Iro(Z()r)l Ikerl(F,Z())l - University of...

Vol. 51, No. 3 DUKE MATHEMATICAL JOURNAL (C) September 1984

STABLE TRACE FORMULA:CUSPIDAL TEMPERED TERMS

ROBERT E. KOTTWITZ

Consider a connected reductive group G over a number field F. For technicalreasons we assume that the derived group of G is simply connected (see [L1]). in[L3] Langlands partially stabilizes the trace formula for G. After making certainassumptions, he writes the elliptic regular part of the trace formula for G as alinear combination of the elliptic G-regular parts of the stable trace formulas forthe elliptic endoscopic groups H of G. The function f/ used in the stable traceformula for H is obtained from the function f used in the trace formula for G bytransferring orbital integrals.

Langlands uses t(G, H) to denote the coefficient of the stable trace formula ofH in the linear combination referred to above. He obtains an explicit formula fort(G, H), which we review in Section 8.

Eventually it should be possible to stabilize the whole trace formula for G. InSection 12 we sketch the stabilization of the cuspidal tempered part of the traceformula for G. To carry this out we are forced to make a number of assumptions,some of which are unlikely to be verified in the near future, but this is not asserious as it appears at first, for our only purpose in Section 12 is to increase ourunderstanding of the formalism that underlies the stable trace formula. In anycase, we are able to prove a small part of what we need, and we put these resultsin Section 11.

Although the stabilization carried out in Section 12 is speculative, it leads us toa new expression for t(G,H). In Section 8 we prove that this expression agreeswith the one found by Langlands. This agreement is encouraging and should,perhapg, be regarded as evidence in favor of the assumptions in Section 12. Ourexpression for (G, H) is

t(G,H) ’rl(G ) ’rl(O) -1. X -1.

The number h also appears in Langlands’s expression for t(G,H); its definitioncan be found in 8.1. The number (G) is given by

Iro(Z()r)l Ikerl(F,Z())l -l.

We need to explain the symbols used in this formula. We write ( instead of/G,Received June 24, 1983. Partially supported by the National Science Foundation under Grant

MCS 82-00785.

611

612 ROBERT E. KOTTWITZ

and we write Z(() for the center of (. The superscript F stands for invariantsunder F, the absolute Galois group of F, and %(Z(()r) stands for the group ofconnected components of Z(()r. The vertical bars are used to denote cardinalityof a finite set. Finally, ker(F, Z(()) is the kernel of

IIn’(ev,Z(6)),where the product is taken over all places v of F. A remark of Shelstad,mentioned on p. 169 of [L3], leads us to expect a connection with Tamagawanumbers. In fact, r(G) turns out to be the relative Tamagawa number of G (thequotient of the Tamagawa number of G by the Tamagawa number of the simplyconnected cover of the derived group of G), as we prove in Section 5. Ourformula generalizes formulas of Ono [O] for tori and semisimple groups and isrelated to a formula of Sansuc [Sa].

This formula for the relative Tamagawa number is a special case of thefollowing general principle. Consider an invariant of connected reductive groupsover a local or global field. Assume that the invariant is trivial for semisimplesimply connected groups. Then it should be possible to compute the invariant ofG from the Galois module Z(G). In this paper we will consider two moreillustrations of this principle. In Section 4 we consider a number field F and theinvariant keP(F, G), the kernel of the Hasse map

H’(F, G)-> II H’(Fv, G).

For groups with no E8 factors we show that there is a canonical bijection fromkerl(F, G) to the dual of the finite abelian group kerl(F, Z(t)). In Section 6 weconsider a p-adic field F and the invariant H(F, G). We show that there is acanonical bijection from HI(F, G) to the dual of the finite abelian group

These results run parallel to results of Sansuc [Sa], who uses Pic 6; and BrG (acertain subquotient of BrG), rather than Z(). In Section 2 we relate the two

points of view by showing that PicG %(Z()r) and BrG H(F,Z()) forany connected reductive group G over an arbitrary field F. Sansuc also studiesthe failure of weak approximation. It is easy to reformulate his results in terms ofZ(G); we leave this to the reader.The proof of the agreement of the two expressions for t(G, H) is an exercise in

Tate-Nakayama duality IT1], IT2]. For the convenience of the reader, we reviewthis theory in Section 3, in the form best suited to our applications.

In Sections 1, 7, 10 we review L-groups, endoscopic groups and admissiblehomomorphisms. It is necessary to have a firm grasp on these before attemptingthe stabilization in Section 12. We have made what we hope are some technicalimprovements in the presentation of this materialfor example, our endoscopicdata are pairs, rather than sextuples, and our L-groups are not rigidified by

STABLE TRACE FORMULA 613

choosing splittings. One effect of these modifications is that it is easier to handleisomorphisms and automorphisms, which works to our advantage in Section 11.

Section 9 gives a new approach to the main construction of Ch. VII of [L3].The problem and its solution are stated in 9.2 and 9.3. In 9.5 we give a newapproach to something else in Ch. VII of [L3]: a global obstruction totransferring a maximal F-torus in a quasi-split group H to an inner form G.Three final comments are needed. First, it should be noted that algebraists will

find little that is new in this paper, since our results on ker(F, G) and Tamagawanumbers are simply restatements of Sansuc’s results, using Z() rather thanPic G and BraG. Those interested in the trace formula, however, may find ourrestatements useful, and it is for these readers that the remarks in 4.4, 5.3 areintended. As the remarks suggest, this paper has been written so as to give thereader a choice between translating Sansuc’s results (this is shorter) andreproving them, by parallel methods, in terms of Z() rather than PicG andBrag (this is longer, but may seem more natural to those interested in the traceformula). In any case, the proofs require all of the important theorems on theGalois cohomology of algebraic groups over local and global fields, and inparticular they give no new insight into the Hasse principle.

Second, given the speculative nature of Section 12, it may be worthwhile toexplain its origin. I wanted to see how Langlands’s numbers t(G, H) would showup in the cuspidal tempered part of the trace formula, so I made some plausibleassumptions about the form of this part of the trace formula and then stabilizedit. This led to the new expression for t(G, H) given above, and the new expressionturned out to be correct. In proving this I was led to express ker(F, G) and "rl(G )in terms of Z(G). After finding out about similar results of Sansuc, I realizedthat it should also be possible to express PicG and BraG in terms of Z(G), and itturned out to be easy to check this. But in writing this paper I have reversed thewhole procedure, so that Pic G, BraG come first and the stabilization comes last.

Third, I would like to thank the referee for pointing out that the generaldefinition of Ext,(A, B) in 2.1 needed to be reformulated.

Notation and conventions. We use F to denote a field (often a number field).Let F be an algebraic closure of F and write Fsep for the separable closure of F inF. We denote by F the Galois group of Fsep over F. If F is a number field, wewrite Ar for the direct limit of the adele rings A, where L runs over the finiteextensions of F contained in F.We consider only discrete F-modules A. This means that Stabr(a) is open in F

for all a A.By an induced F-module we mean a F-module that has a finite F-stable

Z-basis. We say that an F-torus T is induced if X*(T) is an induced F-module.Let S be a 13-torus on which F acts. We say that S is induced if the F-moduleX*(S) is induced.


Let S be a set on which F acts. We write S r for (s S:o. s s for allo F). Let A be a F-module. We write Hi(F,A) instead of Hi(’,A).

Let G be a group. We denote by Z(G) the center of G and by Out(G) thegroup of outer automorphisms of G. For g G we let Int(g) denote the innerautomorphism x -gxg- of G.We write A D for the character group of a locally compact abelian group A.

Sometimes our characters have values in O/Z, in which case we use theexponential mapping x-exp(2rix) from O/Z to G to view them ascomplex-valued characters.

Let G be a connected reductive group over F. For a field extension K of F wewrite GK for the K-group obtained from G by extension of scalars. Let Gdedenote the derived group G, let Gsc denote the simply connected cover of Gaer,and let Gaa denote the adjoint group of G. Let T be a maximal torus of G. Wewill write Tde for the intersection of T with Gder, Tsc for the inverse image of Tdein Gsc, and Taa for the image of T in Ga. We write ( for the connectedLanglands dual group of G, rather than LG. We need the notion of az-extension; for this we refer the reader to [K].We write %(X) for the set of connected components of a topological space X.

We use this only for topological groups G, in which case %(G) is the groupG/G, where G O is the identity component of G.We write IX for the cardinality of a finite set X.Let X be an algebraic variety over F. Following [Sa], we define BraX to be the

following subquotient of the Brauer group BrX of X. Let BrlX denote the kernelof BrX Br(XF p), and let Br0X denote the image of BrFBrX. Then we putBraX equal to (ie quotient BrX/BroX. For an algebraic group G over F wehave the rational point e G(F), and from the corresponding morphismSpecF- G we get a homomorphism BrGoBrF. We write BreG for theintersection of BrG with the kernel of Br G Br F. The obvious homomorphismBr G ---) Br G is an isomorphism.

1. Review of L-groups. In this paper we will not rigidify our L-groups bychoosing splittings. To avoid confusion later we need to state our conventionsregarding L-groups. Our presentation is based on Borel’s [B].

1.1. Let G be a connected reductive group over an algebraically closed field.Consider pairs (T, B) where T is a maximal torus of G and B is a Borel subgroupof G that contains T. Given two pairs (T,B) and (T2,B2), the innerautomorphisms a of G such that a(T)= T2, a(B1)= B2 all induce the sameisomorphism T _7_)T2" In particular, we have canonical isomorphisms X*(T)

X*(T2) and X,(T) X,(T). Let X* (resp. X,) denote the projective limit ofthe groups X*(T) (resp. X,(T)), where the limit is taken over the set of pairs(T,B). Let A* (resp. A,) denote the projective limit of the sets A*(T,B) (resp.A,(T, B)), where z*(T, B) is the set of simple B-positive roots of T, and A,(T, B)is the set of simple B-positive coroots of T. Then (X*, A*,X,, A,) is a based rootdatum, which we will denote by XI’o(G ).


1.2. Let G be a connected reductive group over F. Then F acts on "t’0(Gf).We will denote by ’t’0(G) the based root datum ’t’0(Gf) together with the actionof F.

1.3. A splitting of a connected reductive group G is a triple (T,B,(X)eA,(r,B)), where T is a maximal torus of G, B is a Borel subgroup of G thatcontains T, and X is a nonzero element of the root space Lie(G). The groupAut(G) acts on the set of splittings; the subgroup Gad of Aut(G) acts simplytransitively.

1.4. Let H be a connected reductive group over 13 on which F acts. Then Facts on ,t’0(H ). If F fixes some splitting of H, then we will say that the action isan L.action.

1.5. Let G be a connected reductive group over F. A dual group for G is aconnected reductive group^ ( over (3 together with an L-action of F on ( and aF-isomorphism from xt’0(G to the dual of xI’0(G ). The group ( is the identitycomponent of the L-group/"G and in Borel [B] is denoted by/"G. Since we havenot chosen a splitting of t, our version of the dual group is not rigid. Itsautomorphism group is [(()ad]r. It follows from the next lemma that theseautomorphisms are harmless.

