Arthur's multiplicity formula for GSp 4 and restriction to Sp 4
aa r X i v : . [ m a t h . N T ] J u l ARTHUR’S MULTIPLICITY FORMULA FOR GSp ANDRESTRICTION TO Sp TOBY GEE AND OLIVIER TA¨IBI
Abstract.
We prove the classification of discrete automorphic representationsof
GSp explained in [Art04], as well as a compatibility between the localLanglands correspondences for GSp and Sp . Contents
1. Introduction 12. Arthur’s classification 53. Construction of missing local Arthur packets for
GSpin GSpin Sp and GSpin Introduction
GSp ∼ = GSpin .Later, in [Art13] he proved this classification for quasi-split special orthogonal andsymplectic groups of arbitrary rank, but now with trivial similitude factor. Theclassification stated in [Art04] is important for applications of the Langlands pro-gram to arithmetic. In particular, it is used in [Mok14] to associate Galois repre-sentations to Hilbert–Siegel modular forms, and these Galois representations havebeen used to prove modularity lifting theorems relating to abelian surfaces, for ex-ample in [BCGP]. It is therefore desirable to have an unconditional proof of thisclassification. While it is expected that the methods of [Art13] could be used tohandle GSpin groups, the proofs involve a very complicated induction, which evenin the case of
GSpin would involve the use of groups of much higher rank, sothere does not seem to be any way to give a (short) direct proof of the classificationof [Art04] by following the arguments of [Art13]. T.G. was supported in part by a Leverhulme Prize, EPSRC grant EP/L025485/1, ERC StartingGrant 306326, and a Royal Society Wolfson Research Merit Award. O.T. was supported in partby ERC Starting Grant 306326.
In this paper, we fill this gap in the literature by giving a proof of the classificationannounced in [Art04]. We also prove some new results concerning the compatibilityof the local Langlands correspondences for Sp and GSp . While, like Arthur, ourmain technique is the stable (twisted) trace formula, and we make substantial useof the results of [Art04] for the group Sp , we also rely on a number of additionalingredients that are only available in the particular case of GSp ; in particular, wecrucially use: • the exterior square functoriality for GL proved in [Kim03] (and completedin [Hen09]); • the results of [GT11a]: the local Langlands correspondence for GSp (es-tablished using theta correspondences), and the generic transfer to GSp (with local-global compatibility at all places) for essentially self dual cusp-idal automorphic representations of GL of symplectic type; • the results of [CG15], which check the compatibility of the local Langlandscorrespondence of [GT11b] with the predicted twisted endoscopic characterrelations of [Art04] in the tempered case.We now briefly explain the strategy of our proof, and the structure of the paper.We begin in Section 2 with a precise statement of the results of [Art13] and oftheir conjectural extension to GSpin groups. Roughly speaking, these statementsconsist of:(1) An assignment of global parameters (formal sums of essentially self-dualdiscrete automorphic representations of GL n ) to discrete automorphic rep-resentations of classical groups.(2) A description of packets of local representations in terms of local versionsof the global parameters (which in particular gives the local Langlandscorrespondence for classical groups).(3) A multiplicity formula, precisely describing which elements of global pack-ets are automorphic, and the multiplicities with which they appear in thediscrete spectrum.In Arthur’s work these statements are all proved together as part of a complicatedinduction, but in this paper (which of course uses Arthur’s results for Sp ) we areable to prove the first two statements independently, and then use them as inputsto the proof of the third statement.In section 3 we study the local packets. In the tempered case, the work hasalready been done in [CG15], and by again using that [Art13] has taken care of thecases where the similitude character is a square, we are reduced to constructing thelocal packets in two special non-tempered cases. We do this “by hand”, followingthe much more general results proved in [MW06] and [AMR15].As a consequence of the stabilisation of the twisted trace formula [MW16a,MW16b], we can apply the twisted trace formula for GL × GL to associatea global parameter to any discrete automorphic representation of GSpin (whichis a twisted endoscopic group for GL × GL endowed with the automorphism g t g − ). We recall the details of this twisted trace formula in section 4, whichwe hope can serve as an introduction to the results of [MW16a, MW16b] for thereader not already familiar with them. In section 5 we briefly recall results aboutthe restriction of representations to subgroups, which we apply to the case of re-striction from GSp to Sp . RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp In section 6 we show that the global parameter associated to a discrete automor-phic representation of
