Consequently, we have

E/F c(AIw πH ) = (ξH ◦ βE/F1 )(πH,1).

ω The map ξH ◦ βE/F1 was denoted by β1 at 1.4.11. ω Now, consider the representation πF (πH ). The L-homomorphism ξ1 was deﬁned at 1.4.11 and we have

ω c(πF (πH )) = ξ1 (c(πH,1)).

ω ω We proved at 1.4.11 that β1 and ξ1 are conjugate to one another. This gives

E/F c(πF (πH )) = c(AIw πH ).

E/F Since πF (πH ) and AIw πH are isobaric representations, Theorem 1.3.14 (strong multiplic- E/F ity one) implies that πF (πH ) = AIw πH , as desired.

3.3.3 We will use a special case of the previous proposition as well as earlier global theorems, speciﬁcally (2.2.6) and the computations of 2.5 and 2.6, to deduce some cases of automorphic induction.

Consider the trivial representation 1H of H(AF ). Let 1F be the trivial representation of × AF . Let T be the diagonal torus in G. According to the previous proposition, we have

D−1 AIw1H = πF (1H ) = IT 1F ⊗ ω ⊗ · · · ⊗ ω .

176 D(D−1)/2 The central character of πF (1H ) is ω .

Lemma. Let f ∈ H(D, ωD(D−1)/2) be a function with associated matching function f H ∈

ω H (H, 1E). Then H tr 1H (f ) = tr πF (1H )(f) ◦ AπF (1H ) .

ω Proof. Let φG = φπF (1H ) ∈ Ψ (D) be the parameter

D−1 1F ω ··· ω .

representing π (1 ). Associated to φ is the linear form ID,ω (−). There are equalities F H G disc,φG

IH (f H ) = DID,ω (f) = tr π (1 )(f) ◦ A . disc,φG disc,φG F H πF (1H )

For the ﬁrst equality, see (2.2.6). For the second, see 2.6.8. Consider the linear form

IH (−). It is a sum of linear forms attached to characters that weakly lift to π (1 ). disc,φG F H More precisely, X IH (f H ) = tr π (f H ). disc,φG H AIwπH =πF (1H )

According to Proposition 3.3.2, the trivial character 1H is in the sum. Let πH be another character that occurs in the sum. Then at all but ﬁnitely places, the following condition holds: πH,v is conjugate to 1H,v by an element in AutGv (Hv), the group of automorphisms of

Hv as an ωv datum for Gv. This implies that at almost every place v, the character πH,v is × the trivial character. By applying strong multiplicity one, Theorem 1.3.14, on H(AF ) ' AE, we see that πH is equal to 1H . Strong multiplicity one is overkill here, and the fact that an × automorphic character of AE that is trivial almost everywhere is trivial follows from the following observation. Let πE be the automorphic character that is trivial almost everywhere and let S be the ﬁnite set of places where we do not know that πE,w is trivial. Then Q × × Q × πE,S = w∈S πE,w is a character that is trivial on E . But E is dense in w∈S Ew , so the set S must be empty. In any case, the sum above only contains 1H , and the lemma follows.

177 3.3.4 Consider the ﬁxed extension E0 ⊃ F0 of non-archimedean local ﬁelds. The following case of the local lifting will be used to deduce Theorem 1.7.1 for arbitrary characters.

Proposition. Let π0 be a spherical character of H(F0). Then the ω0-ﬁxed representation

πG,0 = AIE0/F0 π0 exists. In other words, for every function f ∈ H(G(F0), χ0) with matching H function f ∈ H(H(F0), χH,0), there is a nonzero number c such that

H tr πG,0(f) ◦ AπG,0 = c tr π0(f ).

Moreover, if we write π0 = πF0 ◦ NE0/F0 , then

D−1 πG,0 = IT0 πF0 ⊗ πF0 ω0 ⊗ · · · ⊗ πF0 ω0 .

The constant c satisﬁes cc = 1.

Proof. By Lemma 3.3.1, it suﬃces to prove the proposition when π0 = 1H0 , the trivial character. Lemma 3.3.3 gives everything except for the condition on c.

Let φG,0 be the parameter corresponding to πG,0. According to the uniqueness of the

0 0 coeﬃcients in the expansion (3.2.2), we have cφG,0 (π0) = c and cφG,0 (π0) = 0 for π0 6= π0. The −1 orthogonality relations of Proposition 2.8.9 give that cφG,0 (π0)cφG,0 (π0) = m(π0)m(φG,0) .

The number m(π0) is the number of distinct ΓE0/F0 -conjugates of π0, which in this case is 1. Thus we must prove that m(φG,0), the number of parameters for H0 that map to

L L L φG,0 under the homomorphism ξH0 : H0 → G0, is equal to 1. Parameters for H0 can be identiﬁed with characters of WE0 . Then, m(φG,0) is the number of characters of W that, when induced to W , are equal to φ . The trivial character of W , 1 , E0 F0 G,0 E0 WE0

WF0 satisﬁes I 1WE = φG,0. The ﬁber of induction is the ΓE0/F0 -conjugates of the inducing WE0 0 representation, the number of which is also then 1. For some of these facts about parameters and induction, see A.4.

3.4 Main theorems

In the previous sections, culminating in Proposition 3.3.4, we deduced that if π0 is a spherical character of H(F0) then the representation AIE0/F0 π0 exists. The next step is to extend the 178 existence of the local lifting from spherical characters to arbitrary characters. Once we know the main local theorem, Theorem 1.7.1, the main global theorem, Theorem 1.7.8, will follow quickly.

3.4.1 First, we need to slightly reformulate our setup from the previous sections. Let D be a positive integer. We assume Theorem 1.7.1 for extensions of degree a proper divisor of D.

Moreover, we assume that the nonzero number c that appears in Theorem 1.7.1 satisﬁes cc = 1. The base case of the induction is when D = p is prime. In this case, we are not assuming anything.

Let us see what the induction assumption tells us. Let E ⊃ F be an extension of number

ﬁelds of degree D. Suppose v is a place of F which is not inert in E. Recall the discussion from 1.7.6. The group Hv is a product,

Y Hv ' REw/Fv G1,Ew , w|v over the places w of E dividing v. By assumption, Theorem 1.7.1 holds for each factor of the product since Ew ⊃ Fv is an extension of degree dividing D. Let

O πv = πw w|v be a character of Hv(Fv). The representation

O πL = AIEw/Fv πw w|v is a representation of a Levi subgroup L(Fv) of G(Fv) that is ﬁxed by ωv. The representation

ILπL of G(Fv) is a representation of G(Fv) ﬁxed by ωv. It is the representation we will denote by πG,v = AIEv/Fv πv. Let cw be the number occurring in Theorem 1.7.1 for πw and Q AIEw/Fv πw. Let c = w|v cw. Then

H tr πG,v(fv) ◦ AπG,v = c tr πv(fv ), fv ∈ H(G(Fv), χv).

Thus, c satisﬁes cc = 1. Combining this with Proposition 3.3.4, we have a local lifting

179 AIEv/Fv πv whenever v is a place which is not inert and whenever πv is spherical.

Let us just say a quick word about the fibers of AIEv/Fv . As discussed, automorphic induction in general is a combination of automorphic induction for a cyclic field extension and parabolic induction. The fibers of both operations are known, see Theorem 1.7.1 for the fibers of automorphic induction for the field extension, which we are assuming. The

ﬁbers are the equivalence classes under the usual local automorphism group AutGv (Hv) of

Hv Hv as an ωv-datum. The image of the transfer map fv 7→ fv lies in the space of functions which are invariant under this automorphism group and in fact maps onto this subspace, see

1.5.19.

3.4.2 We will use the orthogonality relations, specifically the refinement of Lemma 3.2.3, in conjunction with Proposition 3.3.4, to deduce more about the weak lift of the automorphic representation constructed in Proposition 2.7.3. The automorphic representation constructed in Proposition 2.7.3 was specifically designed so that, outside of the fixed place v0, the local theorems can be proved by hand. Indeed, if v =6 v0 is not inert, we can use the induction hypothesis and the remarks in 3.4.1. At every inert place v 6= v0, the globalization was chosen to be spherical and we can use Proposition 3.3.4.

Proposition. Let π0 be a character of H(F0). Let πH,0 be the automorphic character of

H(AF ) constructed in Proposition 2.7.3. Let πG = AIwπH,0 be the weak lift of πH,0 to

G(AF ). Let v 6= v0. Then πG,v = AIEv/Fv πH,0,v.