1.6. LEMMA. The canonical homomorphism r __> [()ad]I" is surective.Let g be an element of ( whose image in ()ad is fixed b.y F. For o F there

exists zo Z(() such that g =.zog. Choose a splitting (T,B, (X,}) that is fixedby F. The double coset of B in G containing g is fixed by F. Using Lemma 6.2 of

[B], we see that there exists w r such that w normalizes T and g /}w/. LetN denote the unipotent radical of/}. Then g has a unique Bruhat decomposition

--1 o(g uwvt where uNNW/ vN, t T. But g gzo =uwv zo l),and since o preserves 7,/ and w, we see that u, v r and zot. Thus g sanbe written as g ht with h r and 7. Let ; denote the image of T in(8)ad. The image of tin ; lies in r. But X*(;) has a basis preserved by F,namely the set of simple/-positive roots of ;, and therefore ;r is connected.From this it is clear that 7r ;r is surjective, which implies that there existsz Z() such that tz 7r. We see immediately that gz r.

1.7. COROLLARY. Any two splittings of that are fixed by F are in the sameorbit under r.

Let Y l, Y2 be the two splittings, and let a be the unique inner automorphism ofsuch that a .yl y.. Then a .y Y2 for all o F, since Y l, Y2 are fixed by

F. Therefore a [Gad] and the lemma implies that there exists g r such thatInt(g) a.

1.8. Let G, H be connected reductive groups over F. We will say that anF-homomorphism a:G H is normal if a(G) is normal in H and G a(G) isseparable. Choose s,plitti,ngs of and H. Then there is a unique F-hom,omorphis,m c’H G compatible with the splittings and with the map,o(H)- t,0(G ) dual to xt’0(G)- 90(H). Changing the choice of splittings (see


1.7) replacesfi by^ Int(g) o c for some g r. This means that the ( r2conjugacyclass of c H ---) G is canonical. In particular, the homomorphism Z(H) --) Z(G)induced by tl is canonical. Thus we get a contravariant functor G -)Z(G) fromthe category of connected reductive F-groups and normal F-homomorphisms tothe category of diagonalizable C-groups with F-action. Furthermore, if

--) G --) G2 --) G3 ---) is exact, then so is

(1.8.1) 1--) Z(3)- Z(2) --) Z()- 1.

The following facts are also useful.

(1.8.2) X,(Z(t)) X*(G).

(1.8.3) The derived group of G is simply connected if and onlyif Z() is connected, in which case Z()=/, whereD =G/Gder.

(1.8.4) Suppose that G is semisimple, and let C be the kernel of

Gsc- G. Then Z()= X*(C).

2. Ext, Pic and Br. In this section F is an arbitrary field and FGal(Fsep/F). In 2.1, 2.2, 2.3 the group F could just as well be an arbitrary

profinite group.

2.1. Let A, B be F-modules. We define Ext,(‘4, B) to be the direct limit of thegroups Ext,[r/s](As,BS), where N runs over the set of open normal subgroupsof F (As denotes the coinvariants of N on .4).

2.2. LEMMA. Let M be a F-module that is finitely generated as an abeliangroup. Let D be the diagonalizable C-group Homz(M, C ). Then X*(D)= M and

r

Ext.(M, Z) lr0(D r)[H"- I(F,D)

n =0,

n>2.

Furthermore, if M is torsion free, then

Hn(F,X.(D)) Ext,(M, Z).

There is an easy reduction to the case in which F is finite. Then we have aspectral sequence

H (I’, Extz(A, B )) Ext,trl(A, B

for any F-modules .4,B (see [C-E]). In particular, if A is Z-projective or B isZ-injective, then Hn(F, Homz(A, B)) Ext)trl(A, B) for all n > 0. Taking


A M and B Z, we get the last statement of the lemma. Taking A M andB G, we see that Ext,trl(M, (3)= 0 for n > 0. Taking A M and B G, wesee that Ext,trl(M,G)= Hn(F,D). Now consider the long exact sequenceobtained by applying Ext,trl(M,-) to the exponential sequence ---> Z--> (3---> (3

1. We see immediately that Ext.(M, 7)= H 1(1-’, D) for n > 2.Now consider the case n 1. The group Ext[rl(M, Z) is equal to the cokernel

of Homr(M,C)--->Homr(M,C). But Homr(M, C) Lie(D r), Homr(M,C)D r, and the homomorphism between the two is the exponential mapping.

Since exp(Lie(D r)) is the identity component of D r, we get the desired result.Finally we consider the case n--0. We have Ext7_trl(M,Z)- Homz(M,Z)r[Homc.groups(l, D)]r X,(D)r.2.3. COROLLARY. Let - D -- D2- D3 -- be an exact sequence of di-

agonalizable G-groups with F-action. Then there is a natural long exact sequence

X,(D)r X,(Dg.)r X,(D3)F

D D D- H’(F, D,) H’(F, D)-- H’(F, D)

H2(F,D,)- H2(F,D:) H2(F, D3)-

2.4. Let G be a connected reductive group over F. We will constructisomorphisms

(2.4.1) ro(Z(d )r) Pic G,

(2.4.2) HI(F,Z(d)) Br,,G,

functorial in G for normal homomorphisms G G (see 1.8 for the definition ofnormal).

2.4.3. First consider groups G for which Gar is simply connected. FromLemma 6.9 and Corollary 6.11 of [Sa] we see that Pic G 0, where Gand that H(F,X*(G))= PicG, H(F,X*(G))= BrG. Combining (1.8.2) andLemma 2.2, we get the isomorphisms we want, and they are functorial in G.Before considering the general case, we need two lemmas.

2.4.4. LEMMA. Let G - G2 be a homomorphism, and let H -’ Gi (i 1,2) bez-extensions. Then there exists a commutative diagram

H (. H3 H2

G ( G ; G2

in which H3 ---> G is a z-extension.


Take H3 to be the fiber product of H and H2 over G2. The surjectivity ofH2---)G2 implies the surjectivity of H3H1, which in turn implies thesurjectivity of H3---) GI. The kernel of H3---) G1 is the product of the kernels of

H G and H2- G2; in particular, it is an induced torus. It is clear that thekernel of H3 G is central in H3. The preceding statements imply that H isconnected reductive. Finally, since H G factors through H G the derivedgroup of H3 is simply connected.

2.4.5. LEMMA. Let G be a connected reductive group over F. Then there exists acentral extension H---) G of G such that Z(121) is an induced torus. Let y" G-, Gbe a homomorphism, and let ai Hi---) G (i 1,2) be central extensions such thatZ(Hi) (i 1,2) are induced tori. Then there exists a commutative diagram

H ;H , H2

(2.4.5.1) + id lG1 G2 < G2

in which H ---)G is a central extension such that Z(H3) is an induced torus.

First we show that central extensions H G of the required type exist. If wehad such a central extension, then we would get an em,bedding Z(()=- Z().Conversely, given an embedding of Z(t) in a torus Z, we can find a centralextension H G with Z(H.) (see the proof of Proposition 3.1 of [M-S]). It isclear that we can choose Z to be induced.Now we prove the existence of the diagram (2.4.5.1). Since the groups Z(Hi)

(i 1,2) are connected, the derived groups of H; (i 1,2) are simply connected.From the simple connectivity of (n)de it follows that there exists a uniquehomomorphism fl (Hl)der---)(H2)de making the following diagram commute:

(H1)der) (H2)der

G YG2

Let C be the identity component of the center of H, and let CO C (Hl)der.Then H ((Hl)d C)/Co. Now let H (HE C)/Co, where CO is embed-ded in HE C by means of the homomorphism c--(fl(c),c-l). We constructthe homomorphisms needed in diagram (2.4.5.1) as follows:

H- H (hl,Z)((h,),z),

h2 -) (h2,1),


It is not hard to check that these homomorphisms are well defined, that diagram(2.4.5.1) commutes, and that H2 H3 is an embedding. This last fact, togetherwith the fact that (H2)d is simply connected, implies that (H3)d is simplyconnected. By (1.8.3) we have Z(Hi)= Di, where Di Hi/(Hi)de (i 1,2,3),and it is easy to see that D D D2. Therefore Z(H3) is an induced torus.

2.4.6. Now we finish the construction of the isomorphisms (2.4.1), (2.4.2),starting with (2.4.1). By Lemma 2.4.5 we can find a central extension H- G suchthat Z(H) is an induced torus. Let Z denote the kernel of H- G. We get anexact sequence - Z()- Z()-- 1. Using Corollary 2.3 and the exactsequence X*(H)r X*(Z)r Pic G PicH- of Corollary 6.11 of[Sa], we get a commutative diagram with exact rows

X*(H)r, X*(Z)r PicG 0.

Note that %(Z(/-))) 0 since Z() is induced, and hence that Pic H 0 by2.4.3. We define the isomorphism (2.4.1) to be the unique homomorphismmaking the diagram above commute. It is an easy application of Lemma 2.4.5 tocheck that the isomorphisms we obtain are independent of the choice of H Gand are functorial for normal homomorphismsWe will construct the isomorphism (2.4.2) in a similar way. However, this time

we will use a -extension H 6;. Again let Z denote the kernel of H G. SinceZ is induced, we have %(Z"r) --0. Then PicZ- 0 by 2.4.3. Again usingCorollary 2.3 and [Sa, Cor. 6.11 ], we get the following commutative diagram withexact rows:

0 H’(F,Z(()) ) H I(F, Z(/-)) ) H ’(F,)

0 )Br G . BraH Bra,Z

The vertical isomorphisms come from 2.4.3. We define the isomorphism (2.4.2) tobe the unique homomorphism making the diagram above commute. It followsfrom Lemma 2.4.4 that the isomorphisms we obtain are independent of thechoice of z-extension and are functorial for normal homomorphisms G G2.

2.5. There is another way to look at the isomorphisms (2.4.1) and (2.4.2). TheF-group Gm defines a sheaf of abelian groups for the 6tale topology on anyF-variety. In particular, we have 6tale cohomology groups H"(G, Gm) for anyconnected reductive group G over F. Let Hn(G, Gm)e denote the kernel of thehomomorphism Hn(G,(m) Hn(SpecF,m) induced by the identity elemente:SpecFoG of the group G. By a result of Rosenlicht [R] we have


H(G,(m)e X*(G)r. It is well known that H(G,m)=PicG, and fromHilbert’s Theorem 90 it follows that H (G, m)e Pic G as well. By a result ofHoobler [Ho] we have H2(G, 13m)= Br G, and therefore

BraG ker[ n2( G, (m)e ----> n 2( aFsep m)eCombining these facts with Lemma 2.2, we see that

gxt(X*(Z ( t )), Z) H ( G, m)

and

(n =0, 1)

Ext(X*(Z(a )), Z) ker[ H2(G, m)e" n2(aFep,m)e].It seems reasonable to expect that the same relationship holds betweenExt(X*(Z()),X.(T)) and Hn(G,T)e for any F-torus T, but I have notchecked this.

