Automorphic Representations of Low Rank Groups

AUTOMORPHIC REPRESENTATIONS OF LOW RANK GROUPS Y U V A L F L I C K E R AUTOMORPHIC REPRESENTATIONS OF LOW RANK GROUP...

Author: Yuval Z. Flicker

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AUTOMORPHIC REPRESENTATIONS OF LOW RANK GROUPS Y U V A

L

F L I C K E R

AUTOMORPHIC REPRESENTATIONS OF LOW RANK GROUPS

Y U V A L

Z.

F L I C K E R

The Ohio State University, USA

AUTOMORPHIC REPRESENTATIONS OF LOW RANK GROUPS

\jjjp World Scientific NEW JERSEY

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PREFACE This volume concerns two related but independent topics in the theory of liftings of automorphic representations. These are the symmetric square lifting from the group SL(2) to the group PGL(3), and the basechange lifting from the unitary group U(3, E/F) to GL(3,E), where E/F is a quadratic extension of number fields. I initially considered these topics in preprints dated 1981 and 1982, and since then found reasonably simple proofs for many of the technical details, such as the fundamental lemma and the unrestricted equality of the trace formulae. The fruits of these efforts are the subject matter of the first two parts of this volume, which are independent of each other, while the third part concerns applications of the basechange theory for U(3) to the theory of Galois representations which occur in the cohomology of the Shimura variety associated with U(3). The method used relies on a comparison of trace formulae, the same as in my Automorphic Forms and Shimura Varieties o/PGSp(2), which concerns a rank-two situation. Both topics considered in this volume are lower, rank-one cases. They can be viewed as more elementary, certainly more complete. The last part of the volume on PGSp(2), entitled Background, contains many of the (standard) definitions used in this volume too. It is a brief exposition to the principle of functoriality, which predicts the liftings which concern us here, on a conjectural level, in terms of homomorphisms of dual groups. Thus here we consider two rank-one examples of this principle. To describe the first topic, let F be a number field. Denote by A its ring of adeles. Let A be the symmetric square (or adjoint) three-dimensional representation of the dual group H = PGL(2, C) of the F-group H = SL(2) in the dual group G = SL(3, C) of G = PGL(3). We study the lifting (or correspondence) of automorphic forms on SL(2, A) to those of PGL(3, A) which is compatible with A. This lifting is defined by means of character relations. It is studied using a trace formula twisted by the outer automorphism a of G, which takes a representation to its contragredient. Complete results are obtained. We not only demonstrate the existence of the lifting but also describe its image and fibers. Main results include an intrinsic definition of packets of admissible and automorphic representations of SL(2, Fv) and SL(2, A), a proof of multiplicity one theorem for the cuspidal representations of SL(2, A) and of the rigidity theorem for packets of such cuspidal representations, and a determination of the selfadjoint automorphic representations of PGL(3, A). v

VI

Preface

Technical novelties include an elementary proof of the Fundamental Lemma, a simplification of the trace formula by means of regular functions, and a twisted analogue of Rodier's theorem capturing the number of Whittaker models of a (local) representation in the germ expansion of its character. In the second part, locally we introduce packets and quasi-packets of admissible representations of the quasi-split unitary group U(3, E/F) in three variables, where E/F is a quadratic extension of local fields, and determine their structure. We determine the admissible representations of GL(3,E) which are invariant under the involution transpose-inverse-bar. These (quasi) packets are defined by means of both the basechange lifting from U(3, E/F) to GL(3,£) and the endoscopic lifting from V(2,E/F) to U(3, E/F). Globally, we introduce packets and quasi-packets of the discrete -spectrum automorphic representations of U(3,E/F)(A) where E/F is a quadratic extension of number fields, determine their structure, and determine the discrete-spectrum automorphic representations of GL(3, A#) fixed by the same involution. In particular we prove multiplicity one theorem for U(3, E/F), determine which members of a (quasi-) packet are automorphic, establish a rigidity theorem for (quasi-) packets of U(3, E/F), prove the existence of the global basechange and endoscopic liftings, as well as another twisted endoscopic lifting from \]{2,E/F) to GL(3,E), and show that each packet of U(3, E/F) which lifts to a generic representation of GL(3, E) contains a unique generic member. Technical novelties include a proof of multiplicity one theorem and counting the generic members in packets, two elementary proofs of the Fundamental Lemma, and a simple proof of the unrestricted equality of trace formulae for all test functions by means of regular functions. To emphasize, multiplicity one theorem was claimed as proved since 1982, but we noticed that the global proof was lacking and completed our local proof (for all noneven places) only a few years before this local proof appeared in 2004. For more details on the development of this area see the concluding remarks section at the end of part 2. The third part concerns the cohomology H*{SKJ ®E Q, V) with compact supports and coefficients in any local system (p, V), of a Shimura variety SKS defined over its reflex field E, associated with the quasi-split unitary group of similitudes G — GU(3, E/F), where E is a totally imaginary quadratic extension E of a totally real field F. It is a Hecke x Galois bi-module. We determine its decomposition. The Hecke modules which appear are the finite parts 717 of the discrete-spectrum representation •Kf®-K00 of G(kp) such that -K^ has nonzero Lie algebra cohomology. We determine

Preface

Vll

the ^-isotypic part H*(itf) as a Gal(Q/E)-module in terms of the Hecke eigenvalues of 7T/. In the stable case dlm[H*(-Kf)] is 3 ^ ° ] . The dimension is smaller in the unstable case. The cuspidal part of H* (SKS ®EQ> V) coincides with the cuspidal part of the intersection cohomology IH* {S'K ®E Q, V) of the Satake Baily-Borel compactification S'K . Purity for the eigenvalues of the Probenius acting on IH*, using a computation of the Lie algebra cohomology of the 7Too, implies the Ramanujan conjecture for the 7r/ (with the exception of the obvious counter examples un(fj,)n). More precisely we show that the Satake parameters of each local component irv of nf are algebraic, and if 7r / 7r(/i) that all of their conjugates lie on the unit circle in the complex plane. A description of the Zeta function of H* formally follows. This third part uses the results of the second part, and compares the trace formula with the Lefschetz-Grothendieck fixed point formula. This comparison is greatly simplified on using the (proven) Deligne conjecture on the form of the fixed point formula for a correspondence twisted by a sufficiently high power of the Probenius. The underlying idea is used in the representation theoretic parts in the avatar of regular, Iwahori biinvariant functions. It leads to a drastic simplification of the proof of the comparison of trace formulae, on which the work of parts 1 and 2 is based. It was found while working with D. Kazhdan on applications of Drinfeld moduli schemes to the reciprocity law relating cuspidal representations of GL(n) over a function field (which have a cuspidal component) with n-dimensional Galois representations of this field (whose restriction to a decomposition group is irreducible). This work relied on Deligne's conjecture. First representation theoretic applications, inspired by Deligne's insight, were found in the proof with Kazhdan of the metaplectic correspondence, and then to prove basechange for GL(n). However, the higher-rank applications concern only cuspidal representations with a cuspidal component, while in the low-rank case considered here there are no restrictions. I then feel this idea has not yet been fully exploited. It may lead to significant simplifications in the use of the trace formula. Yuval Flicker

CONTENTS PREFACE

v

P A R T 1. O N T H E S Y M M E T R I C S Q U A R E LIFTING INTRODUCTION

1 3

I. 1.1 1.2 1.3 1.4

FUNCTORIALITY AND NORMS Hecke algebra Norms Local lifting Orthogonality

16 17 25 31 41

ORBITAL I N T E G R A L S Fundamental lemma Differential forms Matching orbital integrals Germ expansion

47 47 66 73 80

II. II. 1 11.2 11.3 11.4

III. TWISTED TRACE FORMULA 111.1 Geometric side 111.2 Analytic side 111.3 Trace formulae

83 84 90 94

IV. TOTAL GLOBAL C O M P A R I S O N IV. 1 The comparison IV.2 Appendix: Mathematica program

102 104 121

V.

A P P L I C A T I O N S OF A T R A C E F O R M U L A Approximation Main theorems Characters and genericity

123 123 146 178

VI. C O M P U T A T I O N OF A T W I S T E D C H A R A C T E R VI. 1 Proof of theorem, anisotropic case VI.2 Proof of theorem, isotropic case

186 189 194

V.l V.2 V.3

IX

Contents

X

PART 2. AUTOMORPHIC REPRESENTATIONS OF

1. 2.

T H E U N I T A R Y G R O U P U(3, E/F) INTRODUCTION Functorial overview Statement of results

203 205 205 211

1.1 1.2 1.3 1.4 1.5 1.6

LOCAL THEORY Conjugacy classes Orbital integrals Fundamental lemma Admissible representations Representations of U(2,1;C/R) Fundamental lemma again

222 223 237 244 274 283 293

II. 1 II.2 II.3 II.4 II. 5 II.6 II.7

TRACE FORMULA Stable trace formula Twisted trace formula Restricted comparison Trace identity The cr-endo-lifting e' The quasi-endo-lifting e Unitary symmetric square

311 311 317 320 326 339 346 350

LIFTINGS A N D P A C K E T S III.l Local identity III. 2 Separation III.3 Specific lifts III.4 Whittaker models and twisted characters III. 5 Global lifting III.6 Concluding remarks

352 352 360 369 378 385 392

Contents P A R T 3. ZETA F U N C T I O N S OF S H I M U R A VARIETIES OF U ( 3 ) INTRODUCTION 1. Statement of results 2. The Zeta function I.

xi

397 399 402 409

1.1 1.2 1.3

PRELIMINARIES The Shimura variety Decomposition of cohomology Galois representations

412 412 413 415

II. 1 11.2 11.3 11.4 11.5 11.6

AUTOMORPHIC REPRESENTATIONS Stabilization and the test function Functorial overview of basechange for U(3) Local results on basechange for U(3) Global results on basechange for U(3) Spectral side of the stable trace formula Proper endoscopic group

417 417 418 424 429 434 435

II.

III. LOCAL T E R M S III.l The reflex field III. 2 The representation of the dual group 111.3 Local terms at p 111.4 The eigenvalues at p III. 5 Terms at p for the endoscopic group

436 436 438 442 444 446

IV. REAL R E P R E S E N T A T I O N S IV. 1 Representations of the real GL(2) IV.2 Representations of U(2,l) IV.3 Finite-dimensional representations

447 447 448 459

V.

462 462 465 469

GALOIS R E P R E S E N T A T I O N S V.l Stable case V.2 Unstable case V.3 Nontempered case

REFERENCES

473

INDEX

483

P A R T I . ON T H E S Y M M E T R I C SQUARE LIFTING

INTRODUCTION Let F be a global field, Fv the completion at a place v, A the ring of adeles of F. Let H, or H 0 , be the F-group SL(2), and G the F-group PGL(3). This part studies the lifting (or correspondence) of automorphic forms of H(A) = SL(2, A) to those of G(A) — PGL(3,A). It provides an intrinsic definition of packets of admissible and automorphic representations of SL(2, Fv) and SL(2,A). This definition is not based on relations to representations of GL(2, Fv) and GL(2,A), but rather on character relations and the lifting. This approach applies to groups other than SL(n). The work establishes multiplicity one theorem for cuspidal representations of SL(2,A), proves rigidity theorem for packets of these, computes the multiplicity of a cuspidal representation in a packet of a cuspidal representation, and determines the self-contragredient admissible representations of PGL(3, Fv) and the self-contragredient automorphic representations of PGL(3, A). The lifting is compatible with the symmetric square (or adjoint) three-dimensional representation of the dual group H = PGL(2,C) of H in G = SL(3,C). It is defined by means of twisted character relations. It is studied here by means of comparison of orbital integrals and of twisted trace formulae. The interest in the symmetric square lifting originates from Shimura's work [Sm]. Let f(z) = YJT cne2vmz be a holomorphic cusp form of weight k and character u>, denote by ip a primitive Dirichlet character of Z with ipuj(—l) = 1, and suppose that

E c « " " s = n t d - %>p~s)(i - bpp-'T1n

p

Using Rankin's method Shimura [Sm] proved that the Euler product

7r-3s/2r(s/2)r((s + i)/2)r(i(s - k + 2)) x IJK 1 " V>(p)a£p-8)(1 -
is holomorphic everywhere except possibly at s = k or k — 1. 3

On the symmetric square lifting

4

Since / generates the space of a cuspidal representation n* of GL(2, A) (F = Q, with a discrete-series component 7rooo at oo), this statement can be put in terms of a lifting of automorphic forms compatible with the above dual group homomorphism which takes the diagonal complex matrix diag(a p , bp) to diag(a p , apbp, bp), or rather to diag(a p /6 p , 1, bp/ap) in a normalized (modulo the center) form. To reformulate Shimura's result Gelbart and Jacquet [GJ] put L2(s,TT0v,Xv)

= L(S,KQVXV

£2(s,nov,Xv;i>v)

=S(S,TTOVXV

X

^Ov)/L(s,Xv)

and x n0v;

ipv)/e(s,xv;i>v)

for any representation irov of GL(2, Fv) and character Xv of the multiplicative group F* of the completion Fv of F at a place v. Here 7Tot> denotes the contragredient of TTQV, and tpv is a nontrivial additive character of Fv. The representation TTOV is said in [GJ] to L-lift to a representation nv of Gv = G(FV) if nv is self-contragredient, and for any Xv, L(s,nvXv)

= L2{s,ir0v,Xv),

£{s,-nvXv\^v)

=

£2{s,^0v,Xv;^v)-

If 7r* is an automorphic representation of GL(2, A) and x is a character of A x / F x , put L2(S,-K*,X) = YIVL2(S,TTOV,XV)The main theorem of [GJ] is obtained on adelizing the method of [Sm]. It asserts that for any cuspidal representation IT* of GL(2,A) not of the form 7r*(Ind^(/i*)), see below, the function L2(S,TT*,X) is entire for all x- This refines the statement of [Sm], implies that each component irQv of 7r* L-lifts to some 7r„, and that 7r = ®VTTV is a cuspidal representation of G(A) = PGL(3, A). Our approach to the lifting is different; it is motivated by the ideas of Saito, Shintani and Langlands in the basechange theory. Following Shintani, the local lifting is denned by means of character relations, and following Saito, the global (and local) lifting is studied by means of the (twisted) trace formula. It is shown that the above n* (cuspidal, not of the form 7r*(Indf (//*)), lifts to a cuspidal 7r. This implies the holomorphy of L2{s,w*,x) = £(S)7rx) for all x- As obvious as it might be that the ideas of Saito and Shintani apply in our case too, the techniques required to carry out the work are less obvious. We describe them after we explain our results.

Introduction

5

To describe our work, let L(G) be the space of automorphic forms on G(A) — PGL(3,A). It consists of all right-smooth square-integrable complex-valued functions <j> on G\G(A), where G — G(F). The group G(A) acts on L(G) by right translation: (r(g)<j))(h) = (hg). The irreducible constituents -K of L(G) are called automorphic G(A)-modules, or automorphic representations of G(A) (see, e.g., [BJ]). Each such n is a restricted tensor product ®virv of irreducible admissible representations irv (see [BZ1]) of the local groups Gv = G(FV), which are unramified (contain a nonzero Kv = PGL(3, i?„)-fixed vector) for almost all v. Each irreducible unramified G^-module irv is isomorphic to the unique unramified subquotient of a G„-module I((mv)) normalizedly induced from an unramified character

i

of the upper triangular subgroup. The character (/x»„) is not uniquely determined. Yet we obtain a unique conjugacy class t(irv) = diag(/Ltj„(7r)) (where 7r denotes a generator of the maximal ideal in the ring Rv of integers in Fv) in the dual group G = SL(3,C) of G. The map nv >—• t(nv) is a bijection from the set of equivalence classes of irreducible unramified Gvmodules to the set of conjugacy classes in G. Similar description holds in the case of H = SL(2), where the automorphic representations no = ®v^0v have local components TTQV which are parametrized, in the unramified case, by conjugacy classes £(7To„) in the dual group H — PGL(2, C) of H. A 7To„ is called unramified if it contains a nonzero KQV = SL(2, i?^)-fixed vector. We study lifting of automorphic forms of H(A) to those of G(A), which is compatible with the symmetric square representation Ao = A = Sym : H - • G of H = PGL(2,C) in G = SL(3,C). This is the irreducible threedimensional representation of H. It can be described also as the adjoint representation of H on the Lie algebra of H. It maps the diagonal matrix diag(a, 6) to the diagonal matrix diag(a/6, l,b/a). We say that the automorphic H(A)-module ~KQ — ®V,KQV lifts to the automorphic G(A)-module TT — ®VTTV if t(nv) = Ao(t(7Tot-)) for almost all v (where TTQV and nv are both unramified). Our first global result asserts that each cuspidal H(A)-module lifts to an automorphic G (A) -module. This result is contained in [GJ].

On the symmetric square lifting

6

We obtain more precise results. To state them, we prove a special case of the principle of functoriality, thus we prove the existence of monomial representations for SL(2) and GL(2). Namely, let B b e a quadratic extension of F, put E1 — {z £ Ex; zz = 1} and AE = {z £ AE; z~z = 1} for the kernel of the norm map NE/E ° n Ex and Ag — bar denotes here the conjugation of E over F — and let //' be a character of CE = AlE/El. Denote by Wp the Weil group ([D2], [Tt]) of F. Let Ind B (/z*) be the two-dimensional complex representation of Wp induced from a character /j,* of CE = AE/EX = WEh = WE/E- It factorizes through the quotient WE/F of Wp, an extension of Gsl(E/F) by CE- If the restriction of fi* to CE is //, the image of Indg(//*) in PGL(2, C) depends only on //'. We denote it by Ind E (//)o. It is a two-dimensional projective representation of Wp. At a place v of F where Ev — FV@FV, /j* is a pair (niv, fi2V) of characters of CFV = Fx, the restriction of Indf (/i*) to WFV is the reducible /J,IV ®^2v, and we associate to it the normahzedly induced representation I(/J,IV,^2V) of GL(2, Fv), and to the restriction to Wpv of Ind B (^')o the normahzedly induced representation io(Mif/M2u) of SL(2,FV). At a place v of F where Ev is a field and fi'v is unramified, we associate to the restriction Ind^" (fi'v)o of Ind|J(//)o to WFV the induced IQ(XBV) of SL(2,FV), where xs„ is the character of Fx with kernel NEV/FVEX. If /z* is unramified, or more generally if //* = ~pl, then there is a character fiv of F„x with n*(z) = iiv(z~z) (z £ J3*), and we associate I(fiv,XEvHv) to

Ind£(/0. We prove that for each E and / / 7^ 1 i/iere e:ns£s a cuspidal representation 7To(//')) more precisely 7r 0 (Ind£(//)o), c/SL(2,A), ura£/i t/ie indicated components. Prom this we deduce that /or each E and fx* ^ 7^* (~p*(z) = /i*(2)) i/iere eaisfo a cuspidal representation ir*(n*), or rather 7r*(Ind^ ii*), o/GL(2, A), with the indicated components. Further we prove the existence of analogous local objects. The representations 7To(//) and 7r*(/i*) are called monomial. The existence of the representation 7r*(IndfJ/i*) was proven in [JL] by means of the converse theorem, and that of 7ro(Indf(/i/)o) was deduced from that in [LL]. As our work already contains this existence proof, we do not need to send the reader to study [JL]. It is clear that if n of PGL(3, A) is a lift from SL(2, A) then it is self-

Introduction

7

contragredient, or as we prefer to say: a-invariant. Here o is the involution /o

i\

of G given by a(g) = J*g lJ, J = I -1 I , and an{g) = n(a(g)) is the Vi 0/ contragredient n of n (see [BZ1]). A representation n is called a-invariant i f 7T ~

CT

7T.

Our next global result is a determination of the image of the lifting. Thus we prove that if n is a cuspidal G(A)-module which is a-invariant then it is a lift of a cuspidal H(A)-module TTQ. This no is not of the form 7To(Indf5(//)o) for any E, n'. The cuspidal H(A)-module 7To(IndE(/i')o), fJ-' / 1, lifts to the normalizedly induced, noncuspidal, cr-invariant G(A)-module I(n*(fi"),XE)- Here JJL"{Z) — n'(z/z), z € CE- Note that the central character of 7r*(/z") is XEIf / / = 1 then 7To(/i') is the induced 7O(X.E)> a n d it lifts to the induced I(XE, 1,XE)The trivial H(A)-module lifts to the trivial G(A)-module. This gives a complete description of the image. Indeed, any cr-invariant automorphic G(A)-module which is not in the above list, namely it does not have a trivial component, it is not cuspidal and it is not of the form I{-K*{IJL"),XE)I must be of the form I(ni, 1), namely normalizedly induced from a discrete-spectrum GL(2, A)-module n\ with a trivial central character. Such I(TTI, 1) are not obtained by the lifting. The notion of lifting which we use is in fact a strong one, in terms of all places. Namely we define local lifting of irreducible iJ^-modules to such G^-modules, and show that if no lifts to n, then nov lifts to nv for all places v. The definition of local lifting is formulated in terms of identities of characters of representations. It generalizes the notion of lifting of unramified local representations described above. The character relations compare the twisted character of nv, which is a astable function, with the sum of the characters of irreducible representations nov This sum is a stable function, depending only on the stable conjugacy class of the element where the characters are evaluated. We define the local packet of nov to consist of those representations which occur in the sum. Thus the local lifting asserts that it is not a single if^-module nov which lifts to nv, but it is the packet of 7To„ which lifts. This definition is inspired by our definition of packets of representations of the unitary group in three variables ([F3]) and of the projective symplectic group of similitudes of rank two ([F4]). The packet of an i?„-module 7rov coincides with the set of

8

On the symmetric square lifting

admissible irreducible i7„-modules of the form n^v (g in GL(2,F„)), where ^ovC1) = 'Kov{g~1hg) (h in Hv), and with the set of irreducibles in the restriction to SL(2, Fv) of a representation of GL(2, Fv). Given local packets Pv for each place v of F such that Pv contains an unramified i/„-module TTQV for almost all v, we define the global packet P to be the set of H(A)-modules ®virov with nov in Pv for all v and TTQV equivalent to TTQV for almost all v. We say that the packet is automorphic, or cuspidal, if it contains such a representation of H(A). In the case of G(A), more generally for GL(n, A) and PGL(n, A), packets consist of a single term. We are now in a position to state the main lifting theorem. The lifting defines a bijection from the set of packets of cuspidal representations of H(A) to the set of a-invariant representations of G(A) which are cuspidal or of the form J(TT*(//'), \E), p" ¥" ~P"• This permits the transfer of two well-known theorems from the context of G(A) = PGL(3, A) to the context of H(A) = SL(2, A). The first is a rigidity theorem for cuspidal representations o/SL(2, A). It asserts that if TTQ = „7Tot, and TT'0 = ®VTT'QV are cuspidal representations of H(A) and TXQV ~ ir'0v for almost all v, then TT0 and n'0 define the same packet. The analogous statement for GL(n, A) is proven in [JS]. It does not hold for SL(n, A), n > 3 (see [Bla]). The second application is multiplicity one theorem for SL(2, A). It asserts that each cuspidal representation of SL(2, A) occurs in the cuspidal spectrum of L(H) with multiplicity one. The analogous statement for GL(n) is well known (see [SI]). It holds for PGL(n,A) = GL(n,A)/A x , but not for SL(n), n > 3 (see [Bla]). Since the completion of our work other proofs of this result were claimed, but our technique of the trace formula still remains the most direct and transparent, being a part of a generalizable program. The rigidity theorem holds for packets, but not for individual representations. There do exist two inequivalent cuspidal H(A)-modules which are equivalent almost everywhere. The packets partition the discrete spectrum of SL(2, A). The packets 7To(//), or {TTO(H')}, form the unstable spectrum, and the other packets make the stable spectrum. The reason for these names is that the multiplicity of each irreducible in a stable cuspidal packet is 1. But the multiplicity is not constant on a packet {7r0(/i')}- To describe our formula for the multiplicity,

Introduction

9

note that when Ev is a field, if fi'v — 1, {7To(/0} — IO(XEV) has two constituents; if n'v2 ^ 1 the packet {no{n'v)} consists of two irreducibles; and if fi'v / 1 = /J,'V2 there are 3 quadratic extensions Eiv — Ev, E2v, E3v, and n'iv^l= fi'iv2 on E\v with n'lv = fi'v, and {n0(fJ.'lv)} = { T T O ( / 4 J } = {•7T0(fi'3v)} consists of 4 irreducibles. There are no other relations on the packets. The character relations partition each packet into two subsets 7TO~(AO and TTQ(^,'V) of equal cardinality (note that this partition depends on the characters p!iv when ylv ^ 1 = fi'v2, and TTQ(/J,'V) is unramified if n'v is unramified). Write e(w0v,fi'v) - ±1 ifn0v € ^(fj,'v), ande(7r 0 ,//) = E L ^ O U . A O Almost all factors are 1. When Ev = Fv © Fv, fj,'v is a character of F„* - {(x^-1) e £ * } = El { T T O ( / 0 } is / o K ) . a n d u n l e s s Hv ± 1 = &* this induced is irreducible, in which case TTQV is TTO(^'V) and TTQV is zero, and £(7TO„,AO =

1

-

The multiplicity of no in 7To(/x'), fi'2 / 1, in the discrete spectrum is m(ir0) = - f l + e(7r 0 ,/x')j. If / / / 1 = / / 2 there are 3 quadratic extensions E\ = E, E2, E3, and characters y!{ 7^ 1 = pi2 on Cg = A ^ / E 1 with /x^ = / / and {7TO(M'I)} = {7ro(/X2)} = { ^ ( M S ) } - We have Ili<»<3 e ( 7r 0jA t i) = 1> a n d a n irreducible TTQ in such a packet has multiplicity

m

(*o) = T

1 +

H

£ 71 0

( " '^)

Ki<3

in the cuspidal spectrum. There are no other relations among the packets. Another corollary to the lifting theorem asserts that a cr-invariant cuspidal G(A)-module cannot have a component of the form I(TTIV, 1), where niv is a square-integrable representation of GL(2, Fv). Further, if no is a cuspidal GL(2, A)-module with a local component I(l^iv^'t,l^2vl'vt)> t — 0, normalizedly induced from the character ( J J ) H ^iv(a)^v(b)\a/b\l of the upper triangular subgroup, n\v, ii2v unitary, then we conclude (as in [GJ]) that t < \. The estimate t < | follows from unitarity, and the equality t = 0 is asserted by the Ramanujan conjecture for GL(2,A).

10

On the symmetric square lifting

As a final corollary we note that for cuspidal TTQ which is not of the form 7ro(Ind B (//)o), since the L-function L2(s,ir0,x) is equal to L(s,nx), where IT is the lift of TTQ, we conclude, as noted above, that it is entire for each character x of A x /Fx. An irreducible representation IT of GL(3,F), F local, is said to be essentially self-contragredient if its contragredient it is equivalent to the twist •nx of 7r by a character x '• GL(3,F) - » F X , J H x(det#). If the central character of such a 7r is denoted by w, then IT(OJX)~X has trivial central character and is self-contragredient. Indeed it ~ -KX implies that u>~1 — u>x3, thus x — (<^x)~2> hence fr ~ 7r(wx)~2 and (7r(wx)_1)" — 7r(wx) _1 - The central character of this last representation is w(u>x)~3 = 1- Thus the essentially self-contragredient representations of GL(3, F) are twists by characters of self-contragredient representations of PGL(3, F), characterized by our work. The a-invariant representations TTV of Gv not in the image of the Ao-lifting are of the form I(TTV, 1), where irv is a representation of H\v = Hi(F w ), H j = PGL(2). This lifting, Ai, occurs naturally in our trace formulae comparison. In fact our two liftings, from Hv and from H\v to Gv, are best described as liftings compatible with the natural embeddings of the two elliptic u-endoscopic subgroups H — PGL(2, C) and Hi = SL(2, C) of G = SL(3,C). These a-endoscopic subgroups are simply the cr-centralizers of iT-semisimple elements in G. The character relation which defines the lifting from Hv to Gv takes the form x%{S) = X{Tr0}(N8). Here \% indicates the twisted character of the local representation n. It is a function of tr-conjugacy classes 5 in Gv. The X{TT0} is the character of the packet {no} (sum of characters of the irreducibles in the packet). It is a function of the stable conjugacy classes in Hv. The character of a single representation of Hv is a function of conjugacy classes in Hv, but it may be nonconstant on the stable orbit (rational points in the orbit under SL(2,F^), or under GL(2,F„) in our case). To state the character relation we need a notion of a norm map N. It relates stable a-conjugacy classes in Gv with stable conjugacy classes in Hv. It generalizes the natural norm map diag(a, b, c) i—• diag(a/c, l , c / a ) . A consequence of the existence of the character relation is that the twisted character of the lift ir = A0(7r0) is a stable cr-conjugacy class function,

Introduction

11

namely it is constant on stable u-conjugacy classes. Moreover the lifting relates a packet of Hv, not an individual representation. The simple looking lifting Ai : H\v —> Gv, -K\ i-> IT = I(ni,l), is also defined by means of a natural yet very interesting character relation, which takes the form A(6a)x^(S) = K(5)Ai(N1S)xn1(NiS). Here A and A x are some Jacobians (which appeared also in the case of Ao but were equal to each other in that case). The function Ni : Gv —> H\v is a norm map, relating stable cr-conjugacy classes in Gv with conjugacy classes in H\v. This N\ generalizes the natural norm map diag(a, b, c) H-> diag(a/c, l,c/a) if H\v is regarded as SO(3, Fv). A stable conjugacy class in H\v consists of a single class. However, an elliptic cr-conjugacy class in Gv consists of two cr-conjugacy classes. The character K assigns the values ± 1 to these two classes. It follows from this character relation that the -K which are A i-lifts (of elliptic 7Ti) are cr-unstable, that is, their cr-characters are not constant on the stable cr-conjugacy classes. This surprising fact is interesting and merits an independent local verification. In the last chapter of this part we give an independent, direct computation of the very precise character calculation, by purely local means, not using the trace formula and global considerations. This gives another assurance of the validity of the trace formula approach to the lifting project. The present volume is based on the series of papers [F2;II], . . . , [F2;VI] in our Symmetric Square project, as well as on the papers [FK4] with D. Kazhdan and [FZ1] with D. Zinoviev. In these papers an attempt has been made to isolate different ideas or techniques and make them as independent as possible. The initial results and some of the techniques had been described in [F2;VIII], and the preliminary draft [F2;IX]. The publication of a series of papers could lead to confusion, as to what is the final outcome. Some techniques and results were not known or foreseen at the initial stages. Now that the work reached a stage of completeness, we rewrote it in a unified, updated form. Not all the material in [F2;II], . . . , [F2;VI], [FK4] and [FZ1] is used here. In addition to rearranging the material our foci of interest shifted. For example, §4 of the paper [F2;II] was made redundant by [F2;VII], so we use only the fundamental lemma of [F2;VH] in our section II. 1 here. The second half of [FK4] is no longer needed, as it is replaced by [F2;VII], but

12

On the symmetric square lifting

its first half is used as the basis for [FZ1] in our chapter VI below. In particular the chapters in the present part are labeled I to VI. They are not linearly related to the papers, but chapter I here is related to [F2;III], II to [F2;VII] and [F2;II], III to [F2;IV], IV to [F2;VI], V to [F2;V], and VI to [FK4] and [FZ1]. We refer to the current part as [F2;I]. The contents of the chapters are as follows. The basic definitions of local lifting of unramified and ramified representations are given in chapter I. To study the a-invariant G(A)-modules ir not obtained by the lifting we introduce in section 1.1 the the map Ai: H\ —> G, where Hi = SL(2,C) is the dual group of H i — PGL(2) = S0(3), in addition to the symmetric square map Ao: HQ —> G. We then introduce the dual maps A*: H Q —> Hj from the Hecke algebra H G of spherical functions on Gv to the Hecke algebras Hj of Hiv (i = 0,1). In 1.2 we define a norm map 7 — N6 from the set of stable cr-conjugacy classes of 5 in G to the set of stable conjugacy classes of 7 in H. In section II. 1 it is shown that the stable twisted orbital integral of the unit element of the Hecke algebra of PGL(3, Fv) is suitably related to the stable orbital integral of the unit element of the Hecke algebra of SL(2, Fv). Moreover, the unstable twisted orbital integral of the unit element on PGL(3, Fv) is matched with the orbital integral of the unit element on PGL(2, Fv). Thus these functions have matching orbital integrals. This statement is called the Fundamental Lemma (in the theory of automorphic forms (via the trace formula)). The direct and elementary proof of this fundamental lemma which is given here is based on a twisted analogue of Kazhdan's decomposition of compact elements into a commuting product of topologically unipotent and absolutely semisimple elements. In section II.3 we transfer smooth compactly supported measures fvdgv on Gv to such fovdhv on Hv. The definition is based on matching stable orbital integrals. Similar discussion is carried out for the transfer from Gv to H\v. In chapter III we give the global tool for the study of the lifting, an identity of trace formulae. First we compute the trace formula for G(A) twisted by the outer automorphism a. Since a does not leave all parabolic subgroups of G invariant, we introduced in [F2;IX] a modification of the truncation used by Arthur [Al] to obtain the trace formula. The subsequent computation of the twisted trace formula was carried out in [CLL],

Introduction

13

from which we quote (in section III.2) the contribution from the Eisenstein series. Thus in chapter III we compute explicitly all needed terms in the twisted formula, stabilize it, and compare it with a sum of trace formulae for H(A) = SL(2, A) and H i (A) = PGL(2, A). The formulae in this chapter III are greatly simplified by the introduction of regular functions (see below). In section V.l we give an approximation argument to deduce from the global identity of trace formulae the local (hence also global) results. It is a new argument. It replaces the technique of [L5], which relies on the theory of spherical functions. The new argument is based on the usage of what we call regular functions, which are not spherical but in fact lie in the Hecke algebra with respect to an Iwahori subgroup. Their main property is that they both isolate the representations with a vector fixed by an Iwahori subgroup and their support is easy to control and work with, in contrast to that of a spherical function. The approximation (or separation) argument given here applies in any rank-one situation (since there are only finitely many reducibility points of principal-series representations in this case) and does not use spherical functions at all, except the case where /o„ is the unit element f$v of H 0 and fv is the unit element in H G , which is proven in section II. 1. In deriving the main theorems in section V.2 we use the immediate twisted analogue of Kazhdan's fundamental study of characters [K2]. This is formulated in section 1.4. It is not proven here since the proof is entirely parallel to that of [K2] and requires no new ideas (cf. [F1;II] in the case of any reductive group). The only nonimmediate result needed to twist [K2] is the analogue of [K2], Appendix. This is done in [F1;II], (1.4), in general; the special case needed in this chapter is done here in V.1.7. In III.3.5, together with V.1.6.2, we give a new argument for the comparison of trace formulae for measures fdg = ®vfvdgv such that the transfer fxudhv of fudgv vanishes for some u. This new argument uses the regular functions mentioned above to annihilate the undesirable terms in the trace formula. It replaces the technique of [L5], which relies on the computations of singular and weighted orbital integrals and the study of their asymptotic behavior, and the correction technique of [F1;III]. In chapter IV this argument is pursued to give a simple proof of the comparison of trace formulae for all test measures fdg. Thus in chapter V we can deal with all

14

On the symmetric square lifting

automorphic representations o/H(A). The method of chapter IV establishes — by simple means — trace formulae comparisons also in other rank-one situations. This method may generalize to deal with groups of arbitrary rank and may give a simple proof of any trace formulae comparisons for general test functions, but we do not do this here. It affords a simple proof of the basechange lifting for GL(2) (see [F1;IV]), and its analogues for the quasi-split unitary groups U(2, E/F) and U(3, E/F). See [F3], where the automorphic and admissible representations of U(2) and U(3) are classified, and compared with those of the related general linear groups GL(2) and GL(3), and both rigidity and multiplicity one theorems for U(2) and U(3), are proven. The approach of [F3] — reducing the study of the representation theory of U(3, E/F) to basechange lifting to GL(3, E) — was found by us by direct analogy with the techniques of the present part. Our character relations, in V.2, take the form toirv(fvdgv

x a) = (2m + 1)

y^ytVK0v(fOvdhv),

where the sum ranges over the TTQV in the packet Tro^Ind^1 ((JL'V)O), where -rrv — I(ir*(fJ."),XEv), and m is a nonnegative integer. Multiplicity one theorem for SL(2, A) requires that and follows from: m — 0. We provide two independent proofs that m = 0. One proof is global. It appeared already in [F2;V]. It is based on a remarkable result of [LL], 6.2 and 6.6, essentially derived only from properties of induction, that if 7To is cuspidal and lies in a packet 7ro(lndE(/j,')o), it occurs with multiplicity one in the discrete spectrum. All other cuspidal representations in the packet of such no are TTQ, g € GL(2, F), but those of the form Trg, g e GL(2, A) - GL(2, F)G{n0), G(ir0) = {g € GL(2, A); Trg = n0}, are not automorphic. The complete proof is given in V.2.3-V.2.4. In V.2.5 we give a new, purely local proof that m = 0. It is based on a twisted analogue of a theorem of Rodier, proven in V.3, which encodes the number of Whittaker models of a representation in its character near the origin. Since nv is generic, we conclude that m = 0 and only one -KQV in its packet is ^-generic, for any character Vv The present work can be viewed as the first step in the study of the self-contragredient representations of GL(n). This would lead to liftings of representations of symplectic and orthogonal groups of the suitable index to

Introduction

15

the GL(n) in question. In the present work the twisted endoscopic groups are the symplectic group Sp(l) = SL(2) and the orthogonal group SO(3) = PGL(2). The next work in this project has recently been studied in [F4] in the case of PGL(4), and its twisted endoscopic groups PGp(2) and SO(4). As noted above, chapter VI offers a new technique to compute a special twisted character. The approach of chapter VI is different from the wellknown, standard techniques of trace formulae and dual reductive pairs. It will be interesting to develop this approach in other lifting situations. A first step in this direction was taken in the work [FZ2], where the twisted — by the transpose-inverse involution — character of a representation of PGL(4) analogous to the one considered in chapter VI, is computed. The situation of [FZ2] — see also [FZ3] — is new, dealing with the exterior product of two representations of GL(2) and the structure of representations of the rank-two symplectic group.