Note that, by Proposition 3.3.4 and the discussion in 3.4.1, for v 6= v0 the representation

AIEv/Fv πH,0,v does indeed exist.

Proof. Let φπG be the parameter associated to πG. Recall Corollary 2.3.12, and the reﬁnement H ω of 2.6.8, which states that if f ∈ H(G, χ) and f ∈ H (H, χH ) have matching orbital integrals then

H H I (f ) = c tr(πG(f) ◦ Aπ ). disc,φπG G

(We ignore the precise value of the positive number c on the right side, since we will be collecting various numbers along the way and for now the only thing that matters is when 180 they are nonzero.) The left hand side is a sum of characters πH that weakly lift to πG. The

H ω H ω function f lies in H (H, χH ). In other words, the local function fv lies in H (H(Fv), χH,v), the subalgebra of functions whose orbital integrals and character values are invariant under the local automorphism group AutGv (Hv) of Hv as an ωv-datum for Gv. Thus, the value of H tr πH (f ) depends only on {πH }, the class of representations which are everywhere locally conjugate to πH via AutGv (Hv). We can write

H X H c{πH,0} tr πH,0(f )+ c{πH } tr πH (f ) = tr(πG(f)◦AπG ), f ∈ H(G, χ). (3.4.3) AIw{πH }=πG {πH }6={πH,0}

H Since {πH } represents equivalence classes, the linear forms in f appearing on the left side are distinct. The numbers c{πH,0} and c{πH } are nonnegative numbers. The sum is ﬁnite for H H any function f depending only on the KH -type of f . See the discussion in 2.1.6. Fix a place v. Then we may rewrite the left side of (3.4.3) as   X v H,v H X H  c{πH } tr πH (f ) tr πH,0,v(fv ) + tr πH (f ). {{πH } : {πH,v}={πH,0,v}} {{πH } : {πH,v}6={πH,0,v}}

As in the proof of Corollary 2.4.3, we can ﬁnd a function f H,v so that the ﬁrst sum is nonzero.

Fix this function. With the function ﬁxed away from v, we may rewrite (3.4.3) as

H X H c{πH,0,v} tr πH,0,v(fv ) + c{πH,v} tr πH,v(fv ) = cG,v tr πG,v(fv) ◦ AπG,v . {πH,v}6={πH,0,v} (3.4.4)

The coeﬃcients are the values of linear forms away from v evaluated at the ﬁxed function

H,v H,v f . In particular, the function f was chosen so that c{πH,0,v}, which is the value of the

ﬁrst sum in (3.4.4), is nonzero. The equality holds for any function fv ∈ H(G(Fv), χv). Now

H we regard the left hand side of (3.4.3) as a linear form in fv . The left hand side is a ﬁnite H H sum for any fv in a way that depends only on the Kv-type of fv . Since c{πH,0,v} 6= 0 and

ωv the classes of representations on the left are inequivalent for H (H, χv), linear independence

H of characters implies that the left side is a nonzero linear form in fv . Consequently, the right hand side of (3.4.3) is a nonzero linear form on H(G(Fv), χv). That is, the number

181 v v c = tr π (f ) ◦ A v is nonzero. G,v G πG

Now, assume v is a non-inert place and consider the resulting identity (3.4.4). Let {πH,v} be an AutGv (Hv)-orbit on the left side. Then AIEv/Fv {πH,v} exists. The ﬁbers of the map

AIEv/Fv are precisely the classes {πH,v}. Thus we have

X c0 tr AI π (f )+ c0 tr AI π (f ) {πH,0,v} Ev/Fv H,0,v v {πH,v} Ev/Fv H,v v {πH,v}6={πH,0,v}

= cG,v tr(πG,v(fv) ◦ AπG,v ), fv ∈ H(G(Fv), χv).

The prime is meant to indicate that these are new coeﬃcients that incorporate the nonzero number c that appears in Theorem 1.7.1. This is an equality of traces of distinct, irreducible representations of G(Fv) that is ﬁnite for any choice of fv depending only on the Kv-type

0 of fv. The numbers c and cπ are nonzero. By linear independence of characters, {πH,0,v} G,v c{πH,v} must vanish for {πH,v}= 6 {πH,0,v}. Moreover, we must have AIEv/Fv πH,0,v = πG,v. Assume now that v is inert. We recognize the right hand side of (3.4.4) as the type of expansion at 3.2.2. Due to the uniqueness in the orthogonality relation, we have

−1 0 cπG,v (τv) = cG,vcτv , τv = {πH,0,v}, {πH,v}.

The representation πH,0,v is spherical. By Proposition 3.3.4, the representation AIEv/Fv πH,0,v exists. Since cπG,v ({πH,v,0}) is nonzero, Lemma 3.2.3 implies that πG,v = AIEv/Fv πv.

3.4.5 We continue with a ﬁxed character π0 of H(F0) and πH,0 the character of H(AF ) globalizing π0 = πH,0,v0 constructed in Proposition 2.7.3. We let φG be the parameter representing the weak lift πG = AIwπH,0 of πH,0. Proposition 3.4.2 implies that

AIEv/Fv (πH,0,v) = πG,v, v 6= v0.

0 Recall the discussion of 2.5.7. For a general parameter φG, we cannot yet deﬁne the form H H 0 f 7→ f (φG). However, for the parameter φG, the only place where we do not know the

Hv0 Hv0 local behavior of πG,v is at the inert place v = v0, where the local form fv0 7→ fv0 (φG,v0 ) H H is always deﬁned. Thus, we may consider the form f 7→ f (φG). Moreover, the discussion

182 at 2.6.8 and 2.6.9 tells us that the stable multiplicity formula (2.5.2) holds for φG. This will be our starting point for the next section, where we will relate the local coeﬃcients in the character expansion 3.2.2 to global multiplicities. This is the key step to proving the

existence of AIE0/F0 π0 for an arbitrary π0. H We continue to write Idisc(−) for the trace formula for H, even though there is only a discrete part in this case, if only to indicate how the argument may generalize to the case of M > 1 and so the reader may compare our work here with [Art13, §§6.6-6.7]. Let

Πunit(H, χH ) denote the unitary characters of H(AF ) with central character χH . We may write X IH (f H ) = tr RH (f H ) = n (π ) tr π (f H ). disc,φG disc,φG φG H H πH ∈Πunit(H,χH )

The function nφG (−) is an indicator function on Πunit(H, χH ):   1, if πH is automorphic and AIwπH = πG, nφG (πH ) =  0, otherwise.

Right now, f H is not being regarded as the global transfer of a function f. It just indicates

ω an arbitrary function lying in the symmetric Hecke algebra H (H(AF ), χH ).

We continue to let {πH } denote the class of representations which are everywhere locally equivalent to πH via AutGv (Hv). Consider the modiﬁed coeﬃcient

X n ({π }) = m−1 n (π0 ). φG H φG φG H 0 πH ∈{πH }

The positive integer mφ was defined at 2.5.3, where is was generally denoted mψ. The coefficient is well-defined, in the sense that the sum on the right is finite. Indeed, suppose

0 0 there are infinitely many representations πH in the class {πH } with nφG (πH ) nonzero. Fix a function f H such that tr π (f H ) is nonzero. Then tr π0 (f H ) is nonzero for all π0 in πH H πH H πH H {π }, which is impossible since IH (f H ) is a finite sum for any f H . H disc,φG H H The linear form f 7→ f (φG) appearing in the stable multiplicity formula (2.5.2) is well-defined and equal to tr(πG(f) ◦ AπG ), see 2.5.7. We may rewrite the left hand side

183 SH (−) = IH (−) of (2.5.2) using the modiﬁed coeﬃcient n ({π }) to obtain disc,φG disc,φG φG H

X H H ω nφG ({πH }) tr πH (f ) = f (φG) = tr (πG(f) ◦ AπG ) , f ∈ H (G(A), χ). ω {πH }∈Πunit(H,χH ) (3.4.6)

ω We are using Π (H, χH ) to indicate the classes for the relation of being everywhere locally conjugate.

One of the classes of representations for which nφG (−) does not vanish is the class

{πH,0} of our ﬁxed representation πH,0. Moreover, the previous proposition shows that

πG,v = AIEv/Fv πH,0,v for v 6= v0. Let Av = AπG,v be the normalized intertwining operator N between πG,v and πG,v ⊗ ωv as in 1.6.2. They satisfy v Av = AπG . We continue to use

{πH,v} to indicate the local classes under the conjugacy relation. We have

H H tr(πG,v(fv) ◦ Av) = fv (φG,v) = cv tr{πH,0,v}(fv ), v 6= v0.