3. Review of Tate-Nakayama duality. Duality for tori plays an important rolein the theory of L-indistinguishability. In this application of duality theory onlythe groups Hr(F, T) are needed. Their behavior is simpler than that of thegroups Hr(L/F, T), where L is a finite Galois extension of F. In this section wewill review local and global duality for the groups Hr(F, T).

3.1. Let F be a nonarchimedean local field. The form of duality that we wantcan be found in a book by Shatz [S], and our presentation will follow his.

3.1.1. Let A be any F-module. Then Hr(F,A) 0 for r > 3.3.1.2. Let T be an F-torus. Let H(F, X*( T)) denote the completion of

H(F,X*(T)) in the topology of subgroups of finite index, and let H(F, T)denote the completion of H(F, T) T(F) in the topology of open subgroups offinite index. The pairing X*(T) T-->m induces cup-product pairings

nr(FX*( T)) n2-r(F, T) --- n2(F m) 0/7

under which

(3.1.2.1) the compact group H(F, X*( T)) and discrete groupH2(F, T) are dual,

(3.1.2.2)

(3.1.2.3)

the finite groups H (F,X*(T)) and H (F, T) are dual,

the discrete group H2(F,X*(T)) and compact groupH(F, T) are dual.

3.2. Let F be an archimedean local field. Since F is finite, we have the reducedgroups Hr(F,A) for any F-module A and any r Z. Let T be an F-torus. The


pairing X*(T) T---> G induces cup-product pairings

I(F,X*(T)) IEI2-’(F, T)--) I(F, Gm)--)Q/Z

under which the finite groups Ir(F,X*(T)) and /-2-r(F, T) are dual for allrZ.

3.3. Let F be any local field, archimedean or nonarchimedean, and let 7" bean F-torus. Using Lemma 2.2, we get the following consequences of localduality:

(3.3.1)

(3.3.2)

the finite groups %(7r) and HI(F, T) are dual,

the group H(F, ) is canonically isomorphic to thegroup of continuous characters of finite order of T(F)(this is part of the Langlands correspondence for T).

3.4. Let F be a number field, and let S denote the set of all places of F.3.4.1. Let A be a F-module. Then for r > 3 the canonical homomorphism

Hr(F,A)(vsH(Fv,A) is an isomorphism. Note that H(Fv,A)=O forfinite v, so that the direct sum could just as well be taken over the set of infiniteplaces of F. Let ker(F,A) denote ker[H’(F,A)--)IIvsHr(Fv,A)]. Thenkel"(F, A) 0 unless r 1, 2.

3.4.2. Let T be an F-torus. Let H(F,X*(T)) denote the completion ofH(F,X*(T)) in the topology of subgroups of finite index, and let H(F, T(Ay)/T(ff)) denote the completion of H(F, T(Ay)/T(ff)) in the topology of opensubgroups of finite index. The pairing X*(T)(T(Ay)/T(F))A./F x

induces cup-product pairings

Hr(F,X*(T)) X H2-r(F, r(A)/r(Y))-) H2(F,A / x ) Q/Z

under which

(3.4.2.1) the compact group H(F,X*(T)) and the discrete

group H2(F, T(Ay)/T(ff)) are dual,

(3.4.2.2) the finite groups H(F,X*(T)) and H(F,T(Ay)/T(ff)) are dual,

(3.4.2.3) the discrete group H2(F,X*(T)) and the compactgroup H(F, T(Ay)/T(ff)) are dual.

3.4.3. Let coki(F, T) denote cok[H’(F, T)---) H’(F, T(Ay))]. Note that H’(F,T(Ay))---(vS H(Fv, T) for r > 1. By combining the cup-product pairing of3.4.2 with the inclusion of cok2- r(F, T) in H2- ,(F, T(Ay)/T(ff)), we get a

622 ROBERT E. KOTTWlTZ

homomorphism

H(F,X*(T))cok:-’(F, T)D,

which can also be described as follows" from an element a Hr(F,X*(T)) weget elements a Hr(Fv,X*(T)), and then by means of the local pairings we getelements flv H2-r(Fv, T), which fit together to give a character onH2-’(F,T(Ap)), trivial on the image of H2-r(F,T). This homomorphisminduces

(3.4.3.1) an isomorphism H(F,X*(T)) cok2(F, T)z,

(3.4.3.2) a surjection HI(F,X*(T))cokI(F, T)D, whose kernelis ker(F, X*( T)),

(3.4.3.3) a surjection H2(F,X*(T))--)[%(T(A)/T(F))]z), whosekernel is ker2(F, X*(T)).

3.4.4. The following three subgroups of Hr(F, T) are the same"

(a) ker (F, T(F)),(b) ker[H(F, T)---) 1-Ivs H(Fv, T)],(c) ker[H (F, T) H (F, T(A,))].

We will denote this subgroup by ker’(F, T). There is a pairing

(3.4.4.1) ker’(F,X*( T)) x ker3-r(F, T)--> O/Z

for r 0, 1,2, 3, defined in the same way as for finite F-modules [T1]. For r 0, 3the two groups being paired are trivial by 3.4.1. For r 1,2 the pairing is aperfect duality of finite groups.

3.4.5. Using Lemma 2.2, we can restate all of the results above in terms of 7,so that the duality theorems become statements about the duality of thecohomology of T and 7. Here are two special cases’

(3.4.5.1)

(3.4.5.2)

the groups ker(F, T) and ker(F, 7) are dual finiteabelian groups,

there is a canonical surjection from H (F, 7) onto thegroup of continuous characters of finite order ofT(A)/T(F), and the kernel of this surjection iskerl(F, (this is part of the global Langlandscorrespondence for T).


4. Hasse principle.places of F.

Let F be a number field, and let S denote the set of all

4.1. For an algebraic group G over F we will write ker(F, G) for the kernelof the Hasse map

(4.1.1) HI(F, G)- H H’(Fv, G).

4.2. Let G be a connected linear algebraic group over F, and assume that Ghas no factor of type E8. Sansuc [Sa] has shown that there is a canonicalbijection between kerl(F, G) and the dual of the finite abelian group

(4.2.1) ker Br G --)’ v ,HBr(G. ) ].Now assume further that G is reductive. Then from 2.4 we know that the group(4.2.1) is canonically isomorphic to ker(F, Z(()) (here we are using the notationof 3.4.1). Combining this with Sansuc’s result, we see that there is a canonicalbijection between kerl(F, G) and ker(F, Z(())D. Lemma 8.4 of [Sa] implies thatin the case when G is a torus this bijection is the negative of the bijection givenby Tate-Nakayama duality (3.4.5.1). Renormalize all the bijections by replacingthem with their negatives. Then for every connected reductive group G with noE factors there is a canonical bijection

(4.2.2) kerl(F, G)ker’(F,Z())D,

and these bijections satisfy the following two properties:

(4.2.3) the bijections are functorial for normal homomor-phisms G --) G_,

(4.2.4) if G is a torus, then the bijection is given byTate-Nakayama duality (3.4.5.1).

4.3. In fact, the bijections (4.2.2) are characterized by properties (4.2.3) and(4.2.4). This is an immediate consequence of the following two lemmas.

4.3.1. LEMMA. Let G be a connected reductive group over F having no E8

factors. Assume that Gde is simply connected, and let D G/ Gder. Then(a) ker(F,/)= ker(F,Z()),(b) ker(F, G) kerl(F, D).Part (a) follows trivially from (1.8.3). The proof of (b) is essentially the same as

the proof of Theorem 4.3 of [Sa]. Let fl be the map kerl(F,G)kerl(F,D)induced by G---) D. First we prove that fl is injective. By a twisting argument it is


enough to prove that ker(/3) is trivial. We have an exact sequence

-+ D(F)--) HI(F, GSC) --) HI(F, G)---) H’(F,D ),

as well as the corresponding local sequences. Let x ker(fl). Choosey H (F, Gsc) such that y maps to x. Since x is locally trivial, the localization Yvofy at v is in the image of D(Fv) for all v. Since D(F) is dense in D(F (R) oFI), wecan choose z D(F) whose image y’ in H (F, Gsc) agrees with y at the infiniteplaces. Since H(Fv, Gsc) is trivial for finite v, the elements y, y’ agree locallyeverywhere, hence are equal by the Hasse principle. Therefore x is trivial.Next we prove that fl is surjective. Let x kerl(F,D). Choose a maximal

F-torus T of G such that T is elliptic at some finite place of F. Thenker2(F, Tsc 0. It follows from the long exact cohomology sequence for

1 TscO T--> Do

that x is the image of some element y H 1(F, T). Since x is locally trivial, thelocalization Yv of y at v is in the image of H I(Fv, Tsc) for all v. Since H I(F, Tsc)maps onto I-[vs=Itl(Fv, Ts), where S denotes the set of infinite places of F,we can modify y by an element of H (F, Ts) in such a way that Yv for allv Soo. Let z be the image of y in H I(F, G). It is obvious that z maps to x, andwe will see that z kerl(F, G). Clearly z is locally trivial at the infinite places.Consider a finite place v. Since H(Fv, Gs)= (1), the canonical mapH I(Fv, G)---) 14 (Fv, D) has trivial kernel, and now the local triviality of z at v isobvious.

4.3.2. LEMMA. Let G be a connected reductive group over F, and let H---) G bea z-extension. Then

(a) kerl(F,Z())= ker(F,Z(/)),(b) ker(F, H) kerl(F, G).

Let Z be the kernel of H- G. First we prove (a). We have an exact sequence- Z( ) -- Z(/- ) --) J --) 1,

which yields a long exact sequence (see Corollary 2.3)

---) r0( r) --) H’(F,Z(d))--) HI(F,Z(I)) ---) H I(F, ) --)

globally and locally. Since Z is induced, rro(Z*r) --0 both globally,and locally,and furthermore kerl(F,)=0. It follows easily that ker(F,Z(G))--)kerl(F,Z(O)) is an isomorphism.Next we prove (b). We have an exact sequence

HI(F,Z) --) HI(F,H)---> H’(F, G)---) HZ(F,Z)globally and locally. Since Z is induced, H I(F,Z)= 0 globally and locally, and


furthermore ker2(F,Z)= 0. It follows easily that kerl(F,H)ker(F, G) is anisomorphism.

4.4. Remark. Using the two lemmas above, together with Lemma 2.4.4, onecan easily show the existence of bijections

ker’ (F, G ) - ker’ (F, Z( ))

satisfying properties (4.2.3) and (4.2.4), without using BraG. The first two lemmasforce the definition of the bijection, and the other lemma shows that thebijections are well defined and functorial.

5. Tamagawa numbers. Let F be a number field. For a connected reductivegroup G over F we will write z(G) for the Tamagawa number of G and 71(G ) forthe relative Tamagawa number 7(G)/7(Gsc) of G. Of course we have7(G)-- 7(G) if Gsc satisfies Weil’s conjecture that 7(Gs)= 1.