I. F U N C T O R I A L I T Y AND NORMS Summary The symmetric square lifting for admissible and automorphic representations, from the group H - H 0 = SL(2), to the group G — PGL(3), is denned by means of character relations. Its basic properties are derived: the lifting is proven for induced, trivial and special representations, and both spherical functions and orthogonality relations of characters are studied. The definition is compatible with dual group homomorphisms A0 = Sym 2 : H = PGL(2, C) = SO(3, C) --> G = SL(3, C) and Ai : # 1 = SL(2,C) -> G, where H i = PGL(2). Of course it will be compatible with the computation of orbital integrals (stable and unstable) in chapters II and III.

Introduction In this chapter we define the symmetric square lifting in terms of character relations, and derive its basic properties. This work is required for the study of the lifting of automorphic forms of H(A) to G(A), where H = H 0 = SL(2) and G = PGL(3), by means of the trace formula. The lifting is suggested by the symmetric square, or adjoint, representation A0: H -» G of the dual group H = PGL(2, C) of H in G = SL(3, C). Put lg =transpose of g, and

T{g) = Jtg-1J,

J

^C1-\)'

s=

(~1_1i)-

The group H is a (7-endoscopic group of G (see [KS]). Indeed, H = SO(3, C) is the group Z~(a) = {g € G;ag = g} of points fixed by a in G. It is elliptic (H is not contained in a u-invariant proper parabolic 16

1.1 Hecke algebra

17

subgroup of G). But G has another elliptic cr-endoscopic group, which is H i = PGL(2): Ai: Hi = SL(2, C) = Z~(*r) = {g E G; sa(g)s = g} — G,

* - ( : : ) " * • = ( ; ' ! ) •

Via the Satake isomorphism, the maps Aj formally define the lifting IT — Xi(-Ki) of unramified /^-modules 7I-i to unramified G-modules IT. Moreover, we introduce in section 1 (of this chapter I) the dual maps A*: H —> H* from the Hecke algebra H of G to the Hecke algebra H* of Hi. It follows from the definitions that if /* — A*(/) then the spherical functions / and fi have matching orbital integrals on the split tori. In section 2 we define lifting, denoted 7Tj = Aj(7r), of admissible representations 7Tj of Hi to such representations IT of G, by means of character relations. The definition generalizes the spherical case, and uses packets rather than a single irreducible. Basic examples of the stable lifting A0 are given. These concern induced, trivial, and special representations. Section 3 concerns orthogonality relations for characters, needed in our study of the local lifting. The cases of cuspidal G-modules and Steinberg IT are standard but useful. We also record the twisted orthogonality relation for two tempered G-modules which are not relevant. The proof follows closely that of the nontwisted case by Kazhdan [K2]. It depends on the twisted analogue of the crucial appendix of [K2]; this is proven in [F1;II] for a general group, and in chapter V, (1.8), in our case.

1.1 Hecke algebra 1.1 Dual groups. Let F be a global or local field of characteristic zero. Put G = PGL(3), H = H 0 = SL(2), and H i = PGL(2) = SO(3), viewed as Z-groups. For any field k denote by G(fc), H(fc) and Hi(/c) the group of fc-rational points of G, H and H i . We write G' for the group G'(F) of F-rational points, for any algebraic group G' over F. Fix an algebraic closure F of F. Let G = SL(3, C) be the connected dual group of G (for any reductive group G the connected dual group G is defined in [Bo2], where it is denoted

/. Functoriality and norms

18

by L G°). Consider the semidirect product G' = G xi (a); (a) denotes the group generated by the automorphism a(g) = Jtg~1J of G of order 2. The dual group H of H is PGL(2,C) -^SO(3,C). It is isomorphic to the centralizer of 1 x a in the connected component of 1 in G', and to the cr-centralizer G° — {g in G;g~1a(g) = 1} of 1 in G. The isomorphism is given by / ab\

1 /

°2

ab

^

b2

\

f ° , J I—>" — { acV2 ad+bc bds/2 1

(x = ad — be).

This map will be denoted by A and by Ao: H —• G. The dual group # 1 of H x = PGL(2) is SL(2, C), and the map

embeds i?i in G. The image is the centralizer of s x a in G, where s is the diagonal matrix diag(—1, — 1,1). Equivalently, it is the cr-centralizer G% = {g G G; s a ^ s " 1 = 5 } of s in G. 1.2 Hecke algebra. Let F be a p-adic field, R = {x in F ; |a;| < 1} its ring of integers, and K — G(R) the standard maximal compact subgroup of G. Fix a Haar measure dg on G. The Hecke algebra HI — MQ is the convolution algebra CC{K\G/K) of complex valued compactly supported ii'-biinvariant measures fdg on G. Such fdg are called spherical. Let 7r be an admissible irreducible representation of G on a complex vector space V. A representation n is called smooth if each vector is fixed by an open subgroup of G. It is called admissible ([BZ1]) if it is smooth and if the subspace of V of vectors fixed by any open subgroup is finite dimensional. A smooth irreducible representation is admissible by a wellknown theorem of Bernstein. Put air(g) = 7r(ag) (g in G). Then CT7r is an admissible irreducible representation of G on V. We say that TT is a-invariant if w is equivalent to a7r. In this case there is an invertible operator A: V —> V with ir(o-g) = Ai:{g)A~1 (g in G). Since IT is irreducible and A2 intertwines ir with itself, Schur's lemma ([BZ1]) implies that A2 is a scalar. Multiplying A by 1/vA2, we assume that A2 — 1. Then J4 is unique up to a sign. We put n(cr) = A, and define the operator n(fdg x a) = ir(fdga) = w(fdg)ir(cr) to be the map v >-> / f(g)n(g)Avdg.

1.1 Hecke algebra

19

If fdg is spherical (in MQ) then n(fdg) factorizes through the projection to the space irK of /sT-fixed vectors in (IT, V). If TT is irreducible, dime nK < 1. The representation 7r is called unramified if irK 7^ 0. Then (k € K) acts as the identity on ixK. If n is irreducible, n(fdg) / 0 implies that the image nK of ir(fdg) is one dimensional. If 7r is unramified, it lies in a representation I — I{TJ) of G induced from an unramified character 77 of the upper triangular Borel subgroup B — TN (e.g., [Bo3]). Here N denotes the unipotent upper triangular subgroup, and T denotes the diagonal subgroup. In fact n is the unique unramified constituent in the composition series of / . Fix v in V so that w = n(fdg x a)v is nonzero. Since a{K) = K, Aw is also a .ff-fixed vector, and Aw / 0, since A(Aw) = w ^ 0. Hence there is a constant c with Aw = cw. As A2 — 1, c is 1 or — 1. We replace A by cA to have Aiu = w. This normalization is compatible with the normalization for generic representations, see chapter V, (1.1.1). The character rj is given by rj(6) = /X!(a)/i 2 (bWc) at an element S — diag(a, b, c) in the diagonal torus T of G. Here /Zj are characters of F x with /ii/X2M3 = 1- The induced representation I = I(T]) consists of all (right) smooth cf>: G —> C with (g),

g&G,

n&N,

5eT.

The action is by right translation: (I(g)(f>)(h) = (hg). The value of the factor 6(8) = |det(Ad(<S)|LieJV)|

is

\a/c\2.

Here Lie N denotes the Lie algebra of N. Let IT be a generator of the maximal ideal in the ring R of integers of F. Consider the element t = diag(/zi(7r),/x 2 (fr),/i3M) in the diagonal torus T of G. Then the equivalence class of the unramified representation -K is uniquely determined by the conjugacy class in G of t.

20

I. Functoriality and norms

1.3 Orbital integrals. Fix a Haar measure da on the diagonal torus T. The normalized orbital integral

F(6, fdg) = A(6) J fiaSg-1)^

(g e G/T),

where A(5) = < 5- 1 / 2 (5)|det(l - Ad((5))|Lie JV| =

a—bb—ca—c a b c

depends only on the image of 6 = diag(a, b, c)

in

T/T(R)

~ X , ( T ) = Hom(G m , T)

when fdg is spherical. Indeed, writing g = anik (and dg/da = dndk) and introducing n by n^l5n\ = Sn, changing variables on n in the orbital integral gives the factor

ll-AdtOr^lAd^lll-Adtf)!- 1 . Hence F(6Jdg)=61/2(S)

[ fK(5n)dn, JN

where

fK(g)=

[

/(k^g^dk.

JK

We denote this value of the orbital integral by F ( n , fdg), n being the image of 5 in .X*(T) ~ { ( n i , n 2 , n 3 ) ; n* G Z}/{(n,n,n); n G Z}. For i = diag(ti,i 2 ,i3) in T and n = ( n i , n 2 , n 3 ) in X»(T), we put n(i) = 4nxj.n24.n3 L

l

L

2

L

3

•

The Satake transform (fdg)' of /dg is abbreviated to / and is defined by

f(t) = |T(i?)| Y,F(n,fdg)n(t)

(n G X*(f) ~ X.(T)),

n

where |T(i?)| denotes the volume of T(R) =T(1K with respect to dt. The map fdg — t > / is an isomorphism from the algebra MQ to the algebra C[T] W of finite Laurent series in t G T which are invariant under the action of the Weyl group W of f in G.

1.1 Hecke

algebra

21

Let C£° (G) denote the space of all smooth compactly supported complex valued functions on G. If ir is an admissible representation, for any / in C£°(G) the operator Tt(fdg) = fG f(g)n(g)dg has finite rank. We write tnr(fdg) for its trace. If n is irreducible but not equivalent to °7r, then tv-K(fdgxa) is zero. If n is irreducible and unramified, and fdg is spherical, then n(fdg) is a scalar multiple of the projection on the K-fLxed vector w. If, moreover, -K ~ °7r, then ir(a) acts as 1 on w, and tnr(fdgxcr) = tvir(fdg) is this scalar. Let us compute it. 1.4 LEMMA. Suppose that IT is unramified and t = t{rj) = t(n) is a corresponding element in T. Ifar\ = n, then for any fdg in MQ we have tm(fdg

xa)=

tm(fdg)

= f(t).

P R O O F . Corresponding to g — nak there is a measure decomposition dg = 6~1(a)dndadk. For a test function / e C£°(G) the convolution operator n(fdg) = JGTr(g)f(g)dg maps 4> e n to

f(h-lg)<j>(g)dg

(ir(fdg)4>)(h) = f f{g)4>{hg)dg = / JG

= f

f

JG

f{h-1n1ak)(61/2r])(a)(t)(k)S-1(a)dn1dadk.

f

JNJTJK

The change of variables n\ i—• n, where n is defined by n~1ana~1 — m, has the Jacobian |det(l -*• Ada)|LieiV|. The trace of ir{fdg) is obtained on integrating the kernel of the convolution operator — viewed as a trivial vector bundle over K — on the diagonal h — k € K. Hence trn(fdg)=

I

f

{Arj){a)f(k-1n-1ank)dndadk

f

JK JN JT

= [ n{a) Wa) JT L

f{gag-1)^aa

f JG/T

1.5 DEFINITION. For 5 in T

da.

•

put

^(StT, fdg) = f JT"\G

figto-ig)-1)dg_ da

22

/. Functoriality and norms

Here Ta = {a € T; a(a) = a} is the group of
= ^- 1 / 2 (<5)|det(l -

Ad{5)a)\UeN\.

Note that

>-f K

|det(l-Ad(<J)o-)|LieW| =

jVa^)

=

|a/c|,

hence A(5cr) = Ao(N5), where N5 — diag(a/c,c/a). Here A 0 (diag( a; ,y)) = | ^ - 2 / ) 2 / a ; 2 / | 1 / 2 is the usual A-factor on GL(2). We usually use indices 0, 1 or 2 for objects related to H = H 0 = SL(2), H i = PGL(2), and GL(2), respectively. 1.6

LEMMA.

For any character rjofTwe

tr I{r]; fdgxa)= PROOF.

have

- [77(a) + rj(a))F(ao; fdg) da.

For n = I(rj), we have f(a(h)-1g)4>(g)dg

(n()(h) = [ f(g)ct>(a(h)g)dg = f JG

= f

f

JG 1

f{a{h- )nak){S T])(a)(p(k)S-1(a)dndadk.

f

1/2

JNJTJK

Hence tiir(
[

I

f{a{k)-lnlak){S-l'2r]){a)dnldadk.

[

JK JN JT

We change variables n\ i-> n, where a(n)~1ana~1 = n i , which has the same Jacobian as if naa(n)~1a~1 = n\, which is |det(l - Ad(a)
x cr) = tTir(afdg) = /

n(a)A(aa)da 1

JT/T -"

/ JT"\G

f{^{g)~lag)~. aa

D

1.1 Hecke algebra

23

1.7 Cases of H and H i . Considerations analogous to (1.3), (1.4) apply in the cases of the groups H = H 0 = SL(2) and H i = PGL(2) ~ SO(3), with respect to the maximal compact subgroups Ki = H$(.R). Unramified an representations no, n\ are associated with IQ(HI,(J,2)> hd^,^"1) d their classes are represented by to = diag(zi,z 2 ),

ti = diag(z,z _ 1 )

in H0, Hi. Here z^ = faiir), z = n(ir). For fidhi in the Hecke algebras H, of compactly supported -RTj-biinvariant measures on Hi, the Satake transform is

f0{^g{zi,z2))=\To{R)\YJF{n,fodh0){zi/z2)n, n 1

fi(di&g(z,z- ))=\Ti(R)\Y^F(n,fidhi)zn. n

The symbol |TJ(JR)| denotes the volume of Tj(i?) — TiD Kt with respect to dai. The expression F(n, fidhi) denotes the normalized orbital integral of fidhi at regular elements diag(a, b) in Ti (diagonal subgroup of Hi) with valuations (n, —n) (i = 0) and (1711,7712), mi — 7772 = n (i = 1). It depends on the choice of Haar measures dhi, dai on Hi, TJ; but fi depends only on dhi. The standard computation of (1.3) shows that for spherical fidhi, 7Tj, we have tr TTi(fidhi) = fi(ti) (ti = tj(7rj)). Recall (1.1) that we have maps A;: Hi —» G and ((1.2), (1.5)) classes t$, t in Hi, G for unramified representations 7Tj, IT of Hi, H (i — 0,1). 1.8 DEFINITION. The unramified representation 7Tj lifts to if t = Xi(U). In this case we write 7r = AJ(7TJ).

IT

through Aj

The maps A*: HI —> Mi dual to A; are defined by fidhi = X*(fdg) if fiik) = /(Ai(ti)) for all U in %. Equivalently, fidhi = X*(fdg) if tnri(fidhi) = tm(fdg x a) for all -Ki and n = Xi(ni). Note that 7r = Aj(7Ti) if and only if fi(U) — f(t), where U = Ufai), t = t(w), for all fdg and fidhi =\;(fdg). Note that J 0 (M) = loifJ-, 1), h(n) = hin, H^1) both lift (through A0,Ai) to /(//, 1,/x -1 ).

24

I. Functoriality and norms

There are several formal consequences concerning orbital integrals of measures fdg, fidhi related by fidhi = X*(fdg), as these integrals are the coefficients of / and fi. 1.9

LEMMA.

//

5 = diag(a, b, c),

7 — diag(a/c, c/a),

and

ji = diag(a, c),

then F(Sa, fdg) = F(j, f0dh0) PROOF.

and

F(Sa, fdg) = F ( 7 l , /id/»i).

If t\ — diag(t,t _ 1 ) lies in f\ then \T(R)\

^(m.Wmi-ms=/(Aa(ii))

£ m=(mi

,rri2,7713)

= /i(*i) = i T i ^ l ^ F ^ / i d / n ) * " . Comparing coefficients of £n we obtain \Ti(R)\F(n,

fidhi)

=

Yl

\T(R)\F{m,fdg).

{ m ; m i —7713=71}

A simple change of variables shows that this is the product of |TCT(i?)|, where T°(R) = {teT(R);t = a(t)}, and F(na, fdg) = A(5a) j f{g-l5a{g))

dg,

where 6 = diag(a, b, c),

7 = diag(a/c, c/a),

\a/c\ = |7r|n.

It is clear that the integral depends only on n, but not on the choice of S. In the case of H 0 — SL(2), taking a representative t0 = (t, 1) in T0 we have | T ( # ) | ^ F ( m , / c ^ m ™ = /(Ao(io)) m

1.2 Norms

25

\T0{R)\Y,F{n,fQdh0)tn.

= /o(*o) =

n

Hence F(na, fdg) = F{n, fodh0).

D

(1) We normalize the measures so that |Tj(i?)| = |TCT(i?)|; the groups T* and T" are isomorphic to the multiplicative group G m . (2) Every f\ is so obtained from some / , hence the / i separate the TT\. Every /o is so obtained from some / , hence the /o separate the TTQ. REMARK.

1.2 N o r m s 2.1 Stability. To extend the study of lifting from the unramified case to any admissible er-invariant representation, we need to define norm maps N and N\ to extend the definition suggested by the formal Lemma 1.9 on diagonal matrices. Thus for 8 = diag(a, b, c) we put: N(6) — diag(a/c, c/a)

and

Ni(6) = diag(a, c).

These norm maps will be used to relate orbital integrals and characters, so they should be defined in terms of (twisted) conjugacy classes. More precisely, the norm will be defined to be a map from the set of regular stable cr-conjugacy classes in G to the sets of regular stable conjugacy classes in H and H\. We begin with a description of these classes. Let F be a local or global field of characteristic 0. Fix an algebraic closure F of F. Let G be a reductive group defined over F and G = G ( F ) the group of F-rational points of G. Denote by a an automorphism of G defined over F. The elements 8, 5' of G are called a-conjugate if there is h in G with 6' — h6a(h~l). They are called stably a-conjugate if there is h in G(F) with 8' — hSa(h~1). The term (stable) conjugacy (no mention of a) is employed if a is the trivial automorphism. The stable cr-conjugates of 8 in G are described by the set A(8) of g in G(F) with gSaig-1) in G. The map A(S)^H1(F,ZG(8a)),

g , - {r . - gT =

where ZG(So-) = {geG;g6o-(g-1)

= 8},

g^g)},

I. Functoriality and norms

26

factors through

1 __+ D(S)^H1(F,ZG(Sa))

—

ff^G),

where the double coset space D(S) = G\A(6)/ZG(SCT)(F) parametrizes the cr-conjugacy classes within the stable cr-conjugacy class of 6. The definitions introduced above will be used with G = PGL(3) and the (involution) outer automorphism a{g) = Jtg~lJ, and also with H = Ho = SL(2), H i = PGL(2) = SO(3) and the trivial a. If 7 G H, Z H (7) denotes the centralizer of 7 in H. Similarly, Z H I ( 7 I ) is the centralizer of 71 G Hi in H j .

Note that every conjugacy class of Hi (and of GL(n,F) or PGL(n, F)) is stable. Indeed, the centralizer of a semisimple element 7 in GL(n, F) is a product FT F x , where F j are field extensions of F with J2A^j '• F] = n. We have ^ ( ^ G m ) = {0}, hence .0(7) is trivial for GL(n,F) or PGL(n.F). However, for H = SL(2), the centralizer in if of a nonsplit 7 is E1 = kerNE/F, where E = F(j) is the extension generated by 7. Hence the set of conjugacy classes within the stable conjugacy class of a regular 7 in H is parametrized by Fx /NEX, which is Z/2Z when F is local and 7 is elliptic, and {0} when the eigenvalues of 7 are in F x . For this we need to compute if 1 (F, T) = H1 (Gal(F/F), F x ) where T is G m over E and a ^ 1 in G a l ( F / F ) acts on T ( F ) = Ex by er(x) = :r _ 1 (2: is the conjugate of x in 15 over F ) . Then a cocycle is 6 = 6CT G F x with 1 = 6CT2 — bacr(ba) — 6/6, thus 6 G F x . The coboundaries are 6/, 1) G T ( F ) S L ( 2 , F ) where T(F) is the centralizer of 7 in GL(2, F ) . To realize 7' as g-^g,

g G SL(2,F), we solve ( J ? ) 5 " 1 =

( ^ ) ,

a; = xi + x'2y/A £ E, y = y[ + y'2\fA G F , thus x 2 - y2vl = D. The solutions are x — xi(x2 + 1) + X2\/A~, y = a;2 + 1 + ^p-v^A, provided

1.2 Norms

27

2^2 + 1 = -z&A- We take xx = 0. Then x2 = - | ( f + 1), x =

x2VA,

y = a; 2 + 1 = | ( 1 - ^ ) . Then

satisfies 5
-D/A)^1

where ,

/

1 VA\

/io^(/io) _1 = ( A -

2

V 2v / A

fabA\

n'4 ) '

,~lfa+bA

hogaihog)'1

/

=

0

\, -2AVA D

2A -/A

0

and

so that g satisfies 7' =

g~l^g.

2.2 The Norm. Let <5 be an element of G. The set of eigenvalues of 5a(5) is of the form {A, 1, A - 1 } . Indeed, if A is an eigenvalue of 5 a (6) then there is an eigenvector v with t(5cr(6))v = Xv. Hence X-1v = \8a{5))-1v,

and

A_1(<JJi;) = S^S'1

J{5Jv),

that is, A - 1 is also an eigenvalue. It is clear that A € Fx or that [F(X) : F} = 2. The element 6 of G is called a-regular if the eigenvalues A, 1, A - 1 of 6a(5) are distinct. In this case let N5 be the class in H determined by the eigenvalues A, A - 1 and N±S the class in H\ with eigenvalues of ratio X if H i is viewed as PGL(2), or with eigenvalues A, 1, A - 1 if H i is viewed as SO(3). For any h = (* vt) in GL(2, F) we put

Assume that 6 is cr-regular. Replacing 6a(5) by a conjugate g~l5o~(5)g, hence 6 by a cr-conjugate g~1Sa(g), we may assume that 5
28

/. Functoriality and norms

hi. Since 5J takes A-eigenvectors of \Sa(S)) to A - ^eigenvectors of 5a(5), the assumption 6a(6) = hi implies that 5 J fixes the subspaces ( * ) , ( ° So does S. Hence multiplying by a scalar we have 6 = ai for some a in GL(2,F). Note that if 6 = (ae)i, then N5 — hi; here h = aewta~1ew =

a2. det a

If 5' = (a'e)i and 5' = p-Haifi) [hence 5'a(5') = (3~15a{5)l3 and 0 = h for some b in GL(2, F)], then a'e = b^1aewtb'1w and 1 a' = b^aiewfb^iew)v ; v

'

=

-L-b^ab.

det 6

Hence 5, 6' are (stably) cr-conjugate if and only if a, a' are projectively (stably) conjugate. 2.3 LEMMA. For any given regular 7 in H there is a unique stable aconjugacy class of 5 with N5 = 7. The a-conjugacy classes within such a stable class are parametrized by u in F x / N E X , E — F(5a(5)). A set of representatives is given by 6 = (uae)i. If the eigenvalues A, 1, A - 1 of 5a(5) are distinct then they lie in a quadratic extension of F (or in F) and define a stable conjugacy class N6 in H with eigenvalues A, A - 1 , and a conjugacy class NiS in Hi with eigenvalues A, 1, A - 1 in SO(3, F) or A, 1 in PGL(2, F). Given A there exist a, (3 in F(A) X with a/f3 — —A. Here (3 = a and we use Hilbert Theorem 90 if A ^ F. The pair a, (3 is determined up to a multiple by a scalar u in Fx. The matrix 5a(5) (where 5 = (ae)i) has eigenvalues A, 1, A - 1 iff a has eigenvalues a, (3 so that g ^ a 2 has eigenvalues —a/(3, —(3/a. Hence the norm map is onto the set of regular elements of H, and the 5 in G with a fixed regular NS make a single stable cr-conjugacy class, as a and ua (u in Fx) are projectively stably conjugate. PROOF.

But a and a' — u~1a are projectively conjugate only if u~xa — g^b^ _ 1 a6 for some b in GL(2, F). Then u 2 = det 62, and u = ± det b. If u = — det 6 then —a = b~xab, a has eigenvalues 7, —7 and h = I does not have eigenvalues different than 1. Hence u = det b, a = b~lab and u — det 6 lies in NE/FEX, where E = F(a). •

1.2 Norms

29

Thus the norm map has a particularly simple description in the case where Sa(S) has distinct eigenvalues. Up to a d-conjugacy such S can be assumed to be of the form 5 — (ae)i. Then 7 = NS = (—1/ det a)a2. 2.3.1 COROLLARY. Let F be a global field, u a place of F, and S, 5' stably a-conjugate but non-a-conjugate elements ofG(F). Then there is a place B / « of F such that S, 5' are not a-conjugate in G(FV). 2.4 DEFINITION. If NS is regular put 5 = \[SJ + \SJ)]J. Note that Sa(S) = 1. Hence SJ is symmetric (= t(SJ)). Define K(S) to be 1 if SO(3,SJ) is split and —1 if not. The function K depends only on the c-conjugacy class of S. Indeed if 5 is replaced by fiSJ^J then SJ + \SJ) is replaced by 051*0 + pMp

= P[5J +*(5J)]%

and the form SJ + \6J) splits if and only if 0[5J + XSJ)]lP does. If S, S' are stably cr-conjugate with regular norm, but they are not conjugate, then the forms SJ and S'J are not equivalent, and K(S') — —K(S). Thus if S = (ae)i and 5' = (uae)i, then K(S') = X(«)K(<5), X being the quadratic character of Fx trivial on NEX, E = F(Sa(S)). If NS = 7 is regular in H then Za(Sa) ~ Z H ( T ) - Indeed, if g~15a(g) = S

then

g~15a(S)g — Sa(S);

if 5 = (ae)i then g — b\ and b~1ab = a, since Sa(S) = hi, h = Hence b~1aewtb~1we = a, namely b~xab = a, deto so that det 6 = 1 . It is clear that Zn{^) = Zn{a).

-j^a2.

The norm map can be extended to classes of 5 in G which are not aregular. This is done next. 2.5 I d e n t i t y . We now deal with the (two) cases where all eigenvalues of Sa(S) are 1. If Sa(S) = 1 we write NS = 1 and NiS = 1. Then SJ = \8J) is symmetric, any two symmetric matrices are equivalent over F, hence for each 5' with S'a(S') = 1 there is S in G with SJ = SS'J^, so that 5 = SS'a(S~1), and the S with Sa(S) = 1 form a single stable er-conjugacy class.

/. Functoriality and norms

30

For such S the a-centralizer ZG(6a)

is

(PO(3, o J ) = )

SO(3,SJ),

the (projective =) special orthogonal group with respect to the form 5J. Replacing S by a cr-conjugate u8o-(u~l) or SJ by u5J\i, implies replacing Za(Scr) by its conjugate uZa{5a)u~1. Hence if F is R or p-adic then there are two a-conjugacy classes in the stable a-conjugacy class of the 5 with N5 = 1, corresponding to the split and nonsplit forms 5J. Put K(S) = 1 if ZG{bo) = SO(3,SJ) splits and K(5) = - 1 if it is anisotropic. If we put 7 = NS (— 1) then there is a natural surjection

= SL(2) -+ ZG(<5a) = SO(3,<5J)

with kernel {±1}. The morphism tp is defined over F only if SO(3,5J) split.

is

2.6 Unipotent. If 5a(5) is unipotent but not 1 we check by matrix /i o i\

multiplication that it is a regular unipotent (not conjugate to I o I o I). Vo o 1/ Alternatively, Sa(S)v — v if and only if (5J — \5J))w = 0, where w = t (5J)~1v. Thus the 1-eigenspace of 5a(6) has the same dimension as the zero-eigenspace of the skew-symmetric matrix 5 J — \5J), namely 1 or 3, and 8<J{5) ^ 1 is regular unipotent. Up to stable a-conjugacy we may /i

i 1/2 \

assume that Sa(5) = I o l l I, a a-invariant matrix. Hence 5 commutes Vo o l / /l

x

y\

with a(5) and 6
0\ _ ® (a/13 0 \ (P 0\ (fj/a a) 0 \ 0 l)\0 PJ\ 0

0\ I)'

1.3 Local lifting

31

2.8 Negative unipotent. If 5a(5) = hi and h = —unipotent ^ —I in GL(2, F), then up to conjugacy h = - ( J 2 ° J, hence a = u _ 1 ( J " ) with a £ F x , « 6 F x . But a is equal to

hence it is projectively conjugate to

are (projectively) conjugate only if a/j3 is a square in F x ; they are clearly stably conjugate. Hence the er-conjugacy classes within the single stable (T-conjugacy class of our 5 are parametrized by Fx/F*2. If S=(ae)u

a=(5°),

" ^ 0,

we let NS be the stable conjugacy class of h in H, and define iVi<5 to be the conjugacy class in Hi of elements which generate F ( v / a ) over F, and the quotient of whose eigenvalues is —1. Such an element of GL(2,F) is (

:

:

)

•

1.3 Local lifting 3.1 ORBITAL INTEGRALS. Let F be a local field. Fix a Haar measure dg on G. For any
figSaig)-1)^.

[ JG/ZG(6
OX

If 6 is cr-regular, put &t(5
'£i*(5'
I. Functoriality and norms

32

The sum is over a set of representatives for the (7-conjugacy classes in the stable cr-conjugacy class of 8. If /o is a smooth compactly supported function on H define ${y,fodh)=

[ ln/z„(-y) JH/Z„M

-i.dh foihyh-1)dt

Here dh is a Haar measure on H and dt on the centralizer $st(7,f0dh)

=

ZH{I)-

Also put

J2$(l'Jodh). 7'

If 7 = NS is regular then ZH{I) — Za{Sa). The measures on the two groups are related by assigning the maximal compact subgroup the same volume. The measures fdg and fodh are said to have matching orbital integrals and we write f0dh = X*(fdg) if for all 7, 5 with regular 7 = NS they satisfy the relation $"fr, fodh) = #*(5, fdg). 3.2 Weyl integration formula. Let {T 0 } denote a set of representatives for the conjugacy classes of tori of H over F. The regular set HTee of H (distinct eigenvalues) is the union over {T0} ofInt(H/T0)(T0rez). The Jacobian of the morphism To x H/To - • H,

(t, h) >-• lnt(h)t =

nth'1,

is

D0(t) = |det(l - Ad

t)\Ue(H/T0)\.