GSp by the stable twisted trace formula is of the form pre-dicted by Arthur, by making use of the symplectic/orthogonal alternative for GL and GL , the (known) description of automorphic representations of quasi-splitinner forms of GSpin in terms of Asai representations, and the tensor productfunctoriality GL × GL → GL of [Ram00]. We also make use of [Art13] in twoways: if the similitude character is a square, then by twisting we can immediatelyreduce to the results of [Art13]. If the similitude character is not a square, then thepossibilities for the parameter are somewhat constrained, and we are able to furtherconstrain them by using the fact that by restricting to Sp and applying the resultsof [Art13], we know the possible forms of the exterior square of the parameter.In section 7, we prove the global multiplicity formula in much the same wayas [Art13], as a consequence of the stable (twisted) trace formulas for GL × GL and GSpin , together with the twisted endoscopic character relations already es-tablished.Finally, in section 8 we show that the local Langlands correspondences for Sp established in [GT10] and [Art13] coincide. The correspondence of [GT10] wasconstructed by restricting the correspondence for GSp of [GT11a] to Sp , which bythe results of [CG15] is characterised using twisted endoscopy for GL × GL . Thecorrespondence for Sp obtained in [Art13] is characterised using twisted endoscopyfor GL .In the discrete case we prove this by a global argument, by realising the parameteras a local factor of a cuspidal automorphic representation, and using the exteriorsquare functoriality for GL of [Kim03] and [Hen09]. In the remaining cases theparameter arises via parabolic induction, and we are able to treat it by hand.We are also able to use these arguments to give a precise description in terms ofArthur parameters of the restrictions to Sp of irreducible admissible temperedrepresentations of GSp over a p -adic field.We end this introduction with a small disclosure, and a comparison to otherwork. While we have said that the results of this paper are unconditional, theyare only as unconditional as the results of [Art13] and [MW16a, MW16b]. Inparticular, they depend on cases of the twisted weighted fundamental lemma thatwere announced in [CL10], but whose proofs have not yet appeared in print, as wellas on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time ofwriting have not appeared publicly.The strategy of using restriction to compare the representation theory of reduc-tive groups related by a central isogeny is not a new one; indeed it goes back atleast as far to the comparison of GL and SL in [LL79]. In the case of symplecticgroups, there is the paper [GT10] mentioned above; while this does not make anyuse of trace formula techniques, we use some of its ideas in Section 8, when wecompare the different constructions of the local Langlands correspondence.More recently, there is the work of Xu, in particular [Xu17, Xu16], which alsobuilds on [Art13], using the groups GSp n and GO n where we use the groups GSpin n (of course, these cases overlap for GSp ). However, the emphasis of Xu’s work israther different, and is aimed at constructing “coarse L -packets” (which in the caseof GSp are unions of L -packets lying over a common L -packet for Sp ), and prov-ing a multiplicity formula for automorphic representations grouped together in asimilar way. Xu’s results are more general than ours in that they apply to groups TOBY GEE AND OLIVIER TA¨IBI of arbitrary rank, but are less precise in the special case of
GSp , and our proofsare independent.1.2. Acknowledgements.
We would like to thank George Boxer, Frank Calegari,Ga¨etan Chenevier, Matthew Emerton and Wee Teck Gan for helpful conversations.1.3.
Notation and conventions.
Algebraic groups.
We will use the boldface notation G for an algebraic groupover a local field or a number field, and we use the Roman version G for reductivegroups over C , or their complex points. Thus for example if F is a number field,we will write GL n for the general linear group over F , with Langlands dual group d GL n = GL n , which we will also sometimes write as d GL n = GL n ( C ).For a real connected reductive group G , write g = C ⊗ R Lie ( G ( R )), and let K bea maximal compact subgroup of G ( R ). When working adelically we will sometimesabusively call ( g , K )-modules “representations of G ( R )”. This should cause noconfusion as we will mostly be considering unitary representations in this globalsetting (see [Wal88, Theorem 3.4.11], [War72, Theorem 4.4.6.6]), and distinguishbetween ( g , K )-modules and representations of G ( R ) when considering non-unitaryrepresentations.1.3.2. The local Langlands correspondence. If K is a field of characteristic zero thenwe write Gal K for its absolute Galois group Gal( K/K ). If K is a local or globalfield of characteristic zero, then we write W K for its Weil group. If K is a localfield of characteristic zero, then we write WD K for its Weil–Deligne group, whichis W K if K is Archimedean, and W K × SU(2) otherwise.If π is an irreducible admissible representation of GL N ( F ) ( F local) or GL N ( A F )( F global), then ω π will denote its central character. We write rec for the localLanglands correspondence normalised as in [HT01], so that if F is a local field ofcharacteristic zero, then rec( π ) is an N -dimensional representation of WD F . If F is p -adic then for this normalisation a uniformiser of F corresponds to the geometric Frobenius automorphism.1.3.3.