5.1. Sansuc [Sa] has shown that 7(G) --]PicG Iker(F, G)] -I if G has no Efactors. Using (2.4.1) and (4.2.2) we can rewrite this equality as

(5.1.1) ,( G ) --I0(z( )r)I. Iker’ (F, Z( )) ’"In this form it is not necessary to assume that G has no E8 factors (since thesimply connected form of E has trivial center, any E factor of G is indeed, asthe name suggests, a direct factor of G). For a torus T we have %(7r) H 1(F,X*(T)), and we recover a formula of Ono [O]. For a semisimple group G we haveZ(G)= X*(C), where C is the kernel of Gs-G, and we again recover aformula of Ono [O].

5.2. The function G-7(G) can be characterized by the followingproperties:

(5.2.1) 7(T) 7(T) for every torus T,

(5.2.2) 71(G ) 71(G/Gder) if ade is simply connected,

(5.2.3) for any z-extensionH G with kernel Z.

It is trivial that 7 satisfies the first property; that it satisfies the second twoproperties follows from Corollary 10.5 of Sansuc [Sa]. Note that E factors againcause no trouble. The fact that G-71(G) is uniquely characterized by the threeproperties above is obvious, since z-extensions exist for any G.

5.3. The formula (5.1.1) for 7(G) can be proved directly by checking that thefunction of G defined by the right side of the formula satisfies the threeproperties that we used to characterize 7. The first property is Ono’s result onthe Tamagawa numbers of tori. The second property is obvious from (1.8.3). The


third property follows from part (a) of Lemma 4.3.2 and the exact sequence

X,(Z))X,()o(Z(e,))o(Z())Oobtained by applying Corollary 2.3 to the exact sequence

---) Z(() ---) Z() --- , ---) 1.

6. H for p-adic groups. In this section we will calculate H (F, G), where G isa connected reductive group over a p-adic field F, in terms of Z(G). First weneed to construct certain central extensions.

6.1. LEMMA. Let G be a connected reductive group over a fieM F, let S denotethe identity component of the center of G, let C S f3 Gder, and let C’ denote theinverse image of C under p" Gsc -- Gder. Let i" C’ Z be an embedding of C’ intoan F-torus Z. Then there exists a central extension G’ of G by Z such that Ger is

simply connected.

If char(F)v 0, the statement of the lemma must be interpreted with somecare" C= S X GGder, and therefore C may be a nonreduced scheme. Toconstruct G’ we first define j" C’ --) Gsc S Z by j(c’) (c’, p(c’), i(c’)), andthen we put G’ Gsc S Z/j(C’). Define a" G’ G by a(g,s,z) p(g).s-. It is easy to see that a is well defined and surjective. Furthermore, ker a isisomorphic to Z and is central in G’. We will show that Ge is simply connectedby checking that the homomorphism GscO G’ defined by g-->(g, 1, 1) is anembedding. Clearly ker(Gs---) G’) is a subgroup of ker(Gsc G), which is in turna subgroup of C’. This reduces us to showing that the homomorphism C’ G’defined by c’ -(c’, 1, 1) has trivial kernel. But this homomorphism factorsthrough the subgroup C’ S Z/j(C’) of G’. Call this subgroup S’. If wecompose C’ S’ with the homomorphism S’---> Z defined by (c’,s,z)- i(c’).z- l, we get the embedding i. Therefore C’ S’ has trivial kernel.

6.2. LEMMA. Let F be a p-adic fieM. Then any finite diagonalizable F-group Acan be embedded in an anisotropic F-torus.

Let rn be the order of A. Let K be a finite Galois extension of F such that theaction of F on X*(A) factors through Gal(K/F). Then A can be embedded in afinite product of copies of T, where T ReSK/F(m. Therefore it suffices to treatthe case A Tin, where T,, (t T" m-- ). Let L be an extension of K ofdegree rn (for instance, the unramified extension of degree m). There is a naturalembedding of T in ReSL/F(m The restriction of the norm homomorphismN’ReSL/F(m-->( to the subgroup T is the mth power of the normhomomorphism T--->( Therefore T is contained in kerN, an anisotropicF-torus.

6.3. Let F be a p-adic field. For any torus T over F we have a canonical


isomorphism

(6.3.1) HI(F,T) "’)[%(7r)] D,obtained from local duality (3.3.1). Now consider the two functors G-)HI(F, G)and G--)[%(Z()r)]D from the category of connected reductive F-groups andnormal homomorphisms to the category of sets.

6.4. PROPOSITION.functors

There is a unique extension of (6.3.1) to an isomorphism of

(6.4.1) H(F, G) ) [%(Z()r)] n.We extend the isomorphism of functors in two stages. At the first stage we

extend it to groups G such that Gde is simply connected. Consider such a groupG, and let D G/Gder. In view of the isomorphism Z() =/ the existence anduniqueness in the first stage of the extension will be obvious once we show that

(6.4.2) H I(F, G H (F, D ).

Using a twisting argument plus the vanishing of H for simply connected p-adicgroups [Kn], we see that H l(F, G) ---) H (F, D) is injective. Let T be an ellipticmaximal F-torus of G; such tori exist by a result of Kneser [Kn]. The sequence

---) Tsc---) T---) D---) is exact, and from it we get a long exact sequence

...--),H(F,T)--),H(F,D)--),H2(F, Tsc)

But Tsc is anisotropic, and therefore H:(F, Ts 0 by local duality. This provesthe surjectivity of H(F, T)--), H(F,D), and hence the surjectivity of H(F, G).- H (F, D ).At the second stage we extend (6.4.1) to all groups. Given a group G, we

choose a central extension H--)G such that(a) the kernel Z of H--)G is an anisotropic F-torus,(b) Hd is simply connected.

The existence of such an extension is guaranteed by Lemmas 6.1 and 6.2. Wehave the following commutative diagram with exact rows

H’(F,Z) ) HI(F,H) )H l(F, G) )

,.o(Z(d) ,1

in which the vertical arrows are the bijections coming from the first stage. Thetrivial groups at the right-hand ends of the rows are H2(F,Z) (trivial by local


duality since Z is anisotropic) and [X,()r]n (see Corollary 2.3; X,()r is trivialsince Z is anisotropic). The fibers of H(F,H)- H(F, G) are the orbits for theaction of H (F, Z) on H (F, H). Therefore there exists a unique bijectionH (F,G) r0(Z(()r) that makes the diagram above commute. This proves theuniqueness of the extension (6.4.1). It is easy to check, using a variant of Lemma2.4.4, that the bijections we obtain are independent of the choice of H G andare functorial in G. This proves the existence of the extension (6.4.1).

6.5. Remark. Suppose that G is semisimple, and let C denote the kernel of

Gsc-> G. The exact sequence --> C--> Gsc--> G induces a map H(F, G)--> H2(F, C). Kneser [Kn] has shown that this map is bijective, By local dualityfor finite groups we have H2(F, C)= [X*(C)r], and since X*(C):= Z(G), wesee that Kneser’s bijection can be reformulated as a bijection

H(F, O) ;0(Z()).It is not hard to check that this bijtion is the same as the one provided byProposition 6.4. We choose a central extension H G as in the proof ofProposition 6.4 and write D for H/Ha and Z for kcr(H G). The exactsequence

---) C---) Z Gsc H---)

maps to both of the exact sequences

C---) Gsc---) G 1,

1---) C---) Z--) D--) I.

Using this fact, we see that in order to check the agreement of the two bijections,it is enough to check the commutativity of the diagram

H’(F,D ) ; H’(F,X*(D ))= ro(/r)

cl

in which the vertical maps come from the exact sequence

1---) C---) Z---) D I.

This is easy.

6.6. Remark.form:

Using (2.4.1), we can put the bijection (6.4.1) into the following

H(F, G) "’ ) (Vic G)D.


This is reminiscent of Tate’s result [T1] on H (F,A) for an abelian variety A overa p-adic field F:

H’(F,A) .(A’(F)),where A’ Pic(A), the dual abelian variety.

7. Review of endoscopic groups. In this section, F is a local or global field ofcharacteristic 0, and G is a connected reductive group over F.

7.1. An endoscopic datum for G is a pair (s,p), consisting of a semisimpleelement s of G/Z() and a homomorphism^ p" F--> Out(), satisfying

^0properties (7.1.1) and (7.1.2) below. We will write H for G.

(7.1.1) For o F the element p(o) Out() is induced by anelement of NormL(H) whose image under thecanonical homomorphism/G-> F is o.

Note that p induces an action of i" on Z(). Moroever, the inclusion of Z()in Z() is a F-map (the action on Z(() comes from the action on (). The exactsequence

1 Z()--) Z(I) Z(I)/Z()gives us a long exact sequence (Corollary 2.3)

,n’o(Z(t)v)%([Z(t)/Z()]r) H’(F,Z(r))’’’.We define (s,p) to be the subgroup of %([Z(I)/Z()]r) consisting of allelements whose image in H(F,Z(G)) is

(a) trivial if F is local,(b) locally trivial if F is global.Now we can state the second condition on (s,p). Note that s Z(I)/Z().

(7.1.2) The element s Z(I2I)/Z() is fixed by F, and its

image in %([Z(I)/Z()]r) belongs to (s,p).

^07.2. Let (Si,Pi) (i 1,2) be endoscopic data for G and write H for G,. Anisomorphism from (s,o) to (s_,O) is an element g ( such that

(7.2.1)

(7.2.2)

(7.2,3)

Int(g)(,)#9_--" a ^#, where a denotes the isomorphism Out(H)-- Out(H2) induced by Int(g),

int(g)(s) and s2 have the same image in (s2 02).


We will write Aut(s,0) for the group of automorphisms of (s,p). It is easy tosee that Aut(s, 0) is an algebraic subgroup of G with identity component H. Wewill write A(s, p) for the quotient Aut(s, 0)/H.

7.3. We say that an endoscopic datum (s, ), with associated group (s,is elliptic, if [Z()r] c Z((). For elliptic endoscopic data the third condition inthe definition of isomorphism can be replaced .by

(7.3.1) Int( g)(s ) s2.

7.4. An endoscopic triple for G is a triple (H,s,), consisting of a quasi-splitconnected reductive F-group H (which we will refer to as an endoscopic group forG), an element s Z(H), and an embedding " ---)( of G groups, satisfyingthe following three conditions"

^0(7.4.1) ,/(/-)) G).

(7.4.2) The G-conjugacy class of /is fixed by F.

We can use /to regard Z(() as a subgroup of Z(). By (7.4.2) the F-actionson Z() and Z(H) are compatible. Thus we can define a subgroup (H/F) ofro([Z(H)/Z(G)]r) as in 7.1. Now we can state the third condition.

(7.4.3) The image of s in Z(I21)/Z(() is fixed by F, and its

image in %([Z(I?I)/Z()]r) belongs to (H/F).