We have the Weyl integration formula

/ f0(h)dh='£ IW(To)!-1 / A0(t)2dt / •^-ff

f/r. \

•'To

/oCrtOf-

JH/To

Here W(T 0 ) is the Weyl group of T0 (normalizer/centralizer), and A0(t)2 = Z?o(*)- It is Z/2 if T0 splits over F or - 1 lies in NE/FEX, and {0} otherwise, as the normalizer of T0 ~ E1 is x H-> •£, realized by Int(diag(-1,1)) with the choices of section 2.1.

1.3 Local lifting

33

Let {T0}s denote a set of representatives for the stable conjugacy classes of tori of H over F. It consists of a representative, say the diagonal torus, for the tori which split over F, and elliptic tori, which are parametrized by the quadratic field extensions E of F, where T0 — E1. The Weyl group of T 0 in A(T0) (see section 2.1) is Z/2. Hence

The sum over t' ranges for a set of representatives for the conjugacy classes within the stable conjugacy class of t in To. Next we write an analogue of the Weyl integration formula in the twisted case. We use the observation of (1.9) that each cr-regular element in G is aconjugate to an element 6 = (ae)\ with a in GL(2, F). Recall that 5 = (ae)i and 5' — (a'e)i are cr-conjugate if and only if a' — ( 1 / det 6)6 _1 a6. Hence we may take the a in NZ(E)\TE, where TE ranges over a set of representatives for the conjugacy classes of tori T2 of GL(2) over F. If T2 splits over E (= F or a quadratic extension of F), we denote it by TE- We denote by Z the center of GL(2) and by N the norm map form E to F. Every a-regular element of G has the form gSa(g)-1,

SeT

= T(TE/NZ(E)),

g&G/ZG(Ta),

for some E. Here T(TE/NZ(E))

= {Sa = (ae)i;a e

TE/NZ(E)}.

The cr-centralizer of T in G, ZG(Ta)

= {ge G;g5a{g)-X

= 6,V5 G T } ,

is isomorphic to ZH{NT) where NT = N(T) = TE{= TE n SL(2, F)). The expression is unique up to the action of the cr-normalizer, which consists of the w with w~l5a(w) — 5' = 5'(5) € T for all 5 in T. Then w~15cr(S)w = 5'
-* G,

(5,g) ^

gSerig)-1

/. Functoriality and norms

34

is

A(6a)2 = |det[l -

Ad(6)a]\Lie{G/Ta)\.

The twisted Weyl integration formula is then (put Sa for (ae)i)

*(6a*)2da [

[f(9)dg = lj2[ JG

A

p

JTE/NZ(E)

Let us compute A(da)2

(

Xl

HgSaa(g)-1)^.

>G/Z (Ta) JO G

explicitly. We may assume S is diag(a, b, c). X2 X 3 \

xi x5 x6 J modulo center. Thus we assume that X7 Xg XQ J

x5 = 0 to fix representatives. Note that LieZc^cr) = {diag(x,0, — x)}, since -0~X = J XJ

=

/ x9 -x6 x3 ( —x8 x 5 —X2 X7

/

x - Ad^yx

X1+X9

Xl

X2-§X6

X4 — £ x s \ ( 1 + §)X7

— X4

2X5

(l+f)x3' X6 —^X2

Xg-§X4

Xl+Xg

Recalling that £5 = 0, and noting that in L i e G / Z c ^ c ) the x\ + £9 is a single variable (alternatively, in X we could replace xg by zero and x\ by xi + xg), we conclude that A(6a)2 =

'-! 1+S

a/ \

a

The 4 factors correspond to change of variables on: (x2,xe), x3, (Xi,xs), X7. This A(Sa)2 is then equal to Ao(7) 2 , 7 = NS. Indeed we may assume that 7 = diag(a/c, c/a), and then

2

Ao(7) =

a2-c2

= A(6o-)2

ac

3.3 C h a r a c t e r s . Let F be a local (archimedean or not) field, /$ a compactly supported smooth function on Hi, iti an admissible irreducible representation of Hi, and ni(fidhi) the convolution operator J fi{g)^i(g)dg. This operator has finite rank, see (1.3).

1.3 Local lifting

35

A well-known result of Harish-Chandra ([HC2]) asserts that there exists a complex-valued conjugacy-class function \i = Xm o n Hi which is smooth on the regular set such that for all measures fidhi on the regular set triri(fidhi)

= /

fi{g)Xi(g)dg.

It is called the character of it{. It is locally integrable on Hi. The twisted analogue of [HC2] (see [C12]) asserts that given a a-invariant admissible irreducible representation it of G, there exists a complex-valued a-conjugacy class function x% '• 9 *~* Xvid0') on G which is smooth on the cr-regular set, such that

tm(fdgxo-) = J f(g)X°(g)dg for all measures fdg on the cr-regular set. It is called the twisted character of it. It is locally integrable on G, hence the identity extends to all measures fdg. Note that x% 1S the twisted character of it. It is not the character in the usual sense. We also write Xir(9&) for x£(s)- Note that the (twisted) character is defined only on the ( the sum of Xn' where it'0 ranges over {ito}, is a stable class function. We say that {it0} is a stable set. Similar definition can be made for a set with multiplicities. But in our case it turns out that the stable class functions that we need are all of the form X{7r0}- Note that X{K0} 1S the character of the reducible admissible representation (Bit'0, sum over the it'0 in {it0}.

/. Functoriality and norms

36

DEFINITION. The representation TTO of HQ lifts to the representation tr of G if TT is cr-invariant and there is a stable set {7ro} including TTQ such that whenever 7 = N8 is a regular element of H we have

In this case we write n = Ao(7r0) or TT = A0({7ro}). R E M A R K . This definition is based on the definition of the norm N in (2.2). The norm relates stable a-conjugacy classes in G and stable conjugacy classes in H. To lift, 7 1—> X{TT0}(7) has to be a stable class function. To be a lift of TTQ the twisted character xZ of TT has to be a stable cr-class function, namely xZ($) = xZ(^') if <5 and 5' are stably er-conjugate.

3.4 LEMMA. We have n = Ao(7To) if and only if for all fdg, fodh with fodh = Xa{fdg) we have trTr(fdg x a) = tr{wo}(fodh). P R O O F . Suppose that txir{fdg x a) = tr{7To}(/od/i). We use the Weyl integration formula of (3.2) to write tm(fdg x a) = J f(g)xZ(g) dg as

A0(1)2xZ((ae)1M(ae)1a,fdg)da.

Y,\f z {B}

JNZ(E)\TE

Fix a quadratic extension E of F. Denote by TE the element of {T2} (i.e., a torus in GL(2,i ? )) which splits over E. Take fdg so that its twisted orbital integral $(5a,fdg) is supported on TE, namely on the cr-orbits of the 6a = (ae)i with o i n T ^ . We claim that tr7r(/c?c, x a) = \ f Z

H0{1)2xZ{5)^{5a,fdg)da

(S = ( a e ) x ),

JZ\TE

where $st(5a,fdg) denotes the stable twisted orbital integral of fdg at <5, as in (3.1). To show this, note that the trace tr n(fdg x a) depends only on the stable twisted orbital integral of fdg, since it is equal to tr {no} (fodh). If we take /o — 0, then for each a in TE we have *((uae)i
(u e F -

NE/FE).

Since tr ir(fdg x a) vanishes for such fdg, we have / JZ\TE

A 0 ( 7 ) 2 [ x £ ( M i ) - x;((«ae) 1 )]$((ae)i < 7, / r f 5 ) ^ = 0.

1.3 Local lifting

37

Choosing fdg so that the support of $((ae)icr, fdg) is small, we deduce that tf((ae)i)=x;((uae)i) depends only on the stable c-conjugacy class of {ae)\. follows. On the other hand, tr{Tro}(fodh) = /

Hence the claim

fo(g)X{n0}(g)dg

= ^[W(To)]"1 f

A0(7)2x{,o}(7)$(7,/0^)d7

JTo

{T 0 }

A 0 (7) 2 X{.o}(7)$ s t (7,/o^)^7-

=\ [ JTOE

The last equality follows from our assumption on /o'. the stable orbital integral $ s t ( 7 , fodh) of fodh at 7 is supported on (the stable conjugacy class of) the torus TOE in {^o} which splits over E. Since the map FX\EX —> S 1 by 2: 1—• z/z is a bijection and serves to relate measures from Z\TE to the torus T0E of SL(2,F), and fodh = X^fdg) means $st(8a,fdg) = $ s t ( 7 , fodh) for all 5, 7 with NS = 7, it follows that IT = A0(7r0). The opposite direction is proven by reversing the above steps. • 3.5 Unstable characters. Recall that the norm map N\ of (2.2) bijects the stable u-regular er-conjugacy classes in G with the regular conjugacy classes in H\ = SO(3,F). In each stable cr-conjugacy class of elements 6 such that 5a(S) has distinct eigenvalues there are two
tir(g6
38

I. Functoriality and norms

Note that if 6, 5 are stably <7-conjugate, but not conjugate, then up to cr-conjugacy 5 = (ae)i and 6 = (uae)i with u in Fx but not in NE/pEx, where E/F is a quadratic extension determined by 5. DEFINITION. The representation -KX of Hi = SO(3, F) lifts to the representation 7r of G if x% is a n unstable er-class function and | ( l + 7 ' ) ( l + 7")| 1 / 2 X?(<5)=x. 1 (7i)

(3-5.1)

for all 7x in Hi and S in G such that Zc{8'a) is split and NiS = 71 has distinct eigenvalues as an element of Hi = SO(3,F). Here 7', 7" denote the eigenvalues of 71 which are not equal to 1. Note that xZ($) — — XZ(&') whenever 6, 5' are stably cj-conjugate but not cr-conjugate. We then write 7T = A i ^ ) .

We shall relate orbital integrals on G and on Hi = SO (3, F). 3.6 put

DEFINITION.

If 71 = Nx5 has eigenvalues 1, 7', 7" with 7' ^ 7",

$us(Sa,fdg) = £«(*')*(*'*,/d fl ). If A is a smooth compactly supported function on H\ then for all regular semisimple 71 we put

$(lufidhi)=

[

h

(hllh-^.

We say that Adfti = \\{fdg) if, when the measures c/71, dS used in the definition of the orbital integrals assign the same volume to the maximal compact subgroups of Zn1(ji) and Za(Sa), we have

*(7I,A^I)

= i(i+y)(i+y,)r/2*uB(^./
for all 71 = NiS with distinct eigenvalues. 3.7 LEMMA. We ftawe tm(fdg x a) = trni(fidhi) wii/i /id/ii = ^i(fdg) tf and only ifir = A1(7r1).

for all fdg, Ad/ii

P R O O F . If trtr(fdg x a) = tr7ri(Ad/ii) for /dp, Ad/ii with fidhi — Xl(fdg), then tm{fdg xCT)is equal to / fi(g)xin (d) dg, which by the Weyl

1.3 Local lifting

39

integration formula of (3.2), is Y\

A

o /

i(7i) 2 X*i(7i)^(7i,fidhi)d~/i

Ai(7i)2X7ri(7i)|(l + 7')(l+7")r / 2 ^ U S (^,/^)^7i.

= Y,\I

We write Ai to emphasize that the A-factor is on the group Hi. The sum is taken over a set of representatives for the conjugacy classes of tori Ti of Hi over F. Recall that Hi = SO(3) a PGL(2), and in Hi a stable conjugacy class is a conjugacy class. The element 5, or rather its a-conjugacy class, is uniquely determined by 71 and the requirement that Zc(5&) be split over F. Moreover, $us(5a, fdg) is — us(5a, fdg) if 5, S are stably cr-conjugate but not cr-conjugate. Define x£ by the equation (3.5.1) to be an unstable er-conjugacy class function. Then our sum becomes

&o(7)2x:mus(5o-,fdg)da.

Y,\l {T2}

The sum is over conjugacy classes of F-tori T2 in GL(2,F), S = (ae)i, 7 = (—1/ det a)a2, and a 1—» 71 defines an isomorphism of Z\T% and Ti for tori T2, Ti which share their splitting field. Note that when the eigenvalues of a are u, v, then those of 7 are —u/v, —v/u, we have 2 1/2

Ao(7) =

u

v

V

U

1+H V

u \w

1+V

u

I U

and Ai(7i) =

(u — u) 2

1/2

1/2

1/2

l-H)l-i

v/u

uv

Hence A 0 (7) 2 = A 1 ( 7 i ) 2 | ( l + Y ) ( l + T")I,

7' = 7 " - 1 = - •

The sum is equal to

{TB}

NZ(E)\TE

A0(1yxZ(SMSa,fdg)d6.

/. Functoriality and norms

40

This is

J f{g)xl{9)dg by the twisted Weyl formula (3.2). Hence n = Ai(7Ti) by the definition of • X% and Ai. 3.8 Induced. Let IT = I(rj) denote the representation of G normalizedly induced from the character ?7(diag(a,&, c)) = fJ-(a/c) °f the Borel subgroup B, where fi is a character of Fx. Denote by 7r0 = Io(fJ-) and TTX = h{n) the representations of Ho, H\ normalizedly induced from the characters

(So-O^Mo),

(a0*b)^fi(a/b)

of the upper triangular Borel subgroups. Then the computation of (1.6) and the Weyl integration formulae of (3.2) show that the tr-character x% °f n = I(rj) vanishes at S unless S is diagonal (up to a-conjugacy), where xUS) = A0(7)-1(v(S)+v(S))

(S = J5J,

j = N5).

Similar standard computations show that the XTT* are also supported on the (conjugacy classes of) diagonal elements of Hi. They are given there by X.M = Ao(7)" 1 (M(a) + M ( O ) , 7 = (S 0 - I ) . and X7ri(7l)=A1(7l)-

It follows that if 7r = LEMMA, TT

I(TJ),

1

(M(a)+M(a-1)),

7i = ( S ? ) "

7r0 = -Jo(/•*), ni = hit*), then

— Ao(7To) = A1(7r1), namely Io(fj) and I\{fj) both lift to I(rj).

P R O O F . The characters of n, 7Tj are supported on the split tori, and the stable (j-conjugacy class of an element where x% d ° e s not vanish consists of a single a-conjugacy class. • REMARK.

Here the field F is any (archimedean or not) local field.

3.9 Special representation. Let F be nonarchimedean. Let u denote the valuation character of Fx, thus u(x) = \x\. The composition series of the induced representation I0 = IQ{V) of H consists of the one-dimensional representation 1 0 and of the special, or Steinberg, representation sp, of H.

1.4 Orthogonality

41

Note that sp is irreducible. But by Lemma 3.8 Jo lifts to the representation 7r = I(n) of G, induced from the character n = (v, 1,^ _1 ) of the upper triangular Borel subgroup of G. The composition series of n consists of the trivial representation I3, the irreducible representation 7Tp1(sp(v,l),u~1) normalizedly induced from the representation sp(^, 1) x v~l of the maximal parabolic subgroup P\ of type (2,1), and the reducible representation Ip2(v,sp(l,^-1)) induced from the maximal parabolic P2 of type (1,2). This last representation has composition series consisting of the irreducible TTP2(V,sp(l,^-1)) and the Steinberg representation St. This result is due to Bernstein-Zelevinsky [BZ2]. Now Ip2(v,sp(l,z/-1)) is not cr-invariant, but St, being the unique square-integrable irreducible constituent of I(rj) ~ CT-f(??), is ^-invariant. Hence, •Kp2(u,sp(l,v~1)), as well as 7Tp1(sp(i/, l ) , i / _ 1 ) (for the same reason), is not cr-invariant. The onedimensional representation I3 of G is clearly tr-invariant. Hence trI(n)(fdg

xa) = trSt(f dg x a) + tvl3(fdg

x a).

LEMMA. The trivial and special representations of H lift to the trivial and Steinberg representations of G, respectively. P R O O F . AS the characters of both lo and I3 are identically one, the lemma follows at once from the definition (3.3) of the lifting. D R E M A R K . The only cr-invariant one-dimensional representation TT of G is the trivial one. Indeed, n is given by a character /3 of Fx (namely, n (g) = (3(
1.4 Orthogonality 4.1 Orthogonality relations. For any conjugacy class functions Xi x' on the elliptic set He of H put (X,x')e=

= ^[WiToT'lTo]-1

[

x(h)x'(h)dh

f

A0(7)2x(7)x'(7)d7.

/. Functoriality and norms

42

The sum ranges over a set of representatives T0 for the conjugacy classes of elliptic tori of H over F. [W(T0)] is the cardinality of the Weyl group of T0 (1 or 2). As usual, |T 0 | denotes the volume of T 0 . We write 7 ~ 7' if 7, y are conjugate. The measure dh on He/ ~ is defined by the last displayed equality. The Hermitian bilinear form (x,x')e satisfies the Schwartz inequality (X,X')l < If x, x'

are

(X,X)e-(x',X')e-

stable conjugacy class functions, (x, x')1 is equal to

(X,x')e = \J2 {T 0 } s

WToWol-1

A0(7)2X(7)X'(7)<*7.

j jT

°

Here the sum is taken over a set of representatives To for the stable conjugacy classes of elliptic tori of H over F. [D{TQ)} is the number of conjugacy classes within the stable conjugacy class of T0; it is 2 if T0 is elliptic, 1 if To is split. Tempered (irreducible) representations n, IT' of a reductive p-adic group G are called relatives if both are direct summands of the representation normalizedly induced from a tempered representation of a parabolic subgroup of G (which is trivial on the unipotent radical). The orthogonality relations for characters (see [K2], Theorems G, K) assert that {xinXir')e IS zero unless the tempered n, n' are relatives, and if one of them is square integrable then the result is 1 if -K ~ -K' and 0 if not. Then 4.1.1 LEMMA. Let {710} and {7r0} be stable finite sets of admissible irreducible tempered representations of H which are induced or square integrable. Then (x{7r0}>X{7r'})e *s equal to the number of square-integrable irreducible representations in {TTQ} n {TT'Q}• 4.2 Twisted orthogonality. Let n be a cr-invariant irreducible representation of G. As in (1.2) there is an intertwining operator A from the space of 7T to itself such that a-n{g) = n(a(g)) is equal to An(g)A~1. Since ir is irreducible and A2 intertwines TT with itself, by Schur's lemma A2 is a scalar, which we may normalize (by multiplying A with 1/VA2) to be 1. Extend TT to a representation n' of G' = G xi (cr) by setting n(a) — A. As noted in (3.3), the twisted character \Z> °f 7r ' *s a c-conjugacy class function which is locally integrable on G and is smooth on the subset of G

1.4 Orthogonality

43

which consists of S with regular 7 = N5. Such 5 is called u-regular. Its cr-centralizer ZG(SO) in G is isomorphic to the centralizer Z H ( T ) of 7 in H. For any two cr-conjugacy class functions \G and x'CT on the cr-elliptic (5 with elliptic N(S)) subset G% of G define (xCT,x'CT)e to be

\ £ 1

\Z\TE\~1

f

E

A0(7)\(Sa)x'(S'a)da. JTE/NZ(E)

We write x{^a) f° r Xa($)- The sum defines a measure dg on G%/ ~ , where (5 ~ 5' if 6 is cr-conjugate to 5', for which

(x°,x'a)e=

xa(g)x"T(g)dg.

f JG*/~

If 5 1—> xC^0") is a stable cr-conjugacy class function, the inner product can be written as \ £

\Z\TE\~1

f

A0(7)2X(H £

xWda-

The sum over <5' ranges over a set of representatives for the cr-conjugacy classes within the stable cr-conjugacy class of S. For a in TE we have 5 = (ae)i, and there are two 5' in our case of 5 with compact ZG{SCT) ~ Zjji'y), 7 = W(J). 4.2.1 LEMMA. Given a stable conjugacy class function x on He define XG(<5) = x(N(5)). Given a stable a-conjugacy class function x" on G% define XH(I) = x{&a) for 7 = N(5). Then {X°,X'G)e = (XH,X')eP R O O F . This is clear from the definitions. Note that the inner product on the left is on G, while the one on the right is on H. •

Let 7r be a cuspidal cr-invariant representation. Such n do not exist unless the residual characteristic of F is 2. (This is proven in chapter V using the trace formula.). The orthogonality relations for characters assert in this case the following.

/. Functoriality and norms

44

4.2.2 LEMMA. Let TTI be a a-invariant irreducible admissible representation of G and TT a a-invariant cuspidal representation of G. Suppose that the function 6 \—> x-ni&Q') *s a stable a-conjugacy class function on G°. Then (XwiX£2)e is equal to 0 unless n and 1^2 are equivalent, in which case it is equal to 1. Thus for TT which is cuspidal and cr-stable (by which we mean that \%> is a stable c-class function), (\,x)e (inner product on i ? e / ~ ) is equal to 1, where \ is the stable class function on H defined by x{N5) = Xn'(5a). First suppose that TT2 is equivalent to TT. Put -K[ — w V (i — 0,1), where w is the character of G' which attains the value 1 on G and the value — 1 at a. The representations TT'0, n[ are inequivalent. Put PROOF.

~4>{g)=d{ir){Ti'{g)u,u),

TT •((g)7r'i(g)dg. JG'

Here d(n) denotes the formal degree of TT; u, u are vectors in the space of IT and the contragredient of TT, with (u, u) = 1. By the Schur orthogonality relations for the square-integrable representations 71^ we have tTTT'0(dg) = l,

tlTT[{(/)dg)=0.

Then 1 = trTr'0{dg) - tnr'^cpdg) =2 <j>(ga)xn(g
By the Weyl integration formula (3.2) this is equal to 2 •J E

&oh)2xAt«)da

/

£sJN7,{E)\TE

=2 " \ E /

[ JG/ZG(6
Ao(7)2X.(^)rf«E /

4>{95a{g)-^ aa

W"(S) -1 )$-

Harish-Chandra's "Selberg principle" [HCl], Theorem 29 implies the vanishing of the inner integral if Za(Sa) ~ ZH{I) is a torus of H which splits over F. If it is a compact torus of H = SL(2) over F then the proof of [JL], Lemma 7.4.1, shows that X*(Scr) = d(^) l JG

[(TT'(S

• 6a • g~l)u, u) + {ir'iga • 5a • {ga)~l)u, u)]dg

1.4 Orthogonality = 2d(n)\ZG(5a)\

45

(n'igSaig)-1

f

•

a)u,u)^-. aa

JG/ZG{5a)

Note that 5a(5)a(6)~1 = 5 for the last equality. We obtain ^iZcitaT1 Z

A„(7)2x,N^X,(^)d7.

f

E

JzH(l)

5,

We used the isomorphism Z\TE — Zc(Sa) ~ Zji{"t), and the relation d8 (— da) = dj of measures on the groups ZG(SO-), Z# (7). It remains to deal with the case where 7r and 7T2 are inequivalent. But then (ws7r2)() = 0 for both i, and the lemma follows using the same argument. • 4.2.3 LEMMA. We have that (x£,x£)e is 1 ifn is the a-invariant berg representation.

Stein-

P R O O F . This follows from (4.1) and Lemma 3.9. The orthogonality relation (4.1) for sp follows from the orthogonality relation for the trivial representation of the group of elements of reduced norm 1 in the quaternion division algebra, and the correspondence of [JL]. •

To deal with -K which are not cuspidal or Steinberg, we record a special case of a twisted analogue of [K2], Theorem G. The proof in the twisted case, for an arbitrary reductive not necessarily connected p-adic group, follows closely that of [K2], and will not be given here. Thus, let n, n' be cr-invariant, tempered representations with characters x£> xZ>- Each of IT, IT' defines a unique (up to association) parabolic subgroup and a square-integrable representation p, p' of its Levi factor, such that TT is a subrepresentation of I(p) and w' of I(p'). Then n, n' are called relatives if p is equivalent to p'. Recall that we have the inner product (xff,X,CT)e = V

ITOI" 1

JTE/NZ(E)

E

4.2.4

LEMMA

([K2]). If

A0(j)2x(6a)x'(Sa)da.

/

IT, TT'

are not relatives then (x£,X?')e — 0.

The same result holds also when F is the field of real numbers. In our case of G = PGL(3), a G-module normalizedly induced from a tempered one is irreducible, and we need only the following special case of the lemma.

46

/. Functoriality and norms

4.2.5 COROLLARY. / / TT, TT' are inequivalent a-invariant modules, then (x%,X%')e *s zero-

tempered G-

The methods of [K2] do not afford computing the value {xZ,X%<)e- But in the case of any (
II. ORBITAL INTEGRALS S u m m a r y . It is shown that the stable twisted orbital integral of the unit element of the Hecke algebra of PGL(3, F) is suitably related to the stable orbital integral of the unit element of the Hecke algebra of SL(2, F), while the unstable twisted orbital integral of the unit element on PGL(3, F) is matched with the orbital integral of the unit element on PGL(2, F). The direct and elementary proof of this fundamental lemma is based on a twisted analogue of Kazhdan's decomposition of compact elements into a commuting product of topologically unipotent and absolutely semisimple elements.

II. 1 Fundamental lemma Let F b e a p-adic field (p ^ 2), and F a separable closure of F. Put H = H 0 = SL(2),

G = PGL(3) = GL(3)/Z, /o

where Z is the center of GL(3) and J = I \i

H = H(F),

G = G(F)

(= GL(3, F)/Z,

H i = SO(3, J )

i\

-1

) . Put 0/

Z = Z(F)),

Hx =

Hi(F).

Put a(g) = J-lg-x J for g G GL(3, F ) . The elements 8,5' of G are called (stably) a-conjugate if there is x in G (resp. G(F)) with 8' = x5a(x~1), or 8'a = lnt(x)(Sa) in the semidirect product G x (a). The elements 7,7' of H are called (stably) conjugate if 7' — Int(a;)7 for some x in H (resp. H ( F ) ) ; similar definitions apply to Hi. A norm map TV, from the set of stable a-conjugacy classes in G, to the set of stable conjugacy classes in H, as well as such a map Ni to the set of conjugacy classes in Hi, is defined in chapter I, (2.2)-(2.8). To recall this definition in the crucial, cr-regular case, note that for any S £ G, (8a)2 = 8a(8) G SL(3,F) has an eigenvalue 1 (chapter I, end of (1.8)). Now if 8cr(8) is semisimple, with eigenvalues A, 1, A - 1 , then N8 is the stable class in H with eigenvalues A, A - 1 . If A ^ —1 then N18 is the class in Hi with eigenvalues A, 1, A - 1 . 47

II. Orbital integrals

48

Denote by ZG(5a) the group of x in G with 5a = Int(x)(5a), by ZH(J) the centralizer of 7 in H, and by Z^iji) the centralizer of 71 in H\. For / G C™(G), /o G C™(H), /1 G C ~ ( # i ) , define the or&itaZ integrate $(5a,fdg)

=

f(lnt(x)(5a))^, at

JG/ZaiSrr)

r (-ji, fidhi) =

fi(lnt(x)(ji))

dx — ,

(i = 0,1), where we put f(gcr) = f(g)- These depend on choices of Haar measures, denoted dg or dx or dhi depending on the context. In this section we mostly omit the measures from the notations. The measures on the centralizers are compatible with the isomorphisms Zc{8a) cz ZH{N8) ~ ZHI(NI5) when A ^ i l (in this case 6a,7,71 are called regular). Denote by {6'} a set of representatives for the tr-conjugacy classes within the stable a-conjugacy class of 5 G G; it consists of one or two elements. Define the stable cr-orbital integral of / at 5 with A ^ ± 1 by {6>}

Similarly put $ s t (7,/od/io) = ^ $ ( 7 ' , / o ^ o ) . {7'}

Define A(<5(j) = |(l + A)(l + A- 1 )| 1 /2. Put

K(6')

= 1 if

S0Q[(<5'J) + W ] ) is split, and K(5') = —1 otherwise. Define $us(5a, fdg) to be $(5a, fdg) if A G Fx, but if A £ Fx it is ^Ktf'WS'aJdg). Let R be the ring of integers of F. Put K = G(R), K0 = H(i?), K\ — Hi(R). Denote by / ° the function on G which is supported on K and whose value there is l/vo\(K) — \K\~l. Denote by ff the quotient of the characteristic function c h ^ of Kt in Hi by vol(Ki) = \Kt\,i = 0,1. Recall: p ^ 2.

II. 1 Fundamental

lemma

For A ^ ± 1 we have $st(5o-,f°dg) A(5a)$™(5o-J°dg) = * ( # ! * , / f d / n ) . THEOREM.

49

= $st(N6,f$dh0),

and

This is the fundamental lemma for the symmetric square lifting from SL(2) to PGL(3) and the unit element of the Hecke algebra. A proof of the first assertion — due to Langlands, based on counting vertices on the Bruhat-Tits building associated with PGL(3) — is recorded in the paper [F2;II], §4, but it is conceptually difficult, hence not used in this work. The current simpler proof is based on a twisted analogue of Kazhdan's decomposition [Kl], p. 226, of a compact element into a commuting product of its absolutely semisimple and its topologically unipotent parts, on an explicit and elementary computation of orbital integrals of the unit element in the Hecke algebra of GL(2), and on the preliminary analysis of stable twisted conjugacy classes from section 1.2. For an extension of the Theorem to general spherical functions, and for representation theoretic applications see chapter V. We argue that the (twisted) Kazhdan decomposition of Proposition 2 already reduces all computations to GL(2), and we carry out explicitly these computations. This makes the proof of the fundamental lemma for the symmetric square lifting entirely elementary. Our elementary and purely computational proof extends to prove the fundamental lemma for the lifting from U(2) to U(3), see [F3;VIII], and for the lifting from GSp(2) to GL(4) twisted by an outer automorphism similar to the one considered here; see [F4;I]. We need a twisted analogue of the following definitions and results of [Kl], p. 226. Put F g = R/irR, where n generates the maximal ideal in the local ring R. DEFINITION ([Kl]). An element k e G = GL(n,F) is called absolutely semisimple if ka — 1 for some positive integer a which is prime to p (= residual characteristic of F). A k £ G is called topologically unipotent if kqN —> 1 as N -> oo. 1. PROPOSITION ([Kl]). Any element k e K — GL(n,R) has a unique decomposition k = su — us, where s is absolutely semisimple, u is topologically unipotent, and s,u lie in K. For any k S K and x € G, iflnt(x)k

II. Orbital integrals

50

(= xkx s in G.

1

) is in K, then x lies in

KZQ(S);

here

is the centralizer of •

ZQ{S)

Let a be an automorphism of G of order (., (£,p) = 1, whose restriction to K is an automorphism of K of order L Denote by (K, a) the group generated by K and a in the semidirect product G » (a). The element ka of Ga C G x (a) is called absolutely semisimple if {ka)a = 1 for some positive integer a indivisible by p. DEFINITION.

2. PROPOSITION. Any ka e JCcr /ms a unique decomposition ka = sa • u = u • sa with absolutely semisimple sa and topologically unipotent u. Both s and u lie in K. DEFINITION. This sa is called the absolutely semisimple part of ka and u is the topologically unipotent part of ka.

PROOF. For the uniqueness, if S\a • u\ = s%a • u?, then u\ = u% for a = 0ia2- Since (a, q) = 1, there are integers a^, (3N with a^a + (3NqN — 1. Then

u2u^ = u«»a+e»oNu-a»°-0»"N

= u02NqNu^NqN

-+1

as N —> oo. For the existence, recall that the prime-to-p part of the number n

of elements in GL(n,F q ) is c = f] {q1 — 1). Let {{ka)q

*} be a convergent

i=i

subsequence in the sequence {{ka) qm ; qm = 1 (mod c«)}

in

(K, a).

Denote the limit by sa,s £ K. Then (sa)01 — 1. Define u = m • N Then uq * —> 1 as rrij —> oo, and u 9 —> 1 as iV —> oo.

ka(sa)"1. D

COROLLARY. The centralizer ZQ (sa • u) is contained in Za(sa).

•

3. PROPOSITION. Given k 6 K, ka = sa • u, put a(h) = sa(h)s~l. This is an automorphism of order £ on ZK((SCTY). Suppose that the first l 1 cohomology set H ((a),ZK((sa) )), of the group (a) generated by a, with coefficients in the centralizer Zji((saY) of (sa)e in K, injects in H\(a),ZG((saf)).

II. 1 Fundamental

lemma

Then, any x G G such that k'a = lnt(x)(ka) KZG(sa). PROOF.