Moreover, cv satisﬁes cvcv = 1. See Proposition 3.3.4 when v is inert and the discussion in 3.4.1 for non-inert v. Then we have

v0 v0 v0 H,v0 v0 v0 v0 H,v0 tr πG (f ) ◦ A = f (φG ) = c tr{πH,0}(f ), v 6= v0.

At v0, we have the expansion

X tr(π (f ) ◦ A ) = f H (φ ) = c ({π }) tr{π }(f H ). G,v0 v0 v0 v0 G,v0 φG,v0 v0 v0 v0 ωv0 {πv0 }∈Πunit(Hv0 ,χH,v0 )

H The equality holds for fv0 ∈ H(G(Fv0 ), χv0 ) with matching function fv0 . H ω For f ∈ H (H(AF ), χH ), we rewrite the right side of (3.4.6) as

H H,v0 v0 H f (φG) = f (φG )fv0 (φG,v0 ) X = cv0 · c ({π }) · tr{π }(f H ) · tr{πv0 }(f H,v0 ). (3.4.7) φG,v0 v0 v0 v0 H,0 ωv0 {πv0 }∈Πunit(Hv0 ,χH,v0 )

184 Now, consider the left side of (3.4.6). We may write it as

X X v0 H v0 H,v0 nφG ({πv0 } ⊗ {π }) · tr{πv0 }(fv0 ) · tr{π }(f ). v v ωv {πv0 }∈Πω 0 (Hv0 ,χ 0 ) 0 unit H {πv0 }∈Πunit(Hv0 ,χH,v0 ) (3.4.8)

Both (3.4.7) and (3.4.8) are linear forms in f H,v0 , with (3.4.8) being a ﬁnite sum for any f H,v0 . By linear independence of characters, the only character in (3.4.8) with a nonzero

v0 v0 coeﬃcient is {π } = {πH,0}. Equating the coeﬃcients of these linear forms gives

X = cv0 · c ({π }) · tr{π }(f H ) φG,v0 v0 v0 v0 ωv0 {πv0 }∈Πunit(Hv0 ,χH,v0 )

X v0 H = nφG ({πv0 } ⊗ {πH,0}) tr{πv0 }(fv0 ). ωv0 {πv0 }∈Πunit(Hv0 ,χH,v0 )

H Both sums are linear forms in fv0 . The top sum is ﬁnite due to the coeﬃcients cφG,v0 (−).

The bottom sum is ﬁnite since the coeﬃcients nφG (−) come from the global trace formula. By linear independence of characters again, we obtain

v0 v0 ωv0 nφG ({πv0 } ⊗ {πH,0}) = c cφG,v0 ({πv0 }), {πv0 } ∈ Πunit(Hv0 , χH,v0 ). (3.4.9)

This gives a relation between a global coeﬃcient on the left side and a local coeﬃcient on the right side.

ωv0 v0 Lemma. For any {πv0 } ∈ Πunit(Hv0 , χH,v0 ), the number nφG ({πv0 } ⊗ {πH,0}) is an integer.

−1 v0 Also, m(φG,v0 ) · m({πv0 }) · nφG ({πv0 } ⊗ {πH,0}) is an integer.

Proof. Recall the original deﬁnition:

X n ({π ⊗ πv0 }) = m−1 n (π ). φG v0 H,0 φG φG H v0 πH ∈{πv0 ⊗πH,0}

To show this number is an integer, it suﬃces to show that if nφG (πH ) 6= 0, then the number of distinct Γ -conjugates of π is divisible by m−1. The integer m is a divisor of D. E/F H φG φG

If the number nφG (πH ) happens to be equal to D, then there is nothing to prove. We assume then that nφG (πH ) 6= 0 is less than D. In other words, the representation πH has a nontrivial stabilizer in ΓE/F . We showed in 3.3.2 that if πE is ﬁxed by a group of order k, 185 then the paramater corresponding to the weak lift of πE has multiplicity D/k. But this is the parameter mφG since we are assuming nφG (πH ) is nonzero. Thus mφG = D/k. Since

v0 the orbit of πE has D/k elements, this proves nφG ({πv0 ⊗ πH,0}) is an integer.

For the second claim, we start by proving that m(φG,v0 ) = mφG . We may write

k−1 φG = µ (µ ⊗ ω) ··· (µ ⊗ ω ).

k Here µ is a cuspidal representation of GD/k(AF ) that is ﬁxed by ω . Let φv0 be the parameter of µv0 . Then k−1 φG,v0 = φv0 ⊕ (φv0 ⊗ ω0) ⊕ · · · ⊕ (φv0 ⊗ ω0 ).

Since v0 is inert, the character ω0 has degree D. The parameter φG,v0 is ﬁxed by ω and is therefore induced from WE0 . The integer m(φG,v0 ) is the order of the ﬁber of induction. This is equal to D/k, by the exact same argument we used in 3.3.2. See also the discussion

v0 about parameters at A.4. Thus, using the deﬁnition of nφG ({πv0 } ⊗ {πH,0}), we have

−1 v0 −1 X m(φG,v0 ) · m({πv0 }) · nφG ({πv0 } ⊗ {πH,0}) = m({πv0 }) nφG (πH ). v0 πH ∈{πv0 ⊗πH,0}

Recall that m({πv0 }) is the number of distinct ΓE0/F0 ' ΓE/F conjugates of πv0 . Under this isomorphism, the stabilizer of πH is contained in the stabilizer of πv0 for each πH .

Consequently, m({πv0 }) divides nφG (πH ) for each πH .

3.4.10 We retain all notation from the previous subsection.

Proposition. Let π0 be a representation of H(F0). Then the representation AIE0/F0 π0 exists.

In other words, there is a nonzero number c and an ω0-ﬁxed representation πG,0 = AIE0/F0 π0 of G(F0) such that H tr(πG,0(f0) ◦ AπG,0 ) = c tr π0(f0 )

H for any pair of functions f0 7→ f0 , f0 ∈ H(G(F0), χ0), with matching orbital integrals. Moreover, c satisﬁes cc = 1.

Proof. The proof relies on the previous lemma and the orthogonality relations of Proposition

186 2.8.9. On the one hand, the sum

X −1 v0 v0 m(φG,v0 ) · m({πv0 }) · nφG ({πv0 } ⊗ {πH,0}) · nφG ({πv0 } ⊗ {πH,0}) ωv0 {πv0 }∈Πunit(Hv0 ,χH,v0 ) is a sum of positive integers. On the other hand, using the orthogonality relations, equation

(3.4.9) and the fact that cv0 cv0 = 1, we have

X −1 v0 v0 m(φG,v0 ) · m({πv0 }) · nφG ({πv0 } ⊗ {πH,0}) · nφG ({πv0 } ⊗ {πH,0}) ωv0 {πv0 }∈Πunit(Hv0 ,χH,v0 )

X −1 v0 v0 = m(φG,v0 ) · m({πv0 }) · c cφG,v0 ({πv0 }) · c cφG,v0 ({πv0 }) ωv0 {πv0 }∈Πunit(Hv0 ,χH,v0 )

X −1 = m(φG,v0 ) · m({πv0 }) · cφG,v0 ({πv0 }) · cφG,v0 ({πv0 }) ωv0 {πv0 }∈Πunit(Hv0 ,χH,v0 ) = 1.

v0 Thus, there is a unique class {πv0 } such that nφG ({πv0 } ⊗ {πH,0}) is nonzero. By def-

v0 inition, the class {πH,0} = {π0} ⊗ {πH,0} has nφG ({πH,0}) 6= 0, and it is the only one. 0 0 According to (3.4.9), the number cφG,v0 ({πv0 }) is nonzero only when {πv0 } = {π0}. Thus there is only one nonzero term in the expansion (3.2.2). By deﬁnition, we have πG,v0 =

AIE0/F0 π0. Moreover, since nφG ({πH,0}) = 1, equation (3.4.9) implies that cφG,v0 ({π0}) satisﬁes cφG,v0 ({π0})cφG,v0 ({π0}) = 1.

3.4.11 Finally, we prove the main theorems. This will also resolve the induction hypothesis from the beginning of the section.

Proof of Theorem 1.7.1. We have already shown that every unitary representation π0 of

H(F0) has a lift. All that is left to do is verify the additional properties in Theorem 1.7.1.

Let πG,0 = AIE0/F0 π0 be the representation we showed to exist in Proposition 3.4.10. The representation π0 is a representation of G(F0) = GD(F0) ﬁxed by ω0, a character of degree D.