7.5. Let (Hi,si, /;) (i 1,2) be endoscopic triples for G. An isomorphism from(H,Sl,/l) to (H2,s2,rb) is an F-isomorphism a:HoH2 satisfying thefollowing two conditions:

(7.5.1) / and /2 are -conjugate. This condition makessense because c is well-defined up to Hl-conjugacy.Furthermore, one consequence of this condition is thatc induces a canonical isomorphism (H2/F)"% (HI/F).

(7.5.2) The elements of (Hi/F) defined by s correspondunder (H2/F) -z-) (H,/F).

We will write Aut(H,s, r/) for the group of automorphisms of (H,s, ,/). ClearlyHad(F) is a normal subgroup of Aut(H,s,/). We will write A(H,s,’q) for thequotient Aut(H,s, 1)/ Had(F).

7.6. An endoscopic triple (H,s, q) determines an endoscopic datum (q(s),o),where is the composition

^0F >Aut(/-) ) ’) Aut(G,()) Out(^G"()),


and every endoscopic datum arises from some endoscopic triple. Let (Ht.,si, Ti)(i 1,2) be endoscopic triples, and let 01(si),Pi) (i 1,2) be the correspondingendoscopic data. There is a canonical bijection from the set of isomorphismsfrom (Hl,s l, /I) to (H2,s2, 1), taken modulo inner automorphisms of H2, to the

^0set of isomorphisms from 0/(s), pl to 0/(s), p:), taken modulo Gnts2 Thus thereis a bijection from the set of isomorphism classes of endoscopic triples to the setof isomorphism classes of endoscopic data. Moreover, there is a canonicalisomorphism

(7.6.1) A(H, s, ,/) A(/(s), p).

We will say that (H,s, /) is elliptic if (/(s), O) is elliptic.7.7. There are some related definitions from [L1, L3] that we need to recall.

Let T be a maximal F-torus of G. We denote by (T/F) the setker[H(F, T) H(F, G)]. This set is in one-to-one correspondence with the setof G(F)-conjugacy classes of embeddingsj" T G such thatj is G(F)-conjugateto the inclusion TG. We denote by @(T/F) the image of H(F, Tsc) inH I(F, T). It is always true that (T/F) is contained in (T/F), and the two.sets are equal if F is p-adic.

In the global case we denote by (T/A) the set

ker[H(F,T(A))+ HI(F,G(A))] (T/Fv)

and by @(T/A) the group

im[H’(F, Tsc(A;))---) /-/ ’(F, r(A;))] (r/F),

where the sums are taken over the set of all places of F (it makes sense to takethe direct sum of the sets 5(T/F,) since they are pointed sets). There are obviousmaps (T/F) -+ (T/A) andWe also need to define groups (T/F). From the definition of the Qgroup we

see that Z(() may be identified with a F-submodule of 7. The torus T/Z(G) isdual to the torus T. From the exact sequence

1--) Z()--) 7-- f/Z()--)we get a long exact sequence (Corollary 2.3)

We define (T/F) to be the sub.group of rro([’/Z()]r) consisting of allelements whose image in HI(F,Z(G)) is

(a) trivial if F is local,(b) locally trivial if F is global.


If F is global and v is a place of F, there is a natural homomorphism(T/F) R( T/ Fv). Note the similarity with the definition of R(H/F) for anendoscopic group H. Our definition of R(T/F) agrees with that of [L3] whenT is elliptic, but is slightly different otherwise, because Langlands does not divideby the identity component of [/Z()] r. This difference is not important;dividing by the identity component is simply a technical convenience.

7.8. It is useful to know how (T/F), @(T/F), (T/F) behave underz-extensions. In fact, the behavior is as simple as possible. Let G’---)G be az-extension, let T be maximal F-torus of G, and let T’ be the inverse image of Tin G’. Then the canonical maps (T’/F)- (T/F), (T’/F) (T/F),(T/F)- (T’/F) are all bijections. The proof is not difficult and will be leftto the reader.

7.9. In the local case Tate-Nakayama duality (3.3.1) provides us with anisomorphism

(7.9.1) (T/F) (T/F)D,

since the dual of the image of H(F, Tsc) H(F, T) is equal to the image of%(7r)%([7/Z(()]r). In the global case Tate-Nakayama duality (3.4.3.2)provides us with a surjection

(7.9.2) ( T/F) 3[@(T/A)/im(@(T/F)) D,

whose kernel consists of all the elements of (T/F) which have trivial image in(T/Fv) for every place v of F.

$. t(G,H). Let F be a number field, and let G be a connected reductivegroup over F. To each endoscopic triple (H,s,l) for G, Langlands [L3] hasassociated a constant t(G,H), needed to stabilize the trace formula for G. Thisconstant depends on s and /, as well as H, although the notation may suggestotherwise. First we will recall the definition of t(G,H), and then in 8.3 we willfind a new expression for it.

8.1. We will write ,(H,s, rl), or just , for the order of the group A(H,s,r/)(see 7.5 and 7.6.1); , depends only on the isomorphism class of (H,s,,1).

8.2. Let T be a maximal F-torus of G. Following Langlands [L3], we define anumber t(F, T, G) by putting

ff F, T, G) Iker[e(T/F)-->@(T/A)] I(T/F)I-’.

Let (H, s, /) be an elliptic endoscopic triple for G, and let p: G* ---) G be an innertwisting, where G* is a quasi-split inner form of G. Any maximal F-torus T/ ofH gives rise to a maximal F-torus Ta. of G*, well-defined up to stable conjugacy(see [L3]). Using the fact that (H,s, rl) is elliptic, we see that T/ is elliptic in H ifand only if Ta. is elliptic in G*. In Lemma 8.6 of [L3] Langlands shows that the


numbers

(8.2.2) t(F, TG,, G*) t(F, TH ,H)-’are all equal, as long at T, is elliptic, and he then defines t(G,H) to bewhere a is the common value of the numbers (8.2.2).

8.3. Let (H,s, q) be an elliptic endoscopic triple for G, and letAs in Section 5 we will use r(G) to denote the relative Tamagawa number of G.

8.3.1 THEOREM. The constant t(G, H) is equal to rl(G). rl(H)- I. 2- I.This equality is an immediate consequence of the following lemma (note that

,r,(*) ’(G)).

8.3.2. LEMMA. Let T be a maximal F-torus of G. Let Tsc denote thecorresponding maximal torus of Gsc. Then

/,(F, T, () TI(( ) ’/’I(T) -1. Icok[X*(T)r- X*(Tsc)rl -l.

In particular, if T is elliptic in G, then

,(F, , 6) ,(6). ,(r) -’.

By choosing a z-extension of G and using (5.2.3) and 7.8, we reduce to the casein which Gde is simply connected. Next we apply Corollary 2.3 to the exactsequence

This gives us an exact sequence-- a -- o(Z(( )r)__

ro(Cr)__o(rc) -- H ’(F, Z(()) -- H ’(F, ),

where A cok[X*(T)r+ X*(Tsc)r]. Using the definition of (T/F) and theexact sequence above, we get an exact sequence- .4 - %(Z(( )r)

__%(r)

__(T/F)- ker’ (F, Z(()) - ker’ (F, ) -- B - 1,

where B cok[ker(F,Z())-+ker(F, 7)]. Using (5.1.1), we see that

(8.3.3) IBI" I( T/F)I- z,(G ). ’rl(T) -I" [A 1.

Since Gde is simply connected, we have Z(()=/, where D G/Gder, and byduality (3.4.5.1) the group B is dual to

ker [ker’(F, T)ker’(F, D )]


which, again because Gde is simply connected, is equal to

ker[ @(T/F) --) @(T/A) ].

Combining this fact with (8.3.3), we get the desired result.

8.4. Remark. Note that Lemma 8.6 of [L3] is no longer necessary; we cansimply adopt (8.3.1) as the definition of t(G,H). In any case, that lemma followsfrom our Lemma 8.3.2.

9. Embeddings of tori. Let F be a field, G a connected reductive F-group,and H a quasi-split inner form of G. Choose a F-isomorphism G H; thisdetermines an inner twisting q:H---)G, well-defined up to conjugation by anelement of G. Let T be a maximal F-torus of H. We will say that anF-embedding i" T---)G is admissible (relative to ) if it is of the formInt(g) o qlr for some g G. Any two admissible F-embeddings are stablyconjugate (conjugate under G), but not necessarily G(F)-conjugate.

9.1. Consider the unramified situation. Let F be a nonarchimedean local fieldwith valuation ring 0, and assume that G, H, T are unramified over F (quasi-splitover F and split over an unramified extension of F). Let x0 be a hyperspecialpoint in the building of G over F, and let K Staba(F)(X0)" Then x0 determines asmooth o-structure on G (see [Ti]) for which G(OL)= Staba(L)(X0) for anyunramified extension L of F. Since T is unramified, it extends uniquely to a torusgroup scheme over o. For any unramified extension L of F the group T(o) is themaximal compact subgroup of T(L).

9.1.1. LEMMA. Let i,i2 T G be two admissible F-embeddings, and assumethat 2 are defined over o. Then i, 2 are conjugate under K.

Choose a finite unramified extension L of F that splits T (and hence G). Since(g G Int(g)o i i2) is an F-torsor under T, and since H(L, T) is trivial,there exists g G(L) such that Int(g) 2. We have iI(T(L)) C G(oL), andtherefore x0 belongs to the apartment of il(T). We also have il(T(0t))C g-.G(0L) g, and therefore g-. x0 also belongs to the apartment of i1(T); from thisit follows that g- i(T(o)).G(o), and without loss of generality we mayassume that g G(OL). Then (g-l. go)oGal(L/F is a 1-cocycle of Gal(L/F) inT(L) f3 G(oL)= T(oc). Since Hi(L/F, T(o/))= 0, there exists T(o.) suchthat g-. gO t-. o, or, in other words, gt- G(o)= K. This proves thelemma, since gt- conjugates i into i.

9.2 Now take F to be a number field, and assume that G has no E factors,so that Gs satisfies the Hasse principle. Let S denote the set of all places of F.For v S we will write Gv instead of GF. Suppose that we are given anadmissible Fv-embedding i(v)’Tv--> Gv for each v S. When can we find anadmissible F-embedding i" To G such that (iv)v s and (i(V))v s are conjugateunder G(,6,F)9. Here T G is the Fv-embedding obtained from by extension


of scalars from F to Fv. This question is answered in Ch. VII of [L3]. First of all,for almost all v the situation must be unramified. More precisely, let K be acompact open subgroup of G(AfF). Then for all but a finite number of places ofF, the groups G and T are unramified and K f3 G(Fv) is a hyperspecialmaximal compact subgroup of G(Fv). For these places we get Ov-Structures on

Gv, Tv. Any family (i(v)) coming from a global must satisfy the followingproperty:

(9.2.1) i(v) is defined over o for almost all v S.