51

is in Ka,

must lie in

Put k'a — s'a • v!. Then s'a = \im{k'a)qTni

= Int(a;)lim(fca) 9mi =Int(a:)(s
Hence {s'a)1 = Int(x)(scr)^, and by Proposition 1 there is y G K with {saf = Int(y)(sV)* =

(ta)e,

where t — ys'a(y~1). Replacing x by yx and k! by yk'a(y~1), assume that y — 1. Put a(l) = 1, and for 0 < r < I, a(ar) = s'a(s') • • • Then a{ar) G ZK((sa)e),

r 1 CT

- (s')ffr_1(*)~1'

we may

• -^(a)-1*-1.

and

a(au)au(a(ar))

= a(au+r)

(0 < u, r < £).

Hence a = { a(ar)} G

H\(a),ZK{(sa)1)).

Of course, s' = xsa(x~1) implies that a(a) = s's-1

= xa(x~1),

= icr(a; - 1 )s hence a is trivial in

H\(a),ZG((saY)). The injectivity assumption then implies that s's-1 bsa(b~1)s~1, and s' = bsa(b~l), with b G ZK({sa)e). Int(6)(scr) = s'a = Hence b~xx G Zc(sa),

= a(a) is ba{b~l) — It follows that

Int(x)(sa).

and x G bZc{sa) C KZc(sa),

as asserted.

•

R E M A R K . Let us verify the injectivity assumption of Proposition 3 in the case considered in the Theorem. We use the fact (chapter I, end of (2.1)) that if A is an eigenvalue of sa(s) then so is A - 1 . Thus the semisimple element scr(s) in K is the identity, or has eigenvalues — 1, 1, — 1, or

77. Orbital integrals

52

A, 1, A - 1 , A2 ^ 1. In the first case Zx(sas) — K, and / = kak implies ksJ = \ksJ). This represents a quadratic form in 3 variables over R (= ring of integers in F), and these are parametrized by their discriminant, in Rx /Rx2. If the form splits over F, thus the discriminant lies in Fx2, and in Rx, then it lies in Rx2, and the form splits already over R. The injectivity follows. In the second case, replacing s by a er-conjugate (see (2.7)), we may assume that sa(s) = diag(—1,1, —1), and s = diag(—1,1, —1). Then an element of ZQ(S(T(S)) has the form a\ (a in GL(2, F), entries of ai indexed by (i,j), i+ j = odd, are 0), and aa\ = ((deta) _ 1 a)i. So 1 = aiaai means a2 — det a, and a is a scalar, in Rx. Taking any h £ GL(2, R) with deth = a, we get h\a{h^1) = a\. In the third case, H1 {{&), ZK{{S<JY)) is trivial (as in the second case) if A e Rx, so let us consider the case where F(X) is a quadratic extension of F. As in chapter I, (2.2), we may assume that T = ZG{SCF(S)) consists of b\, b G GL(2,F), and s = (ae)\. Since scr(s) = (—(deta) _ 1 a 2 )i, ai lies in

T, and a(t) =

sJ^Js'1

= (aew^wea-1)!

= ((detfe)- 1 afea- 1 )i = ((detb)"^)]..

Hence 1 = ta(t) means that b is a scalar, in Rx.

The image in

H\(a),ZG((saY)) is trivial when &i = cio^cf 1 ) = (detc)i, where c\ € T, hence b = detc lies an in the norm subgroup NF^)/FFWX•> d in Rx, hence in Np(A)/F-R(A)X, where R(X) denotes the ring of integers of F(X). We conclude that c can be taken in GL(2,i?), and c\ in ZK(SCT(S)), as asserted. D 4. PROPOSITION. If the elements ko — so • u and k'a — s'a • u' of

Ka

are stably conjugate, then sa and s'a are stably conjugate. If s = s', then u,u' are stably conjugate in ZG(SO-). P R O O F . Suppose that k'a = lnt(x)(ka) for some x G G = GL(n,F), where F is a finite Galois extension of F (in the course of this proof). We have the /^-decomposition

s'a • u' = lnt(x)(sa)

• Int(x)u

II. 1 Fundamental

lemma

53

in G. The uniqueness of the if-decomposition in G implies that s'a = lnt(x)(sa), namely sa,s'a are stably conjugate. If sa,s'a are conjugate, we may assume that s'a — sa, then sa • u' = Int(x)(sa) implies that x £

Z-Q{SG)

• lnt(x)u

and Int(x)u = u', as asserted.

•

To prove the Theorem, decompose ka — sa • u (in our case a{x) = J • *x~x • J"1). Then ka(k) = sa(s) • u2. We shall consider three different cases, depending on whether sa(s) is the identity / , or it is diag(—1,1, —1), or it is regular (its eigenvalues A, 1, A - 1 are distinct). In all cases put Il{u)=

J

f(lnt(x)(sa-u))dx

JG/ZG(s<j)

= [

f(Int{x)(sa-u))dx

=

\K/Kr)ZG(sa)\f(sa-u),

JK/KnZG(sa)

(4.1) where the second equality follows from Proposition 3. Note that / ^ ( l ) = $(sa,f°dh). Then $(ka,f°dg)=

f

f(Int(x)(ka))dx

JG/Za(k
= [

fl(Int(x)u)dx

= $(u,fldx).

(4.2)

JZa(sa)/ZG(sa-u)

Here $(u, f®adx) denotes the orbital integral of the characteristic function f°a of the compact subgroup ZK(SCT) = K n ZG{SCT) of Zo(sa) (multiplied by \ZK{SCT)\~1) at the topologically unipotent element u in ZK{SCT)As a useful example we compute explicitly the orbital integral of the characteristic function IK of the maximal compact subgroup K = GL(2, R) in G — GL(2, F ) , where — as usual — F is a local field of odd residual characteristic with ring R of integers. Normalize the Haar measure on G to assign K the volume \K\ = 1. Put IT for a generator of the maximal ideal in R, q for the cardinality of the residue field R/nR, | • | for the normalized (by \n\ = q~x) absolute value on F. Let E be a, quadratic extension of F; then E = F(\/6) for some 6 with |0| equals 1 or q~x. The torus

54

II. Orbital integrals

in G is isomorphic to Ex, it subgroup RT — T C\ K is isomorphic to R^, the group of units in Ex, via 7 — i > a + by/9. 5.

PROPOSITION.

For a regular (6 7^ 0) 7 in i?T ; <^e orbital integral /

lx(Int(a;)7)d:r

JG/r

is e^wa/ £0 g-1

q-1

Here e = e(E/F) is the ramification index of E over F. Note that b = (7 — 7 ) / 2 v # , where 7 — a — b\/6. P R O O F . One has the disjoint decomposition G = \J K[0 -m ) T, and m>0 ^ " '

Here = {a + bV9; \b\ < \n\m, \a\ < 1} = R + 7rmRE = R + RirmV6.

RE(m)

For any function / e C^°(G/T)

we then have

f(g)dg=Y,{RE--RE(rn)x][

/ •'G/T

m>0

/(fc ( J ,_° m ) ) dfc, JK

\

J

and so /

J2lRE--RE(mr]lK(Kbna_m^abe)

l*(Int(*) 7 )ds =

JG T

I

m>0

= £ lRE--RE(m)*}, 0<m
if |b| = |TTB|- Recall that 7r = 7r|; and qE = q2^e for the uniformizer nE and residual cardinality qE oi E. Since [ J R B (m) x :l+irmRE}

= [Rx : Rx n(l+nmRE)}

=

{q-l)q,m—l

II. 1 Fundamental

lemma

55

and [R*:l+nmRB}

(qE-l)qeEm-\

=

we have that [Rg : i?£;(m) x ] is qm if e = 2, while if e = 1 it is 1 when m — 0 and ( 1. The proposition follows on taking the sum over 0 < m < B. O P R O O F OF THEOREM; STABLE CASE. We deal separately with the three cases, where the eigenvalues A, 1, A - 1 of sa(s) (sa is the absolutely semisimple part of 6a G Ka) have: I. A ^ ± 1 ; II. A = —1; III. A = 1. Of course, if $a(5a, f°dg) ^ 0, then we may assume that 6 € K.

Case I. Here 5a = sa • u, and sa(s) has distinct eigenvalues A, 1,A _1 . If 5a, 5'a in Ka are stably conjugate but not conjugate, then so are their absolutely semisimple parts sa, s'a. Indeed, if sa, s'a are conjugate (in G), then they are so in K by Proposition 3, hence we may assume that s = s'. If S'a — Int(x)5a then x G Zc(sa), and u,u' G Zc(sa). As Zc(sa) is a torus, u' = u. Since A, A - 1 are absolutely semisimple and distinct, neither A nor —A are topologically unipotent (as this would imply A = ± 1 , and these are cases II, III). It follows that F{\) is not ramified over F. Indeed, if it is, A = a + bVe,

where

|0| = \ir\,

\a\ = 1,

|6| < 1,

and 1 = AA = a 2 - b26 = a2(l -

6{b/af).

But (1 - 6{b/af)qN

-> 1

as

N -> oo.

Hence 1nN

aq —> 1, and ± a = l+7rc, \c\ < 1, for some choice of a sign. Then ±a, and consequently ±A, is topologically unipotent. For the same reason, if \ = a + by/6,

\0\ = 1,

OeF-F2,

and F(\)/F is unramified, then \b\ = 1 and \a\ < 1. A set of representatives for the set of cr-conjugacy classes within the stable <x-conjugacy class of 5 is given (see (2.3)) by 5y — (yhe)\, e=

{~o°i)>

hGGL(2,R)

(ifff=(°;)thenffl=(oio)),

56

II. Orbital integrals

as y ranges over a set of representatives of Fx /NEX, E — F(X). Note that 5a(5) = (ainr^ 2 )i- Take y — 1 to represent one class. When F(X)/F is unramified, the second representative y is not a unit, hence 5y £ K, and the stable orbital integral is the sum of a single integral (same conclusion if A<EF X ): <S>st(5a, fdg)

= $(5a, fdg)

= \K/K n ZG(sa)\f°(sa

= \KnZG(sa)\-1

=

• u)

\ZK(sa)\-1.

The same reasoning implies in our case (A ^ ±1) that $st(~f,fdh) = x $(7, fdh), and A 6 F or F(X)/F is unramified, in which case 7 can be taken to be represented by ( ° ), |6| = 1 > \a\. A stably conjugate, but not conjugate, element, is of the form 7' = Intf Q V7), with y £ F — NE, E = F(A). In particular y is not a unit, and the conjugacy class of 7' does not intersect KH (by Proposition 3, and since the eigenvalues of the absolutely semisimple part s 7 of 7 are distinct). Hence $ s t ( 7 , fdh)

= $(7, f§dh) = j

f°(lnt(x)(s7Ul))dx

JH/ZH(I)

= \Ko/K0 n zH (S7)|/0°(S7u7) = \K0 n z H (s 7 )r 1 . Since ZG(SCT) ~ Zn(s-y), and the measures are chosen in a compatible way, we conclude that $st(<5<7, f°dg) = $st(N6, fodh) when 5a — sa-u, and scr(s) has distinct eigenvalues A, 1, A - 1 . The stable assertion of the theorem is proven in case I. • Case II. Here 5a = sa-u, sa(s) has eigenvalues —1,1, —1. All such s € G make a single c-conjugacy class. Suppose that S'a = s'a • u' =Int(x)(6
<S>st(5a, fdg) = J ] $(s<7 • u', fdg) = £ *(«', J t ^ ) = *St(«. / i ^ ) ,

Ill

Fundamental

lemma

57

where Za{scr) = H and f°a — / $ . This we compare with $st(u2,f°dh). Using the explicit computation of Proposition 5, it suffices to note that for topologically unipotent /i, the value of |(/z — fJ-^1)2]1^2 is equal to that of \(fj,2 — / i - 2 ) 2 ! 1 / 2 , since |(//, + A*_1)2| — 1. This completes the proof of $st(5a, f°dg) = $st(N5, f§dh) in Case II. D Case III. By (2.5), there is one stable conjugacy class of 5 € G with (5a)2 — I, and it consists of two conjugacy classes, represented by a and by s'a (s' G G). The centralizer Zc(a) of a in G is the split form SO(2,1) = PGL(2,F), while that of s'a, Zc{s'a), is the anisotropic form SO(3) = PDX, D = quaternion algebra over F. 6.

PROPOSITION.

The orbit Int(G)(s'a) does not intersect (o

PROOF.

The element s" — I

Ka.

i\

e

\ir

I, where e is a nonsquare unit, lies in 0/

Int(G)(s'a), since the Witt invariant of s" J = diag(l, — s,ir)

is

(£,TT) = —1-

Note that the quadratic form associated to d i a g ( a i , . . . , an) represents zero precisely when its Witt invariant fj(ai,aj)

is

(-1,-1);

j
(•, •) denotes the Hilbert symbol. If s' lies in K, and s'JVJ = 1, namely s'J = Xs'J), then there is x € K such that xs'Jlx is diagonal, of the form diag(ui,U2> W3), in K. Its Witt invariant is

n(ui,u,-)=i=(-i,-i). j
Hence s'J ^ zgs"Jfg for all g e G.

D

We conclude that at 5a = a • u,u £ K topologically unipotent, u € SO(2,1) = ZG(a) ~ PGL(2,F), we have $st(au,fdg)

= <S>(au,f°dg) =

$(u,~fadhx).

Recall that the eigenvalues of ua(u) — u2 are /z, l , / / - 1 . Hence those of u are / / , l , / / - 1 , where /i' is topologically unipotent in R^, with i/' 2 = fi.

II. Orbital integrals

58

Since fi'fi' = 1, we have / / = v/v for some topologically unipotent u in R%. Via the isomorphism SO(2,1) ~ PGL(2), u can be regarded as an element of PGL(2, R) with eigenvalues v, V. The integral $(u, f°dh) is then computed in Proposition 5. It has to be compared with the orbital integral $st(v,f$dh) on SL(2,F), where v is an element of K0 = SL(2,i?) with eigenvalue /z, pTl. The stable orbital integral of a function /o on SL(2,F) coincides with its orbital integral over GL(2,F), where / 0 is extended to a C£°-function on GL(2,F). This too is computed in Proposition 5. We are reduced then to comparing |(i/-I7)2/^|*=|(l-F/i/)(i//l7-l)|i=|(l-/i')(l-M'-1)l' = |(l-/i)(l-M-1)|i

with

These are equal since v, / / , / i are topologically unipotent. This completes the proof of $st(6cr, f°dg) = $st(N5,fgdh) hence in all stable cases.

in Case III, •

Note that if A, 1, A - 1 are the (distinct) eigenvalues of the regular 5
|(1 + A)(1 + A - 1 ) | < 1 . In Case I, as noted in the discussion of the stable case, F(X) is F or is unramified over F, the unstable integral is a sum of a single term, and since A(6a) — 1, if N\5 = 71 is the regular class in Hi with eigenvalues A, 1, A^1, and s 7 l is its absolutely semisimple part, we have A(5a)$us(6cT,f0dg)

=

$(6a,fdg)

= lifnZcHr 1 = iKi/KtnZH^Mi-n). The tori ZQ{S(J) and ZHX (s 7 l ) are isomorphic. The measures are chosen to be compatible with this isomorphism. •

II. 1 Fundamental lemma

59

In Case III, by Proposition 6 (and since A(Sa) = 1) we have the first equality in A(5a)$as(8
= $(au,fdg)

= $(u,/°a7ii) = ^ u . / f d / n ) .

Here 5a = a • u, u being topologically unipotent. The second equality follows from (4.2), and / ° = f° by (4.1). Note that / ° is the characteristic function of Kx = K n ZG((u2, fxdh{) = $(u,f°dhi). Hence A(5a)^s(5a,

fdg)

= *(«, /°d f l ) = $(u 2 , /°d/n) = $(iV^, /°dfci).

n In Case II,

<5CT

= so • u G ifu,

so-(s) = d i a g ( - l , 1, - 1 ) ,

s = d i a g ( - l , 1,1),

and u G SL(2,i?) = ZK(SCT) has eigenvalues 7 , 7 - 1 . Then <5
^J^)

=

fchKl((l7)(ll)(ll))dX F

= / ^ ( ( ; " ) ( ; F

( l _

i

A ) B

) )

d a :

=

1

'

where / ° — \K1\~1 chx1, c h ^ is the characteristic function of K\ in H\. Indeed, —A = v2 is topologically unipotent, hence 1 — A (and A) are units va.R. If A ^ F, it lies in a quadratic extension F{\/6), 6 £ F — F2, and we may assume \6\ = 1 in the unramified case, and \9\ = \K\ in the ramified case.

60

II. Orbital integrals

Since vv — 1, we have v — a + bVd, with a,b e R. Since v is topologically unipotent, we have a = 1 (modir), and \b26\ < 1. Then

\ = -v/v = vVe/(vV9),

vVe = be + aVe,

and 71, as an element of Hi = PGL(2, F), is represented by ( ™ a® J, with eigenvalues 60 ± a\/#. In the ramified case, the determinant b282 — a 2 0 does not belong to RxFx2, hence $ ( 7 i , / f d / i i ) = 0. In the unramified case, 71 = siUi, where the absolutely semisimple part s i ( e PGL(2,i?)) has eigenvalues whose quotient is —1. Hence \ZKM\~1

$(7i,/idfci) = | ^ i / ^ ( s i ) | / i ( 7 i ) =

by Proposition 1 (the integral ranges over the quotient of KIZH1(SI) ZH1{H), and ZHl{si) = ZHl(-yi) is a torus in H i ) . Let us compare this with A(<5cr)$us(&7, f°dg). If v £ Rx, then ^s(6aJ0dg) = f

=

by

^(sa-u,fdg)

f°{lnt(x){scj-u))dx=

I

JG/ZG(so-u)

f$(Int{x)u)dx. JH/ZH(U)

Here H = Za(sa), and we used Proposition 3 in the last equality, noting that /°(1) = \K\~~1 and / o ( l ) = l-^o] -1 - We may represent u by diag(i/, ^ _ 1 ) , to get

X-MGrK^OOO)* =X^((:„!.)(;(,T"))'fa=^rI. since |1 — v~2\ = \v — i/ _ 1 | = A(<S<j), and v is a unit. \iv $. Fx, then the stable conjugacy class of u in H contains a second conjugacy class u', represented by Int(<7)u, where g € H = GL(2,F) has detg e F - NE, E = F(u); here 7V.E = Norm E / i r JS. Then $us(<5fr, fdg) = $(SCT • «, fdg) - $(s
f(lnt(x)u)dx-

f

f(Int(gx)u)dx JH/ZH(U)

II. 1 Fundamental

lemma

61

is zero when F(u) is ramified over F, since g can be chosen in K0, with detg in RX~RX n NE, in this case. When F(v) is unramified over F, we have that NEX = TT2ZRX D Rx. Since H/ZH(u) is open in H/Zs(u), the measure on H/ZH(U) defines one on H/Zfj(u), and if K denotes the character of Fx whose kernel is NEX (this is the unramified character of Fx of order exactly two), then 3>us{5aJ°dg) = f

f${Int(x)u)K(detx)dx.

JH/Z6(U)

We may represent the topologically unipotent element u by f " x

R

J, 6 £

x2

~R . It is important to note that Sa = sa • u = u • sa with /be

-a\

5J = usJ=(

-i

i

,

-[{5J) + %SJ)] = dia,g(b9,-l,-b). \a -b) 2 The quadratic form associated to diag(ai,... ,an) represents 0 precisely when rL a .j) i s equal to (—1, - 1 ) , (•, •) is the Hilbert symbol. Hence K(S) is 1, and SO(diag(60, —1, —6)) splits, precisely when (—b,9) = 1. In our unramified case this happens precisely when b £ ir2ZRx. Hence n{b) = 1. Note that A(Sa) = |(1 - v2)(l - v-2)\1'2 = \{v-V)2\xl2 Put t = \K0/ZKo(u)\.

= \(u -

u-1)2^2

= \4b2B\1l2 = \b\.

Then, with |6| = |7Tn|, OO

A(Sa)^(Sa,f°dg)

= t ]T <W^-m)|6|/0°((^m

be

f

))

m=0 oo

1

= *«(6)M((^))+*(l + ?- )E' 6 ( hr " m )l hr " m |^(( h r-»T)) m=l n

= [(-i)nq-n + (i+g- 1 ) X)(-i)n-m
Since ZH(U) and ^Hi(7i) are isomorphic tori, and the measures are chosen in a compatible way, the theorem follows in the unstable case II as well, as asserted. •

II. Orbital integrals

62

I I . l . l SL(2) to tori We shall use below the theory of endoscopy for H = SL(2). We then prepare here the theory of transfer of orbital integrals from H to the proper endoscopic groups of H. For this, note that the connected component of the centralizer of a noncentral semisimple element s in H = PGL(2, C) is the diagonal subgroup A, up to conjugacy. The centralizer is connected, hence gives a nonelliptic endoscopic group, unless s = diag(l, —1), in which case ZJJ(S) — A xi (w), w = I 1. Consequently the nonelliptic endoscopic groups of H are T ^ , where F is a quadratic extension of F, LTE = TE x WE/F, the Weil group WE/F acting via its quotient G a l ( F / F ) on TE = (C x x C x ) / C x (C x embeds diagonally in C x x C x ) by a(x,y) = (y,x). The embedding eE • LTE —> LH is (x, y) H-> diag(x, y), a H-> wo. The group T B is the F-group {(x,y);xy = 1} (= G m ) with G a l ( F / F ) action r(x, y) — (rx, ry) if T\E = 1 and T(X, y) — (ry, TX) if T\E ^ 1. Then TE(E) = {{x,x~l);x e Ex} and TE = TE(F) = {{X,
6

M if x = a + by/0, where F = F(y/6), 6 G i? - R2 if

-E/F is unramified, 0 is 7r if F / F is ramified. A character \i' : TE — E1 —> C x is parametrized by a map W^E/F -* L r E , F x 9 2 H-> (//(z/l),].), a H-> cr. Recall that the relative Weil group W B / F is generated by z E F x and a with -> diag(^*(.z), (J.*(z)), o H-» wo, namely the image Ind(//; WE/E, WE/E)O in PGL(2, C) of the two-dimensional representation lnd(/j,*;WE/F,WE/E) induced from any extension /x* to WE/E = Ex of our / / . We denote the twodimensional representation also by Ind|J(/i*), and the image in PGL(2,C), which depends only on the restriction / / of fj,* to E1, by Ind E (/i')oThis Indg(/i*) is reducible if jU* = JI* as a character of F x , that is /u' = 1 on F 1 , in which case there is a character // of F x with /z*(.z) = ^(zz). Indeed in the direct product LH = PGL(2,C) x WE/p the image eEW))

= ("*0W ^-z))wa

conjugate to ( M „

of the class t ( ^ ) = ( " ^ / ^ ) a of M ' is

_ ?,-•, ) a, and Ind||(/z*) is the reducible representation

A* ® MXB of WF/p. Here XB is the character of W V / F = Fx of order 2 whose kernel is the norm subgroup NE/FEX. The character on WE/F

II. 1 Fundamental

lemma

63

with Ex 3 z i-> /i*(2:) = /x(-zz) factorizes via WE/F —• WV/F> 2 ^ -N2, Gal{E/F) 3 a •-> cr2 £ Fx - i V £ x , and / i : W F / F = F x -> C x , i ^ M (x). The group TE is compact. Hence its spherical functions are the constants. The unstable orbital integral of fodh in C£°(H) at 7 which generates the quadratic extension E over F is

$m(~fJodh)= [

foih-yh-^dh- f

foWh-^dh J

JH/TE

H/T'E

fo{gig~l)K{g)dg.

= / JHITK

Here 7' is stably conjugate but not conjugate to 7. Hence there is g S H = GL(2, F) with determinant in F — NE, where n(g) = n(detg) and K is the isomorphism of Fx /NE* with {±1}. Recall that A(7) = \2by/9\. It is |6| if E/F is unramified and p ^ 2. 7. LEMMA. Lei .E/F be unramified. Then the normalized unstable orbital integral «(6)A(7)$us(7,/o^) of the unit element fodh — f$dh of the Hecke algebra of H is equal to 1. P R O O F . The computation is as in Proposition 5, except that in the sum we need to insert the factor «;(7rm) = (—l) m . We get

K(b)\b\ (l-(q+l)f^ ^

m=l

(-q)™-1) = (-l)B\b\[l -(q+ l ) ^ ^ " 1 ] = 1. '

^

D Other spherical functions of H (E/F unramified) are generated by fM =

(-l)M\nM\ch(Kdi&g(irM,Tr-M)K),

M > 1. Then

x m ,r 1 2: ^l = ^M ](-D /«(( T .-»»)) m>l

II. Orbital integrals

64

= (q + l)q^\TtM\\K-Mb^\{-l)B

=[l

+ t j A( 7 )- 1 «(6)

as the only term in the sum is indexed by m = M + B. For general measures fodh G C£°(H), using the same decomposition it is easy to see that K ( 6 ) A ( 7 ) $ U S ( 7 , fodh) is a locally constant measure on TE, and any locally constant measure on Tg is of such form for some fodh e C?(H). For the global case we need to consider also places which split in the quadratic extension, namely E = F ® F. There K = 1, 7 = diag(a,a~ x ), its stable conjugacy class consists of a single conjugacy class, F^Jodh)

= \a- a" 1 ! / ^(n'^dn

= \a\ f

JN

fQK{in)dn

JN

implies that / A (7) = F{^, fodh) is locally constant and compactly supported on the diagonal torus A, it is the characteristic function of \a\ = 1 if /o = /o 1 a n d spherical if f0 is. Globally, fix 7 0 G TE with eigenvalue xo = do + bo\/0. Then \bo\v = 1 for almost all v, and note that n(b) = K( JZ2 ). The embedding e^ : LTE —> LH defines a lifting of representations in the unramified case. In this case the unramified character of TE is fi' = 1. The class parametrizing / / = 1 is £(//) = (1, l)er, whose image in LH is e(£(//)) = w<7, which is conjugate to diag(l, — l)cr in the direct product LH = PGL(2, C) x WE/F- Thus the endoscopic e^-lift of / / — 1 is 7r = I(n, HXB) where fj, = 1. Working with GL(2, F), and the corresponding eE and TE in GL(2,F), if (J,*(z) = (J,(zz), namely //* = /I*, then eE(t(fJ.*)) is conjugate to t(I(n,nxE)) in GL(2, C). In terms of the Satake transform we have t r ^ ( / r B d t ) = /^(t(M*)) = /o V («(%*))) = /o V ( ( " J 0 - , V ) ) " ) =

tT«J*,HXE-,fodh).

Working with SL(2,F) we replace n* by its restriction / / to E1, and /i by 1. At the places where the global quadratic extension E/F splits, the local component of the global character /i' of A^/E1 is a pair of characters (ilt fi2 = (i^1 of the local Fx, and the endoscopic, e^-lift to H is the induced representation I({j,i,fi2)- In the unramified case we have tvfi'(fTEdt)

=

ftBW))

=

fXieW)))

II. 1 Fundamental

lemma

= /oV((Mf\2 (.))')

65

=trI(ni,Mfodh).

In section 1.4 we considered orthogonality relations for characters x o n H = SL(2, F) and for twisted characters x" °n G — PGL(3,F), and their relationship. We need analogous investigation of the relations between character relations on H and on an elliptic torus TE = E1 of H, where F is a local field. Thus we view TE as the group of t — ( a J with det(i) = 1, a, b G F. Denote by t! an element stably conjugate but not conjugate to t. Put t = ( ^b ~ J. Let frEdt be a measure on TE. Let y! be a character on TB. 8. PROPOSITION. If fTE{t)dt = K{b)A0{t)[§(t, f0dh) obtained from the measure fodh on H, then fjf(fTBdt)=

[

- $(*', f0dh)}

is

fi'(t)fTE(t)dt

JTE

is equal to (Xn','${fodh))e where '$(t,f0dh) = | Z # ( i ) | _ 1 (£, fodh) and Xfi' is the unstable (x^'(i') = ~Xju'W) function on H which is zero on the regular set of H except for the stable conjugacy classes of t € TE where

XM'(t) = agj(M'(t)+ /*'(*))• We have that (X/J'> X// )e *s zero unless y!, \J!X are characters on the same E1 and n' equals y!x or / ^ i - 1 , in which case the inner product is 4 if n'2 = 1 and 2 if fi'2 ^ 1. Note that //(i) = / / ( £ ) _ 1 = ~P-'{t) where the first bar in conjugation in E over F, and the last is complex conjugation. Note that t is conjugate to t by d i a g ( - l , 1). We distinguish two cases. If —1 lies in NE/FEX, then diag(—1,1) can be realized in H, in N o r m ^ T ^ ) — TE (it is in HZQ^2,F)(TB))Hence K ( - 1 ) - 1, fTE{t) = frE(t) and the Weyl group W(TE) has [W{TE)\ = 2 elements. Then PROOF.

AfrEdt) = £ / Ao(*)3 • ^ { „ } JTB

EE^1^)]"1 /

2

^

• < f ^ * ( * « , fodh)* A

0W

Mt)2xAtu)Htu,hdh)dt

= (Xll>Mfodh))e

66

II. Orbital integrals

Here u ranges over the two-element group such that {tu} is {t, t'}, and K is the nontrivial character on {u}. If —1 $ NE/FEX then t is stably conjugate but not conjugate to t, so we choose t' = t. Then K ( - 1 ) = - 1 , thus «(*>(?)) = -«(&(*))> /r E (<) = / r B ( i ) and [ W ( T E ) ] = 1. Then A 0 (i) 2 • (ix'(i) +

M'(/r B *) = \ f

IT/

'(i))^M[$(i;

/od/l)

_ $(I)/od/l)]di

&0(t)2Xls(tMt,fodh)dt=(xti>,'$(f0dh))e.

2-\Y,l We used:

M

JTEI

f (x/(t) + //(*))«(&(*))£,,(*)*& /odfr)* = /" (tl'(t) +

fl'(t))K(b(t))A0(t)$(t,f0dh)dt.

JTE

For the last claim of the proposition, since «(&) € {±1} and /s(u) e {±1}, we see that (x^s X/V )e ls z e r 0 unless /x' and /x^ are characters on the norm one subgroup of the same quadratic extension E — E' of F. Since x^'X^' is a stable function and 2 • | = 1, the inner product is \TE\~1

[

(/i'(t) + /*'(*))(#(*) + Pi(*))dt.

We are done by the first comment in this proof.

•

II.2 Differential forms 2.1 T h e r e g u l a r set of H . To compare orbital integrals on different groups we need to compare Haar measures, or invariant differential forms of highest degree. Let G a the trace map. If 7 e H has distinct eigenvalues 71, 72 = 7] - 1 , then the differential d(, of ( at 7 is given by dji d( = d7i + d-y2 = ^7i

dji T

7i

= 7i 7i

-id-fi 7i 7i

. .dji = (7i - I2) , 7i

II.2 Differential forms

67

and it is nonzero. At a neighborhood of 7 — la J with 71 — 72 we may assume that a ^ 0, d = (1 + be)/'a (or that d ^ 0, a — (1 + bc)/d; this case is analogously treated). Then £(7) = a + d has the differential c 6 (1 - o~ 2 (l + 6c))da + -db + -de. a a It vanishes only if a 2 = 1 + be, b — 0, c = 0, namely at 7 = ±7. The subset H r e g of H where d£ is nonzero is called the regular set. Fix (nonzero invariant) differential forms u>n and \i (of highest degrees 3 and 1) on H and on G a . Then \x defines a nonzero invariant form w7(/z) on Z H ( 7 ) (which is independent of WH). If \x — dx then u;7(/z) = - ^ if 7 is regular, or = dx if 7 = ± ( *

J. If 7 is stably conjugate to 7' then wy(fj,)

is obtained from w 7 (/i) by the induced isomorphism of Z H ( T ) and Z H ( 7 ' ) The fibers of £ are the stable conjugacy classes in H r e g . The quotient of ojji by w7(/z) defines an invariant form on the fibers of £ in H r e g . The trace map £ extends to a map £ from GL(2) to X — G 2 , defined by C(T)

=

( t r 7 , d e t 7 ) = (a + d, ad — be).

It has 2 x 4 differential diag(da dbdedd)-*^]

_°c °b *J ,

which is nonsingular if one of a — d, 6, c is nonzero. The singular set consists of the scalars. 2.2 The cr-regular set of G. Similarly, let £ : G —• G a be defined by £(5) = tr7V<5. To compute its differential note that £(<5) + 1 = tr(5J*5 _ 1 J). Then d£ is the trace of the differential of the map 5 1—> 5Jt5~1J, which is d5-Jt5-lJ

+

5J-d{t5~l)-J.

But 0 = dl = dOW -1 ) = d<5 • (T 1 + 5 • d
1

=-S-1

• d5 • S-1,

II. Orbital integrals

68

and tr[«JJ -'cT 1 • d^S) -'(T 1 • J] = triJS-1

• dd • S'1^}

= tr[dS • S'1 • Jft.AT 1 ].

So d£ = ixdS[j(S) -

S^aiS-^S-1].