The representation π0 is supposed to be tempered. In fact, every ω0-ﬁxed representation of G(F0) is tempered, so there is nothing special to prove here. Indeed, let φ0 be the

187 parameter of π0. The same proof as in Lemma 3.1.1 shows that

k−1 φ0 = µ0 ⊕ (µ0 ⊗ ω0) ⊕ · · · ⊕ (µ0 ⊗ ω0 )

for µ0 and irreducible representation of WF0 . In particular, every such π0 is ω0-discrete and π0 is induced from supercuspidal representations. It is also unitary and generic. The classiﬁcation of such representations implies that πµ0 , the supercuspidal representation associated to µ0, must be unitary and thus π0 is tempered.

Next, we need to prove that the lifting does not depend on the choice of generator ω0 for the group of characters of ΓE0/F0 . Let us recall brieﬂy how πG,0 was constructed. Given

π0, we constructed a globalization πH,0 in Proposition 2.7.3 that had nothing to do with ω0.

Then, πG,0 was constructed as the local component at v0 of AIwπH,0. Since weak lifting is deﬁned in terms of unramiﬁed conjugacy classes, it is independent of any choices. Thus πG,0 is independent of ω0.

Now we need to show that every ω0-ﬁxed representation πG,0 is of the form AIE0/F0 π0. Let

φ0 be the parameter of πG,0 and consider the expansion (3.2.2). There is some representation

0 π0 of H(F0) such that the coeﬃcient cφ0 (π0) 6= 0. Let φ0 be the parameter associated to 0 AI π . Then c 0 (π ) 6= 0. According to Lemma 3.2.3, we must have φ = φ and thus E0/F0 0 φ0 0 0 0

πG,0 = AIE0/F0 π0. 0 0 Suppose π0 and π0 satisfy AIE0/F0 π0 = AIE0/F0 π0. The construction of AIE0/F0 π0 shows that the class of π0 is the unique class in the expansion (3.2.2). Since the same is true of

0 π0, they must belong to the same class, which were deﬁned to be the conjugates under

ΓE0/F0 .

3.4.12 The only thing left to do is prove the global theorems.

Proof of Theorem 1.7.8. Let πH be an automorphic character of H(AF ). Recall that we set

O 0 AIE/F πH = AIEv/Fv πH,v, v and the first claim is that AIE/F πH is automorphic. By definition, the representation AIwπH is automorphic. It suffices to show that, for every place v, we have (AIwπH )v = AIE/F πH,v. 188 We proved exactly this in Proposition 3.4.2 when v was non-inert and we assumed Theorem

1.7.1 by induction. Since we now have Theorem 1.7.1 for any place v, the same proof implies that AIE/F πH = AIwπH and in particular AIE/F πH is automorphic. H H Next, we discuss the fibers of AIE/F . The linear form f 7→ f (φG) is now defined for any φG, see the discussion in 2.5.7. By the discussion at 2.6.8 and 2.6.9, formula (2.5.2) holds for any φG. Fix φG and let πG be the corresponding representation of G(AF ). As usual, IH (−) is the sum of characters that weakly lift to π . According to the previous disc,φG G paragraph, there is a unique class {πH } of automorphic characters with πG = AIE/F πH . Let m(πH ) denote the size of this class. Any such πH satisfies

H tr(πG(f) ◦ AπG ) = c tr πH (f ), f ∈ H(G(AF ), χ).

The number c is the product of the numbers appearing in Proposition 3.4.10. In particular it satisﬁes cc = 1. We have

H mφG tr(πG(f) ◦ AπG ) = mφG f (φG) (deﬁnitions in 2.5.7)

= IH (f H ) (Theorem 2.5.1) disc,φG

X H = nφG (πH ) tr πH (f ) (uniqueness of the class {πH })

πH ∈Πunit(H,χH )

H = m(πH ) tr πH (f )

−1 = c m(πH ) tr(πG(f) ◦ AπG ).

−1 Thus mφG = c m(πH ). This implies c is a positive number satisfying cc = 1 and thus c = 1. Consequently,

mφG = m(πH ).

According to 3.3.2, the number mφG is equal to the number of ΓE/F -conjugates of πH whenever πH has a nontrivial stabilizer. If πH has a trivial stabilizer in ΓE/F , then mφG must be equal to D, also by 3.3.2. Thus, the ﬁber of AIE/F is precisely these ΓE/F -conjugates.

Moreover, 3.3.2 shows that πG is cuspidal if and only if m(πH ) = D.

Suppose πG is an arbitrary ω-stable representation. There is some representation πH

189 that weakly lifts to πG. But such a weak lift is actually a strong lift according to the ﬁrst part of this proof, so πG is in the image of AIE/F . Lastly, we discuss independence of the choice of character ω determining E ⊃ F . We N already showed that the local lift AIEv/Fv is independent of ωv. Since AIE/F = v AIEv/Fv , this implies that AIE/F is independent of ω too.

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200 Appendix A Odds and ends

A.1 Restriction of scalars and endoscopy

The next lemma is used a few places in the main text. Let E ⊃ F be a cyclic extension of degree D.

Lemma A.1.1. Let GE be a connected reductive group deﬁned over E and let GF = RE/F GE.

Let E(GE) and E(GF ) denote the (equivalence classes of) ordinary endoscopic data for GE and GF , respectively. There is a bijection

E(GE) ←→ E(GF ).

A.1.2 In the notation of [KS99] or 1.2.1, an ordinary datum is one for the triplet (GE, id, 1). Let ρ be the L-action of W on G used to deﬁne LG . Then IWF (G , ρ ) is a GE E bE E WE bE GE dual group for GF . The group carries an action of WF by right translation. Fix coset

D−1 representatives 1, wσ, . . . , wσ for WE in WF , where wσ 7→ σ under WF ΓE/F = hσi. The coset representatives induce an isomorphism

f 7→ (f(1), f(w ), . . . , f(wD−1)) : IWF (G , ρ ) → G × · · · × G σ σ WE bE GE bE bE which we use to transfer the action of WF and form the L-group

L GF = GbE × · · · × GbE o ρGF WF .

201 Both realizations of GbF , as a space of functions and as a direct product of complex groups, are occasionally useful. In what follows, it is often the realization as a space of functions that is best.

A.1.3 Let (HE, HE, sE, ξE) represent an ordinary endoscopic datum for GE. We need to define an ordinary datum (HF , HF , sF , ξF ) for GF that corresponds to (HE, HE, sE, ξE). Define H = R H . As above, the dual group H can be taken to be IWF (H , ρ ) or F E/F E bF WE bE HE can be identified with HbE × · · · × HbE. Let

1 → HbE → HE → WE → 1 be the split exact sequence associated to the datum for GE with

cE : WE → HE ⊂ Aut(HbE)

the associated splitting. There is by deﬁnition an isomorphism HE ' HbE ocE WE that we

ﬁx. Let (HbE, cE) be the group HbE equipped with the action of WE given by cE. Then, WF acts on IWF (H , c ). The resulting semidirect product IWF (H , c ) W is our choice WE bE E WE bE E o F D−1 for the group HF . As above, using the coset representatives 1, wσ, . . . , wσ we ﬁx the isomorphism ∼ Y IWF (H , c ) → H ' H WE bE E bE bF Q and thus obtain an action ρHF of WF on HbE. We obtain the alternate realization

H = (H × · · · × H ) W . F bE bE oρHF F

Q The element sE lies in GbE. Using the isomorphism GbF ' GbE, we can take sF to be the semisimple element that corresponds to (sE, . . . , sE). Equivalently, it is the function

i which sends wσ to sE for each i.

A.1.4 We still need to deﬁne the map ξF . It is supposed to be an L-homomorphism from

L HF to GF . We will use the realization of these groups as spaces of functions, so that H = IWF (H , c ) W and LG = IWF (G , ρ ) W . F WE bE E o F F WE bE GE o F 202 Consider the map ξE, now viewed as a WE-homomorphism

ξ : H W → G W . E bE ocE E bE oρGE E

Let f : W → H be an element of IWF (H , c ). Then we set F bE WE bE E

(ξF f)(wF ) = ξE(f(wF )).

Then ξ is a map from IWF (H , c ) to IWF (G , ρ ). To show that it extends to an F WE bE E WE bE GE L L-homomorphism from HF to GF , it suﬃces to show that ξF commutes with the actions of WF . More precisely, we need to show that

0 0 ξF (wF f) = wF ξF (f).

This follows immediately, since the actions on both spaces are by right translation. Thus ξF

L extends to an L-homomorphism from HF to GF .