Now consider a family (i(V))v s that satisfies (9.2.1). Langlands constructs anelement of (T/F)n whose vanishing is a necessary and sufficient condition forthe existence of an admissible F-embedding i" To G such that (iv)v s and(i(V))v s are conjugate under G(AF). This construction, which involves cocyclecalculations, is rather complicated, and the purpose of this section is to give asimpler construction. We will again get an obstruction lying in (T/F), andundoubtedly the two obstructions are the same, at least up to a sign, but we willnot stop to check this. It would not be unreasonable simply to reformulate theglobal hypothesis on p. 149 of [L3] in terms of the new obstruction.Now we will see how to get our obstruction. Let --0 be the set of pairs (i, g)

satisfying

(9.2.2) is an admissible F-embedding

(9.2.3) g (gw)wg Gsc(Ap) (here S is the set of places of F),

(9.2.4) iw Int(gw)o i(V)w for every v S and every w Slying over v.

In (9.2.4) the subscript w on and i(v) denotes extension of scalars to Fw. Wehave three actions on -0:

(i) F acts on --0 by (i, g) (i, gO) (o F),(ii) Gsc(F) acts on -0 by x. (i, g)= (Int(x)o i, xg) (x Gs(F)),(iii) Ts(A) acts on -0 by (i, g). (i, i(t ). g) (t Ts(A)).

We have the following compatibilities:(a) [x. (i, g)] x. (i, g) for x Gs(F), o F,(b) [(i, g). t] (i, g)- for Tsc(A), o F,(c) the actions of Gs(F) and Ts(A) commute.

Let -- denote the quotient of --0 by Gs(F). We have induced actions of F andTs(A) on , and it is not hard to check that if- is an F.torsor underTs(A)/Tsc(F).

9.2.5. LEMMA. The following are equivalent"(i) There exists an admissible F-embedding i" To G such that (i,,),(i(v)) are


conjugate under Gsc(AF).(ii) The set - is nonempty.(iii) The torsor - is trivial.

It is obvious that (i) and (ii) are equivalent and that (ii) implies (iii). We willnow check that (iii) implies (ii). Assume that - is trivial. The if-r is nonempty.Choose an element of --r, and let - be the fiber of -0-->- over thatelement. Then - is an F-torsor under Gsc(F). To finish the proof we must showthat ff- is trivial. Choose (i, g) --. The associated 1-cocycle (xo) of T’ inGsc(F) is determined by the equation

(i, g)O= xo .(i, g).

In particular, we have x go. g-l with g Gsc(A). Hence (xo) is trivial inH(F, Gs(A)). Since we are assuming that G has no E8 factors, the Hasseprinciple holds for Gs, and we see that :- is trivial.

9.3. We retain the notation of 9.2. Associated to o- is a cohomology classc H (F, Ts(Ap)/ Ts(ff)). By Tate-Nakayama duality (3.4.2.2) and Lemma 2.2this cohomology group is canonically isomorphic to (Tsc/F)D. If G Gs, thenthe element c, viewed in (Ts/F), is the desired obstruction. In the generalcase we have (T/K)c (Tc/F) and hence a canonical surjection

(9.3.1) ( Tc/F)--)(T/F)

The image of c in (T/F)D is the obstruction we want, but before we can provethis we need to establish some preliminary results.

9.4. We retain the notation of 9.2 and 9.3. It is easy to find a 1-cocycle of Fin Tsc(Ap)/T(F) that represents the cohomology class c. We choose (i, g) --and extend i" T- G to an P-isomorphism i" H-7+ G (recall that H is a quasi-splitinner form of G). Then is an inner twisting (up to conjugation by G it is thesame as k" H G); therefore we can choose, for o F, an element h H(F)such that i- o Int(ho). Then (Int(ho))oer represents the cohomology class inH (F, Hd) associated to the inner twisting " H G. An easy calculation showsthat c can be represented by the 1-cocycle (to), where to denotes the image inT(Ap)/T() of the element i-(g.g-o), ho of T(Ap).

9.4.1. Remark. Let c’ denote the image of c under the homomorphism

H’(F, T(A)/T(;))-- HE(F,

induced by the exact sequence

--> Tsc(F ) --> Tsc(A.) ----> T,:(A)/Tse(F ) --> I.

By direct calculation we see that c’ is equal to the image of (Int(ho)) under

H’(F, Had)--> HE(F,Z(Hsc))-’> H:(F, Ts).


9.4.2. We will need to know the effect on c of a change in the family (i(v)).Use the family i(v) to build a homomorphism i(A): T(A)-G(Ap). Thenconsider a 1-cocycle (do) of F in T(Ap) whose image under

H ’(F, T(A)) --> H ’(F, G(A))is trivial. Then there exists x (Xw)wg G(A) such that i(A)(do)= x -. xfor all o F. Let v S. We get an admissible Fv-embedding j(v):Tv- G byputting j(v)= Int(xw)o i(v) for w lying over v (this is independent of w). TheG(AF)-conjugacy class of (j(v))v s is independent of the choice of x, and themap (do)-(j(v)) induces a bijection from

(9.4.2.1) ker[ H l(F, T(A))--> H l(F, G(Ap))to the set of G(AF)-conjugacy classes of families (j(V))v s satisfying (9.2.1).There is a bijection between the set of admissible embeddings of T in G and theset of admissible embeddings of Tsc in Gsc; thus, applying the discussion above toTs, G, we also obtain a bijection from

(9.4.2.2) ker[ H ’(F, Tsc(A)) ---> H ’(F, Gsc(A))to the set of G(A)-conjugacy classes of families (j(V))v s satisfying (9.2.1). Letd be an element of the group (9.4.2.2) and let (j(v)) be the corresponding family.Let Cnew be the element of H (F, Tsc(A)/ Ts(ff)) associated to (j(v)) in the sameway that c is associated to (i(v)). Using the method given above to find1-cocycles representing c and Cnew, we see that Cnew is the sum of c and the imageof d under

H ’(F, T(Ap)) - H ’(F, T(A)/T( )).9.4.3. From 9.4.2 we see that in order to prove that the image of c in

(T/F)n is the obstruction we want, it is enough to prove the equality of thefollowing two subsets of H (F, Tsc(Ap)/Ts(ff)) (Ts/F)D:

(a) the kernel of the homomorphism (9.3.1),(b) the image in H(F, T(A)/T(ff)) of the subset A of HI(F, Tc(A))

consisting of elements that have trivial image in both H(F,T(Ap)) andn (F, Gse(Ap)).The first step in proving that the sets (a) and (b) are equal is to note that

neither set changes when G is replaced by a z-extension G’ of G (see 7.8). Thus,without loss of generality, we may assume that Gdr is simply connected. LetD G/Gder.The next step is to show that the set (b) is equal to the image in

H (F, Tsc(A)/Tc(ff)) of B, where

B ker[ H I(F, Tc(A))---> H ’(F, T(A))].Since A c B, it is obvious that this set contains the set (b). To check the reversed


inclusion one uses the long exact cohomology sequence for

TscT D---)

and the fact that D(F) is dense in D(FThe last step is to show that (a) is equal to the image of B. Consider the

diagram

H ’(F, Tsc(A)) a__; H ’(F, Tsc(A)/Tsc(ff ))

H’(F, r(Ar)).

Taking duals, we get the diagram

D

(Ts/F ,,oz, ; II( Tsc/Fv

where Fv is the absolute Galois group of Fv. By definition, (T/F) is the inverseimage under a n of im(/3 z), or, in other words,

(T/F) ker[ ft( T/F) cok( tip )1"Therefore (T/F) is equal to

cok[ ker fl --) (T/F)n ],which shows that the image of B ker fl in H(F, Ts(A)/ Ts()) is equal tothe kernel of

( Ts/F)’-->( T/F),which is the definition of the subgroup (a).

9.5. Let G,H, T be as in thee beginning of the section. We will say that Ttransfers to G (with respect to G- H) if there exists an admissible F-embeddingT--) G. Now consider the case in which F is a global field and T transfers to Gfor every place v of F. As before, we assume that G has no E8 factors. Langlands[L3] constructs an obstruction which vanishes if and only if T transfers to G. It iseasy to understand this in terms of the results of this section. Since T transfers toG locally, we can choose a family (i(v))v s as in 9.2. We can assume that thefamily satisfies (9.2.1). This gives us an element c H(F, Tsc(A,)/Tsc(ff)). It isobvious that T transfers to G if and only if there exists a family (j(v)) s for


which c + e vanishes, where e is the image in HI(F, Tsc(A)/Tsc()) of theelement d Hi(F, Tsc(A,)) corresponding to (j(V))vS (see 9.4.2). In otherwords, T transfers to G if and only if belongs to the image inHI(F, Tsc(Af)/ Tsc()) of the group (9.4.2.2). It is easy to see that this image isthe same as .the image of the whole group Hi(F, Tsc(Af)) (use the fact thatH(F, Tsc) maps onto 1-IvsooHl(Fv, Tsc), where So is the set of infinite places ofF). Using the long exact cohomology sequence for

--> Tsc(F ---> Ts(A. ---> Ts(A.)/Tsc(F ----> 1,

we see that T transfers to G if and only if the image c’ of c in H -(F, Tsc) is trivial.We have seen in Remark 9.4.1 that c’ can be calculated directly from the class inH(F, Ha) that corresponds to the inner twisting " H--)G. In fact, thisalternative definition of c’ can also be used directly to show that T transfers to Gif and only if c’ is trivial.

10. Admissible homomorphisms. In this section F is a local or global field andWF is the Weil group of F/F. For a connected reductive group G over F wewrite ZG for the Weil form WF of the L-group of G. In the global case, givena place v of F, we write G for GF. We consider continuous splittingsfD" WFIG of ZG---> WF In the global case, given a place of v of F, we have ahomomorphism 0 WFc WF, canonical up to conjugation by an element of Wr(see [T3]), and 0 induces a homomorphism C(Gv)IG. There is a uniquehomomorphism %" WF l(Gv such that

WF )IG

commutes. As in (2.1) of [T3] we set that --v induces a well-defined mapfrom the set of homomorphisms qg" WFG as above, taken moduloconjugation by G, to the analogous local set.We say that tp" WF--->IG is admissible if(i) tp(w) is semisimple in/G for all w WF (an element of/G is said to be

semisimple if its projection in the algebraic group Gal(K/F) ( is semisimplefor every finite Galois extension K/F that splits G),

(ii) tp is locally relevant: in the local case this means that if q0 factors through aLevi subgroup of a parabolic subgroup of /G, then the parabolic subgroup isrelevant; in the global case this means that v is relevant for every place v of F.

It is obvious that % is admissible if is admissible.10.1. Let q" WFIG be an admissible homomorphism. We will write C for

the centralizer of tp(WF) in .10.1.1. LEMMA. The group C is reductive (not necessarily connected).