Then dl; is zero for all d5 only if Sa(6) = (5a(S))~1, thus 5u{5) has square 1, hence has eigenvalues 1 or — 1. Since Sa(S) also has determinant 1, it is semisimple and N5 is ±1. We conclude that the a-regular set GCT"res of G, defined to consist of the S with d£ ^ 0, consists of all 5 with N5 ^ ±7. The fibers of £ on the regular set Ga'res are stable cr-conjugacy classes. We fix an invariant differential form WQ of highest degree on G. As above /U determines an invariant form uJs(fJ') of maximal degree on ZG(5CT). If 5' is stably a-conjugate to 5 then ZG(5'O) is isomorphic to ZG(5(J) over F and u>s (M) transforms to a form uis' (M) of ZG (S'a) via this isomorphism. 2.3 Differential forms on G. Suppose that 8 x a is semisimple in G x (a) (namely {5a)2 = &T(<5) is semisimple, hence 7 = 7V<5 and 71 = N16 are semisimple in H and #1). Here F is a local field and as usual G = G(F). Choose a neighborhood X^ of the trivial coset ZG(SCT) in ZG(8&)\G, a section s : ZG(5 G, and a a-invariant neighborhood Y5 of the identity in ZG(5(T) (all defined over F) so that the morphism

YjxX^G,

by

(s,g)^s(g)-1£6a{s(g)),

is an immersion (its differential is nonsingular at each point). For the F-rational points we have that the map Y$ x X$ —> G is an analytic isomorphism onto an open subset of G. The neighborhoods X 7 , Y 7 , X 7 l , Kyj can be introduced for 7 in H, 71 in H i . Let 0(e) be the determinant of the transformation Ad(s5)a — 1 on the Lie algebra Lie(ZG(S(r)\G) of ZG(5a)\G. Locally the invariant form U>G on G can be taken to be 0(e)a;JA u> , where u\ is an invariant form of maximal degree on ZG{SO~), and w2 is a highest degree invariant form on ZG(5O-)\G. LEMMA.

2

P R O O F . TO compute the differential we introduce an extension F(n) of F, the quotient of the polynomial ring F[x] by the ideal (a;2). For any algebraic group J over F there is an exact sequence

0 -> Lie J -> J(F(r?)) - • J -> 1,

II.2 Differential forms

69

with maps X i-> J + r}X, h(I + rjX) i-> h. To study the map (e,h) i-> h~l • e5 x a • h (e in Z G ( ^ C ) , /I in Z G ( 5 O - ) \ G ) , we replace h by (7 + r?y)/i,

where V is in Lie(ZG,(5cr)\G), and £<5 x a by (J + r]X)(s6 x cr). Note that (J + rfY)'1 —I — r]Y, and aYa"1 — Ad(a)Y. Then / i _ 1 • s6 x <j • h becomes / i _ 1 ( J - r?F)(/ + r/X)(e5 x a)(7 + rjY)h = h~l(I + rj(X - Y)){I + r] • Ad(e6)a)Y l

= h' {I + rj(X -(I-

• e5 x a • h

Ad(£<5)cr)y)] • eS x a • h.

Then CJG(X

+ Y)= ^(X) =

A co2([Ad(eS)a - I]Y) 0(e)-L»1(X)Au)2{Y),

as asserted.

•

2.4 LEMMA. Let 3 be a linear algebraic group defined over a local field F, contained in the matrix algebra M . Suppose that S is in J and s in the centralizer Zj(5) of 6 in J. If e is near 1, then Zj(e6) C Zj(5). P R O O F . The group J acts on M by inner automorphisms. Enlarge F to include all eigenvalues A of S. Let M(A) be the corresponding eigenspace. Then M — ©M(A). The group Z3(S) is the intersection of J and M ( l ) . Since e lies in Zj(S), e5 leaves each M(A) invariant. If e is near 1 all fixed vectors of eS lie in M ( l ) . Indeed, if v lies in M(A), then v = e5 • v — Xe • v and A - 1 is an eigenvalue of e. This is impossible if A ^ 1 and e is near 1. But then Z3(e5) c J f l M ( l ) = Zj(S), as asserted. •

Applying the lemma with J = G » {1, a} and 5 in G, we have: COROLLARY.

If e is in ZG(SO)

near 1 then ZcisSa)

C

ZG{5O).

2.5 LEMMA, (i) If N5 = 1, e e Zc(Sa) is near 1 and N(eS) has distinct eigenvalues, then K(SS) —K(S). (ii) If N8 = —I; e, e' in ZG(5<J) ~ H are stably conjugate but not conjugate, and N(e6) has distinct eigenvalues, then K(SS) = —K(E'S). PROOF,

(i) Note that

edJ + %e8J) = eSJ + \SJk~1)

= eSJ + e-u(SJ)

= (e + e - 1 )(JJ.

Hence the value of K(eS) is 1 if and only if ZQ{\{e+£~1)5a) splits. But this is contained in ZG(SO) by Corollary 2.4. Hence the two special orthogonal

II. Orbital integrals

70

groups split together. (ii) We may assume that ^ = e i ) e = ( ^ 1 i ) > a n d then e = a±, e' = a[, with a, a' in SL(2, F). The elements e5 and e'5 are cr-conjugate (and define equivalent forms) if and only if a and a' are conjugate (not only projectively conjugate, since N(e5) has distinct eigenvalues). • 2.6 Jacobians. Let f : ZG(5a) ->
Similarly we have wH = % X A w 2 ,

w H l = <M?7i)w7l Aw 2 ,

where 6{rj) and #1(771) are the functions det[Ad(r?7) - /] L ieZ H (7)\H. on Zii(-y) and

ZH^TI)C'(T?)

are used to define

W'(/J.),

det[Ad(r7i7i) - /] L i e z H l (7i)\H 1 ,

The maps = tr(777), w'(fi),

<^0) = %V,,7(/i),

Ci(»7i)=tr(»7i7i) and we have w^(/i) = ^ ( r ^ u ^ t / i ) .

If 7 = JV<5, 71 = NiS and e is in Zc(Sa), then £(5
A^etf) = T7l7l

(771 e Z H l ( 7 l ) ) .

To compute 0(e), #(??), #i(?7i), we may assume that e, hence 77, 771 are semisimple, since these functions depend only on the semisimple parts of £, 77, 771 in their Jordan decomposition. Further, we can work over the algebraic closure F, and take 6 to be the diagonal matrix diag(o, b, c). Then £ can also be taken to be diagonal; hence e = diag(cf, 1, d"1) since it lies in 2 ZG(8(J). If the eigenvalues of N(e6) are denoted by 0i(= ad /c), (32 = /3f\ then it can be checked that:

II. 2 Differential forms

71

2.7 T w i s t e d J a c o b i a n . If 7 = 7 then #(77) = 1 and since a 3 x 3 matrix X — crX has the form

(

Xl X2

0

\

X4 0 X2 1 , 0 X4 —x\ J

we have 9(e) = (1 + d)(l + d~l){\ + d2){\ + d~2). If 7 = —J, take (5 = d i a g ( - l , 1,1), then Q(rj) = 1 and since a 3 x 3 matrix

(

Xl 0

X3

000

\

I, we have

x7 0 - x i /

G(e) = (l + d2)(l + d-2). If 7 7^ ± 7 then 6^) = (1 - /3X)(1 - /32), and since X = Ad(5)trX has the form diag(a;i,0, — £1), we have 9(e) = (1 - /3?)(1 - /?£),

9(V) = (1 - /3?)(1 - /?22).

2.8 Pullback. The map ip : Zu(l) —• ZQ,{5U) of 1.2 can be used to pull back the form co$(fi) to a form ^ ( ^ ( / x ) ) on ZH(J). The comparison is given by 2.8.1

LEMMA.

The form ip*(ujs(fj,)) is equal to u>7(/x).

The trace map Ci : Hx = SO(3) —> Ga is smooth on the regular set H^eg of 71 with distinct eigenvalues, and w7l(/z) can be introduced for such 71. Note that the centralizer ^Hi(7i) of 71 in H j is isomorphic to Za(Sa). The pullback of LJg(fJ.) to Z\ix (71) is denoted again by ^ ( / u ) . 2.8.2 LEMMA. 7/71 = NiS has distinct eigenvalues 1, 7', 7" = 7 / _ 1 (see 1.2.3) t/ien w 7l (/i) = ( l + 7 ' ) ( l + 7 ' > i ( M ) . P R O O F . To verify the lemmas it suffices to take the standard form \x — dx on Ga- If NS has distinct eigenvalues then ZQ,(8O) is abelian, onedimensional, and isomorphic to Z H ( T ) and to Zn^i). As in (2.1) we compute

(0*(M) = de' = ( / 3 i - A ) ^ Pi

II. Orbital integrals

72

But cjg = e-jp- for some constant e. It is the product of w'£{n) and the quotient w]/(|')*(/i) = e/(/?i - P2) of one-forms on ZG(5a) and G a . The same computation yields the same value for w^(/i) and w' (/x). So it remains to note that Q(e)/0(rj) — 1 and that e(e)/0i(7 7 i) = (l + /? 1 )(l+/? 2 ), and Pi = 7J when £ = 7, to deduce the lemmas for 6 with N6 ^ ±7. It remains to complete the proof of lemma 2.8.1. If 7 = NS is I or —I then the epimorphism

^ = (2 0 ? , ) e Z H ( 7 ) = S L 2 A e = ( '

/-I

, °)

gZ G (fe) = SO(3)

ON

If 7 = —7 we may take 5 = I 1 I and V 0 1/ r?! = ( »

o°1)

£ Z H (7) = SL2 &e = ^

: "^

e ZG(^)

Given e near 1 we may choose 771 near 1. Then Z H ( ? 7 I 7 ) = Zu(r]l) a n d ZG(e<&r) - ^(Z H (r?i7)). It remains to show that (p* (CJ'E((J,)) = 7n2u^(/u) at a unipotent e in ZG(5a), for then e(e)y>*(wea(/i)) =

m29{ri)ujm{n)

and at e = 1, 7(/z) (since #(77) = 1 and 6(e) —> m 2 as £ —> 1). Let Ov, Om, Oe be the conjugacy classes of 77, 771, e. Since we have a commutative diagram Z H ( 7 7 I 7 ) \ Z H ( 7 ) - °m

^

—I ¥> ZG(e5a)\ZG{5a)~Oe

<->

^H(T)

i f1 ZG(<Ja)

II.3 Matching orbital integrals

73

the pullback tp*(w'e(n)) of the form w'E(ii) on ZG{E5CF) is a form on Z H ( J ? I 7 ) denned by the function £'oip : Z H ( T ) —> G a and the form <^*(wj) on Z H ( 7 ) Define V(??i) = 7?J". Then £'(
Z H (r?7)\^H(7) - 0 „

^

^H(7)

— ^H(7)

hence y>*(w^(//)) = V*(wJ,(/x)). But

- ' W ^

-

(l-|8?)(l-/3g)

2

is equal to m as f3i —> 1. This completes the proof of lemma 2.8.1.

•

II.3 Matching orbital integrals 3.1 Definitions. Let F b e a local field. All objects below are defined over F. A highest degree invariant differential form U>G determines a Haar measure dg — dag = do on G. A maximal degree F-rational invariant form UJS on ZG{S(T) determines a measure dg — dgt on ZQ{$'
= [

f(g-^Sa(g))% d

JZG(6
UNS^l

6t

put $st(5a, fdg) = $st(5a, / ; d5, dG) = £ 5'

$(S'a,

fdg).

II. Orbital integrals

74

The sum is over a set of representatives for the cr-conjugacy classes in the stable cr-conjugacy class of S. If N5 — 1 put

$st(5a,fdg) = J2<S'M6'°Jdg). 6'

If /o is a smooth compactly supported function on H define

and $st(7,/o^)-*st(7,/o;d7I^) = ^$(7/,/o^). 7'

Here d 7 is a measure on ZH (7), and d/f is a measure on H. If 7 = iV5 then there is tp : ZH(I) -» ZG(<5
$st(6
for all 7, 5 with 7 = NS, we write fodh — X*(fdg). 3.2 PROPOSITION. For each fdg there is fodh with f0dh For each fodh there is fdg with fodh — X*(fdg).

=

\*(fdg).

P R O O F . Applying partition of unity and translating, when passing from / to /o (resp. /o to / ) we may assume that / (resp. /o) is supported on a small neighborhood of a fixed semisimple element <5o (resp. 70). The proposition is proved by dealing with the various possible 70, 60. If SQ and 70 are such that 70 = NS0 is nonscalar then the proof is simple, and it remains to deal with 70 = —/ and 70 = / . Suppose that 70 = —/. Fix a section s of Zc(S0a)\G in G. Given / and 771 in Z/f (70) = H, put £ = (p(r]i). For 77 in some fixed neighborhood of I define

Mrno) = fo(Vi),

fo(Vi) = /

Hg^eSoaig))^-.

(3.2.1)

II.3 Matching orbital integrals

75

Here ip : H —> H, 771 H-> 77 = 77™ (m = 2) has analytic inverse for 77 near 1, and we put 771 = i})~l{rj). Put foivio) — 0 otherwise. Note that vp(ff) = ZG{50a), that (p(ZH(v'i)) = ZzG(50a){£) = ZG(e'S0a) if rji is near 1 and e' =
Hence * st (r?7o,/o;^,d 7o ) = W ^rJzHW)\H = £

/

=£ / ,

V

fo(h-%hfa

/

Mh-Wlohfa a r, =

^(rh,fcdm,dl0)

fig-Mh'Wi^Soaig))

dj0

dp

JZH{TI[)\HJZG{SOO)\G

/

d

-1_/JC _ / „ \ \ G _ / /(
d£50

Here r] £ H is near 1, and rj' £ H ranges over a set of representative for the conjugacy classes within the stable conjugacy class of 77. The element 77' can be taken to be near 1. The same comment applies to 77^ — T / J - 1 ^ ' ) . Then e'S0 (e' = y ? ^ ) G ZG(S<J)) ranges over a set of representatives for the cr-conjugacy classes within the stable cr-conjugacy class of e
= fi>(m)P(9),

where (3 is a smooth compactly supported function on ZG{5Q<J)\G with P{9)dg = 1. / .ZG{S0a)\G Next we deal with the case where 7 = J. We replace H by an inner form H' if necessary, so that

Zc(5a) be defined over F. Then

ZG(5(T) is a local isomorphism and (3.2.1) defines a function /o on H'. If 771 ^ I then ZZa(Sa)(£) = Zc(£<5cr), but its square. Here we take 771 near ±1.

II. Orbital integrals

76

Hence dm = j^ip*(de$0). We have taken dl0 = T%
$st(e60(T,f;d£sB,dG).

=

Both sides are 0 when 771 is not close to ±1. Since ip : 771 1—> 77' — 77™ (m = 4) has an analytic inverse on H' in a neighborhood of I, we may define a function f{{ on H' by /O'(T/) = /o(»?i)As is well known, the orbital integrals of ftf can be transferred to H. This is clear if H' is isomorphic to H over F. Otherwise there exists /o on H with

when 77 in H is regular and corresponds to 77' in H', and with ^st(vJo;dT1,dH)=Q if 77 has distinct eigenvalues in Fx or it is a scalar multiple of a nontrivial unipotent. In this case fo(±I) = —/Q(±I). This is the required /o- The passage back from /o to / is done as in the case of 7 — —I, but we have to choose ($0 with N5Q = I and K(5O) — 1. O 3.3 COROLLARY. / / / , /o are compactly supported smooth functions on G, H with $st(7,/o;dT,dff) = $st(6o-J;ds,dG) for all 7 = N5 with distinct eigenvalues, then X*(f) = foP R O O F . Choose /Q with /Q — X*(f). Then the stable orbital integrals of /o — /o are 0 on the regular semisimple set, hence identically 0, since the germs of $ s t ( / ) at u — ±1 are scalar multiples of fo{u) and D

* • * ( « ( ; ; ) •*»)•

3.4 Unstable lifting. Analogous discussion has to be carried out for the transfer of functions from G to H\ = SO(3, F). If 7 = N\S has eigenvalues 1, 7', 7 " with 7' 7^ 7", put $us(6o-,f)

= $us(5o-,f;ds,dG)

=

^K(5')$(5'a,f;ds,dG). 5'

II.3 Matching orbital integrals

77

If / i is a smooth compactly supported function on Hi then *(7,/id/n) = *(7,/i;d7,dif1)= /

hih-^h)^-,

^z H l ( 7 )\ffi

for all regular semisimple 7. We say that /1 = \\{f)

if

*(7,/id/ii) = | ( l + 7 ' ) ( l + 7 " ) | 1 / 2 ^ U S ( ^ , / r f 5 ) for all 7 = N\5 with distinct eigenvalues, where d1 = ip*(ds) and : Z ffl (7)AZ G (<5 CT ). PROPOSITION.

fdg, with fidh1 =

For eac/i fdg there is fidhi, \\{fdg).

and for each fidhi

there is

P R O O F . This is easily verified for a function / with support near 5Q and a function /1 with support near a fixed element 70, if 70 = iVi<5o n a s distinct eigenvalues, due to Lemma 2.8.2. The difficulty is when N5Q is —I, for then there are several conjugacy classes in H\ of elements 70 with eigenvalues 1,

— 1, — 1. For each quadratic extension of F there is such a 70 in Hi (with representative (° e J in GL(2, F), 9 in F but not in F 2 ) . The proposition defines ^(,y,fi;dy,dH1) at any 7 in Hi with distinct eigenvalues; it is 0 unless the eigenvalues of 7 are close to those of 70. It has to be shown that the function $ ( 7 , fidhi) is smooth at 70 to use the classification theorem of orbital integral on Hi to deduce the existence of / j . Namely, we have to establish the smoothness at 70 of the sum Y^K(e'6)^(e'60o-,f;ds,dG)

=

^2K(e'50)$(rii>fo,dm>d'ro)

of the proof of (3.2), multiplied by |(l+7')(l + 7")|1/2H7'f/2|l+7'lHere ZG(SCT), and the product is smooth, since the eigenvalues 7', 7 ' - 1 of 7 = N(E5Q) are near —1. • 3.5 Unstable germs. It was noted above that there is a natural bijection between the conjugacy classes of 7 in Hi with eigenvalues 1, — 1, — 1 and the quotient Fx/Fx2. The cr-conjugacy classes of S in G with N5 equals the product of —1 and a nontrivial unipotent are also parametrized by Fx/Fx2. The Hilbert symbol defines a pairing, which we denote by

78

II. Orbital integrals

PROPOSITION. / / 7 in H\ has eigenvalues 1, - 1 , - 1 and fidhi \\ifdg), then lim |(l +

7l)(l+7r)|

1/2

=

*(7i,/i;rf7l(M),^1)

71—»7

= ^<7,*>*(<Ja)/;d4(M)JdG). TVie sum is over a-conjugacy classes of 6 in G with N5 — — 1 times a nontrivial unipotent. The eigenvalues of 71 are 1, 7J, 7". PROOF.

As in (3.4) the expression on the left is

\(l+yi)(l+^)\1/2^(11J1;d1,(fx),dHl)

^s(S1aJ;ds/(^,dG)

=

where 5i = e50 and N5i = ji. If (^(77^) = e',

Zc(Soa), by Lemma 2.8.1 this is equal to (the sum is over the conjugacy classes rj^ in the stable class)

J2 "(vfaiWSfai, fd; d^ (JJ), dH). Here r][ is a regular element of H, and lies in some torus T. The right side

{6;NS=-uniP7i-I}

is equal to X^T> v(»?i)<So)$(f?i. /o; d m (A0> d # ) , where <5 = f(rji)5o, and the sum ranges over the nontrivial unipotent classes 771 in H. It suffices to show the equality of the two sums only for / supported on a small neighborhood of 6' = (p(r)[)5o, where 5' is close to 6 = (p(r]i)8o, where 771 is a nontrivial unipotent in H. So we may assume that S0={

1 \ 0

, 1/

(J =

1

Uo,

*i =

V o l /

where a; £ F * , 771 = ( * * J, £ is near 0, ^ — ("e

1 \ <*e

Uo, a /

a

*) where a 2 ( l - e x ) = 1

since 1 — ex 6 F x 2 as e is small; we may assume that a is also a square,

II. 3 Matching orbital integrals

79

since it is close to 1. It has to be shown that: when N5\ = 71 —> 7, and 5i is near 5, namely r][ lies in the centrahzer ZH{I) of 7 in H (as Ndi — d ~ L 77/f), and it is near 771, then K(5') = (7,5). But 1

/ xa

i[<5'J + VV)]= ^

0

\

-1 \ 0

, -eaj

hence K(5') — (x,—e). The centrahzer ^ ( 7 ) of 7 splits over F(A) with A2 — c = 0 for some c in F x , hence (7,8) = (c,x). But r)[ lies in ZH(J) only if (A — l ) 2 — ex = 0 splits in -F(A), namely if ex/c is a square in Fx. Hence (7, S) = (x, c) = (x, ex) = (x, - e )

=

K{5I),

as asserted.

•

3.6 PROPOSITION. If \\{fdg) = f1dh1 then / i ( l ) = |2| £$(<5<7,/ds), where the sum is over the a-conjugacy classes of 5 with N6 — 1. If"y = N5 is a nontrivial unipotent then $ ( 7 , / i ; d 7 ( / x ) , ^ 1 ) = \2\*(6
PROOF.

(3.6.1)

If N5 = 1 and /£ is defined by (3.2.1) then $us(e5
where ip : H —> Z G ( < W ) , r/i is near 1 with ip{n{) = e, hence K(S5) — K(6) by Lemma 2.5. The factor |(1 + 7 ' ) ( 1 +j")\1^2 is smooth for 7' near 1, the asymptotic behavior permits the application of [L5], Lemma 6.1, hence /1 satisfies A ( l ) = «(<J)|2|/^(1). When K(S) = 1 the right side of (3.6.1) is the limit of A ! (771) $(771, fqdh) as 771 —• 1, and the left side is the corresponding limit of A ! $ ( / i ) as N{sS) = e2N8 = e 2 = 77? -> 1; 771 can be taken in the split set.

•

77. Orbital integrals

80

II.4 Germ expansion This section is not used anywhere else in this work. We sketch the wellknown germ expansion of orbital integrals (cf. Shalika [SI], Vigneras [Vi]), from which one can deduce that the fundamental lemma of II. 1 implies the matching result of II.3. For any g in G, the centralizer Zo{g) of g in G is unimodular (see, e.g., Springer-Steinberg [SS], III, (3.27b), p. 234). By Bernstein-Zelevinski [BZ1], (1.21), it follows that there is a unique (up to a scalar multiple) nonzero measure (positive distribution) on every Int(G)-orbit O. By Rao [Ra] for a general G in characteristic zero, and Bernstein [B], (4.3), p. 70, for G — GL(n) in any characteristic, this extends to a unique (nonzero) Int(G)-invariant measure 3>e> on G whose support is the closure O of O in G ($e> is the orbital integral over O; it is a linear form on C^°(G) — not only C£°(0) — which takes positive values at positive valued functions). Let s be a semisimple element in a p-adic reductive group G. Its centralizer ZG{S) in G is reductive, and also connected when the derived group of G is simply connected ([SS], II, (3.19), p. 201). Lemma 19 of HarishChandra [HC1], p. 52, can be used to reduce the G-orbital integrals near s to Za{s)-OYh\ta\. integrals near the identity. The set X of the elements in G whose semisimple part is in Int (G)s is closed (see, e.g., [SS], III, Theorem 1.8(a), p. 217). There are only finitely many Int(G)-orbits O in X (see Richardson [Ri], Proposition 5.2, and Serre [Se], III, 4.4, Cor. 2). Since O is open in O, and d i m C < dimO for every orbit O' C0,0' ^O (see Borel [Bol], 1.1.8 ("Closed Orbit Lemma"), and Harish-Chandra [HC1], Lemma 31, p. 71), there are fo £ C™{G) with ^o(fo') = So,o' for all orbits O, O' in X. In fact, the O can be numbered Oi (1 j) to have ^Oi(fOj) = 0 also for i > j . LEMMA. For every f e C%°(G) there exists a G-invariant neighborhood Vf of the identity in G, such that the orbital integral $ ( 7 , / ) of f is equal to E o $ o ( / ) $ ( 7 , fo) for all 7 inVf. The germ r 0 ( 7 ) of $(7, fo) at the identity in G is independent of the choice of fo • PROOF. The function f

= f - E o * o ( / ) / o satisfies * o ( / ' ) = 0 for

II. 4 Germ expansion

81

all O C X. Denote by C£°(X)* the space of distributions on X, and by C™(X)*G the subspace of Int(G)-invariant ones. Denote by G ~ ( X ) 0 the span of h-g-h(h£ C™{X),g G G), where g • h(x) = ^ ( I n t ^ - 1 ) ^ ) . Then ,G C~(X) = (C™{X)/C™(X)o)*. The $ o span CC°°(X)*G. Hence / ' is annihilated by any element of ( C ~ ( X ) / C ~ ( X ) 0 ) * . Then the restriction / ' of / ' to the closed subset X (see [BZ1], (1.8)) is in C™(X)0. Hence there are finitely many hi in C%°(X), and & G G, with / = 52(fij ~~ 5» ' ^ ) Extend (by [BZ1], (1.8)) ^ to elements ht of Q ° ( G ) . Then /-$>o(/)/o-$]>i-ii) O

i

is (compactly) supported in the (G-invariant) open set G — X. Hence there is a (G-invariant) neighborhood Vf of the identity in G where / = $ > 0 ( / ) / 0 + $ > i ~ Si •/>*), O

i

and the lemma follows.

•

The fundamental lemma of II. 1 can be deduced from the matching theorem of II.3 on using the following homogeneity result of Waldspurger. Let G be any of the groups considered in [W2] (these include all the groups considered here) g its Lie algebra, K a standard maximal compact subgroup (i.e. the fixer of each point of a fixed face of minimal dimension in the building of the reductive connected F-group G whose group of Fpoints is G), and t its Lie algebra (which is a sub-i?-algebra of g). Denote by ch/<- and chj the characteristic functions of K in G and 6 in g. Then [W2] defines an isomorphism e : gtn —> Gtu from the set gtn = {X G g; lim XN = 0} of topologically nilpotent elements of g to the set Gtu = JV—>oo

{u G G; lim uq

N

= 1} of topologically unipotent elements in G, named the

N—*oo

truncated exponential map. Let Onu denote the set of nilpotent orbits in 0. For each O G C?nii fix a G-invariant measure on O, and denote by 3>e>(/) the orbital integral of / G G£°(fl) over O. Fix a maximal F-torus T, let t be its Lie algebra, and denote by r r e g and t reg their regular subsets. For each O G Onii there exists a unique real positive valued function T^ on t reg satisfying the homogeneity relation

Yl{fH)

=

\n\-^°YT0{H)

82

II. Orbital integrals

for all fi G FX,H G t reg , and such that for each / G C%°(g) one has that the orbital integral

$f(H) = f

f(lnt(x)H)

JG/ZG{H)

is equal to

Yl ^''o(H)^o(f)

f° r each H in a neighborhood of 0 in i r e g .

060nil

Waldspurger's fundamental coherence result — which is not used in our proof— is the following (see [W2], Proposition V.3 and V.5). PROPOSITION

([W2]). For a sufficiently large p, for any H in t reg Cigtn,

we have <&(e(#),ch K ) =

J2 r5(ff)*o(ch0oeo n i ,

III. T W I S T E D TRACE FORMULA Summary. A trace formula — for a smooth compactly supported measure fdg on the adele group PGL(3, A) — twisted by the outer automorphism a — is computed. The resulting formula is then compared with trace formulae for H = Ho = SL(2) and H i = PGL(2), and matching measures fodh and fidhi thereof. We obtain a trace formula identity which plays a key role in the study of the symmetric square lifting from H(A) to G(A). The formulae are remarkably simple, due to the introduction of a new concept, of a regular function. This eliminates the singular and weighted integrals in the trace formulae.

Introduction The purpose of this chapter is to compute explicitly a trace formula for a test measure fdg — ®vfvdgv on G(A), where G = PGL(3) and A is the ring of adeles of a number field F. This formula is twisted with respect to the outer twisting a{g) = Jtg-lJ,

J = (° - i ' Vi o

and plays a key role in the study of the symmetric square lifting. We also stabilize the formula and compare it with the stable trace formula for a matching test measure fodh — ®vfovdhv on H(A), H = SL(2), and the trace formula for a matching test function f\dh\ = ®vf\vdh\v on H i (A), H i = PGL(2). The final result of this section concerns a distribution I in fdg, fodh, f\dh\ of the form

i = i + \r

+

\i»

+

\rx-

'o + j E ' S - ^ - ^ + K E

E

E

where each / is a sum of traces of convolution operators. The result asserts: (3.5(1)) I — 0 if fdg has two discrete components; 83

84

III. Twisted trace formula

(3.5(2)) 1 is equal to a certain integral if fdg has (i) a discrete component or (ii) a component which is sufficiently regular with respect to all other components. The result (3.5(1)) is used in the study of the local symmetric square lifting in chapter V. The result (3.5(2)) can be used to show that X = 0 and to establish the global symmetric square lifting for automorphic forms with an elliptic component. The vanishing of I for general matching functions is proven in chapter IV. Our formulae here are essentially those of the unpublished manuscript [F2;IX], where we suggested, in the context of the (first nontrivial) symmetric square case, a truncation with which the trace formula, twisted by an automorphism a, can be developed. This formula was subsequently computed in [CLL] to which we refer for proofs of the general form of the twisted trace formula. Our formulae here are considerably simpler than those of [F2;IX]. This is due to the fact that we introduce here a new notion, of a regular function, and compute only an asymptotic form of the formula for a test function with a component which is sufficiently regular with respect to all other components. For such a function / the truncation is trivial; in fact / vanishes on the G(A)-cr-orbits of the rational elements (in G) which are not (T-elliptic regular, and no weighted orbital integrals appear in our formulae. In chapters V and IV we show that this simple, asymptotic form of the formula suffices to establish the symmetric square lifting, unconditionally. Similar ideas are used in [F1;IV] to give a simple proof of basechange for GL(2), and in our work on basechange for U(3) (see [F3]) and other lifting problems.

I I I . l Geometric side 1.1 The kernel. Let F be a number field, A its ring of adeles, G a reductive group over F with an anisotropic center, and L the space of complex valued square-integrable functions

H = PGL(2, C). When y! — 1 on E1, ip = (XE © l)o o n WF factorizes via F x , and Cv = (wo,A), where A is the diagonal subgroup, w is the antidiagonal matrix, and index 0 means image in PGL(2,C). Hence Cv is Z/2. If fi'2 ^ 1 then CV = {w0). If p!"1 = 1 ^ \J! then C v = (iy 0 ,diag(-l, 1)0) is Z/2 x Z/2. If {TTQ} is a global packet containing a cuspidal representation, which is associated with a homomorphism ^.H", then the local packet {7Tot,} is associated with the restriction ipv of ip to the decomposition group WFv, C^ = Z(v>(W^)> L #) C C„„ = Z( V (W0rJ, L H), C% C C£v induce For 7To in {%o} let (s,7To) be Ylv(s,^ov)- Then the multiplicity m(7To) of 7To in the discrete spectrum is IC^I -1 X)sec (•s>7ro)> at least where we know to associate {7To} with ip, namely in the monomial case. Unipotent, nontempered representations, their quasipackets and multiplicities in the discrete spectrum, are described by conjectures of Arthur [A2]. However, for our group SL(2) these are only the trivial representation.

V.3 Characters and genericity In this section we reduce Proposition 2.5.2 to Proposition 2.5.1 for G (not for H), so we begin by recalling the main lines in Rodier's proof in the context of G. Fix d = diag(fl- r + \-7r- r + 3 ,... ^r~l). Put Vn = dnGnd~n n n and ipn(u) = ipn(d- ud ) (u £ Vn). Note that 0(d) = d, 6{Gn) = Gn, 9{Un) — Un, 6ipn = ipn, and that the entries in the j t h line (j / 0) above or below the diagonal of v = (vij) in Vn lie in 7r^ 1_2;, ' n i? (thus Viti+j G is a, ^-invariant n(i-2j)nR i f j > 0 ) a n d a l s o w h e n j < Q) T h u s Vnf)U

V.3 Characters and genericity

179

strictly increasing sequence of compact and open subgroups of U whose union is U, while Vn D (UH) — where UH is the lower triangular subgroup of G — is a strictly decreasing sequence of compact open subgroups of G whose intersection is the element / of G. Note that ipn = ijj on Vn D U. Consider the induced representations indy^ rj)n, and the intertwining operators K

• indyn ^n - • ind£ m V>m,

(A?
*
i>m{uMu-lg)du

(g in G,

302

/. Local theory

Let Mx = p £ \ Ni = V^PE™1 with m = 2mi + o(i>i) > 0. Then Nx/Mx ~ 0E/irniOE. Now we have to look for lattices M23 C V2 © V3 with the properties: a) JV23 = ViiMh => ^ 2 3 and A W M 2 3 =* b) ^23-^23 = M 2 3 and i 2 3 = id on N23/M23.

0E/*niOE; Note: £23-^23 = M 2 3 =>

^23-^23 = -^23-

The lattice M 2 3 is given by lattices p£ 2 c p ^ 2 C V2,

p ^ 3 C p ^ 3 C V3

and an isomorphism p^Vpjg 2 — P J J V P J ? 3 - We must have m 2 — m'2 = m 3 — m 3 > 0. The isomorphism in question corresponds to elements of (0E/irm*-m2 0B)x, irm'i + p£ 2 ^ uirm3 + p™3. The lattice JV23 = i/^Maa is given by u^pT"'2 _1

i/ 3 pg

m3

13

^PT '

1

and the isomorphism i/J 7r~

from ^ P ^ M " ^ "

4

m2

onto

+

C

I/^PE™ l

I^PS™2*

2

x

^PT'3

C

x

+

m

>-> -v3 u~ -K~ ^

m

v3 p~ ^/u^p^.