The discrete spectrum.
Let G be a connected reductive group over a numberfield F . Write G ( A F ) = (cid:8) g ∈ G ( A F ) (cid:12)(cid:12) ∀ β ∈ X ∗ ( G ) Gal F , | β ( g ) | = 1 (cid:9) , so that G ( F ) \ G ( A F ) has finite measure. Let A G be the biggest central splittorus in Res F/ Q ( G ), and let A G be the vector group A G ( R ) . Then G ( A F ) = G ( A F ) × A G . We write A ( G ) = A ( G ( F ) A G \ G ( A F )) = A ( G ( F ) \ G ( A F ) )for the space of square integrable automorphic forms. This decomposes discretely,i.e. it is canonically the direct sum, over the countable set Π disc ( G ) of discreteautomorphic representations π for G , of isotypical components A ( G ) π which have finite length.If χ G is a character of A G , we could more generally consider the space of χ G -equivariant square integrable automorphic forms A ( G ) = A ( G ( F ) \ G ( A F ) , χ G ) . RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp Since we can reduce to the case χ G = 1 considered above by twisting, we willalmost never use this more general definition.2. Arthur’s classification
GSpin groups.
We now recall the results announced in [Art04] for
GSp ,as well as those for Sp proved in [Art13]. In fact, for convenience we begin byrecalling the conjectural extension of Arthur’s results to GSpin groups of arbitraryrank, and then explain what is proved in [Art13].We work with the following quasi-split groups over a local or global field F ofcharacteristic zero: • The split groups
GSpin n +1 . • The split groups Sp n × GL . • The quasi-split groups
GSpin α n .Here we can define the groups GSpin n +1 and GSpin α n as follows. If α ∈ F × / ( F × ) , we have the quasi-split special orthogonal group SO α n , which is de-fined as the special orthogonal group of the quadratic space given by the directsum of ( n −
1) hyperbolic planes and the plane F [ X ] / ( X − α ) equipped with thequadratic form equal to the norm. We have the spin double cover0 → µ → Spin α n → SO α n → , and we set GSpin α n := ( Spin α n × GL ) /µ where µ is embedded diagonally. Note that GSpin α n is split if and only if α = 1.We define the split group GSpin n +1 in the same way. This expedient definitionis of course equivalent to the usual, more geometric one (see [Knu91, Ch. IV, § g, λ ) λ . It is convenient to let GSpin = GSpin = GL .The corresponding dual groups are as follows. G b GGSpin n +1 GSp n ( C ) Sp n × GL GSO n +1 ( C ) = SO n +1 ( C ) × GL ( C ) GSpin α n GSO n ( C )Let µ : GL → Z ( G ) be dual to the surjective “similitude factor” morphism b µ : b G → GL ( C ). Note that in the case G = Sp n × GL , µ : GL → Z ( G ) isthe map x (1 , x ), and it is the only case where it is not injective. Moreover theimage of µ is Z ( G ) except in the case G = GSpin α .We set L G = b G ⋊ W F , where the action of W F on b G is trivial except in thecase that G = GSpin α n with α = 1, in which case the action of W F factorsthrough Gal( F ( √ α/F ) = { , σ } , and σ acts by outer conjugation on GSO n . Moreprecisely, in this case we identify b G ⋊ Gal( F ( √ α/F ) with GO n ( C ) as follows: ifSO n is obtained from the symmetric bilinear form B on C e ⊕ · · · ⊕ C e n givenby B ( e i , e j ) = δ i, n +1 − j , then 1 ⋊ σ is the element of O n ( C ) which interchanges e n and e n +1 and fixes the other e i .We have the standard representationStd G : L G → GL N ( C ) × GL ( C ) , TOBY GEE AND OLIVIER TA¨IBI where N = N ( b G ) = 2 n if G = GSpin α n or G = GSpin n +1 , and N = 2 n + 1if G = Sp n × GL . In the first two cases the representation is trivial on W F , andis given by the product of the standard N -dimensional representation of b G andthe similitude character. In the final case it is given by the product of the naturalinclusion O n +1 ( C ) ⊂ GL n +1 ( C ) and the identity on GL ( C ). The standard rep-resentation realises G as an elliptic twisted endoscopic subgroup of GL N × GL ,as we will explain below.We set sign( G ) = 1 if G = GSpin α n or GL × Sp n , and sign( G ) = − G = GSpin n +1 (equivalently, we set sign( G ) = − b G is symplectic).2.2. Levi subgroups and dual embeddings.