Let WF denote the kernel of II’WF R+, and let A (resp. B) denote the


Zariski closure of p(q0(WF)) (resp. p(q0(WF))) in Aut(d), where p" LG- Aut(()is the homomorphism determined by the action of G on G by conjugation. SinceWF is compact (see IT3]), B is reductive. There is a continuous homomorphismR+ A/B, whose image is Zariski dense in A/B and consists of semisimpleelements (by part (i) of the definition of admissible homomorphism). ThereforeA/B is a torus, which implies that A is reductive. The group C is equal to ,the group of fixed points of A in . The fact that C is reductive is aconsequence of the following lemma.

10.1.2. LEMMA. Let k be an algebraically closed fieM of characteristic O, G aconnected reductive group over k, and ,4 a reductive subgroup of Aut,(G). Then thegroup GA offixed points of A in G is reductive.

The first step is to note that if k’ is an algebraically closed extension field of k,then (GA)(k’) G(k’)A(k’). Using this, we reduce to the case in which k is finitelygenerated over O; using it again, we reduce to the case that k (3.Now we assume that k (3. There exists a real structure on A for which A (R)

is compact and Zariski dense in A(C) (see [HI). We have G= GK, whereK= A(R).

It is now sufficient to prove the following statement: if K is a compactsubgroup of Aut(G) then Gr is reductive. Choose a real structure on G for whichG(R) is compact. For this real structure the group Aut(G)(R) is a maximalcompact subgroup of Aut(G). Replacing K by a conjugate under Aut(G), wemay assume K is contained in Aut(G)(R). Then G is defined over R.Furthermore, Gr(R) is compact, for it is a closed subgroup of G(R). ThereforeGr is reductive.

10.2. Let 0: WF---)LG be an admissible homomorphism. We define a group

D containing C (see 10.1) and Z() by putting

D {s dis- (w). s-. (w)- z(d) for all w WF ).0 have the same image in the adjoint group of , as one seesThe groups C, D

by considering Lie algebras, and therefore

0 z(d).(10.2.1) C. Z(d)= O.It follows from Lemma 10.1.1 that D is reductive.

For s D and w WF define an element z Z((r) by

Then (z) is a continuous 1.cocycle of Wr in Z((), and s--(Zw) is a

hom,omorphism from D to the group of continuous 1-cocycles. Let H(WF,Z(G)) denote the group of continuous 1-cocycles modulo 1-coboundaries. Weget a homomorphism

(10.2.2) D, --> H ’( WF Z(d )).


Let S denote the subgroup of O consisting of elements s whose image inHI(WF, Z(()) is

(a) trivial if F is local,(b) locally trivial if F is global, or, in other words, an element of

kerl(W,Z(G)), the kernel of H(W,Z(G))--)IIH(W,Z(,)).Finally, we define a group (R) by putting

(10.2.3) --- S/S Z(a)- ro(S/Z())Using (10.2.1), we see that

(10.2.4) C. Z(()= S. Z().In the local case we have (R) C/C Z(()r, which agrees with the usualdefinition. In the global case we have an exact sequence

(10.2.5) ---) C/C Z(()r-- ---)ker’(Wr,Z(())and the definition of w is the obvious generalization of the definition in [L-L].An admissible homomorphism q: WFLG is said to be tempered if the image

of cP(WF) in Gal(K/F)< G is relatively compact in Gal(K/F)< for everyfinite Galois extension K/F that splits G. Let eP:WF-LG be a temperedadmissible homomorphism. In the local case, Langlands [L3] conjectures that thegroup (R) controls the L-packet H(tp) (see [Sh] for the archimedean case). In theglobal case, Labesse and Langlands [L-L] conjecture that the group (R) controlsthe automorphic representations in the L-packet I[().

10.3. Let tp: WroLG be an admissible homomorphism. We will say that tp iselliptic if q factors through no proper Levi subgroup of/G. If is elliptic andtempered, then one expects that lI(q) consists of cuspidal representations in thelocal case, and one expects that the automorphic elements of H(tp) are cuspidalin the global case.

10.3.1. LEMMA. The following are equivalent:(i) is elliptic.(ii) C c Z(G),(iii) S c Z(().The equivalence of (ii) and (iii) is obvious from (10.2.4). The equivalence of (i)

and (ii) follows from Lemma 3.5 of [B] and the fact that C is reductive (Lemma10.1.1), from which it follows that C^c Z(() if and only if every torus in G thatcommutes with is contained in Z(G) (note that the centralizer in/G of a torusthat commutes with must project onto WF, since it contains (WF)).

10.4. Let ,2: WF---)LG be admissible homomorphisms. We say that l,are equivalent if there exists g ( such that ---(Int(g) o 02). z for somecontinuous l-cocycle z (Zw) of WF in Z(() whose class in H(WF, Z(r)) is

(a) trivial if F is local,(b) locally trivial if F is global.


We will use [(p] to denote the equivalence class of 99. It is not hard to see that theproperties of being tempered or elliptic depend only on the equivalence class ofan admissible homomorphism. Note that S can be thought of as the group ofself-equivalences of qo. If (l,q02 differ by a 1-cocycle of WF in Z(G), thenS, S2. If qOl,qO2 are equivalent, then by choosing g G as above we getInt(g): S2-% S,, well-defined up to inner automorphisms of S,. The sameremarks apply to (R),, (R).

11. Admissible homomorphisms for endoscopic groups. Let F be a numberfield and let G be a connected reductive group over F. In this section we willstudy the relationship between elliptic admissible homomorphisms for G andelliptic admissible homomorphisms for the elliptic endoscopic groups of G. Wewill use these results when we discuss the stabilization of the trace formula for G.

11.1. Let (H, s, r/) be an endoscopictriple for G. Then Z() is a F-submoduleof Z() (see 7.4), and we can use Z(G) to define a new equivalence relation onthe set of admissible homomorphisms : WF-->LH. We say that two admissibleembeddings ,9. for H are Z(G)-equivalent if there exists h such thatff (Int(h)o 2)" z for some continuous 1-cocycle z =(Zw) of Wr in Z()whose class in H(Wr, Z()) is locally trivial. It is obvious that Z()-equivalence implies equivalence. We will use [[]] to denote the Z()equivalence class of .

It is easy to understand why we have introduce Z()-equivalence. Consideran L-homomorphism T/’ :CH-->CG extending H--> G. Then we have a mapb->T/’ b from the set of admissible homomorphisms for H to the set ofadmissible homomorphisms for G. If ,2 are Z(G)-equivalent, then T/’oTI’o are equivalent. Thus Z()-equivalence is suitable for the study of therelationship between admissible homomorphisms for H and admissible homo-morphisms for G.

Let " W-->ZH be an admissible homomorphism. We denote by GS thegroup of self-equivalences of for Z(G)-equivalence; in other words, GS is thesubgroup of H consisting of all h H such that int(h)o and differ by acontinuous 1-cocycle of W in Z(() whose class in H(WF, Z(G)) is locallytrivial. It is obvious that S contains Z(). We define (R) in analogy with

(11.1.1) os,/os z(O)

11.2. Let (H,s,/) be an elliptic endoscopic triple for G, and let p: WF---)IHbe an elliptic admissible homomorphism for H. Recall that we use rl(G) todenote the relative Tamagawa number of G (see Section 5).

11.2.1. PROPOSITION. The number of Z()-equivalence classes contained in theequivalence class of is equal to

"r,(G). ’r,(H) -. I %1" I%1


Before proving this proposition, we need to prove a lemma. Let kerl(WF,Z(/)) denote the kernel of HI(WF, Z(I)) IvHI(WF,Z(I)).

11.2.2. LEMMA. The inflation homomorphism ker(F, Z(i))--) ker( WF, Z(I))is an isomorphism.

The injectivity of the map is obvious. We will now verify the surjectivity. Let abe a continuous 1-cocycle of WF in Z(H) whose class is locally trivial. Let K/Fbe a finite Galois extension that splits H. Then WK acts trivially on Z(/), andtherefore the restriction of a to WK is a continuous homomorphism fl:Wr- Z(H). Usin class field theory, we view fl as a continuous homomorphismAc/K - Z(H) whose restriction to Kx is trivial for every place w of K, Sincethe sum of the groups Kwx is dense in A, we see that fl.is trivial. Therefore a isthe inflation of a 1-cocycle a’ of Gal(K/F) in Z(H). The class of a’ inH (F,Z(I)) is locally trivial and maps to a.

11.2.3. Now we prove Proposition l.2.1. Our problem is to count thenumber of Z()-equivalence classes [[q/I] contained in the equivalence class [q].It is enoug,h to consider q/of the form q. z where z is a continuous 1-cocycle ofWF in Z(H) whose class a H I(WF, Z(I-)) belongs to kerl(Wr, Z(JQ)),. Fromnow on we use Lemma 11.2.2 to replace kerl(Wr, Z(t?I)) by ker(F,Z(H)). It iseasy to see that q. z is Z(G)-equivalent to + if and only if a belongs to the sumof the following two subgroups of kerl(F,Z())

(11.2.3.1) im ker’(r, Z( ))- kerl(r, Z(/))],(11.2.3.2) im[(R) --> ker(F, Z())(see 10.2 for the definition of (R),,joker (F,Z(H))). We need some temporarynotation. We will write A for cok’[kerl(F,Z())--->kerl(F,Z(H))] and B for thecokernel of the composition

(R) ker’(F, Z()) --) A,

where ker(F,Z()) A is the canonical map. From the previous discussion wesee that the number of Z(()-equivalence classes contained in [q] is equal to IBI.The definition of B gives us an exact sequence

(11.2.3.3) (R)+ ---> A --> B ---> 1.

Consider the homomorphism a(R)- (R)q induced by the inclusion aS S. Aneasy calculation shows that

(11.2.3.4) imI(R)- (R)] ker[(R)--> A ].Another easy calculation, which uses the fact that (H,s,,/) and p are elliptic,shows that

(11.2.3.5) ker[a --> (R)] (H/F)


(see 7.4 for the definition of (H/F)). Putting together (11.2.3.3), (11.2.3.4),(11.2.3.5), we get an exact sequence

(11.2.3.6) (H/F)---)(R) (R), --)A B 1,

which implies

(11.2.3.7)

Consider the exact sequence

1- Z(r ) ---) Z (l ) -’-) Z(121)/Z( 6 ) -’) 1,

and apply Corollary 2.3. As in the proof of Lemma 8.3.2 we get an exactsequence

o(Z( )r) %(Z( I?t lr) --) (H/F)

ker’(F,Z(t))ker’(F,Z(I))A 1.

Note that the injectivity of

)is a consequence of our assumption that (H,s, q) is elliptic. The formula (5.1.1)for relative Tamagawa numbers, together with the exact sequence above, implies

(11.2.3.8) IAI I(H/F)I-’ ,(G). ,(H)-’.