Property a) means that M 2 3 should have the elementary divisors 7T™1 and 1 with respect to N23- The exponents of the elementary divisors are m 2 + m'2 + m3 + m'3+ o(z/2) + o(i/3) and min[m 2 + m'2 + o(f 2 ), rn3 + m 3 + 0(^3), 0{u3^m'3NB,F{u)

+

u2TTm2+<+<-m3)]

[use, e.g., the basis (7rm2,7rm3u), (0, u m 3 ) of M 2 3 and the basis (i/a" 1 *-" 1 , - j / g - ^ - ^ u - 1 ) , ( O , ^ 1 * - ^ )

of 7V23]. Thus a) means m 2 + m 2 + m 3 + m 3 = 2m 1 + 0(2/1) - 0(^2) - t)(^ 3 ), min[m 2 + m 2 + o(f 2 ), m 3 + m 3 + 0(1/3), v(v3Tt2m'*NE/F{u)

+ ^Tr2™2)] = 0.

Consider property b). We have t23M23 — -M23 <=> t23M23 C M23 <=> (t 2 7r m 2,t 3 7r m 3u) e M 2 3 o o(i 2 - t 3 ) > m 3 - m 3 . Moreover (£23 - 1)^23 C M23 &• v(h - 1) > m 2 + m 2 + 0(2/2), 0 ( i 3 - l ) >m3+m'3

+ 0(2/3),

1.6 Fundamental lemma again

(t2 - l)v^-2m^NE/F{n)

303

+ (t3 - IK 1 *- 2 ™ 3 G OE.

Put Tij = 2m j + o(i/j). It follows from 7712 — m 2 = 7713 — m 3 and a) that m

2 = 2^ni

_ n

3 - °(jy2)),

m'3 = - ( n i - n 2 - 0(1/3))

and properties a) and b), together with m2 — m'2 > 0, are: ra2 + 7 i 3 > n i ,

ni+n3>n2,

^(n 2 + n 3 - ni) < o(£2 - £3), \{ni + n2-n3) NE/F{U')

\{ni + n3-

n2) < o(i 3 - 1),

< t>(t2 - 1), 1 n2 n3

G -v2v- n - -°^+'°^

(+nn2-niOE (t 2 - l)NE/F(u)

ni+7i2>n3,

+irn*-niOE,

if ni +n2 > n 3 and m + n3 > n 2 ), + (t3 - l) I y 2l y 3 - 1 7r ^-"3-o(^) + t.(^3)

e

nn2(DE

We have Tij = o(^) mod2. The r/j satisfy 0(1/1) + 0(^2) + 0(2/3) € 2Z. Here u is to be considered as an element of (0E/'T2(-n2+n3~ni'>0E)x. We compute Yl Card{<^} • Cardju}. (For Card{y>}: see 3.1 above). Tii,n2,ri3

3.2 (Computation of Card{u}). We may take i/* = 1 or ir, so that v ^ 1 ^ ^ ^ ^ = 1. If n 2 + n 3 = n i , the conditions are: 0 < n2 < v(t2 — 1), 0 < n 3 < 0(^3 — 1) and n2 — 0 or 713 = 0. There is one u. Assume n2 +n3 > Tii. The congruence NE/F(u) G -Tr™2"™3 + T T T 1 2 - ™ 1 0 ^ (resp. T T " 2 " " 1 ^ ) . If ni + n 2 = 713 or ni + 713 = n 2 , then 711 = 0, n 2 = n 3 > 0. The congruence NE/F{u) = —lmod^r™2 has qn2~1(q + l) solutions modulo 7r"2. If ni + 7i2 > n3 and n\ + n3 > n 2 , we get NE/F(u) G —n-™2-™3 + n2 ni ir ~ OF. We have the following cases. ni > 713. Then n2 = n3. This gives 0 < n\ < n2 — n3, NE/F(u) G -l+irn2-niOF. n\ > 77.3. Then n\ = n2. This gives 0 < 113 < ni — n2, u arbitrary. 1*1 = n3. Then ni > 712. This gives 0 < n2 < n\ — TI3, u arbitrary, and ni = n 2 = n 3 > 0, NE/F(u) ^—lmodpp. The c o n g r u e n c e (t2 - 1)NE/F(u) + (t3 - l ) ^ " 2 - " 3 G irn20EIf o(£2 — 1) > n2 and 0(^3 — 1) > n 3 , M is arbitrary. If X)(t2 — 1) > n2 and 0(^3 — 1) < n3, or o(*2 — 1) < n 2 and o(i3 — 1) > n3, there is no solution.

304

I. Local theory

If t>(*2 - 1) < n 2 and o(i 3 - 1) < n 3 , we must have t)(t2 - 1) - t)(*3 - 1) = n 2 — TI3. T h e n

NE/F(u)

= --2-— ^ - " 3 ^2

—

mod

-^o

£

C2 — -L

•!•

is equivalent to f \

tt(t3

" 1) + *>(*2 - t3) > n3,

NE/F{u)

= -irn*-n3t£Er

^modTr"2"0^-1).

[|^T ^ = f^T + g f l - We have o(t 2 - <3) > | ( « 2 + n3 - m ) > 0, so f (*2 + £3) = 0. The right hand side is the congruence for NB/F(u) is an element of Op]. The inequality o(t 3 - 1 ) + 0(^2 — £3) > n 3 is a consequence of the inequalities for t)(*2 — £3) and o(i 3 — 1) (see 3.1). If b(£2 — 1) < n 2 and 0(^3 — 1) < n 3 , the two congruences together give the following. I) m = 0, n 2 = n 3 > 0. Then x>(t2 - 1) = t>(£3 - 1) < n 2 < t>(*2 - *3)Further, JV(u) = -lmod7T n 2 ,

and

N(u) = J

3

~ l t 2 + t3 m o d T r " 2 - " ^ - 1 ) . t 2 — 1 2i 3

The element u is to be taken mod7r n2 . Prom |a=i i ^ a _ j = ^ g l ^ and t>(i2 - *3) > "2 we see that the second congruence for N(u) is a consequence of the first one. So there are qn2~1(q + 1) solutions for u. II) 0 < ni < n 2 = n 3 . Then - n i < o(i 2 - 1) = t>(*3 - 1) < "2, Further N(u) = - l m o d i r n 2 _ n i , N{u)

f(<2 - *s) > n2 -

^ -lmod7r"2-ni+1,

= _ ^ I *1±*?_ modx»a-«.(*a-D.

The element u is to be taken modulo

TT™ 2 - 2 " 1 .

-nx.

1.6 Fundamental lemma again

305

a) If ra i) there is no solution unless 0 (t2—13) > n2 and in that case the conditions for u are N(u) = -lmodTr" 2 - ™ 1 , ^ - l m o d 7 r ™ 2 - n i + 1 . There are qn2~2(q2 - 1) solutions. b) If t)(i2 — 1) < n i , necessary for solvability is that 0(^2 — £3) — o(*2 — 1) = n2 — ni and then only the last congruence for N(u) is left. There are qe(t2-t3)-i(q + -Q solutions. III) 0 < n 3 < n 2 = n i . Then \n3 < o(t 3 - 1) < n 3 , n 2 - \n3 < o(i 2 - 1) < Ti2i 0 (t2 — 1) — °(^3 — 1) = Ti2 — TI3. We have only the second congruence. There are g 0 (* 3-1 ) -1 (qr + 1) solutions for •umod7T2Tl3. IV) 0 < n 2 < n 3 = m . Then | n 2 < t»(t2 —1) < "2, n3-\n2 < tj(f 3 -l) < TI3, o(*2 — 1) _ "(*3 — 1) = n 2 — TI3. Again only the second congruence counts. There are g B ( t 2 - 1 ) - 1 (g + 1) solutions for umod7T2™2. V) ni = n 2 = n 3 > 0. Then \n\ < v(t2 - 1) = t>(i3 - 1) < nx. Further,

K

'

t2-l

2h

and N{u) ^ —lmod7r. The element u is to be taken modulo 7T5"1. Necessary for solvability is that ^j *22+*3 ^ lmod7r, i.e. o(t 2 — £3) = o(t 2 - 1). Then «f C 2 " 1 ) " 1 ^ + 1) solutions. If o(t 2 — 1) > Ti2, D(£ 3 — 1) > n 3 , the number of u is in the different cases: n\ = 0,n 2 = n 3 > 0

9" 2 - 1 (9 + l)

0 < n i < ri2 = n3

0 < n 3 < n\ — n2

0 < n2 < ni — n3

n\ — n2 = n 3 > 0

qni-2(q+l)(q-

-2)

3.3 Notations: A = t>(i2 - £3), -B = t>(*i - t3), C = v(h - t2), M = min(A,B,C), N = max(A,B,C). If A > B, then B = C, etc. F{v,t) = Y^, Card{y>} • Card{V} is the sum of the following sums (where always t»l,»2,l3

ni = » ( f j ) m o d 2 ) .

306 1)

/. Local theory ^3

1= 1

if all o(i/j) = 0, otherwise 0.

ni=n2=n3=0

2 —1

2)

J2

qni-1(q+l)

= q{q

\

2

_[X'

if all u(i/«) = 0,

n2=0,0
=

3)

5]

—

if o(i/ 2 ) = 0, 0(1/1) = o(i/ 3 ) = 1-

gWl~1(g + l ) =

2]—1 \ 1 ^

if all t>(^) = 0,

ri3=0,0
if 0(2/3)^0,0(2/1) = 0(2/2) = 1. g-1

4)

J]

gna~1(g + l) = g ( g l i

if all 0(2/,) ^ 0,

0=ni
g

5)

5]

"J - 1

" " V

if

f("i) = °. " M = 0(2/3) = 1.

g" 1+n2 ~ 3 (g + l ) ( g 2 - l )

0
_ g(g + D(g 4 W-i)

g(g 2 M-i)

g* - 1

g- 1 if all o(z/j) = 0,

g4(g + D(g4^]-i) g4 - 1

gV^]-P g- 1

if 0(2/1) = 0,

0(i/ 2 ) = e(i/ 3 ) = 1.

1.6 Fundamental lemma again

6)

Y,

307

0
n3
2 g

ffl+yM-i)

g 3 ( g + 1 ) ( g 4[f]_ 1 }

q-l

q* - 1 if all o(z/i) = 0, 2

g'W^W-i)

g

(g + i)( 9 4 W-i)
« - 1 if 0(1/3) = 0,

7)

JI

qni+n2-3(q

0(1/1) = 0(2/2) = 1.

l)(q2~l)

+

0
n2
,2W+1(,2W-i)

«3(9 + i ) ( 9 4 M - i ) g4 - 1 if all o(z/i) = 0,

q-l 2 g

[^]((?2ffl-l)

g 2 (g + 1 ) ( g 4 [ M ] _ 1 }

g4 - 1

g- 1 if 0(2/2) = 0,

8)

£

0(2/1) = 0(2/3) = 1.

g 2 "- 3 (g-2)(g + l) 2

0
=

, ( , - 2 ) f a +4 i ) y M - i )

itallo(l ,, )s0 .

g - 1

_

v

y

g 2 [f]+ 1 ( g 2 W- 2 M-l)

,

ni=0,B
if all B, 2

2

2

g ^]^ ^]- ^]-!) q - l if 0(2/1) = 0,

0(2/2) = 0(2/3) = 1, A >

B.

308 10)

/. Local theory £

qni+n*-3(q2-l)(q+l)

0
2 (?

[f]+ 1 ( g 2 M-l)(g 2 M- 2 [f]-l) q-1 if all o(fi) = 0 , A > B,

2 g

2

2

[^](g ffl_l)(g [^]-2[^]-l) if o(i/i) = 0, o(i/2) = o(i/3) = 1,

ID

£

«ni+A-2(
gJV+2M

A> R

^+1)^2[^1]-1) q

Bni

if 0(1/1) = 0,

o(z/2) = o(z/3) = A-B,

A> B.

^-^x1,_d?W)(d^) Bm £

13)

£

g ^ + B - 2 ( g + l)2 ifo(z/3)=0,

Q

o(i/i) = o(j/2) = C - B ,

S
^+c-2(g+l)2

Cn2

q-1 if o(z/2) = 0, o(z/i) = o(z/3) = B-C,

B>C.

/I
if all O(fj) = 0,

A = £ = C.

1.6 Fundamental lemma again

309

If x>{v{) = 0, 0(i/2) = o(i/3) = 1, F(v,t) is the sum of (4) + (5), (9) + (10) (if A > B) and (11) (if A ^ B and A > B). If o(i/2) = 0, 0(1/1) = 0(1/3) = 1, ^ ( M ) is the sum of (2) + (7) and (13) (if5^CandB>C). If o(i/3) = 0, o(z/i) = 0(1/2) = 1, F(v,t) is the sum of (3) and (6) and (12) (if B ^ C a n d O B ) . We can make the symmetry in the answer explicit by some computations.

+ i)(g4M-i) (4 ) + (5) = ™ T ^ 4 V ~ ^ g(g

if all „(„<) = 0,

9 -i

if 0(1/1) = 0,

(9) + (10) = 2m(

=

¥_w

^——

2[«±I]-2[-±I]

}

J

if all o(i/i) = 0 and M / A,

jf ^

^

q-1

(2) + ( 7 ) ^ ( 9 +

0j

^

= 0 ^ 3 ) = 1 and M ^ A.

l)(^-D

= idem + -

o(i/2) = o(i/3) = 1-

i f M = B>

^

^

if M + B,

q-1 if all 0(1/,) = 0;

(2) + (7)=fa + 1)ff = idem + 2

-^ ^

if o(i/2) = 0, 0(1/1) SE 1, 0(1/3) = 1.

ifM = B) ^

if Af ^ B,

/. Local theory

310

(3) + (6) = same formulas, but the different cases are M — C (resp. M / C) and all x>{vi) = 0 (resp. t>(z/3) = 0, t)(vx) = x>(v2) = 1).

The final result is: If x>(vi) = 0, v(u2) = o(f 3 ) = 1, then F(v,t) is equal to 4 [M±il

(g + 1 )

9

_

^ _ ~

idem + g 2 M -

idem +idem +

HM = A,

—

if M ^ A and M = 7Vmod2,

JV+2 J g

[ £] (g + 1)^

'_

~

1

if M / A, M £ N{2).

If 0(^2) = 0, 0(1^1) = 0(^3) = 1: the same formulas, read B instead of A. If 0(1^3) = 0, 0(^1) = 0(^2) = 1: the same formulas, read C instead of A. If all v(ui) = 0, then F(v, t) =

l + q(q3 + lV +

^+2[f](g

qi_1

+

l)g

+q4^i+iq'J"

1

~

x

(M^iV mod 2).

The last term occurs when M = N mod 2 only.

II. T R A C E FORMULA II. 1 Stable trace formula 1.1 Let F b e a global field with a ring A — Ap of adeles. Denote by E a quadratic field extension, and by A 1 the group of ideles of E whose norm from E to F is 1. The center Z(A) of G(A) = U(3, E/F)(A) is isomorphic to A 1 . Fix a character w of Z(A)/Z (Z is Z(F)). Denote the action of (a ^ 1 e) Gal(E/F) on the idele x in A£ by x. Then UJ'(X) = to(x/x) defines a character of the center Z'(A) = A^ of G'(A) — G(AE), which is trivial on EXAX. For each place v of F, let /„ be a smooth (this means locally constant in the nonarchimedean case) complex-valued function on Gv = G(FV), which satisfies fv(zx) — cjv(z)^1fv(x) for all z in Zv, x in Gv, where wv is the component of CJ at v. Further, the support of fv is compact modulo Zv. At v which splits in E we have Gv = GL(3,.F„). If v is nonarchimedean let Ry be the ring of integers in Fv and REV that of Ev = Fv ®F E. Let Kv be the hyperspecial maximal compact subgroup G(RV) of Gv. That is, it is the group of Gal(£/F)-fixed points on G(REV)At almost all v the character UJV is unramified, and we take /„ to be the function / ° , which attains the value cuv(z)~1/\Kv/Kv D Zv\ at zk in ZVKV and 0 elsewhere. Here \KV\ denotes the volume of Kv with respect to a Haar measure fixed below. Put / — ®/„. Let L = L2 be the space of complex valued functions tp on G\G(A) with ip(zg) = u>(z)ip(g) (z € Z\Z(A)) which are square integrable on GZ(A)\G(A). The group G(A) acts on L by right translation, thus (r(g)ip)(h) — ip(hg). Each irreducible constituent of the G(A)-module L is called an automorphic G(A)-module (or representation). Fix a Haar measure dg = ®dgv on G(A)/Z(A) such that FJ„ \KV/KV D Zv\ converges. Let / be any smooth complex valued function on G(A) which transforms by w _ 1 under Z(A) and is compactly supported on G(A)/Z(A). Let r(fdg) be the (convolution) operator on L which maps ip to

(r(fdg)1>)(h) - J f{g)^(hg)dg 311

(g e G(A)/Z(A)).

II. Trace formula

312

This is f(h-1g)iP(g)dg=

/ JG(A)/Z(A)

f

K(h,g)i>(g)dg.

JGZ(A)\G(A)

Hence r(fdg) is a convolution operator with kernel K(h,g)=Kf(h,g)= £

/(h^g).

(1.1.1)

ieG/z

The theory of Eisenstein series provides a direct sum decomposition of the G(A)-module L as Ld © Lc. The "continuous spectrum", Lc, is a direct integral of irreducibles. The "discrete spectrum", Ld, is the sum of the irreducible submodules of L. It splits as the direct sum of the cuspidal spectrum LQ and the residual spectrum Lr. It is a direct sum © ^ T O ^ ) ! ^ of irreducible G(A)-modules {-K,LV) occurring with finite multiplicities m(7r). If {(p'i} is an orthonormal basis of Lv then the kernel of r(fdg) on La is

Kd(k,g) = J2mW

E

[ f(h-lk)ff(h)dh-tf(g),

h in GZ(A)\G(A). Indeed, (r{fdg))(g) = £

m( 7 r)(r(/d 5 ),^} • <%(g)

= 5>(TT) J\r(fdg)4>)(h)^f(h)dh- fi(g) = VmW

/" /"

f(k)4>(hk)dk-^f(h)dh • tf(g)

JhJkeG(A)/Z(A)

f ^m(Tr) f /(h-'WUVdh-tf ro?) (k)dk. Jk L A The trace of r(fdg) over the discrete spectrum is the integral of .FQ over the diagonal k = g in Z(A)\G(A): £ E m 0 0 / f W{h)f{h-lgWi{g)dhdg = E E m W / [W(h)f(g)4>Uhg)dgdh = E E m W / [r(fdg)fi](hfflW>fo = E m w E ^ ( ^ ) ^ ' ^ ) = EmWtr7r(^)'

II. 1 Stable trace formula

313

where ir(fdg) denotes the restriction of r(fdg) to ir. The contribution to the trace formula from the complement of Ld in I? is described using Eisenstein series; we describe this spectral side below. This side will be used to study the representations n whose traces occur in the sum. The Selberg trace formula is an identity obtained on (essentially) integrating the spectral and geometric expressions for the kernel over the diagonal g = h. To get a useful formula one needs to change the order of summation and integration. This is possible if G is anisotropic over F or if / has a cuspidal component and a component supported on the regular elliptic set, or is regular in the sense of [FK2]. In general one needs to truncate the two expressions for the kernel in order to be able to change the order of summation and integration. We now turn to the geometric side of the trace formula. The geometric side of the trace formula is obtained on integrating over the diagonal g = h e Z(A)G\G(A) the kernel of the convolution operator r(fdg) on L2: f(h-1g)(g)dg

{r{fdg)4>){h) = f JG(A)/Z(A)

= 'Z(A)G\G(A) / f E w1™)
-reG/z l.TTTir,

We consider only the subsum

Ke(h,g)=

£

f(h~lx9)

XEG../Z

over the set Ge of semisimple, regular and elliptic elements x in G. A semisimple element x of G is called regular if its centralizer Z Q ( X ) in G is a torus, and x is called elliptic if it lies in an anisotropic torus. In our global case anisotropic means that T(A)/TZ(A) is compact, and in the local case it means that Tv/Zv is compact, where T = T(F) and Tv = T(FV). If x is elliptic regular, T is an elliptic torus. The integral over h — g in Z(A)G\G(A) of Ke(g,g)dg is the sum over a

II. Trace formula

314

set of representatives x for the conjugacy classes in Ge/Z of orbital integrals:

u

f(g~1xg)dg

ZG(X)Z(A)\G(A)

= J2voldt[ZG(x)Z(A)\ZG(x)(A)} [

Hg-'xg)^-.

J ZG(x)(A)\G(A)

dt

1.2 The conjugacy-class analysis of 1.1 is motivated by the appearance in the trace formula of the absolutely convergent sum that we just obtained: Y/5(x)-1\T(A)/Z(A)Tmx,fdg)

(1.2.1)

X

over all conjugacy classes x of regular elliptic elements in G modulo Z. Here 5{x) is the index \ZQ/Z{X) : T/Z] of T/Z in the centralizer ZG/Z{X) of x in G/Z, and T is the centralizer ZG{x) of x in G. The volume |T(A)/Z(A)r| of the quotient (with respect to a Tamagawa measure) is finite since x is an elliptic regular element. Wefixdifferential forms of highest degree defined over F on G / Z and T / Z , and define Haar measures dg and dt on Gv/Zv and Tv/Zv at all v. The factor $(x,fvdgv) is the orbital integral x 1 / fv{d 9~ )dg/dt (over Gv/Tv) if x is regular with centralizer Tv. We put $(x, fdg) = J[ <E>(:r, fvdgv) for regular x in G (with centralizer T). 1.3 The sum (1.2.1) can be written as a sum over the conjugacy classes in G of elliptic tori T, and a sum over the regular x in T/Z. But we have to note that 5{x) equals the number of w in the Weyl group W(T) of T in G with wxw~l = zx for some z in Z, and the conjugacy class of x in G / Z intersects T/Z precisely [W(T)]/5(x) times. So we have

where J2X indicates sum over regular elements. This is equal to

T

I

V yj

x

6 in

B(T/F)

Here X ] T indicates sum over (a set of representatives for the) stable conjugacy classes of elliptic T. The group W'(T) is the Weyl group of T in

II. 1 Stable trace formula

315

A(T/F). The element xb is b 1xb, where b is a representative of b in G(F). Note that $(xb, fdg), as a function of b, depends only on the projection of b in B(T/F). 1.4 For a fixed regular x the sum over b is finite. The pointed set B(T/F) is a subset of the group C(T/F). We extend the sum to C(T/F), setting $(xb, fdg) = 0 if b lies in C(T/F) - B(T/F). Note that if the image in C(T/A), of b in C ( T / F ) , lies in B(T/A), then 6 lies in B{T/F). Since 6 b ^(a: , fdg) — Yiv $(x , fvdgv), it depends only on the image of b in C(T/A). It remains to note that in our case the map C(T/F) —> C(T/A) is injective (in general the kernel is finite). DEFINITION.

(1) If

KV

is the restriction of

K

to C(T/FV)

*""(!,/„d f f „) = Y,Kv(1>WxbJvdgv)

we put

(b in C(T/F„)),

where we set <&{xbJvdgv) = 0 if b lies in C(T/FV) - B{T/FV). Let $K(x,fdg) be the product over all places v of the local sums (which are almost all trivial). (2) When K is trivial, put $st(x,fdg) for ^(x,fdg), and $st(x,fvdgv) for $lv(x,fvdgv). These are called stable orbital integrals. (3) Let k{T) be the finite group of characters of C ( T / A ) / C ( T / F ) . We obtain a sum

[k(T)]~lJ£^(x,fdg). Here K ranges over the finite group k(T), which is described in 1.1. 1.5 The group k(T) is trivial unless T is quadratic, when [/c(T)] = 2, or T splits over E, when [fc(T)] — 4. We obtain the sum of y

,

|T(A)/Z(A)Tlr>

t

[W"(T)p(r)] and

iy-"

|T(A)/z(A)r|

VV'

*«(*/*,)

(i«y

Y!T ranges over the T with even [k(T)\, where we put [fc'(T)] = [k{T)}/2. Consider the stable conjugacy class of the elliptic T which splits over E. Fix K / 1 .

II. Trace formula

316 LEMMA. PROOF.

We have Y!x

$K'(Z,

fdg) = E x $ " 0 , fdg) for any

K!

^ 1.

The group W'(T) acts (transitively) on the group Im[JT- 1 (j;.Y.(T" c )) - IT-^F.X.CT))],

hence on its dual group k(T). For 6 in B(T/A) (bw)T — (bw)~1T{bw) = w~1bTw • wT If Kw({bT}) = Kilw^br-w}),

and u> in W'(T), we have (wT =

W~1T(W)).

then

$(xw, fdg, KW) = £ ^ { u r ^ u ; } ) * ^ , /d 5 ) 6

6

The last equality follows from the triviality of K on C(T/F). O 1.6 Note that there is a bijection between the stable conjugacy classes of T in (1**)", and the stable conjugacy classes of elliptic tori in H = U(2) ~ U(2) x U ( l ) / Z (where U(l) ~ Z ~ E1). If T is quadratic (its splitting field is a biquadratic extension of F), then [fc'(Tfl')] = 1, and [W'(Tfj)] = 2 is the cardinality of the Weyl group of TJJ in A(Tn/F) with respect to H. If T splits over E, there are three K / 1 in (1**)", [fc'(T)] = 2 and [W(T)] = 6. With respect to H, [k(TH)] = 2 and [W'(TH)] = 2. Hence we can write (1**)" in the form ly-„

|Tg(A)/rH|

y^,

now

indicates the sum over the stable conjugacy classes of elliptic TH in H. The groups W'(TH) and k(Tu) are defined with respect to H, and J2'x is the sum over all regular x in T with eigenvalues not equal to 1. The character K is nontrivial. J2TH

The Fundamental Lemma and the Matching Orbital Integrals Lemma of 1.2 imply that we can put (1**)' in the form lv-„

|Tg(A)/rH|

^ „

t

,

II.2 Twisted trace formula

317

Indeed, we choose a global character K of AE/EXNAE A x is nontrivial. At v which splits in E we take 'f[x) = fM{x)lK{-{e-l){e:

whose restriction to

-\)).

As usual, / M is denned by / M ( " I ) = 5p{m)1/2 fN f(mn)dn where P — MN is the standard parabolic subgroup with Levi factor M and unipotent radical N, m E M has eigenvalues 1, e, e', and Sp(m) = | det(Ad(m)|n)|. The sum of (1**) is the stabilized elliptic regular part of the trace formula ofH(A).

II.2 Twisted trace formula 2.1 Analogous discussion has to be given in the twisted case. Again F is a global field, and £ is a quadratic field extension. We fix a character u>'(x) — co(x/x) on Z'(A)/Z', namely on AE/EX, which is trivial on A x . We use a test function <j> = ®(j>v on G'(A) — G{AE) = GL(3, A^), where G' = R B / F G. The component (f>v is smooth, transforms under Z'v = 2i'(Fv) = Z(EV) by OJ'V~X, and is compactly supported modulo the center. For almost all v the component cf>v is (/>°, the function supported on Z'VK'V, whose value on K'v — G'{RV) is the volume \K'vIK'vr\Z'v\~x. When v splits we take 4>v = (/„, / ° ) if fv is spherical; otherwise / ° is a measure of volume one with fv = fv* f°. So for almost all split v, we have 0° — (/°, / ° ) . The trace formula, twisted by a, is developed in close analogy with the nontwisted case. Let L' be the space of complex valued functions tp' on G'\G'(A) which transform under Z'(A) via u>', and are square integrable on G'Z'(A)\G'(A). The group G'(A) acts on L' by right translation, thus (r(g)tp')(h) — ip'(hg). Each irreducible constituent of the G'(A)-module V is called an automorphic G'(A)-module (or representation). Let a be the involution of G(A#) given by a(g) = ^^^J. This is the group of points avatar of the algebraic involution i(x,y) = (y,x) of the F-group G' = R E / F G . Put G"(A) = G'(A) xi (a) for the semidirect product of GL(3,A E ) and the group G a l ( £ / F ) = (a). Thus G " = G ' xi (t). Extend r to a representation of G"(A) on L' by putting (r(a)tp')(h) — tp'(a(h)). Fix a Haar measure dg' = ®dg'v on G'(A). Let be any smooth complex valued compactly supported modulo Z'(A) function on G'(A) which transforms under the center according to a / - 1 . Let r(
II. Trace formula

318

operator on L' which maps ip' to (r(cj>dg')ij')(h) = J4>{gW{hg)dg'

(g e G'(A)).

Then r(dg')r(cr), which we also denote by r((j)dg' x a), is the operator on L' which maps ip' to (r((j)dg')r(tj)ip')(h) = [

<j>{g){r{o-W){hg)dg' =

JG'(A)/Z'(A)

U{g)V\a{hg))dg'

J

4>{h'1a{g)W{g)dg'

= [ fth-igWivWW = / J

JG'(A)/Z'(A)'

= /

K4h,g)1,'(g)dg',

JG'Z'(A)\G'(A)

where K4h,g)=

Yl ^xo-ig)). xeG'/z'

(2.1.1)

The (T-twisted trace formula is obtained on integrating over the diagonal g — h in G'(A) the geometric and spectral expressions for the kernel of our convolution operator r{<j)dg' x a), and changing the order of the summation and integration. For this change we need to truncate both expressions for the kernel. However, the truncation does not affect the cr-regular elliptic part of the geometric side (nor does it affect the discrete part of the spectral spectrum). Thus as in the nontwisted case, we begin by analyzing the aelliptic regular part of the geometric expression for the kernel, namely its integral over the diagonal. Thus we begin with a sum J2 d'(x)-1\T(A)/TZ(A)mxa,

cf>dg'),

X

over the c-conjugacy classes x of cr-regular tr-elliptic elements in G'/Z'. The group T is the cr-centralizer of x in G'; 5'(x) is the index of TZ' in the acentralizer of x in G'/Z'. Here $(xdg') is the integral J cf)(yxa(y~1))dy over G'(A)/T(A)Z'(A). As <\> transforms by w / _ 1 , we have (zzx) = 4>(x) for z in Z'(A) ~ A^. The orbital integral dg') is a product of local orbital integrals $(xa,<j)vdg'v).

II.2 Twisted trace formula

319

LEMMA. Let J2T indicate the sum over the stable conjugacy classes of elliptic T in G, and J2X ^e sum over ^ e Te-9u^ar x' * n T/Z. Then our sum is

E' ^ ^ p E ' E •«-A«rt. T

L

V ;j

x>

b in

B'(T/F)

The sum over 6 is defined to be 0 unless there is x in G' with Nx = x'. If Nx = x', we let W'(a;') be the set of g in G/Z^x') with PROOF.

gx'g~l = zx' for some z in Z; and W'(a;) the set of g in G /FXZQ>(XL) -1 with gx^g ) = za;' for some z in Z'. Here Z c ( x t ) is the i-centralizer of x in G', and F x is the group of (z, z~l), z in Fx. It is clear that the map W'(x) —> W'(a;'), by g = (g',g") i-> 5', is an isomorphism. Also we put W(x) for the 5 in G'/Z'ZG>(xi). It is clear that 5'(x) = [W(aO], and that W(:E) —> W(a;) is injective. Further we note that the stable conjugacy class of x' intersects T/Z in [W"(T)]/[W(a;)] points. If 6"{(xL)b) is the number of b' in B'(T/F) with (xt)b conjugate to z(xt)b for some z in Z', it remains to show that [W'(x)] is S'{(xi)b)6"((xi)b), or 5"(XL) = [W'(x) : W(z)], as we can take 6 = 1 . But this is clear. Note that it suffices to deal only with x so that W'(x'), W'(x) are trivial, by virtue of our assumptions below about the support of <j>. • 2.2 The sum over 6 can be replaced by the quotient by [k"(T)] of the sum over K in k"(T) of $*(x, dg'). The group k"(T) is the dual group of the quotient of B'(T/A) by (the image of) B'(T/F), computed above. Note that [k"(T)] = [k(T)]. Hence we obtain the twisted analogue of (1*) and (1**)', namely

^

[W'(T)][k(T)} L.

l V "

|Tg(A)/TH|

$

^ ^ 9 ) ,

(2)

and ^ ,

$K(

,dl)

(2„y

The notations in (2**)' are taken with respect to H. 2.3 The twisted and nontwisted stable terms (1*), (2*) are related by the basechange map. The twisted unstable sum (2**)' can be related to the

II. Trace formula

320

stable sum of the elliptic terms in the trace formula of H, as in the case of the nontwisted unstable sum (1**)'. For that we need both the matching and the fundamental Lemmas of 1.2. Assuming that <j> and ' are global matching functions, (2**)' can be put in the form ly^„

|Tg(A)/r|

y ^

^t(xl(f)dh)

( 2 n

This is the stabilized elliptic part of the trace formula for H(A) and '4>dh.