A.1.5 Having defined the tuple (HF , HF , sF , ξF ), we need to verify that it satisfies the properties of Definition 1.2.1. Since restriction of scalars preserves the property of being quasi-split, see 1.1.4, the group HF is quasi-split.

The group HF is a split extension of HbF by WF by deﬁnition. We need to show that the L-action of WF on HbF induced by HF is the same as ρHF . Said diﬀerently, we would Q like to show that ρHF and ρHF , viewed as maps from WF → Out( HbE), coincide. We have

a k WF = WEwσ.

Let f be a function W → H in one of the induced spaces IWF (H , ρ) for ρ equal to ρ F bE WE bE HE m or ρHE . Then if w = wEwσ ,

k k m wf(wσ) = f(wσwEwσ )

k −k m+k = f(wσwE · wσ )

k −k m+k = ρ(wσwEwσ )f(wσ ).

203 k −k m+k k −k m+k Since ρHE and ρHE are conjugate, ρHE (wσwEwσ )f(wσ ) and ρHE (wσwEwσ )f(wσ ) Q are conjugate in HbE. When we identify f with (bhk) the results are conjugate in HbE.

The fact that sF is semisimple follows from the assumption that sE is semisimple.

We have seen that ξF is an L-homomorphism. We need to know that it satisfies the −1 properties of Definition 1.2.1. The first is to show that sF ξF sF = aF ξF where aF : WF →

Z(GbF ) is a 1-cocycle which is equivalent to the trivial cycle if F is local and is everywhere locally trivial if F is global. This follows from the fact that ξF is deﬁned in terms of ξE and the assumption that ξE transforms under conjugation according to a cocycle.

Lastly we check that ξF is an isomorphism between HbF → GbF,sF . This follows, since ξE

maps into the centralizer of sE.

Thus the original datum (HE, HE, sE, ξE) for GE induces a datum (HF , HF , sF , ξF ) for

GF . If gE is an isomorphism of endoscopic data for GE, then (gE, . . . , gE) is an isomorphism of the corresponding data for GF and we obtain a map E(GE) → E(GF ).

A.1.6 Assume now that (HF , HF , sF , ξF ) is a datum in E(GF ). We need to ﬁnd a datum

(HE, HE, sE, ξE) in E(GE) associated to it. Let

I = {f ∈ IWF (G , ρ ): f is trivial outside of W ⊂ W }. 1 WE E GE E F

−1 Then I1 is a normal subgroup of GbF isomorphic to GbE. Then ξF (Ib1 o 1) ⊂ HbF ⊂ HF is a reductive algebraic group.

We have that ρ (w )ξ−1(I 1) ⊂ ξ−1(I 1). Indeed, assume ξ(h 1) = (g, 1) ∈ I 1. HF E b1 o b1 o bo b b1 o

Let cF : WF → HF be a section of

1 → HbF → HF → WF → 1.

Then

−1 −1 −1 ξF (c(wE)(bh o 1)c(wE) ) = (wbE, wE)(g,b 1)(wbE, wE) = (wbE(wEgb)wbE , 1).

−1 −1 Since I1 is normal and preserved by WE, we have ρHF (wE)ξF (Ib1 o 1) ⊂ ξF (Ib1 o 1). Then,

204 set

H = ξ−1(I )c(W ) = ξ−1(I ) W . E F 1 E F 1 oρHF E

Consider the map

−1 WE → Aut(I1) → Out(ξ (I1))

−1 provided by ρHF . Recall also that Out(ξ (I1)) is the same as the group of automorphisms −1 which ﬁx a given splitting. Thus, we can form the group ξF (I1) o WE using the L-action −1 provided by WE → Out(ξ (I1)). −1 There is a quasi-split group HE deﬁned over E such that ξF (I1) is a dual of HE, we −1 now write ξF (GbE) = HbE, and such that HbE o WE as above is an L-group for HE. See [Spr94].

Note that I is a direct factor of IWF (G , ρ ) and let π be the projection. We take 1 WE bE GE 1 −1 sE to be an element in ξF (I1) such that π1(ξF (sE)) = π1(sF ).

Let ξE be given by

π ◦ ξ : H = ξ−1(I )c(W ) → IWF (G , ρ ) W = LG → I W , 1 F E 1 E WE bE GE o F F 1 o F

L noting that the image actually lies in I1 o WE = GE.

We’ve created a tuple (HE, HE, sE, ξE). That it satisﬁes the conditions of Deﬁnition 1.2.1 is similar to the previous argument.

A.2 The Langlands classiﬁcation

This exposition is taken from [SZ14]. Similar results hold in the archimedean case, for which we refer the reader to [Lan89].

A.2.1 Let F be a non-archimedean local field of characteristic 0. Let G be a connected reductive group which is defined over F . Let A0 be a maximal F -split torus. Then its centralizer L0 is a minimal F -Levi subgroup of G. Let P0 be a fixed minimal F -parabolic subgroup of G which has L0 as a Levi subgroup. Thus P0 = L0N0 where N0 is the unipotent radical of P0. An F -parabolic subgroup P of G is called semi-standard if it contains L0.

205 The group P is called standard if it contains P0. If P is semi-standard, there are uniquely determined subgroups NP , LP and AP . These are the unipotent radical of P , the uniquely determined Levi subgroup of P which contains L0 and the maximal F -split torus in the center of LP . Note that there is a surjective map

P 7→ LP from the set of semi-standard F -parabolic subgroups onto the set of semi-standard F -Levi subgroups.

An unramiﬁed character of L(F ) is a continuous homomorphism

× L(F ) → C

∗ which is trivial on all compact subgroups of L(F ). We denote X (L)F the group of rational characters

χ : L → GL(1) which are deﬁned over F . Such a character χ induces a homomorphism

L(F ) → F ×

∗ which we will also denote by χ. By convention the abelian groups X (L)F are written additively. The group Xur(L(F )) of unramiﬁed characters can be given in terms of F -rational characters as

∗ × ∗ C ⊗ X (L)F C ⊗Z X (L)F Xur(L(F )), where the first arrow takes s ⊗ χ to (qs)⊗χ and the second arrow takes λ⊗χ to the unramified character m 7→ λνF (χ(m)). Note that the second surjection has finite kernel. Using the normalized absolute value on F , the combined map is

s s ⊗ χ 7→ (m 7→ |χ(m)|F ) .

s The latter character is often denoted by |χ|F . It is unramiﬁed because the rational character χ : L(F ) → F × takes compact subgroups of L into the subgroup of units in F .

206 × The subgroup Xur(L(F ))+ of positive, real-valued, unramified characters α : L(F ) → R+ is a the subgroup of unramified characters defined via the isomorphisms

∗ ∗ × ∗ aL = R ⊗ X (L)F ' (R+) ⊗ X (L)F ' Xur(L(F ))+.

s Note that s 7→ q is a bijection from R to R+ so that, unlike the corresponding map of complex spaces, this is indeed an isomorphism. We will write this map as

∗ ν ∈ aL 7→ χν.

If L is semi-standard then restriction from L(F ) to L (F ) provides an injection a∗ ⊂ a∗ . We 0 L L0 can consider these spaces as Euclidean spaces with respect to a ﬁxed Weyl-group invariant pairing h−, −i on a∗ . Here we are thinking of the relative Weyl group W G(A ). L0 0 A standard triple (P (F ), σ, ν) for G(F ) consists of P (F ), a standard F -parabolic subgroup, an irreducible tempered representation σ of the standard Levi subgroup LP (F ) and a real parameter ν ∈ a∗ = a∗ which is regular, that is, properly contained in the chamber P LP ∗ which is determined by P . More precisely, let ∆(P ) ⊂ X (AP ) be the simple roots for the adjoint action of the torus AP on the Lie algebra of NP . Via restriction we may identify

∗ ∗ ∗ ∗ ∆(P ) as a subset of aP : aP = R ⊗ X (LP ) = R ⊗ X (AP ) ⊃ ∆(P ). The condition for ν to be regular is that

hν, αi > 0 for all α ∈ ∆(P ). Such regular characters may also be called positive relative to P or be said to lie in the positive cone associated to P .