Combining (11.2.3.7) and (11.2.3.8) finishes the proof.11.3. Assume that Gde is simply connected. Choose a set o of representatives

for the isomorphism classes of elliptic endoscopic triples for G, and for each(H,s,l) eg choose an L-homomorphism /’" H---)tG that extends r/" --)(.Our assumption that Go is simply connected guarantees that such extensionsexist (see [L1]). Let :WFZH be an admissible homomorphism, and let

/’ . Then r/(s) S, Write for the image of ,/(s) in (R) and [e] for theconjugacy class of in . Then the map q -(,) induces a map

(11.3.1)

from the set of Z(()-equivalence classes [[qJ]] of admissible homomorphisms forH to the set of pairs ([],[e]), where [] is an equivalence class of admissiblehomomorphisms for G and [e] is a conjugacy class in (R) (note that if q0’ alsobelongs to [], then there is an isomorphism (R) (R),, canonical up to innerautomorphisms, and therefore the set of conjugacy classes in (R), is independentof ’ [1).


11.3.2. PROPOSITION. Let q be an elliptic admissible homomorphism for G, andlet be an element of (R). Write ((R)) for the centralizer of in (R). Then

I(%),1-’= I(R)l -I,

where the first sum is taken over (He s, 7) o and the second sum is taken over aset of representatives q for the Z(G)-equivalence classes of admissible homomor-phisms for n that map to ([p],[])under (11.3.1).

We will use the proposition when we discuss the stabilization of the traceformula. The remainder of this section will be devoted to a proof of theproposition.

11.3.3. Let A denote the group of continuous 1-cocycles of WF in Z()whose class in H I(WF, Z()) is locally trivial. The group A acts on the set ofadmissible homomorphisms q for H. Let (q) denote the orbit of q under A. Thegroup A also acts on the set of admissible homomorphisms p for G. Letdenote the orbit of p under A.

Let U denote the set of quadruples (S, Oo,,(q)) satisfying the followingconditions:

A0(a) s is a semisimple element of /Z(G), and O0 is an L-action of I" on Gsuch that (s, O) is an elliptic endoscopic datum for G, where O is the composition

I’ Aut(() Out(().(b) /j" LH---)LG is an L-homomorphism that extends the inclusion ---)

0where is G and LH is the semi-direct product of WF and H provided by O0.(c) +:WFH is an admissible homomorphism such that o q is elliptic

for G.Let V denote the set of triples (S, Oo, O satisfying (a) and (b). There is an

obvious map p: U V.Let Y denote the set of elliptic endoscopic data (s, #). There is an obvious map

g: V---) Y.Let X denote the set of pairs ((P),0, where is an elliptic admissible

homomorphism for G and (R) (note that (R) is independent of therepresentative of ()).We define a map f: U--)X by sending (S, Oo,,(q)) to (( o q),s), and we

define a map q"X Y by ending ((p),) to (,p), where p is the uniquehomomorphism F ---) Out(G) such that the diagram

0WFq

)Norm,(G)

^0r , o t(a. )commutes.The group /Z() acts on U, V,X, Y. The four maps we have defined are all


t/Z(()-maps and give us a commutative diagram

(11.3.3.1)u.f x

V >Y

It is not hard to check that this square is Cartesian (i.e., it makes U into theproduct of V and X over Y). The map g is surjective, since we are assuming thatGder is simply connected. Proposition 11.3.2 now follows from Lemma 11.4.1below.

11.4, Let G be an abstract group. Consider a Cartesian square (11.3.3.1) ofG-sets. Assume that g is surjective and that the stabilizer Gx of x in G is finite forall x X. Suppose that we are given, for each y Y, a subgroup H(y) of finiteindex in g__y For v V we write H(v) for H(g(v)) and use the bijectionP-l(v) "7-> 1;(g(v)) induced by f to transport the action of H(v) on q-(g(v))over top- (v).

11.4.1. LEMMA. For an), x X and an), cross section

composition V--> Y--> G \ Y we haveVo C V of the

v Vo ue Ux.v

where Ux,v n(v)\ u p- (v) f(u) G. x ).

We reduce immediately to the case in which V Y, U X, f idx, g id r,and G acts transitively on X and Y. Then V0 has only one element, and withoutloss of generality we may assume that V0 {y), where y q(x). Then theformula of the lemma is just a reflection of the decomposition

g

where g runs over a set of representatives for H(y)\Gy/Gx.11.4.2. To apply the lemma to the proof of the proposition, we need to

choose ,subgroups n(y) of finite index in ((/Z(t))y for y (s,p) Y. But(/Z(,G))y Aut(s,p)/Z(), since (s, p) isellipt (see 7.3), and we take H(y)to be H/Z(G). Let x (<q), ) X. The (G/Z(G)) ((R)), since is elliptic.Let v (s, P0, 0 V. Then p-l(v) can be identified with the set of (@) such that@ is an admissible homomorphism WF-CH and o @ is elliptic. The action ofH(v)-- I?I/Z(() on p-(v) is the obvious conjugation action. Let u (s, p0,,<>) p-(v). Then H(v)u (R), since the fact that/j @ is elliptic implies that

S C Z(G). We leave the verification of the remaining details to the reader.

12. Stable trace formula. This section is purely speculative. We will try tounderstand the stabilization of the contribution to the trace formula made by


automorphic representations corresponding to elliptic tempered admissiblehomomorphisms. It is impossible for us to prove anything without making anumber of assumptions, whose correctness seems impossible to verify at thepresent time. Nevertheless, the theory of automorphic representations is difficultenough that it is not without interest to guess how some of its parts fit together.In our discussion we will assume that the reader is familiar with Langlands’swork on the stable trace formula [L3].

Let F be a number field, and let G be a connected reductive group over F.Assume that Gde is simply connected, so that ---) can be extended to LH--)LGfor every endoscopic group H for G [L1]. Let 0Z be a closed subgroup of Z(A),where Z Z(G), such that oZ. Z(F) is closed and oZ. Z(F)\Z(A) is compact.Assume that oZ 1"Iv oZv Let X be a character of 0Z, trivial on oZ (3 Z(F). Weget L2x(G(F)\ G(A)), and we have a trace formula for suitable functions f 1Iv fvon G(A).

Let be a set of representatives for the isomorphism classes of ellipticendoscopic triples for G, and for each (H,s, q) choose an L-homomorphism/" CH---)CG that extends ,/" ---)(.Under the assumption that there exists, for each (H,s, r/) , a function fH

on H(A) with suitable orbital integrals, Langlands [L3] proves a formula

(12.1) Te(f) ,(G,H)STe(fH ).H,s,rl)

Here Te(f) is the elliptic regular part of the trace formula for f, and STe(f) isthe elliptic G-regular part of the stable trace formula for f.We would like to have the same result for the other side of the trace formula.

That is, we want to have

(12.2) Tc(f) ,(G,H)ST(f ),H,s,rl)

where To(f) is the tempered cuspidal part of the trace formula forf and STc(f)is the tempered G-cuspidal part of the trace formula for fH. In order to accountfor all of the tempered cuspidal automorphic representations of G, Langlands[L2] has suggested the existence of a group even larger than the Weil group WFof F. We will consider a variant LF of this conjectural group, and we suggest thatthis group be named the Langlands group. The group LF should be an extensionof WF by a compact group. It should satisfy axioms (Wl), (W2), (W3) of [T3] (butnot (W4), of course). For the local fields F it is known that LFt. should be:

(a) LF WF if v is archimedean,(b) LF WFt SU(2, Iq) if v is nonarchimedean.

There should be a natural bijection between the irreducible n-dimensionalcomplex representations of LF (resp. LFc and the cuspidal automorphicrepresentations of GLn(A (resp. irreducible admissible representations ofGLn(Fv)). There should be homomorphisms LF.---)L, canonical up to


conjugation by elements of LF. The definitions and results of Sections 10 and 11can be extended to LF (if one grants the properties above), since the properties ofWF that we used are shared by LF.Following Labesse-Langlands [L-L], we will guess that the tempered cuspidal

part of L2x(G(F)\ G(A)) is isomorphic to

(12.3) D m(, rr)- r,[1

where m(, r)= I(R)1- (, rr). This expression requires some explanation.The symbol [] stands for the equivalence class (10.4) of an admissiblehomomorphism : LFLG, and the first direct sum is taken over all equivalenceclasses of elliptic tempered such that the associated character X of Z(A) agreeswith X on oZ. Note that y, depends only on []. The symbol 1-I() stands for theL-packet of . These L-packets should be defined so that rr t)v rrv belongs toI-l() if and only if rrv I-l(v) for all v and % is Kv-spherical for almost all v(K 1-I K is some maximal compact subgroup of G(A)). We should have

(12.4) (e, rr) 1"I (ev, rrv),

where % is the image of under (R)(R). The local functions (,):(R) 1-I(v)---)13 are somewhat noncanonical, but the global function ( iscanonical. One expects that the complex-valued functionsare characters of nonzero (possibly reducible) finite dimensional complexrepresentations of (R)v" We will need to normalize the local functions ( ) sothat for almost all v we have (%, vrv) whenever rr is Kv-spherical. Assumingall of this, we see that m(rr, ) is a nonnegative integer.

Using (12.3), we get the following formula:

(12.5) T (f)[]

We also assume that

(12.6) STc(f)= l l -l (1,r>trr(f).[1 ,en()

Let (H,s,) . Recall that we have chosen an extension ,/’" LH---)LG of "q. Let’LF---)LH be a tempered admissible homomorphism, and let /’ o q. Thenthe functionfn should have the property that

(12.7) (1,r)trr(fn)= (,r)trr(f),r l-I(q) r l’I()

where s is the image of /(s) Sw in (R). Thus the stabilization (12.2) that we are


trying to obtain is equivalent to the equality of

(H,s,))6 [ff] r H()/’ )

and

<, r>tr r(f)

(12.9) l(R)l-’ , (,r>trr(f).(q) r l-I(qo)

The class [’ o tp] is not determined by the class [p]. This is why we introducedZ(t)-equivalence (11.1). We write [[tp]] for the Z(()-equivalence class ofWhether or not rl’ o tp is elliptic should only depend on [tp]. In (12.8) we sum onlyover [tp] such that r/’ o tp is elliptic. Using Proposition 11.2.1 and Theorem 8.3.1,we can rewrite (12.8) as

(12.10) h(H,s,,))-’ I1-’ <,r>trcr(f).(H,s,r/)6 [[p]] r H(,/’ +)

Let be an admissible homomorphism LF-->LG such that [] appears in (12.9),and let e (R). Let [e] denote the conjugacy class of in , and let ((R)),denote the centralizer of e in (R). The contribution of [],[] to (12.9) is theproduct of

and

<, r>trr(f)rH()

(12.11)

The contribution of [],[e] to (12.10) is the product of

<, r>tr r(f)rH()

and

(12.12) E h(H,s, 1)-’ E IG,I ’,(H,s,n)

where the last sum is taken over those [[pl] that map to ([9],[e]) under (11.3.1).However, Proposition 11.3.2 tells us that the (12.11) and (12.12) are equal.Therefore (12.9) and (12.10) are equal. This finishes the stabilization of thetempered cuspidal part of the trace formula for G.

[BI

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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WASHINGTON, SEATTLE, WASHINGTON 98195

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