II.3 Restricted comparison 3.1 The theory of Eisenstein series decomposes the module L' — L(G') of automorphic forms into a direct sum of two submodules, L'd and L'c. The G'(A)-module L'd is the submodule of L' consisting of all G'(A)-submodules II of L'. Each such II appears with finite multiplicity in L'd C L', and is called discrete-series representation. The G'(A)-module L'c decomposes as a direct integral. The G'(A)-module L'd further decomposes as the direct sum of the space L'0 of cusp forms, and the space L'r of residual forms. The theory of Eisenstein series provides an alternative, spectral expression for the kernel of the convolution operator r((f>dg')r(a) of section II.2. The Selberg trace formula is an identity obtained on (essentially) integrating the two expressions for the kernel over the diagonal g = h. To get a useful formula one needs to change the order of summation and integration. This is possible if G is anisotropic over F or if / has some special properties (see, e.g., [FK2]). In general one needs to truncate the two expressions for the kernel in order to be able to change the order of summation and integration. The discussion above holds for any automorphism a of finite order of a reductive connected F-group G. When a is trivial, the truncation introduced by Arthur involves a term for each standard parabolic subgroup P of G. For a ^ 1 it was suggested in our 1981 IHES preprint "The adjoint lifting from SL(2) to PGL(3)" (in the context of the symmetric square lifting) to truncate only with the terms associated with cr-invariant P , and to use a certain normalization of a vector which is used in the definition of truncation. The consequent (nontrivial) computation of the resulting

II.3 Restricted

comparison

321

twisted (by a) trace formula is carried out in [CLL] for general G and a. We proceed to record the expression of [CLL] for the analytic side of the trace formula, which involves Eisenstein series. Let Po be a minimal cr-invariant F-parabolic subgroup of G, with Levi subgroup M 0 . Let P be any standard (containing P 0 ) -F-parabolic subgroup of G. Denote by M the Levi subgroup which contains Mo and by A the split component of the center of M . Then A C Ao = A ( M 0 ) . Let X*(A) be the lattice of rational characters of A, AM = A P the vector space X*(A) M — Hom(X*(A),M), and A* the space dual to A. Let Wo = W(AQ,G) be the Weyl group of A0 in G. Both a and every s in WQ act on AQ. The truncation and the general expression to be recorded depend on a vector T in Ao = AM0- I n the case considered in this part this T becomes a real number, the expression is linear in T, and we record further below only the value at T = 0. PROPOSITION [CLL]. The analytic side of the trace formula is equal to a sum over (1) The set of Levi subgroups M which contain Mo of F-parabolic subgroups ofG'. (2) The set of subspaces A of Ao such that for some s in Wo we have A — As^a, where As^a is the space of s x a-invariant elements in the space AM associated with a a-invariant F-parabolic subgroup P of G'. (3) The set WA{AM) of distinct maps on AM obtained as restrictions of the maps s x a (s in Wo) on Ao whose space of fixed vectors is precisely A. (4) The set of discrete-spectrum representations r o/M(A) with (s x cr)r ~ r. The terms in the sum are equal to the product of

L ^ ( d e t ( l - s x cOU^/^) - 1 and

f

tT{M^(P,\)MpWiP)(s,0)IPiT(\;d9'

x a)]\d\\.

JiA'

Here [W0M] is the cardinality of the Weyl group W^ = W(A0, M) of A0 in M; P is an F-parabolic subgroup of G' with Levi component M; Mp\a(P) is an intertwining operator; M\{P, A) is a logarithmic derivative of intertwining operators, and IP>T(X) is the G'(A)-module normalizedly induced from the M(A)-module m t—» r(m)e( A ' f f ( m ^ (in standard notations).

322

II. Trace formula

REMARK. The sum of the terms corresponding to M = G' in (1) is equal to the sum / = ^trll(dg' x a) over all discrete-spectrum representations II of G'(A) which are cr-invariant. We write tr Il((pdg') for the trace of the trace class convolution operator U((j)dg') = / 4>(g')U(g')dg' (g' in G'(A)/Z'(A); dg' is a Tamagawa measure, often omitted from the notations), for an admissible II. The spectral side of the nontwisted trace formula for G(A) is described by the Proposition above, where a is replaced by the identity and G' by G. The trace formula for G(A) and the trace formula for G'(A) twisted with respect to a, are compared in II.4 below for measures fdg and , '/, ' whose components at u, u' have (twisted in the case of ) orbital integrals which vanish on the (resp. a-) regular split set. An element is called split if its conjugacy class intersects the diagonal torus nontrivially. Further, we fix a nonarchimedean place u", and require that the (resp. a-) orbital integral of the component at u" be zero on the (resp. a-) semisimple singular set. These conditions imply that the geometric expression for the kernel contains only terms indexed by (resp. a-) conjugacy classes of rational (resp. a-) elliptic elements in G (resp. G').

Under the above restrictions at u, u', u" on the test function / on G(A) (and the matching

All sums here are absolutely convergent. The new sum extends over all characters n of Ag/ExNAg. The diag1 onal subgroup A (A) of G(A) consists of diag(a, 6, a"" ), o in A^, b in A 1 . Any character of A (A) /A whose restriction to Z(A) is w, is of the form diag(a, b, a - 1 ) H-> fi(a)(w//j,)(b), where /z is a character of Ag/Ex. We denote the G(A)-module normalizedly induced from the character of A(A)

II.3 Restricted

comparison

323

by 1(H). We shall also use the analogous notations in the local case. The intertwining operator M(fi) is defined in the theory of Eisenstein series. 3.2 The twisted trace formula of our group G'(A) is to be discussed next. The center Z'(A) of G'(A) = GL(3,A E ) is isomorphic to Ag. The norm map N takes z in Z'(A) to z/~z in Z(A). The restriction to A x of the character w' = w o N of Z'(A) is trivial. Let L(G') once again be the space of complex valued functions -ip on G'\G'(A), which transform under Z'(A) by u/, and are absolutely square integrable on G'Z'(A)\G'(A). The group G'(A) acts on L(G') by right translation. The irreducible constituents II are called automorphic. The discrete and continuous spectra are invariant under the action of <7, which maps ip to aip, where ((x)If.{x)Il{o-)d'x (x in G'(A)/Z'(A); d'x is a Tamagawa measure, often omitted from the notations). As usual normalizedly induced G'(A)-modules are denoted by I(rj). Here T] — (/z, / / , /i") is a character of the diagonal subgroup A'(A) of G'(A). The /x, / / , fi" are characters of Ag/Ex. For each element w of the Weyl group W of A in G, there is an intertwining operator M(w, rj) from I(rj) to I(wrj), where (wrj)(a) = r)(waw~l). The I(r)) which appear in the trace formula are those whose central character \iy! y!' is equal to u/. Suppose T is an irreducible H'(A) = H(A£)-module, where H ' is RE/FXJ{2,E/F), thus H' = GL(2,E). Denote by I(T) the G'(A)-module normalizedly induced from the H'(A) x / asserts the following. (For a similar case see [F2;I].) LEMMA. The sum J^ tr Ii(4>dg' x a) over the a-invariant representations II o/G'(A) in the discrete spectrum is equal to the sum of (2*), (2**), and

324

II. Trace formula

- ^ Y,tr J fa' ^dg' xCT)+ g I Z t r 7^' ^ ' x (J) + g J2tr 7^' ^ ' xCT)_ 2 ^ t r 7 ( T ' ^d5'x

CT

^'

A// G'(A)-modules I{rf), I(T) here are a-invariant. The characters n, y!, n" in rj are trivial on EXNA^. The first sum is over all unordered triples of pairwise distinct fi, y!, //'. The second is over all (fi,y',fi), yl ^ y. In the third y — y! = y". The I(rj), I{T) here are all irreducible. In fact the way in which the l(rj) appear in the trace formula is as

±Y,tTM((13),v)I(r,dg',cf>xo-)

+±

£

JtrMK^.^i/xff).

w=(12),(23)

r)

The nonzero contributions are given by the rj for which rj, acted upon by a and then the reflection w is equal to rj. Thus the first sum is over the 77 with y, y!, y" trivial on NA^; the others are over the n with y = y' — y", y trivial on NAg. The intertwining operators M(w,r}) can be written as local products m(w,r]) ®v R(w,riv) (see [Sh]). Here R(w,rjv) are the local normalized intertwining operators. They are trivial in our case. The normalizing factors m(w,r]) are given by m((12),?7) = L(l, y'/y)/L(l,y/y'), m((23),V)=L(l,y"/y')/L(l,y'/y"), and m((13),7y) is [L(l, y"/y')/L{\,

H'/H")] [ i ( l , l*"/ri/L(l,

/ V / O ] [HI, fJ.'/l*)/L(l,

y/y%

If at least two of the y's are equal, m(w, rj) has to be evaluated as a limit; the value is —1. If the y are all distinct, then m((13),J?) is 1. Indeed, L(l,y) = L(l,~y), and here ~y = y_1. Up to equivalence each I{rf) appears in the first sum 6 times if the y are distinct, 3 times if exactly two of the y are equal, and once if y = y! = y". Whence the expression of the lemma. 3.3 The character y oiA^/Ex defines a character of the diagonal subgroup 'A(A) of H(A), by diag(a, a - 1 ) 1—»y(a), and an induced representation I(y). Under the usual restriction on '/ at u, u' and u", the trace formula for H(A) and '/ asserts the following (see [F3;II]).

II.3 Restricted

comparison

325

The sum Yln(p)tT{p}{'fdh) over all automorphic packets {p} o/H(A), is equal to the sum o/(l**) (times 2), and \ ^ t r I(p,,'fdh). The sum over p, is taken over all characters of Ag/EXAX. LEMMA.

The automorphic, and local, packets of H(A)-modules, and the global multiplicities n(p) (= 1 or 1/2), are defined in [F3;II]. 3.4 We now obtain an identity of trace formulae. Let E/F be a global quadratic extension, and tydh, 4>dg', fdg, 'fdh matching measures on H(A), G'(A), G(A), H(A). We assume that the (twisted) orbital integrals of the components at u, u' are 0 on the (
In the above notations, we have

]TII trn v(v d 9v

x

°~) + 2 S I I t r / ( ^ ; ^vdgv

~ 2^£2ntTl(Vv'Vv,Hv);vd9'v = ^W^^v{fvd9v)

-

x CT

)

x o-) ^^2n(P)Yltr{pv}('fvdhv)

+ 2 5 Z " ^ I I te{Pv}('vdhv) - ^2 m(r)) J J tr R(pv)I{pv, +

g^2~[[tTH^v/fvdhy)

fvdgv) -

-^2Y[trI(nv,'(f)vdhv).

The products FJ are taken over all places v of F. It is useful to fix a finite set V of places, which includes u, u', u", the archimedean places and those places which ramify in E/F, such that '<j>v, cpv, fv, 'fv are spherical outside V. Then the components II„, nv and pv are unramified, and correspond to the conjugacy classes t'v xcr,tvx cr, 'tv x a in the dual groups LG', LG, LH, by the definition of the Satake transform. For each v outside V we fix tv x a, and let t'v x a be its image under the basechange map LG —> LG', 'tv x a the pullback via the endo-map LH —> LG, and 't'v x a the pullback of t'v x a via

326

77. Trace formula

the (T-endo-map LH —> L G''. A standard approximation argument, based on (1) the fact that the sums in the Proposition are absolutely convergent, and (2) the matching result of 1.2 and 1.3, for corresponding spherical functions, implies the following COROLLARY. Fix {tv x a; v outside V}. Then all products in the Proposition extend over V. The sums range over H, -K, p whose component at v outside V is parametrized by tv x a.

The rigidity theorem for G' = GL(3) of [JS] implies that at most one of the first five sums involving G'-modules is nonempty, and this sum consists of a single G'-module by multiplicity one theorem.

II.4 Trace identity S u m m a r y . The identity of trace formulae is proven for arbitrary matching functions, under no restriction on any component. The method requires no detailed analysis of weighted orbital integrals, or of orbital integrals of singular classes. 4.1 I n t r o d u c t i o n Let E/F be a quadratic extension of global fields. Put G' for G(E) = GL(3,.E). Denote by G = G(F) the quasi-split unitary group in three variables. It consists of all g in G' with ag = g, where we write ax = Jtx~1J for x in G' : x is (x~ij) if x = (x^j), the bar indicating the action of the nontrivial element of the Galois groups Ged(E/F), and

' = ( X ) > »=(-°iJ)Similarly we put ax = wtx~1w~1 for x in H', and introduce H' — 11(E) = GL(2,E) and H = H ( F ) = {g £ H';ag = g}. Then G = \J(3,E/F), H = U(2, E/F). We use the following smooth complex-valued functions. (1) '/ = ®'fv and '4> — ® '4>v are compactly supported on H(A) (A = Kp indicates the ring of adeles of F). (2) f — ®/„ on G(A) transforms under the center Z(A) (~ A^ : ^-ideles of norm 1 in A£) of G(A) by a fixed character w _ 1 , where a; is a character

II. 4 Trace identity

327

of Ag/E1 (E1 = {x G E;NE/Fx = 1}); / is compactly supported modulo Z(A). (3) (j) — ®(j)v is a function on G'(A) = G ( A E ) which transforms under the center Z'(A) = Z(A B ) (~ A£) of G'(A) by w / _ 1 , where J{x) = ui(x/x), x in the group A^ of ideles. The local components of 'fdh, fdg, <$>dg',tydhare taken to be matching, namely their orbital integrals are related in a certain way, specified in II.2. Our purpose here is to prove the following: T H E O R E M . Let 'fdh, fdg, 4>dg', '4>dh be matching measures. Then the identity displayed in Proposition II .3.3 holds.

We abbreviate this identity to: ^ m ( n ) t r l l ( < £ d < / x a) - ^ n ( p ) t r {p}('^d/i) {/>}

n

= 5>(7r)tr7r(/d 9 ) - ^5>(p)tr{p}('/d/i). n

{p}

Here the sum over II ranges over various automorphic a-invariant G'(A)modules and m(II) is 1 if II is discrete spectrum, \ if II = I(T) and \, — | or — | if II = I(rj). The IT are automorphic G(A)-modules which may be discrete spectrum or induced and the m(-7r) are integers, \ or \. The {p} are automorphic H(A)-packets, and the n({p}) = n(p) are again rational numbers. Proposition II.3.3 asserts the theorem under the additional assumption that two local components of (each of) '/, / , , f, etc., rather than the measures ,... on two (or more) groups G, G', . . . , appears already in (Chapter 16 of) [JL]. But attention to the problem was drawn by Langlands' study [L5] of the first nontrivial case, namely the comparison needed for the completion of the cyclic basechange theory for GL(2), initiated by Saito and Shintani. Langlands proved the reauired identity for GL(2) on (1) computing the weighted orbital integrals and orbital integrals of singular classes which

328

II. Trace formula

appear in the trace formulae, (2) analyzing the asymptotic behavior of the weighted integrals, (3) applying the Poisson summation formula, and so on. The method presented here is entirely different. The principle is that it suffices to check the identity of the Theorem only for a small class of convenient test functions, and then use the fact that we deal with characters of representations to conclude that the identity holds in general. It is not necessary to deal with arbitrary /,,... at the initial stage. In fact, it is shown below that for a suitable choice of test functions (whose definitions we leave to the text itself), the weighted orbital integrals and the orbital integrals at the singular classes are equal to zero. In particular they need not be further computed and transformed. The proof turns out to be rather simple, once the right track is found. The present method applies also in the case considered in [L5] to yield a simple and short proof of the trace identity needed for the comparison of basechange for GL(2). It makes a crucial use of the existence of a place u of F which splits in E. The observation underlying our approach is that the subgroup Fx of rationals is discrete in the group A£ of ideles. That this simple fact can actually be used to annihilate the undesirable terms in the trace formula was suggested by Drinfeld's use of spherical functions related to powers of the Probenius, in the course of the work, [FK2], [FK3] with D. Kazhdan, on the Ramanujan conjecture for automorphic forms with a cuspidal component of GL(n) over a function field. In the present section admissible spherical functions are used to establish the theorem by our simple approach. This technique is developed in [FK1] to establish the metaplectic and simple algebra correspondences in the context of arbitrary rank and cusp forms with a single cuspidal component. A different variant of the approach, based on the use of regular Iwahori biinvariant functions, is applied in [F1;IV] to give a simple proof of cyclic basechange for GL(2) with no restriction on any component, in [F2;VI] to prove the absolute form of the symmetric square lifting from SL(2) to PGL(3), and in [F1;V] to establish by simple means cyclic base change for cusp forms with at least one cuspidal component on GL(n). To complete this introduction we now sketch the proof which is given below. We deal with four trace formulae for test functions / , <j>, '/, ', '/, '). Each

II. 4 Trace identity

329

trace formula is an equality of distributions in the test function. These distributions are as follows. 01 involves "good" orbital integrals, on the set of rational regular elliptic elements. WI involves "bad" orbital integrals, on the set of rational elements which are not regular elliptic; these "bad" integrals are mostly weighted and noninvariant as distributions in the test function. RD is a (discrete) sum of traces of automorphic representations; these occur with coefficients which may be negative when the representation is not cuspidal. RC is an integral (continuous sum) of traces of induced representations; these traces are often weighted, and the distributions which make up RC are mostly noninvariant. The trace formula takes the form I — R, where R = RD + RC is the representation theoretic side, and I = 01 + WI is the geometric side (orbital integrals) of the formula. We shall be interested in a linear combination of the four formulae. Put RD(q) =

RD(/)~RD('/)

RDM-^RD(ty)

and introduce OI(q), RC(q) analogously. From now on we always choose the four components of q to have matching orbital integrals. This choice implies the vanishing of 01 (q). Hence RD(q) = WI(q) - RC(q). In these notations, the Theorem can be restated as follows. THEOREM.

For any quadruple q of matching functions we have

RD(q) = 0. Fix a nonarchimedean place u of F which splits in E. Then G(F„) = GL(3, Fu),

G'{FU) = GL(3, Fu) x GL(3, Fu),

H(FU) = GL(2, Fu). Fix a quadruple q" = (/", cf>u, '/", '(j>u) of the components outside u of q. Put RC(q u ) for RC(q„ ® q"), where Qu

=

yjuityui

Jui

fyu)-

As the first step in the proof we explicitly construct for any /„ a quadruple Qu = q(/u) which has the property that RC(q(/„)) depends only on the orbital integrals of fu.

330

II. Trace formula

For the second step of the proof, we say that a function f'u on G(FU) is n0-admissible (for some n 0 > 0) if it is spherical and its orbital integrals on the split regular set vanish at a distance < no from the walls [namely, on the orbits with eigenvalues of valuations n\, n2, n 3 such that \rii — rij\ is at most n 0 for some i ^ j (i, j = 1, 2, 3)] We prove: For any quadruple q" of matching / " , ",'/", '", which vanish on the adeles-outside-u orbits of the singular-rational elements, there exists an integer no = no(q") such that WI(q(/^)) = 0 for every no-admissible f'u. Note that in this case all of four components of q(/„) are spherical. To prove this we show in the Proposition that, given / " which vanishes on the G(A u )-orbits of the singular set in G(F), there exists n 0 = n 0 ( / u ) > 0, such that for every no-admissible /„ there exists a function /„ with the same orbital integrals as f'u with the property that / " /„ is zero on the G(A)orbits of all "bad" rational elements. In particular W I ( / U ® fu) — 0. The function /„ is obtained by replacing f'u by zero on a small neighborhood of finitely many split orbits where the orbital integral of f'u is zero. Choosing no sufficiently large, depending on q", and noting that the construction of q ( / u ) is such that its components are zero on the image of the split regular orbits where /„ is zero, we conclude that for every no-admissible f'u there is fu with orbital integrals equal to those of f'u such that WI(q(/ u )) = 0. Consequently WI(q(/„)) = 0 for every no-admissible fu, since WI(q(/„)) - RD(q(/„)) + RC(q(/ u )) and RC(q(/ u )) depends (by Step 1) only on the orbital integrals of /„ (which are equal to those of f ' u ) . The third step asserts that since RD(q(/^)) = - RC(q(/£)) for every no-admissible f'u we have RD(q(/^)) = RC(q(/^)) = 0 for every spherical f'u. This follows from the final Proposition in [FK2], where this claim is stated and proven in the context of an arbitrary reductive p-adic group. Fix a nonarchimedean place v! of F which splits in E. It follows from Step 3 that for any q„/ whose 4 components vanish on the singular sets, we have RD(q" <8> q u ') = 0 for all q u . The fourth step is to show that this holds for any spherical qu<. The proof is the same as in III.1.2. We will recall the argument here. Write RD(q) as a sum 5^ x RD(q, x) o v e r a u infinitesimal characters x, of the partial sums R,D(q, x) of RD(q) taken only over those automorphic

II. 4 Trace identity

331

representations whose infinitesimal character is x- Since the archimedean components of q axe arbitrary, a standard argument of "linear independence of characters" implies that since RD(q) = 0, for every x w e have RD(q, x) — 0 if q u ' = 0 on the singular set. Fix q" , and consider RD(q, x) as a functional on the space of Iwahori quadruples q„' (i.e., quadruples whose components are biinvariant under the standard Iwahori subgroups). There are only finitely many automorphic representations with a fixed infinitesimal character, fixed ramification at each finite place ^ u', whose component at u' has a nonzero vector fixed under the action of an Iwahori subgroup. Hence as a functional in the Iwahori quadruple q u /, RD(q, x) is a finite sum of characters. As it is zero on all q u / which vanish on the singular set, and our groups are GL(2) and GL(3), it is identically zero. In particular RD(q, x) vanishes on the spherical quadruples q„/, from which the Theorem easily follows. This completes our outline of the proof of the Theorem. 4.2 Conjugacy classes Let v be a place of F. Denote by Fv the completion of F at v, and put Ev = E ®p Fv. If v stays prime in E, then Ev/Fv is a quadratic field extension. If v splits into v',v" in E, then Ev — Ev> x Ev", where Evi ~ Ev" ~ Fv. In this case G'v = G(EV) = GL(3,F„) x GL{3,FV), and Gv = G(F„) = {(g, has an unramified component u>u at u. Replacing w by its product with an unramified (global) character we may assume that uiu = 1. Then fu(zg) = fu(g) for g in Gu, z in the center Zu of Gu, and /„ is compactly supported on ZU\GU. Let F{9ifu) = A(g)3>(g, fu) be the normalized orbital integral of / „ . Let Ru be the ring of integers in Fu. Put Ku = G(RU); it is a maximal compact subgroup of Gu. A spherical function is a K„-biinvariant function. The

II. Trace formula

332

theory of the Satake transform implies that a spherical fu on Gu is determined by its orbital integral on the split set. Let | • | be the (normalized) valuation on Fu, put q = qu for the cardinality of the residue field of Fu, and val for the additive valuation, defined by \a\ = <jrval(a) for a in F£. Let n = (ni, 712, 713) be a triple of integers. Let f'u be the spherical function on Gu for which F(g, f'u) is zero at the regular diagonal element g = (a, b, c), unless up to conjugation and modulo the center we have (val a, val b, val c) = n, in which case we require F(g,f'u) to be equal to one. Embed Z in Z 3 diagonally. The symmetric group S3 on three letters acts on Z 3 . Denote by Z 3 /S3Z the quotient space. Then / „ depends only on the image of n in Z 3 jS^L. We write f'u — f'u(n) to indicate the dependence of fu on n. DEFINITIONS. (1) The function fu on Gu is called pseudo-spherical if there exists a spherical function f'u with F(g,fu) = F{g,f'u) for all g in Gu. We write /„(n) for / „ if f'u = /£(n). (2) Let no be a nonnegative integer. An element n = (ni, ri2, 713) of Z 3 /S I 3Z is called no-admissible if \rii — rij\ > no for all i ^ j ; i, j — 1, 2, 3. We also fix a place u' of F which stays prime in E such that Eui /Fu> is unramified, and a positive integer n'. Let S = S(u',n') be the set of g in Gui which are conjugate to some diagonal matrix diag(a, b,a~x) with \a\u> = q™', (and |6|„' = 1); a e E*, and b € E\,. We shall assume from now on that the component fu> is a (compactly supported, locally constant) function on Gu* such that F(g, fui) is the characteristic function of S. Since S is open and closed we may and do take /„/ to be supported on S. There exists an integer no > 0 depending on fu, such that for any no-admissible n there is a pseudo-spherical fu = /u(n) with the property that f = fu ® / „ satisfies the following. If 7 lies in G(F), x in G(A), and f(x~lryx) ^ 0, then 7 is elliptic regular. PROPOSITION.

If /„/ (x~lryx) ^ 0 then 7 lies in S, hence it is regular in Gu* and also in G. If 7 is not elliptic, then we may assume that it is the diagonal element diag(a,6,a - 1 ) with a in Ex and b in E1 = {b in Ex; bb = 1}. Modulo the center we may assume that 6 = 1 . Also we have aa ^ 1. At the split place u we have a = (a,P), with a, ft in F u x . Hence 7 is diag(a, l , / ? - 1 ) in Gu. Since / " is fixed, there are Cv > 1 for all v ^ u, with Cv = 1 for almost all v, such that C~l < \a\v < Cv for all v ^ u PROOF.

II. 4 Trace identity

333

if / " ( x _ 1 7 x ) 7^ 0 for some x in G(A). Here \a\v = \NE/Fa\v. Since a x x lies in E*, and NE/Fa in F , the product formula on F implies that |a/5|u = \NE/Fa\u — \a\u lies between Cu = Y\v^u CV and C " 1 . We take n 0 with q™° > Cu. Consider an n 0 -admissible n and the spherical fu = fu(n). If f'u{x~1"fx) ^ 0 for some x in Gu, then there is some C'u > 1 such that \a\u and |/3| u are bounded between C'u and C'u~l, so that a lies in the discrete set Ex and in a compact of A^, hence in a finite set. Hence 7 lies in finitely many conjugacy classes modulo the center; let 7 1 , . . . ,7t be a set of representatives. Put 7i = diag(aj, l,ft _ )• By definition of fu, if F{lii f'u) 7^ 0 t n e n w e n a v e t n a t l a ift| or \ctif3i\~1 is bigger than <7™°, hence f{x~lr)ix) — 0 for all x and i. We conclude that F(ji, fu) = 0 for all i. Let Si be the characteristic function of the complement of a small open closed neighborhood of the orbit of 7, in Gu. Then the function / „ = f'u \\i Si on Gu has the required properties. • Let L(G) denote the space of automorphic functions on G(A); these are the square-integrable functions on Z(A)G\G(A) which transform on Z(A) by uj and are right invariant by some compact open subgroups; see [BJ] and [Av]. The group G(A) acts on L(G) by right translation: (r(5)^)(/i) = ty(hg). Then r is an integral operator with kernel Kf(x, y) — ^ / ( £ _ 1 7 y ) , where 7 ranges over Z\G. In view of the Proposition, the integral of Kf{x,y) on the diagonal x — y in G(A)/Z(A) is precisely the sum (1.2.1), which is stabilized and analyzed in II. 1. The remarkable phenomenon to be noted is that for / with a component /„ as in the Proposition, the only conjugacy classes which contribute to the trace formula are elliptic regular. The weighted orbital integrals and the orbital integrals of the singular classes are zero, for our function / . Moreover, the truncation which is usually used to obtain the trace formula is trivial, for our / . Each component vdg'v on G'(A) = G(A#) is taken to be matching fvdgv in the terminology of 1.2. In particular we take 4>udg'u to be (fudgu, fudgu), where f®dgu is a unit element of the Hecke algebra. Namely the pseudo-spherical fudgu is biinvariant under some cr-invariant compact open subgroup /„ of Gu, where a(g) — Jtg~1J, and f° is taken to be the characteristic function of ZUIU, divided by the volume of IUZU/ZU. Then fudgu = f°dgu * fudgu = fudgu * f°dgu. An immediate twisted analogue of the proof of the Proposition establishes the following.

334

II. Trace formula

PROPOSITION. / / n is n0-admissible, 5 lies in G', x in
G{&E)

o-nd

Here N denotes the norm map from the set of stable cr-conjugacy classes in G" (and G ( A B ) ) onto the set of stable conjugacy classes in G (and G(A)) (see 1.1.5 and [Kol]). Again we can introduce the space L(G') of automorphic functions on G'\G'(A) which transform on Z'(A) by u>' and the right action r' of G ( A E ) on L(G'). The Galois group Gal(E/F) acts on L(G') by (r'({x^1Sa(y)) (6 in Z'\G'). The Proposition shows that the integral of Kj, along the diagonal x = y in Z(AE)\G(AE) is precisely the sum which is stabilized and discussed in II.2. The functions '/ and ' on H(A) are taken to be matching with / and , as defined in 1.2. Their components at u can be taken to be pseudospherical, and the Proposition and its applications hold for '/ and '4> as well. It remains to consider the contribution to the trace formulae from the representation theoretic side. 4.3 Intertwining operators For brevity we denote by J the difference of the two sides in the equality of our theorem. Then J is the difference of the two sides in the equality of II.3. These are the invariant representation theoretic terms in our trace formulae. The work of II. 1 and II.2 concerns the stabilization of the orbital integrals on the elliptic regular conjugacy classes which appear in the trace formulae. It implies that for arbitrary matching functions '/, / , , '4> the difference J can be expressed as a sum of integrals of logarithmic derivatives of certain intertwining operators, which we momentarily describe, and weighted and singular orbital integrals which vanish for functions as considered in 4.2. In II.3 we concluded from this that J = 0 if the functions f,,... have two elliptic (= discrete) components. To deal with the case of arbitrary / , ( / > , . . . we now record an expression for J, excluding the weighted and singular orbital integrals, as follows. The expression consists of four terms, one for each of , f, '/, 'cp. These are the terms involving integrals (over iR) in the trace formulae. They are analogous to the terms (vi), (vii), (viii) of [JL], p. 517. We use the notations of II.3.1, which are standard notations. The term J(
II. 4 Trace identity

335

three expressions, equal to each other. The coefficient [W()Vf]/[Wo](det(l — s x CF\AM\A) of H.3.1 (and [A2], Thm 8.2, p. 1324) is ^ (here M = M 0 is the diagonal subgroup A; the Lie algebra A is one-dimensional). Hence we obtain

J{4>dg') = \Y,J

^[M{\ 0, -X)IPoT ((A, 0, -A); <j>dg' x a)}dX.

The sum is over all connected components (with representatives r = (/ii,/X2,Af3)) of characters of A(AE)/A(E), with ar = r. More precisely, let v be the character v(x) = \x\ of A^. Note that A ~ G^,. The connected component of r consists of T\ — (/zii'A,/X2,At3I/_A)! ^ m *^- The fij are unitary characters of AE/EX, and /ii^2A*3 = u/. We put Jp0]1.((A, 0, —A)) for the G(A£)-module normalizedly induced from T\\ T\ is regarded as a character of the upper triangular subgroup Po(A) which is trivial on the unipotent radical of Po(A). The action of a takes r to (JI^1, Jl^1, Pi1), where ~p(x) = n(x). Hence ar = T implies r = (fi,w'/l///,/!-1), where fi = Hi.

The operator M. is a logarithmic derivative of an operator M — m®vRv, where Rv denotes a local normalized intertwining operator. The normalizing factor m = m(A) = m(A,r) is an easily specified (see [F2;I], C2.2) quotient of L-functions, which has neither zeroes nor poles on the domain iM. of integration. Then the logarithmic derivative M. is m'(A)/m(A) + ( s i ^ 1 ) —(®ij„), and we obtain J(4>dgl) = Jl{(j)dgf) + Y^v Jvi^dg'), where dX T

7m

'

*" V

and Jv{4>dg') is i ^y'[tril r „(A)- 1 iJ T B (A) , / T „(A;0„d 5 J l x
The abbreviated notations are standard. The sum over v is finite. It extends over the places v where 4>v is not spherical, since when <j>v is spherical

II. Trace formula

336

the operator ITv(X;(f>vdg'v x a) factors through the projection on the onedimensional subspace (if rv is unramified) of Kv = GL(3,i?„)-fixed vectors, on which RTv (A) acts as the scalar one, so that RTv (A)' = 0. Next we have to record the analogous term J(fdg) of the trace formula for G(A). Again we use the notations of 3.1, with a = 1. This rank-one nontwisted case is well known (see [JL], pp. 516-517). We take M — Mo, and A = AM is one dimensional. The element s of the Weyl group is s = id; it lies in WA(AM)The Weyl group W0 has cardinality two, and [Wo1] = 1, and AM/A = {0}. Hence the coefficient of J(fdg) is \, and J d

(f 9) = \Y.

[trM(\)I(fi®\;fdg)d\.

The sum ranges over all connected components with representatives fi, where fi(a, b, a - 1 ) — n{a/b)oj(b). Here a lies in A^, b in A^, \i is a character of Ag/Ex, and the connected component of [i consists of fi ® A, where H is replaced by /J,VX, for A in iR. The induced G(A)-module I(fi ® A) lifts (see Proposition 1.4.1) to the induced G(As)-module IT{X), where r — (n,uj'jl/n,'p~1). This relation defines a bijection /j, <-> r between the sets over which the sums of J((j>dg') and J(fdg) are taken. Here A4(X) is again a logarithmic derivative of an operator M = m®vRv, and J(fdg) is the sum of J'(fdg) and ^2 V Jv{fdg), where

^(/ds) = ^E/^[n t r / (^® A ;/» d fl«) and Jv(fdg)

i2 ^

d\

is

[ti[Rllv(\)-1Rltv(\yi(iiv®\;fvdgv)]-

J J t r / ( ^ ® X; fvdgv)dX.

-

^

Note that here the normalizing factors m(A) depend on fi, while those of J'(cf)dg') depend on r. It is clear (see Proposition 1.4.1) that for matching measures fvdgv and vdg'v we have tiI(nv®X;fvdgv)

= ti ITv{X;4>vdg'v x a),

if

r„ =

(nv,uj'lJIv/ij.v,-p^1).

It can be shown directly that 2m'(X, fi)/m(X, fi) = m'(A, r)/m(A, r ) , and hence that J'(fdg) = J'(4>dg'), but we do not need this observation. The fundamental observation which we do require is the following.