Theorem A.2.2 (Langlands classification). In G we fix the data (P0,L0) and the Euclidean pairing h−, −i on a∗ . Then there is a natural bijection between the set of all standard triples L0 {(P, σ, ν)} and the set of equivalence classes of irreducible representations of G(F ). It takes a triple (P, σ, ν) to the representation j(P, σ, ν), which is the unique irreducible quotient of the normalized parabolically induced representation σ ⊗ χν from LP to G extended to P . A different bijection between standard triples and irreducible representations can be realized by taking the unique irreducible subrepresentations of the representation induced

207 from σ ⊗ χ−ν. If this is denoted by s, then

s(P, σ, ν) = ˜j(P, σ,˜ ν) and

j(P, σ, ν) =s ˜(P, σ,˜ ν).

A.3 Global induced representations

The goal in this section is to define and discuss some different realizations of the global induced representations needed for the trace formula, as well as the global intertwining operators between induced representations. The material is extremely well known, but since different realizations are used, somewhat implicitly, in the main text, we opted to write a few things down. In this appendix, we fix a number field F and a general group G defined over F . Let L ⊂ G be a Levi subgroup and let P = LN be a parabolic subgroup of G with

L as a Levi subgroup. We ﬁx a central character datum for G. We also have the standard space a∗ . L,C

A.3.1 Consider the space of χ-equivariant functions

2 V (χ) = Ldisc(L(F )\L(A), χ).

The group L(A) acts on V (χ) by right translation RL:

0 0 0 (RL(m)ψ)(m ) = ψ(m m), m, m ∈ L(A), ψ ∈ V (χ).

For λ ∈ a∗ , we let R denote the twist of R by λ. The representation R is also on L,C L,λ L L,λ V (χ). The action is given by

0 λ(HL(m)) 0 0 (RL,λψ)(m ) = e ψ(m m), m, m ∈ L(A), ψ ∈ V (χ).

We let HP (χ, λ) denote the space of functions

1/2 λ(HL(m)) {f : G(A) → V (χ): f(mng) = δP (m) e RL(m)f(g)}.

208 Thus f can be viewed as a function on G(A) × L(A) satisfying

0 1/2 λ(HL(m)) 0 f(mng, m ) = δP (m) e f(g, m m).

The action of G(A) on HP (χ, λ) is by right translation. Note that the action is independent of λ but the space HP (χ, λ) depends on λ. We write the pair of action and space as

(RP (χ), HP (χ, λ)). It is out ﬁrst realization of the induced representation.

A.3.2 Let HP (χ) denote the space of functions φ : G(A) → C satisfying the following two properties.

1. φ is N(A)-invariant:

φ(ng) = φ(g), n ∈ N(A), g ∈ G(A).

2. The function

(m, k) 7→ φ(mk): L(A) × K → C

lies in V (χ) ⊗ L2(K).

The space HP (χ) carries an action we denote by RP (χ, λ):

0 0 0 0 (λ+ρP (g g)) −(λ+ρP )(HP (g )) 0 (RP (χ, λ)(g)φ)(g ) = φ(g g)e e , g, g ∈ G(A), φ ∈ HP (χ).

The situation with (RP (χ, λ), HP (χ)) is the opposite of the situation with (RP (χ), HP (χ, λ)); namely, in this realization of the induced representation the action depends on λ but the underlying space does not.

A.3.3 There is a third realization of the induced representation, whose underlying space is

HK (χ) = {fK : K → V (χ): f(mnk) = RL(m)f(k), m ∈ L(A) ∩ K, n ∈ N(A) ∩ K, k ∈ K}.

This space is also independent of λ, and thus the action RP,K (χ, λ) must depend on λ. The action is deﬁned as at 1.6.7, using the global Iwasawa decomposition.

209 A.3.4 We have described three realizations of an induced representation. The local map which gives the equivalence between (RP (χ), HP (χ, λ)) and (RP,K (χ, λ), HK (χ)) is discussed a bit at 1.6.7, albeit with slightly diﬀerent notation. The global equivalence is deﬁned similarly. We will write down the other relevant equivalence and give some of the details, just to get a sense of how the proofs go.

Lemma. There is an equivalence of representations of G(A):

(RP (χ), HP (χ, λ)) ' (RP (χ, λ), HP (χ)).

Proof. Let φ be a function in HP (χ). Deﬁne a function

fφ : G(A) × L(A) → C

by setting

0 (λ+ρP )(HP (g)) 0 fφ(g, m ) = e φ(m g).

We claim that fφ lies in HP (χ, λ). We must ﬁrst show that fφ(g, −) lies in V (χ) for any g. So, consider the function

m0 7→ e(λ+ρP )(HP (g))φ(m0g).

Write g = mnk. According to the second property deﬁning HP (χ), we can write

n 0 0 X 0 0 φ(m k ) = fi(m )gi(k ). i=1 Then n n 0 0 X 0 X 0 φ(m g) = φ(m mnk) = fi(m m)gi(k) = gi(k)(RL(m)fi)(m ). i=1 i=1

But RL(m)fi lies in V (χ), so the claim holds.

Next, we check that fφ satisﬁes the right translation property:

0 (λ+ρP )(HP (mng)) 0 fφ(mng, m ) = e φ(m mng)

= e(λ+ρP )(HP (m)+HP (g))φ(m0mg)

(λ+ρP )(HP (m)) 0 = e fφ(g, m m).

210 Thus fφ does indeed lie in HP (χ, λ). Note that φ 7→ fφ depends on λ, as it must. Thus we λ will write φ 7→ fφ. Next, we check that λ φ 7→ fφ : HP (χ) → HP (χ, λ) intertwines the actions RP (χ, λ) and RP (χ). Thus we must show that the diagram

λ φ7→fφ HP (χ) / HP (χ, λ)

IP (χ,λ,g) IP (χ,g) HP (χ) / P (χ, λ) λ H φ7→fφ

Thus we must check that

0 (λ+ρP )(HP (nmk)) 0 fRP (χ,λ,g)φ(nmk, m ) = e (RP (χ, λ, g)φ)(m nmk)

0 0 = e(λ+ρP )(HP (nmk))φ(m0nmkg)e(λ+ρP )(HP (m nmkg))e−(λ+ρP )(HP (m nmk)) is equal to

0 0 (RP (χ, g)fφ)(nmk, m ) = fφ(nmkg, m )

= e(λ+ρP )(HP (nmkg))φ(m0nmkg).

It suﬃces to show that

0 0 HP (nmk) + HP (m nmkg) − HP (m nmk) = HP (nmkg).

Let nmkg = n00m00k00. Then

0 0 0 00 00 00 0 HP (nmk) + HP (m nmkg)−HP (m nmk) = HP (nmk) + HP (m n m k ) − HP (m nmk)

0 00 0 = HL(m) + HL(m ) + HL(m ) − HL(m ) − HL(m)

00 = HL(m )

= HP (nmkg), and the claim holds.

211 One can construct an inverse to the map φ 7→ fφ. Given f, we set

−(λ+ρP )(HP (g)) φf (g) = e f(g, 1).

Then

−(λ+ρP )(HP (g)) φ(fφ)(g) = e (fφ)(g, 1)

= e−(λ+ρP )(HP (g))e(λ+ρP )(HP (g))φ(g)

= φ(g).

This completes the proof of the lemma.

A.3.5 Let π be a discrete spectrum representation of L(A) with central character χ. Thus

π can be realized as (RL,Vπ) ⊂ (RL,V (χ)). Then π deﬁnes a subspace of HP (χ, λ):

HP (π, λ) = {f ∈ HP (χ, λ): f(g, −) ∈ Vπ for all g ∈ G(A)}.

Then (RP (χ), HP (π, λ)) is a realization of the representation induced from πλ. We have

M (RP (χ), HP (χ, λ)) = (RP (χ), HP (π, λ)). π Similar remarks hold for the other realizations of the induced representation. For example, the subrepresentation π together with its realization Vπ deﬁnes a subspace

2 HP (π) = {φ ∈ HP (χ):(m, k) 7→ φ(mk) lies in Vπ ⊗ L (K)}.

Then we have a decomposition

M (RP (χ, λ), HP (χ)) = (RP (χ, λ), HP (π)). π When dealing with groups where multiplicity one in the discrete spectrum is not known, one ought to be a bit more careful with the notation. Then φ 7→ fφ deﬁnes an intertwining map:

φ 7→ fφ :(RP (χ, λ), HP (π)) ' (RP (χ), HP (π, λ)).

There are similar decomposition and isomorphisms in the model (RP,K (χ, λ), HK (χ)).