As in our description of the dualgroup SO n above, we may realise the groups SO α n and SO n +1 as matrix groupsusing an antidiagonal symmetric bilinear form (block antidiagonal with a 2 × SO α n with α = 1). Let B be the Borel subgroup consisting ofupper diagonal elements (block upper diagonal in the case of SO α n ). Let T be thesubgroup of diagonal (resp. block diagonal) elements. This Borel pair being given,we can now consider standard parabolic subgroups and standard Levi subgroups.(We recall that we only need to consider Levi subgroups up to conjugacy; indeed,given a Levi subgroup L of a parabolic P , we obtain an L -embedding L L ֒ → L G ,which up to b G -conjugacy is independent of the choice of P .)It is well-known that the standard Levi subgroups are parametrised as follows.Consider ordered partitions n = P ri =1 n i + m , where m > G = SO α n with α = 1, and m = 1 if G = SO n . Such a partition yields a standard Levi subgroup L of G isomorphic to GL n × · · · × GL n r × G m where G m is a group of the sametype as G of absolute rank m . Explicitly, an isomorphism is given by(2.2.1) ( g , . . . , g r , h ) diag (cid:0) g , . . . , g r , h, S − n r t g − r S n r , . . . , S − n t g − S n (cid:1) , where S n denotes the antidiagonal n × n matrix with 1’s along the antidiagonal.For G = SO n and m = 0 and n r >
1, there are two standard Levi subgroups of G corresponding to the partition n = P ri =1 n i : the one described above and its imageunder the outer automorphism of G . This completes the parameterisation of allstandard Levi subgroups of special orthogonal groups. Standard Levi subgroups of Sp and GSp admit a similar description.Denote G ′ = GSpin α n if G = SO α n and G ′ = GSpin n +1 if G = SO n +1 .Parabolic subgroups of G ′ correspond bijectively to parabolic subgroups of G , andthe same goes for their Levi subgroups. Consider L as above, and let L ′ be itspreimage in G ′ . An easy root-theoretic exercise shows that there exists a uniqueisomorphism GL n × · · · × GL n r × G ′ m ≃ L ′ lifting (2.2.1) such that for any 1 ≤ i ≤ r , the composition of the induced embed-ding of GL n i in G ′ with the spinor norm G ′ → GL is det. Alternatively, theembeddings GL n i → GSpin n i can be constructed geometrically using the defi-nition of GSpin groups via Clifford algebras (see [Knu91, Ch. IV, § L ′ easily follows. The conjugacy class of L ′ under G ′ ( F )is determined by the multi-set { n , . . . , n r } .Dually, this corresponds to identifying the dual Levi subgroup b L of b G = GSO n or GSp n with GL n × · · · × GL n r × d G ′ m via the block diagonal embedding:( g , . . . , g r , h ) diag (cid:0) g , . . . , g r , h, b µ ( h ) S n r t g − r S − n r , . . . , b µ ( h ) S n t g − S − n (cid:1) RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp Endoscopic groups and transfer.