II. 4 Trace identity For our choice of fudgu, Ju{dg') = Ju(fdg). LEMMA.

337

4>udg'u = (fudgu, f°d9u)

we have

P R O O F . This is precisely Lemma 16, p. 47, of [F1;III], in the case I = 2. Note that the proof of this Lemma 16 is elementary and self-contained. To see that this Lemma 16 applies in our case, recall that we choose / ° to be the characteristic function (up to a scalar multiple) of ZUIU, where /„ is a cr-invariant open compact subgroup of Gu- Then f>9)

= f°(9),

a

-Ku{fu)=*u{fu)

and

fu =

fu*°fu=fu*fu

in the notations of [F1;III], (1.5.2), p. 42, /. 7. In fact this Lemma 16 of [F1;III] asserts that tr RTu

(T)"

1

^ {T)'ITU

(7; 4>udg'u xa)=£ti

R^ {j)'I(pu

® 7; fudgu)

in our notations, where I = 2. This is precisely the factor needed to match the \ of Ju{(f)dg') with the \ of Ju(fdg). Our lemma follows. • It remains to deal with the terms of J('fdh) and J('(f>dh). Since this case of U(2) is well known (see [F3;II]) we do not write out the expressions here, but simply note the following. (1) We may assume that the place u is such that the component KU of the character K on Ag/ExNAg is unramified. (2) We may and do multiply K by an unramified (global) character to assume that KU = 1. (3) If 'fvdhv and '4>vdhv are matching measures on Hv in the notations of 1.2, and pv = 'I{Hv),

p'v -

'I{/J,VKV)

in the same notations, then trp v ('f v dh v ) = trp'v('(j>vdhv) by Proposition 1.4.1. (4) At the split place u we take the components 'fu and '4>u to be defined directly by the same formula (of 1.4.4) in terms of / „ ; they are equal to each other. We conclude: LEMMA.

In the above notations, we have Ju('fdh)

= Ju('
P R O O F . This follows from (3) and (4). Indeed, the sets of p. parametrizing the sums which appear in J('fdh) and J('<j>dh) are isomorphic. The

II. Trace formula

338

isomorphism ('I{n) —>' I((I,K)) is defined by the dual group diagram and by Proposition 1.4.1. • R E M A R K . J'('fdh) and J'('(pdh) are given by precisely the same formulae, hence they are equal to each other by (3). We do not use this remark below.

4.4 A p p r o x i m a t i o n We conclude that for / — fu / „ with fixed fu and / „ = / u ( n ) where n is no-admissible for some no = n0(fu), we have the identity J = J'{4>dg') - J'(fdg) + J'{'fdh) - J'('dh)

+ Y}Jv(dg') - Jv{fdg) + M'fdh) - Jv('dh)}.

(1)

V

The sum over v is finite and ranges over v / u. On the left J represents a sum with complex coefficients (depending on / " but not on / „ ) of traces of the form tnru(fudgu), tr Uu((f>udg'u x a), tr{p u }('f u dh u ) or tr{pu}('cf)udhu). This is an invariant distribution in fudgu; it depends only on the orbital integrals of fudgu. On the right we have a sum over the connected components (represented by /J,U) of the manifold of characters mentioned in§2, of integrals over iR. The integrands are of the form c(A) tr 7(/xu ® A; fudgu). The right side of (1) is therefore also an invariant distribution in fudgu, depending only on the orbital integrals of fudgu. We conclude The identity (1) holds with the pseudo-spherical function fu — /„(n) replaced by the spherical function f'u — /^(n). LEMMA.

PROOF.

By definition / u ( n ) and /^(n) have equal orbital integrals.

•

From now on we denote by /„ a spherical function of the form /4(n) with no-admissible n. The identity (1) holds for our f = fu ® fu- Since /„ is spherical, tr nu(fudgu) ^ 0 only when wu is unramified. The theory of the Satake transform establishes an isomorphism from the set of unramified irreducible G u -modules nu, to the variety Cx3/S3'. the unordered triple z = (zi,Z2,zz) of nonzero complex numbers corresponds to the unramified subquotient 7ru(z) of the G u -module J„(z) normalizedly induced from the unramified character (a,ij) i—• f|i zja of the upper triangular subgroup.

II. 5 The a-endo-lifting e'

339

The central character of TT„(Z) is trivial if and only if z\z?,zz = 1. For z in C x and z in C x 3 we write zz for {z\z, Z2, z^z*1). We conclude that there are (a) t* in C x 3 / 5 3 (i > 0) and zt in C x 3 (i > 0) with tiitata = 1, Zi\ZiiZiz — 1 and \z^\ = 1, and (b) complex numbers Cj, and integrable functions Ci(z) on |z| = 1, such that (1) takes the form ^2citr(iru(ti))(fudgu)

= ^2

/

cj(z)tT(TTu(zjz))(fudgu)dxz.

(2)

The Satake transform /„ H-> / V , defined by / ^ ( z ) = tx(-Ku{z)){fudgu), in an isomorphism from the convolution algebra of spherical functions fu on Gu to the algebra of Laurent series f^ of z in C x 3 / 5 3 with Z\Z-LZ$ = 1. Then (2) can be put in the form

!>•#(**) = E

/

Cj(*)fi(*j*)dxz.

(3)

Our aim is to show that Ci — 0 for all i > 0. For that we note that all sums and products in the trace formula are absolutely convergent for any / „ , in particular for the function with f^ = 1. Hence £V \ci\ 1S finite, and S i / IciWII^I i s finite. Moreover, let X be the set of z in C x 3 /S l 3 with Z1Z2Z3 — 1, z _ 1 = z, and q~x < \zt\ < q for each entry Zj of z. Since all representations which contribute to the trace formula are unitary, the tj and z;z lie in X. But then the case where n = 3 of the final Proposition in [FK2], where the analogous problem is rephrased and solved for an arbitrary reductive group, implies that all Cj in (3) are zero. The theorem follows. •

II.5 T h e
II. Trace formula

340

from U(3) to GL(3,£). Basechange quasi-lifting is also proven. Our diagram is: LG

_ ^

eT H —>

L

LQt

» T H'

L

V

\ e' <-LH b"

5.1 Quasi-lifting The notion of local lifting in the unramified case is defined in 1.4. A preliminary, weak, definition of global lifting, is given next in terms of almost all places. Let J, J' be a pair of groups as above (H, G, etc.) for which the local notion of lifting is denned in the unramified case. If TT = ®TTV and -K' = ®~K'V are automorphic J (A)- and J'(A)-modules, and irv lifts to n'v for almost all v, then we say that -rr quasi-lifts to IT'. DEFINITION.

We shall later define the strong notion of global lifting, in terms of all places. This has been done in [F3;II] in the case of the basechange liftings b' and b". The map i is simply induction. Our aim in this section is to study the local and global lifting in the case of the cr-endo-lift e'. This, or the alternative approach of 5.3, will be used in II.6 for the study of the quasi-endo-lift e, and the basechange lift b. Our first aim is to study the local lifting e'. Let Ew/Fw be a quadratic extension of p-adic fields. 5.1.1 PROPOSITION. Suppose that TW is the stable basechange b" lift of an irreducible Hw-module pw. Then for any matching measures 4>wdg'w and '4>wdhw, we have tr I(Tw;wdg'w x a) =

tr{pw}('
P R O O F . This is shown in 1.4.1 for induced representations. The case of the one-dimensional .ff^-module follows from the case of the Steinberg representation, as its character is the difference of the characters of an induced and the Steinberg representation. Suppose then that pw is a discrete-series i? w -packet (consisting of discrete series .H^-modules). Fix a global totally imaginary extension E/F

II. 5 The a-endo-lifting e'

341

whose completion at w is the chosen local quadratic extension. At two finite places v = u, u', say u splits and v! does not split in E/F, we choose cuspidal representations pu and pu>. Let V be a finite set containing w, u, u'', and the places which ramify in E/F, but no infinite places. It is easy to see (using the trace formula) that there is a cuspidal H(A)module p whose components at w, u, u' are the given ones, which is unramified at all finite v outside V, and its components at the v in V are all discrete series. We choose a sequence {tv; v outside V} so that p makes a contribution to the sum in the trace formula for H(A), which is associated with 'cj). Then the trace formula identity of II.3 asserts Y[trI(Tv;(j)vdg'v

x a) = Y[tr{pv}('(j>vdhv)

+

2^\\_tx-K v {f v dg v )

The products extend over the finite places in V. The {pv} are the packets of the components of our p. But by [F3;II], p lifts via the stable basechange map b" to an automorphic H'(A)-module r . Rigidity theorem for G'(A) = GL(3,A) (see [JS]) implies that I(r) is the only contribution to the terms involving

I(r). Moreover, since u' is a nonsplit place, and the character of {pu>} (namely sum of characters of the members in the packet) is nonzero on the elliptic set, we may choose 'ur supported on the regular Hu>-elliptic set with tT{pui}('dhui) ^ 0. Then the matching ui can be chosen so that its stable cr-orbital integrals are 0. Namely we can take / „ ' = 0, and '/«' — 0. Consequently

Y[tTl(Tv;vdg'v xa) = Y[ti{pv}('4>vdhv).

(5.1.1)

We can repeat the same discussion with an automorphic H(A)-module p' which is unramified outside V, its components at all finite v ^ w in V are in the packets {pv}, and at w the component is induced. In this case we obtain the identity (5.1.1), in which the product extends over all finite v ^ w in V. Since there are < ' />„ supported on the regular set, with tr{pv}('(f>vdhv) ^ 0, for discrete-spectrum representations pw and nonarchimedean Ew/Fw, the proposition follows.

342

II. Trace formula

Moreover, the proposition holds also when Ew/Fw is C/K, and {pw} is unitary. It suffices to consider discrete-series pw, and take F = Q and an imaginary quadratic E. Repeating the proof of (5.1.1), the proposition follows in this case too. • Note that in the proof of the Proposition above, besides the identity of trace formulae we have used only the (generalized) fundamental lemma, but we do not need to transfer general test functions. It suffices to work with test functions supported on the regular set. These are easily transfered. 5.1.2 COROLLARY. The twisted character of the representation I(TW) induced from the stable basechange lift TW of an irreducible Hw -module pw is unstable. Here unstable means that if 8 and 8' are distinct cr-regular (a-elliptic) cr-conjugacy classes in G'w which are stably cr-conjugate, then xauT )(^') —

-X?(r.)(*)5.1.3 PROPOSITION. Let E/F be a quadratic extension of local fields. Let II be a square-integrable a-invariant representation of GL(3,E). Then its a -character is not identically zero on the a-elliptic regular set, and it is a a-stable function on the a-elliptic regular set. P R O O F . The cr-character is not identically zero on the u-elliptic regular set by the twisted orthonormality relations. We need to show that the vdg'v of such a Il„ is a tr-unstable function. Hence for a test measure <j)dg' = ®v(f)vdg'v with such components at the specified v, spherical components at all other finite places, and /<» vanishing on the cr-singular set, only

II. 5 The a-endo-lifting e'

343

cr-unstable orbital integrals occur in the geometric side of the a-twisted trace formula for GL(3,Ae). AS usual, the choice of the spherical and the archimedean components can be used to reduce the geometric sum to a single stable a-elliptic regular conjugacy class. In the trace formulae identity we may choose / = 0 on U(3, E/F)(A), that is, fv = 0 at the places v where dg' xa) = tr ir(fdg) for all matching measures 4>dg' on GL(3, E) and fdg on V{3, E/F). P R O O F . If II is Steinberg, so is 7r, SO we may assume that II is cuspidal and E/F is nonarchimedean. The cr-character of II is a er-stable function on the cr-elliptic regular set. We can view our local extension as the completion of a global totally imaginary one, and use a global cuspidal a-invariant representation of GL(3,A£) whose component at this place is our II, at a few additional places, including those which ramify, the component be Steinberg, and all other components at finite places be unramified. The trace formula identity for such a global representation (having fixed almost all components by "generalized linear independence of characters") will contain only n on U(3, E/F)(A) on the twisted side. Using the (known) lifting for Steinberg representations and at the archimedean places, we get an identity

tr U((j)dg' x a) = y^m(-7r)tr7r(/dg) TV

for all matching <j>dg' and fdg. Using the orthonormality relations for twisted characters of square-integrable representations we see that the sum reduces to a single term, with m{ir) = 1. •

II. Trace formula

344

5.2 Alternative approach In the proof of Proposition 5.1 we used only the a-endo-transfer e' of the unit element <j>° in the Hecke algebra of G' to the unit element '° in the Hecke algebra of H; and the transfer of spherical functions with respect to e' : LH —> LG', which follows from the statement for (°, '(f>°) by a global method. This is needed only at places where E/F, K, LJ are unramified. At the other places it suffices to transfer functions supported on the regular set, and this is easily done We shall now give an alternative approach, whose purpose is to show that the character of I(TW) is an unstable function, namely that tr I(TW; wdg'w x a) depends only on 'cf>wdhw. We shall not use the fundamental lemma for e', and conclude that complete local, and some global, results about the endo-lifting e can be obtained without using any knowledge of the cr-endotransfer e'. We use the results of Keys [Ke] concerning the reducibility of induced G-modules recorded in 1.4.4. We shall also make use of the following result of [F3;II]. A local module is called elliptic if its character is nonzero on the elliptic regular set. Put CF=FX if F is a local field, and CF = AF/FX if F is a global field. 5.2.1 PROPOSITION. (1) If T is an elliptic {resp. discrete-spectrum) ainvariant local (resp. global) H'- (resp. H'(A))-module, then its central character is trivial on CF. (2) Such r is the basechange lift of a unique elliptic or discrete-spectrum H- or H''(A)-module p, either through b' or through b", but not both. A proof of (1) in a more general context is given in [F1;VI]. The second statement here implies, in the global case, that if I(T) is the only term on the left side of the trace formula identity II.3.3, then precisely one of the sums involving '/ and '<j> on the right is nonzero, and it consists of a single term. Note that the elliptic (local) p are the one-dimensional, Steinberg and cuspidal, and also the components of a reducible tempered induced Hmodule, which make a packet. We shall now prove a special case of Proposition 5.1, but without using the fundamental lemma for e' stated in 1.2. 5.2.2

PROPOSITION.

Let rw be the stable basechange lift of the elliptic

II. 5 The a-endo-lifting e'

345

Hw-module pw. Then trI(Tw;wdg'w matches '(/)wdhw rvnA '4> A\ w rth 4e O and dhw is 0. P R O O F . We deal with the one-dimensional case first. Let p be a onedimensional iiZ-module, and r its basechange lift. Then p is a constituent of an induced '/(/i^ 1 / 2 ), and r of I{p,vxl2,/w-1/2). We choose '<j>w = 0, so that < ' /> = 0, and no term involving '<j> appears in the trace formula identity II.3.3. We choose a sequence {tv} so that our I(T) is the only contribution associated with . The only other possible terms in II.3.3 are of the form tnr(fdg) since we can choose '/„, = 0, thus ' / = 0. The local components of any such -K are almost all of the form / ( / i ^ 1 ' 2 ) . In any case, we conclude that for any v ^ w, if t?I(rw\4>wdg'w x a) ^ 0 then tr I(TV; (f>vdg'v x a) depends only on fvdgv. More precisely, there are Gu-modules ^v a n d complex constants C(TTV) with

tTI(Tv;vdg'v x a) =

J~]c(Trv)trirv{fvdgv)

for all matching (j>vdg'v, fvdgv. Taking matching functions whose orbital integrals are supported on the conjugacy classes of the diag(a, b, a - 1 ) , \a\ / 1, the Deligne-Casselman [CI] theorem implies that tr I{Tv)A{4>vAda'v x a) =

'Y^c(irv)tTirvA(fvAdav).

Here 11^, IT A denote the modules of coinvariants of II, 7r (see 1.4.4) with respect to the upper triangular parabolic subgroup with Levi subgroup A, tensored by 5~xl2, where <5(diag(a, 6, a - 1 ) ) = \a\2 (resp. J(diag(a,6, c)) = |a/c| 2 ) is the modulus function on Gv (resp. G'v = GL(3,FV)), and 4>VA, fvA are the functions on A, A' defined by /„,4(diag(a,6,a - 1 )) = \a\ I

/ fv(k~1ank)

dndk,

JK JN

4>vA(dia,g(a,b,c)) = \a/c\ /

/ cj)v(a(k)~1ank)

dndk.

JK JN

Since the functions fvA, <J>VA are arbitrary, and the module I(TV)A of coinvariants consists of a single (increasing) a-invariant exponent, we conclude from the Harish-Chandra finiteness theorem [BJ], and linear independence of characters on A, that on the right there should be a single nv with

346

II. Trace formula

nonvanishing nonunitary TTVA, and then TTVA should consist of a single exponent which lifts to I(TV)A- Here we used the fact (see 1.4.3) that if the irreducible irv and TT'V have nonunitary characters in TTVA and -K'VA which are equal, then nv and n'v are equivalent. Hence our nv is a subquotient of I = I(pv1/2). But / is irreducible (see 1.4.3), hence TTV — I, and WVA has two exponents, one increasing and one decaying. This contradiction establishes the proposition when pw is one dimensional, hence also when it is special. To deal with the cuspidal pw, it suffices to construct a cuspidal p with this component, and a component pv which is special. If '
II.6 The quasi-endo-lifting e 6.1 Cancellation

The results of II.5 concerning the cr-endo-lifting e' can be used to simplify the identity 1.3.3 of trace formulas. First the terms tr I(T; 4>dg' x a), where r is a stable basechange lift of an H(A)-module p, are canceled with the terms tv{p}('(f>dh). Indeed, if a discrete-spectrum {p} basechanges to a discrete-spectrum r, then n(p) = 1 according to [F3;II]. When n(p) ^ 1, it is equal to 1/2, and p is of the form p(9) in the notations of [F3;II], p. 721, (where it is denoted by n(9)). According to Proposition 1 there, p(9) lifts to an induced H'(A)-module r = 1(6'K, 9"K), where 9', 9" are distinct characters of CE/CF related to the character 6 (of CE x C^). There is no need to elaborate on this result. We simply note that the tr{p}('. The induced representation I(p) lifts via e' to the G'(A)-module I(p, p, w'/p2). If w' ^ p3 then we obtain a cancellation with the term tr I((p, p',n); 4>dg' x a), which also appears with coefficient —1/8. If u>' = p3 then we obtain a partial

II.6 The quasi-endo-lifting e

347

cancellation, which replaces the coefficient —3/8 by —1/4, in the twisted side of the formula. 6.2 I d e n t i t y So far we eliminated all terms which depend on '(/>. Let us record those terms which are left. We denote by fi any character of CE trivial on Cp. Put $ i =^2Y[tTllv(vdg'v

x a),

$ 2 = ^2Y[trI(Tv®Kv;vdg'v

x a).

In $ i the sum is over all (equivalence classes of) a-invariant discretespectrum G'(A)-modules. In $2 the sum is over the cr-invariant discretespectrum H'(A)-modules r which are obtained by the stable base change map b", namely r ® K is obtained by the unstable map b'. Further, $3 = 5 ^ ] ^ t r J((/i,/x',/z");<Mff{,

x

°)

(distinct /z,//,//'),

and

*5 = JZ I I t r J((A». A*. A*)? ^wdfli xcr)On the other hand, we put Fl

=

^mWYlteKvifvdgv). •K

The sum is over the equivalence classes IT in the discrete spectrum of G(A). They occur with finite multiplicities m(7r). Further, F

*=

E

HHPvKfvdK).

The sum ranges over the automorphic discrete-spectrum packets of p of H(A), which are not of the form p(0, '6). In this case n(p) = 1 (see [F3;II]). Also, Fs=

Yl p=p(8,'8)

H^{Pv}(%dhv).

77. Trace formula

348

Here the sum ranges over the packets p = p(6,' 0), where 0, 6' and ui/6'6 are distinct. In this case n{p) = 1/2. Finally we put F4 = ^2m((j,K)Y\tr F5 = ^2

m

I(/J,VKV,

fvdgv)

( ^ ) I I t r / ( ^ « ' fvd9v)

+ -^2Y[te

'I(pv,'fvdhv),

(M3 = w')>

•Fe = 5^m(/x) JJtri2(/x„)/(/i„,/„dp„) n

^2Y[tr{pv}(%dhv). P

3

In ^6, the first sum is over all p with p ^ OJ' . The second is over the packets p = p(9,u>/92), where 03 ^ u>. We deduce from the identity II.3.3 of trace formulas the following 6.2.1

PROPOSITION.

The identity of trace formulas takes the form

To simplify the formula we first note that the normalizing factor m which appears in F4 and F$ can be evaluated as a limit. It is equal to — 1. The representations I(pvnv), I(pv) of Gv = G(FV) in F4 and F5 are irreducible, and Proposition 1.1.4 asserts the following. In the notations of F4 and $ 4 we have at each v tr I(pv,'fvdhv)

= tTl(pvKv,fvdgv)

= tr/((«;

);
In the case of F5 and $5 we have tr I(pv,fvdgv)

= tr I((pv,pv,pv);(j>vdg'v

x
Hence $4 = —2F4 and $5 = — F5, and these terms are canceled in the comparison of the Proposition. The G(A)-modules in F 4 and F5 are irreducible, and their characters are supported on the split set. If / has a component fv such that the orbital integral &(fvdgv) is supported on the elliptic set, we can conclude that F4, F5 are equal to 0. The normalizing factor m(p) of F6 can be shown to be equal to 1, and FQ can be shown to be equal to 0, but this will not be done here. However, it is clear from Proposition 1.4.1 that p = p(9,uj/92) with 83 ^ u> quasi-lifts to I{p), where p = 6 o NE/F- I*1 a n y c a s e the trace identity takes the form

II.6 The quasi-endo-lifting e

349

6.2.2 PROPOSITION. At each v, let 4>vdg'v, fvdgv, 'fvdhv be matching functions. Fix unramified irv, namely the corresponding Satake parameters tv. Then 2

4

2

4

4

The terms consist of products over a finite set of places, and at most one of the terms on the left is nonzero, consisting of a single nonzero representation. We conclude 6.2.3 T H E O R E M . Every discrete-spectrum automorphic H(A)-module p with two elliptic components quasi endo lifts to an automorphic G(A)module. It is clear from Proposition 1.4.1 that if p appears in F3 then there is a nontrivial term in $3, but if p appears in Fi then there is a contribution in $2- So we apply the identity with a function

C with tp(ugk) = tp(u)tp(g), u e U, g € G, A; in a compact open Kv depending on tp), then A is normalized by Aip = > , °ip(g) = ¥>Mff)). The twisted character x% IS a complex valued cr-conjugacy invariant function on G (its value on {hga(h)~1} is independent of h & G) which is locally constant on the a-regular set (g with regular gcr(g)), locally integrable ([C12], Thm 1, p. 153) and defined by trir(fdg)A = JG x%{g)f{g)dg for all test measures / dg. Define tpE '• E —* C 1 by IPE{X) = i>{x + %)• Define a character i/> : U —> C 1 on the unipotent upper triangular subgroup U of G by ip((uij)) = 'tpEiJ^KjKr uj,j+i)- This one-dimensional representation has the property that ip(a(u)) = i>(u) for all u in U. Note that if>(u) = V*H(U 2 ) at u e UH = UCiH. There is only one orbit of generic a-invariant characters on U under the adjoint action of the group of cr-invariant diagonal elements in G. Write Gn = exp(jj n ), where Qn = 7r"flo- For n > 1 we have Gn = t UnAnUn, where Un — U n Gn, and An is the group of diagonal matrices in Gn. Define a character ipn : G —> C 1 supported on Gn by ipn^bu) = wnere i'E(J2i<j C is defined by Vn(expX) = chfln(X)V>£(tr[X7r-2"/?]) where /? is the r xr matrix with entries 1 at the places (j,j — l),l<j< r, and 0 elsewhere. The (j-twisted analogue of Rodier's theorem of interest to us is as follows. Let chga denote the characteristic function of G£ = {g — crg;g € Gn} in Gn. 3.5.2 PROPOSITION. For all sufficiently large n the multiplicity dime Home/ (Ind^ ip, n) — dime HomcCIndy ip, ir) is equal to |GSr 1 tr7r(^„ch G «j dg x a) = \G°n\~l f

xZ(g)ri>n(g)dg.

III. 3 Specific lifts

375

The proof of this is delayed to the next section. P R O O F OF PROPOSITION 3.5. The identity

tnr(fdg

x a) — (2m + 1) Y^tr7TH(/ffrf/i). •KH

for all matching test measures fdg and fndh implies an identity of characters:

for all S e G = GL(3,E) with regular norm 7 G H = \J(3, E/F). Note that S H-» x%{$) i s a stable a-conjugacy class function on G, while 7 H-> J2n XTTH (7) ls a stable conjugacy class function on H. We use Proposition 3.5.2 with G = GL(3,E) and H = Ga. Then Gan = Hn. On 5 e G£, the norm JV<5 of the stable cr-conjugacy class of 5 is just the stable conjugacy class of S2. Hence x°M = (2m + 1) £ „ w ( < $ 2 ) at 5 e G£ = # „ . If <J = expX, I g C ; = W n , then i/,E(ti[Xir-2n0\) = V(tr[2X7r- 2n /3 ff ]). Indeed (3 = fin and V'-EKE) = ^>(x + x), thus i/in^bu) = IPE((X + y)^~2n) if M =

0 1 y 1. This is = i/'(2(x + x)7T~2™) if y = x, while ipH^bu)

=

\o 0 1/

^((a;-l-x)7r~2") at such u £ UH (thus with y = x). Hence ^n(<5) = i/>H,n(52) for <5 e G£ = ifn. Also d(# 2 ) = dg when p 7^ 2. It follows that 1 = dime Home(Ind^V, 1") = (2m + 1) ^

dime HornH(Ind^ H

J>H,^H)-

•KH

Hence m = 0 and there is just one generic TTH in the sum (dime / 0, necessarily = 1). • 3.6 PROPOSITION. In the notations of Proposition 3.4, £ij£2j£3j = 1P R O O F . Again we use the trace formula, and global notations. We study the situation at a place w. We may and do assume that E/F are totally imaginary. At three finite places v — vm (7^ w; m = 1, 2, 3) which do not split (and do not ramify) we choose 6V, '9V so that pv, p'v, p" are cuspidal. Since eijvS2jv£3jv is independent of j , then for each v there exists j = j(v) so that EijV is independent of i. Since EijV can attain only two values, and we have three v at our disposal, we can assume that Si1jlVl = £i2j2v2 > where jm — j(vm), and both sides are independent of i\, i
III. Liftings and packets

376

We now construct global characters 6, '9 with the chosen components at vi, V2 and our place w, which are unramified at each place which does not split in E/F (we can take 9V = '9V at the v which ramify). It is clear that px = p{9,'9), pi = p(9,u>/9 • '6), p3 = p('9,u>/9 • '9) are cuspidal and distinct. All three appear in the trace formula together with I(T (g> K) = I(9','9',cu'/0' • '9'), and with coefficients n(p) = \ (see [F3;II]). Namely, we obtain

n

+En

'^trirjv{fvdgv)

/ j £ijv t r

TTJV(Jv(igv)

= 4^2m{i:) Y[ trnv(fvdgv). The product ranges over v = w, v\, v2. At v — vm (m = 1,2) we take fvdgv to be a coefficient of Wjv, where j = j(v) was chosen above. Then the product II can be taken only over our place w. Hence, for every j , we have 0 (mod 4). xyw

'+£'

This holds only if EijW — 1 for an odd number of i, and the proposition follows.

•

3.7 Sum up twins To sum up our case (3.3), fix 0, '9 so that px = p{9,19), p2 = p(9, u/0 • '9) are disjoint cuspidal -ff-packets. Denote by n the induced G'-module I{9','9', ui'lff •'&). COROLLARY. There exist four cuspidal G-modules TTJ (1 < j < 4), so that 7Ti has a Whittaker model but Wj (j ^ 1) do not, so that

tiU(dg' x a) =

^tiITj(fdg),

and trpi('fdh)

= trTTi(fdg) + trTTi+i(fdg)

- trw^ifdg)

-triTi>,(fdg).

The indices i', i" are so that {i + 1, i', i"} = {2,3,4}. We write ir+{pi) for {7TI,7T, + I}, and ir~(pi) for {7Tj',7rj»}.

III. 3 Specific lifts

377

3.8 p(0,o>/02) The next special case of interest is that of the packet associated with p — p(9,ui/92), where 93 ^ u>, so that {p} consists of cuspidals; in fact {p} consists of a single element, and this is clear also from the comments below. The associated G'-module is the cr-invariant tempered induced II = I(9',w'/9'2,9'). It is the basechange lift of the reducible G-module -K = 1(9'). The representation IT is the direct sum of the tempered irreducibles 7r+ and 7T~. Then we have tTU((j)dg' xa) = tnr(fdg)

= tr ir+(fdg) +

tvn'(fdg),

and also ixpCfdh) = tTir+(fdg) +

-

tin'(fdg),

+

for a suitable choice of TT . Namely ir has a Whittaker vector, while n~ does not. In particular 2[{p}] = [{7r+,7r_}] = 2, so that {p} consists of a single element, as asserted. 3.9 P a c k e t s With this we have completed the description of all tempered packets {n} of G. The packets are in bijection with the tempered cr-stable G'-modules IT. If II is a square-integrable cr-invariant G'-module, then it is cr-stable, and the packet {IT} consists of a single element (this has been shown already in [F3;III(IV)]). If II is induced from a square-integrable //'-module, and it is cr-stable, then it is of the form I(T ® «), where r is the stable basechange lift of a square-integrable packet {p} of H. The associated G-packet {7r} consists of 2 = 2[{p}] elements, each occurring with multiplicity one. If II is induced from the diagonal subgroup, and it is not simply the basechange lift of an induced G-module I(p) (in which case the packet {IT} consists of the irreducible constituents of 1(H)), then II is of the form 1(9', '9', w'/9' -'9'), where the three characters are distinct, and trivial on Fx. In this case the packet {n} consists of 4 = 2[{p}} elements, where p = p(9, '9). Using this, and the related character identities between p and the difference of members of {it}, we can use the trace formula to describe the discrete spectrum of G.

378

III. Liftings and packets

III.4 Whittaker models and twisted characters We shall reduce Proposition 3.5.2 to Proposition 3.5.1 for G (not H), so we begin by recalling the main lines in Rodier's proof in the context of G. Fix d = diag(7r- r + 1 ,7r- r + 3 ,... . r - 1 ) (bar over the last [r/2] entries). Put Vn = cTGnd-n,

i/jn(v) = ijn(d~nvdn)

(v e Vn).

Note that cr(d) = d, and that the entries in the j t h line (j ^ 0) above or below the diagonal of v — (vtj) in Vn lie in IT^-^^RE (thus vi}i+j G -K^~2^nRE if j > 0, and also when j < 0). Thus Vn fl U is a cr-invariant strictly increasing sequence of compact and open subgroups of U whose union is U, while Vn D (*C/G) — where tUG is the lower triangular subgroup of G — is a strictly decreasing sequence of compact open subgroups of G whose intersection is the element / of G. Note that ipn = t/> on Vn n U. Consider the induced representations Indy tj)n, and the intertwining operators ^ I n d ^ V - ^ I n d ^ m , « V ) ( c > ) = {{\Vm\-llVm^m)

*
^m{u)iP{u-1g)du

(g in G,

n > 1 we have (AZ
*
rl>{u)iP{u-1g)du.

Hence AemoA% = Aen for £ > m > n > 1. So ( I n d ^ Vn, A™ (m > n > 1)) is an inductive system of representations of G. Denote by (/, An : Indy n if>n —> I) (n > 1) its limit. The intertwining operators n : Indy n V>„ —> Indjy V, (n(lu * {u)ip{vT*g)du, Ju satisfy 4>noA™ = n > 1. Hence there exists a unique intertwining operator cj> : I —> Ind^ V with (j>o An — (f>n for all n > 1. Proposition 3 of [Rd] asserts that

III.4 Whittaker models and twisted characters 4.1.1

LEMMA.

379

The map is an isomorphism of G-modules.

4.1.2 LEMMA. There exists n 0 > 1 such thatipn *tpm *i/>n = \Vn\\Vm D Vn \4>n for allm> n> no. P R O O F . This is Lemma 5 of [Rd]. We review its proof (the first displayed formula in the proof of this Lemma 5, [Rd], p. 159, line -8, should be erased). There are finitely many representatives ui in U fl Vm for the cosets of Vm modulo Vn fl Vm. Denote by e(g) the Dirac measure in a point g of G. Consider (e(«i) * t M v m n v J ( f f )

= / e{ui)(gh-1)(ipnlVmnVn)(h)dh

=

iJ)n(ur1g)=,l>m(ui)-1'il>m(g).

JG

Note here that if the left side is nonzero, then g G Ui(Vm fl Vn) c Vm. Conversely, if g G Vm, then g G Ui(Vm n Vn) for some i. Hence ipm = Y^i1>m{ui)e(ui) *ipn^vmnvn, thus ll>n*ll>m*ll>n

= ^2^m{Ui)lpn

* z(Ui)

* t M v m n V „ * V»n-

Since V»nlymnvn * Vn = |Vm n KlV'n, this is = ^ V ' m ( W t ) | V r m n K | ^ n *£(Uj) *Vni

But the key Lemma 4 of [Rd] asserts that ij)n * e(u) *ipn ^ 0 implies that u € Vn. Hence the last sum reduces to a single term, with w, = 1, and we obtain = \Vm fl Vn\lj}n * Vn = |Vm D V„||V m |Vn. This completes the proof of the lemma. 4.1.3

LEMMA.

•

For an inductive system {/„} we have Hom(j(lim/ n , w) =

limHom G (/n,7r). P R O O F . See, e.g., Rotman [Rt], Theorem 2.27. Let us verify this in our context as in [Rd]. Our Lemma 2, which is Lemma 5 of [Rd], implies Proposition 4 of [Rd], that A^o A^ = \VmnVn\\Vm\~l • id(Ind£ n ipn) if m > n > no- This implies that A™ is injective, A7^ is surjective, that An

III. Liftings and packets

380

and o An are injective, and by duality that for any G-module -K the maps Hom G (Ind£ m ipm,Tr) -+ Hom G (Ind£ n ipn, ir), Hom G (Ind°rp,IT) -> Hom G (Ind£ n il)n,ir),

ip^tpoA™,

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