212 A.3.6 We can now consider the global intertwining operator. We will only consider the operator in the model that is actually used explicitly in the main text. Let P,P 0 ∈ P(L) be parabolic subgroups of G. Consider the integral

(MP 0|P (χ, λ)φ)(g) =

Z 0 e−(λ+ρP 0 )(HP 0 (g)) · φ(n0g)e(λ+ρP )(HP (n g)) dn0. NP 0 (A)∩NP (A)\NP 0 (A) One checks that it formally satisﬁes

IP 0 (χ, λ, g)MP 0|P (χ, λ) = MP 0|P (χ, λ)IP (χ, λ, g).

It is known that the integral converges for λ whose real part lies in a certain chamber.

Moreover, the resulting holomorphic function of λ has analytic continuation to a meromorphic function of λ and we obtain an operator

MP 0|P (χ, λ): HP (χ) → HP 0 (χ)

that intertwines (IP (χ, λ), HP (χ)) and (IP 0 (χ, λ), HP 0 (χ)). The values at λ that are purely imaginary are known to be unitary.

A.3.7 The intertwining operator MP 0|P (χ, λ) has some more reﬁned behavior. Consider a discrete spectrum representation π of L(A) on the space of functions Vπ. Then we have the associated induced representation (RP (χ, λ), HP (Vπ)). We also have the induced representation (RP 0 (χ, λ), HP 0 (Vπ)).

Lemma. The restriction of the intertwining integral MP 0|P (χ, λ) to HP (Vπ) formally intertwines (RP (χ, λ), HP (Vπ)) and (RP 0 (χ, λ), HP 0 (Vπ)). Thus the integral intertwines the two representations when λ is in the cone of absolute convergence. By analytic continuation, the operator MP 0|P (χ, λ) intertwines the two representations for any choice of λ.

Proof. Suppose φ lies in HP (Vπ). Really the only thing to show is that the function

m 7→ (MP 0|P (χ, λ)φ)(m): L(A) → C

213 lies in Vπ. So, for the moment regard m as ﬁxed in L(A) and consider the integral

Z 0 e−(λ+ρP 0 )(HP 0 (m)) · φ(n0m)e(λ+ρP )(HP (n m)) dn0. NP 0 (A)∩NP (A)\NP 0 (A) We would like to make the change of variables n0 7→ mn0m−1. The set of roots associated with the integration space is

ΣP 0|P = ΣP 0 r (ΣP 0 ∩ ΣP ).

Let ρP 0|P denote half the sum of the roots in ΣP 0|P and let ρP 0∩P denote half the sum of the roots in ΣP 0 ∩ ΣP . Then

ρP 0|P = ρP 0 − ρP 0∩P .

The change of variables introduces a factor e2ρP 0|P (HL(m)). Doing the change of variables gives the expression

Z 0 2ρ (H (m)) e−(λ+ρP 0 )(HP 0 (m)) · φ(mn0)e(λ+ρP )(HP (mn ))e P 0|P L dn0. NP 0 (A)∩NP (A)\NP 0 (A) Using the Iwasawa decomposition relative to P , we have

0 0 0 0 n = LP (n )NP (n )KP (n ).

We may substitute this into the expression above and use NP -invariance to obtain

e−(λ+ρP 0 )(HL(m))

Z 0 0 0 (λ+ρP )(HL(mLP (n ))) 2ρ 0 (HL(m)) 0 · φ(mLP (n )KP (n ))e e P |P dn . NP 0 (A)∩NP (A)\NP 0 (A) Next, we claim that

ρP + 2ρP 0|P − ρP 0 = ρP + ρP 0 − 2ρP 0∩P = 0.

Suppose that α lies in ΣP . If α also lies in ΣP 0 , then α lies in ΣP 0 ∩ ΣP and the sum contains

α + α − 2α. If α does not occur in ΣP 0 , then −α lies in ΣP 0 and α does not lie in ΣP 0 ∩ ΣP .

214 The sum contains α − α. Thus the claim holds. Consequently the integral reduces to

Z 0 0 0 (λ+ρP )(HL(LP (n ))) 0 φ(mLP (n )KP (n ))e dn . NP 0 (A)∩NP (A)\NP 0 (A)

0 Fix n ∈ NP 0 (A). It suﬃces to show that the function

0 0 0 (λ+ρP )(HL(LP (n ))) m 7→ φ(mLP (n )KP (n ))e

2 lies in Vπ. This follows by expanding φ as a function in Vπ ⊗ L (K), similar to the proof of Lemma A.3.4.

A.3.8 The global intertwining operator that appears in the trace formula is a certain self-intertwining operator associate to a parabolic subgroup and a Weyl group element.

Now, we discuss how to convert the standard intertwining operator MP 0|P (χ, λ) into a self-intertwining operator.

Consider the relative Weyl group

G W (L) = W (L) = NG(AL)/L.

Let w ∈ W (L) be a Weyl element and let we be a representative of w in G(F ). In general, it is not possible to choose a representative of w in G(F ) ∩ K, as one can in GL(N). Set

0 −1 −1 P = w P = we P w.e

0 0 Suppose φ lies in HP 0 (χ) and consider the function l(w)φ deﬁned by setting

0 0 −1 (l(w)φ )(g) = φ (we g).

0 We claim that l(w)φ lies in HP (χ). First we check the invariance. If NP 0 denotes the

0 −1 unipotent radical of P , we have NP 0 = we NP we. Let n ∈ NP be a ﬁxed element. Then

215 0 0 −1 there is an n ∈ NP 0 such that wne we = n. Then,

0 0 −1 (l(w)φ )(ng) = φ (we ng) 0 −1 0 −1 = φ (we (wne we )g) 0 0 −1 = φ (n we g) 0 −1 = φ (we g) = (l(w)φ0)(g).

−1 0 0 We may write we = nwmwkw according to the Iwasawa decomposition relative to P . Now, suppose n 0 X 2 φ (mk) = fi(m)gi(k), fi ∈ V (χ), gi ∈ L (K). i=1 Then,

0 0 −1 (l(w)φ )(mk) = φ (we mk) 0 −1 −1 = φ (we mwewe k) 0 −1 0 = φ (we mwne wmwkwk) 0 −1 = φ ((we mwme w)(kwk)) n X −1 = fi(we mwme w)gi(kwk). i=1

−1 Now, we just need to note that m 7→ fi(we mwme w) lies in V (χ) and k 7→ gi(kwk) lies in L2(K).

Given a representation π of L, the representation wπ is deﬁned by

(wπ)(m) = π(w−1mw).

It acts on the same space as π. Suppose for example that (π, Vπ) is an explicit discrete spectrum representation. Let wVπ be the subspace consisting of functions wf for f ∈ Vπ, deﬁned by the equation

(wf)(m) = f(w−1mw).

216 Then the following diagram commutes:

Vπ / wVπ

wRL RL Vπ / wVπ

Thus (RL, wVπ) is the explicit realization of the discrete series representation wπ. Note that the function fi above lies in wVπ if fi lies in Vπ. Thus l(w) deﬁnes an operator

l(w): HP 0 (Vπ) → HP (wVπ).

We claim that the following diagram commutes:

l(w) HP 0 (χ) / HP (χ)

IP 0 (χ,λ) IP (χ,wλ) HP 0 (χ) / HP (χ) l(w)

Here wλ is the character that acts by

wλ(X) = λ(w−1Xw).

On the one hand, we have

0 0 0 0 (wλ+ρP )(HP (g g)) −(wλ+ρP )(HP (g )) RP (χ, wλ, g)l(w)φ(g ) = l(w)φ(g g)e e

0 0 −1 0 (wλ+ρP )(HP (g g)) −(wλ+ρP )(HP (g )) = φ(we g g)e e .

On the other hand,

0 −1 0 l(w)RP 0 (χ, λ, g)φ(g ) = RP 0 (χ, λ, g)φ(w g )

−1 0 −1 0 −1 0 (λ+ρ 0 )(H 0 (w g g)) −(λ+ρ 0 )(H 0 (w g )) = φ(we g g)e P P e e P P e .

We need to prove that the exponential factors are equal.

0 −1 −1 We are continuing to assume that P = we P we. In particular, we have NP 0 = we NP we. −1 First, note that wρP 0 = ρP . Indeed, if α is a root of AL in NP , then w α is a root of

AL in NP 0 . Next, consider an arbitrary element g ∈ G(A). According to the Iwasawa

217 decomposition relative to P , we can write

g = NP (g)LP (g)KP (g).

Then we have

−1 −1 −1 −1 HP 0 (w g) = HP 0 (we NP (g)we)(we LP (g)w)(we k) −1 −1 = HP 0 (we LP (g)w)(we KP (g)) −1 −1 = HP 0 we LP (g)we + HP 0 (we ).

We apply this to the relevant terms above:

−1 0 −1 0 −1 0 −1 HP 0 (we g g) − HP 0 (w g ) = HP 0 (we LP (g g)we) + HP 0 (we ) −1 0 −1 − HP 0 (we LP (g )we) − HP 0 (we ) −1 0 −1 0 = HP 0 (we LP (g g)we) − HP 0 (we LP (g )we) −1 0 0 = w HP 0 (LP (g g)) − HP 0 (LP (g )) we −1 0 0 = w HP (LP (g g)) − HP (LP (g )) we −1 0 0 = w HP (g g) − HP (g ) we

Finally, we have

−1 0 −1 0 −1 0 0 (λ + ρP 0 ) HP 0 (we g g)) − HP 0 (we g ) = (λ + ρP 0 ) w HP (g g) − HP (g ) we −1 0 0 = (wλ + wρP 0 ) w HP (g g) − HP (g ) we −1 0 0 = (wλ + ρP ) w HP (g g) − HP (g ) we .

0 −1 A.3.9 We continue to assume that P = we P we. Let π be a discrete spectrum representation of L(A) with central character χ and let (RL,Vπ) be its concrete realization inside 2 Ldisc(L(F )\L(A), χ). Let us now collect what we’ve been discussing in a commutative

218 diagram:

MP 0|P (χ,λ) l(w) HP (Vπ) / HP 0 (Vπ) / HP (wVπ)

RP (χ,λ) RP 0 (χ,λ) RP (χ,wλ) HP (Vπ) / HP 0 (Vπ) / HP (wVπ) MP 0|P (χ,λ) l(w) The intertwining operator in the left hand box is only provided by an integral for certain

λ, but the analytic continuation in λ provides the diagram for each λ. In particular the operator

l(w) ◦ MP 0|P (χ, λ):(RP (χ, λ), HP (Vπ)) → (RP (χ, wλ), HP (wVπ)) is a well-deﬁned intertwining operator, and in fact intertwines the larger representations

(RP (χ, λ), HP (χ)) and (RP (χ, wλ), HP (χ)). In particular, l(w) ◦ MP 0|P (χ, λ) maps HP (χ) back to itself but need not map the irreducible summands of HP (χ), namely the HP (Vπ), back to themselves. Consider the value of the composition at λ = 0. In this situation we will write RP (χ) = RP (χ, 0). We obtain a unitary intertwining operator

MP (w) = l(w) ◦ MP 0|P (χ, 0) :(RP (χ), HP (Vπ)) → (RP (χ), HP (wVπ)).

On the larger space HP (χ) we obtain a self-intertwining operator of the standard induced representation RP (χ). This is the operator, restricted to suitable subspaces of HP (χ), that occurs in the trace formula.

A.4 More on the Local Langlands correspondence

Let E ⊃ F be a cyclic extension of local ﬁelds of characteristic 0 of degree D. Let GN = GN,F be the group GL(N) regarded as an F group. Let HM = RE/F GL(M). Then HM is an

F -group that satisﬁes HM (F ) ' GM,E(E). The group HM represents the unique elliptic datum in the set of ω-endoscopic groups Eω(N).

A.4.1 Let us be a bit more precise about the isomorphism HM (F ) ' GM,E(E). The group

HM comes with an isomorphism HM (A) ' GM,E(A ⊗F E) for each F -algebra A. The family of isomorphisms is functorial in A. The family of isomorphisms is not unique. For example, if 219 σ is an element of the Galois group ΓE/F acting on HM and {iA} is the family of isomorphisms, then {iA ◦ σ} is also a family of isomorphisms. More generally, any F -automorphism α of

HM gives a family of isomorphisms {iA ◦ α}. Conversely, any two families {iA} and {ιA} differ by an F -automorphism of HM . We have already suggested one natural family of automorphisms of HM , namely the D automorphisms coming from ΓE/F . There is also the standard automorphism of transpose-inverse. It turns out that the 2D automorphisms obtained from ΓE/F and transpose-inverse are the only F -automorphisms of HM . See [nfd] for a thorough discussion. Any fixed family of automorphisms induces a bijection between the irreducible, admissible representations of HM (F ), denoted by Π(HM (F )), and the irreducible, admissible representations of GM,E(E), denoted by Π(GM,E(E)). Let Π(HM (F ))/ΓE/F denote the classes of ΓE/F -conjugate representations, and define Π(GM,E(E))/ΓE/F similarly.

Fix a family of isomorphisms {iA}. For any σ, the family {iA ◦ σ} induces the same bijection between Π(HM (F ))/ΓE/F and Π(GM,E(E))/ΓE/F ; composition with σ and transpose-inverse gives a second bijection. Thus, given some initial choice of {iA}, perhaps coming from a basis of E over F , we obtain two bijections between the ΓE/F -orbits of representations. The former, namely the bijection obtained without using transpose-inverse, is preferable though, as it preserves, for example, the central character of the representations. We ﬁx some such bijection between Π(HM (F ))/ΓE/F and Π(GM,E(E))/ΓE/F .

A.4.2 We cannot expect to obtain a local Langlands correspondence for HM (F ), but only a correspondence up to ΓE/F . Let Φ(HM (F )) denote the set of Langlands parameters

L φF : LF → HM .

Let Φ(HM (F ))/ΓE/F be the equivalence classes under the action of ΓE/F . Consider the map

L ξHM : HM → GL(N, C).

Then the ΓE/F -classes are the ﬁbers of the induced map

φF 7→ ξHM ◦ φF : Φ(HM (F )) → Φ(GN (F )).

220 Let Φ(GM,E(E)) denote the Langlands parameters for GM,E(E). They are L-homomorphisms

φE : LE → GL(M, C).

Let Φ(GM,E(E))/ΓE/F be the classes under ΓE/F .

A.4.3 To deduce a local Langlands classiﬁcation for HM (F ), up to the action of ΓE/F , it suﬃces to show that there is a bijection between the set Φ(HM (F ))/ΓE/F and the set

Φ(GM,E(E))/ΓE/F . For this, it suﬃces to show that there is a bijection between two sets of L L-homomorphisms φF : WF → GL(N, C): those that factor through HM and those that are induced from WE. Consider ﬁrst a parameter of the form

ILF φ : L → GL(MD, ). LE E F C

We must show that this parameter factors through the map ξH . If VE is the space on which

LE acts via φE, then the induced representation of LF acts on the space

VF = {f : LF → VE : f(wEwF ) = φE(wE) · f(wF )}.

Let wσ ∈ LF generate the quotient LF /LE. Then VF decomposes under the action of LE into a direct sum of subspaces

i Vi = {f ∈ VF : f is supported on LEwσ}.

QD Thus the image of LE under the induced representation lies in Hb = i=1 GL(M, C). Since `D i LF = i=1 LEwσ and wσ permutes the spaces Vi cyclically, the ﬁrst part of the claim follows.

For the second inclusion, consider a parameter φ of the form

L φ : LF → H → GL(MD, C).

We must show that φ is induced from LE. It suﬃces to show that φ has the same character

221 as an induced representation. We have

i(w) φ(w) = wAb σ for wb ∈ ξH (Hb) and 0 ≤ i(w) ≤ D − 1. Then the trace tr(φ(w)) vanishes unless i(w) = 0.

That is, w projects into WE under the natural map LF → WF . If this is the case the trace is just the trace of the image of w in Hb under the projection Hb × WE → Hb. Composing

φ|LE with Hb × WE → Hb gives a direct sum of D representations of LE, say φ1, . . . , φD. Consider the induced representations ILF φ . The character of this representation is given by LE 1 the Frobenius formula and vanishes on all w in LF outside of LE. If w lies in LE we get the sum of the traces of the φi which is the same as before.

A.4.4 There are a few related observations about representations of WE that are well- known but useful at a few points in the main text. Let φE be an irreducible, M-dimensional representation of LE. Let ΓE/F1 be the stabilizer of φE in ΓE/F . If ΓE/F = hσi and [E : F1] D/k has order k, then ΓE/F1 is generated by σ . Since φE is ﬁxed by ΓE/F1 , there is an

M-dimensional representation φF1 of LF1 whose restriction to LE is equal to φE. Let ω be a degree D character determining the extension E ⊃ F . Then ωk is a degree D/k character determining the extension F1 ⊃ F . Then,

LF LF LF k−1 LF I φE = (I φF1 ) ⊕ (ω ⊗ I φF1 ) ⊕ · · · ⊕ (ω ⊗ I φF1 ). LE LF1 LF1 LF1

The computation is similar to that in 1.4.11.

222