Before stating the conjectural param-eterisation, we need to recall some definitions and results about endoscopy. Webegin by recalling that an endoscopic datum for a connected reductive group G over a local field F is a tuple ( H , H , s, ξ ) (almost) as in [KS99, § • H is a quasi-split connected reductive group over F , • ξ : b H → b G is a continuous embedding, • H is a closed subgroup of L G which surjects onto W F with kernel ξ ( b H ),such that the induced outer action of W F on ξ ( b H ) coincides with the usualone on b H transported by ξ , and such that there exists a continuous splitting W F → H , • and s ∈ b G is a semisimple element whose connected centraliser in b G is ξ ( b H )and such that the map W F → b G induced by h ∈ H 7→ shs − h − takesvalues in Z ( b G ) and is trivial in H ( W F , Z ( b G )).Note that we modified the notation slightly: in [KS99] H is not contained in L G and instead ξ is an embedding of H in L G . We choose this convention because incontrast to the general case where z -extensions are a necessary complication, in allcases that we will consider the embedding ξ : b H → b G will admit a (non-unique)extension as L ξ : L H → L G . Of particular importance are the elliptic endoscopicdata, which are those for which the identity component of ξ ( Z ( b H ) Gal F ) is containedin Z ( b G ).For G belonging to the three families introduced in Section 2.1 the groups H willbe products whose factors are either general linear groups, or quotients by GL ofproducts of groups of the form considered in Section 2.1. At this level of generalitywe content ourselves with specifying the group H , for each equivalence class ofnon-trivial ( s Z ( b G )) elliptic endoscopic datum of G . They are as follows. • If G = GSpin n +1 , then H = ( GSpin a +1 × GSpin b +1 ) / GL with a + b = n , ab = 0, and the quotient is by GL embedded as z ( µ ( z ) , µ ( z ) − ). • If G = Sp n × GL , then H = ( Sp a × GL × GSpin α b ) / GL ∼ = Sp a × SO α b × GL , where a + b = n , ab = 0, and α = 1 if b = 1. • If G = GSpin α n , then H = ( GSpin β a × GSpin γ b ) / GL , where a + b = n , βγ = α , β = 1 if a = 1, and γ = 1 if b = 1.In this paper we will also need one case of twisted endoscopy. Recall [MW16a, § I.1.1] that if F is a local field of characteristic zero (in the paper we will also take F to be a number field), and G is a connected reductive group defined over F , thena twisted space e G for G is an algebraic variety over F which is simultaneously aleft and right torsor for G . Consider the split group GL n × GL over a local orglobal field of characteristic zero F , and let θ be the automorphism of GL n × GL given by θ ( g, x ) = ( J t g − J − , x det g ), where J is the antidiagonal matrix withalternating entries − , , − , . . . (that is, J ij = ( − i δ i,n +1 − j ). The reason fordefining θ in this way is that it fixes the usual pinning E of G consisting of the upper-triangular Borel subgroup, the diagonal maximal torus and (( δ i,a δ j,a +1 ) i,j ) ≤ a ≤ n − .Then e G = GL n × GL ⋊ { θ } is a twisted space which happens to be a connectedcomponent of the non-connected reductive group GL n × GL ⋊ { , θ } .There is a notion of a twisted endoscopic datum ( H , H , s, ξ ) for the pair ( GL n × GL , θ ), for which we again refer to [KS99, § ω there to be equal to 1, aswe will throughout this paper, and using the same convention as above for ξ ) and TOBY GEE AND OLIVIER TA¨IBI [MW16b, § VI.3.1]. We will explicitly describe all of the elliptic twisted endoscopicdata (up to isomorphism) in the case n = 4 in Section 4.2 below. In the presentsection we shall only need the fact that if H is one of the groups considered inSection 2.1 (denoted G there), then H is part of an elliptic twisted endoscopicsubgroup of ( GL N ( b H ) × GL , θ ). Remark 2.3.1.
The definitions in [MW16a] and [MW16b], using twisted spacesrather than a fixed automorphism of G (not fixing a base point), are more generalthan those used in most of [KS99], due to an assumption in [KS99] that is onlyremoved in (5.4) there. Note in particular the notion of twisted endoscopic space[MW16a, § I.1.7]. In the cases considered in this paper, where e G is either G (stan-dard endoscopy) or G ⋊ θ where θ ∈ Aut( G ) fixes a pinning E of G (defined over F , i.e. stable under Gal F ), this notion simplifies and we are under the assumptionof [KS99, (3.1)]. Namely, the torsor Z ( e G , E ) under Z ( G ) := Z ( G ) / (1 − θ ) Z ( G )defined in [MW16a, I.1.2] is trivial with a natural base point 1 ⋊ θ , and so for any en-doscopic datum ( H , H , ˜ s, ξ ) for e G , the twisted endoscopic space e H := H × Z ( G ) Z ( e G )is trivial with natural base point 1 ⋊ θ , where θ now acts trivially on H . For thisreason we can ignore twisted endoscopic spaces in the rest of the paper, and simplyconsider endoscopic groups as in most of [KS99].We now very briefly recall the notion of (geometric) transfer in the setting ofendoscopy. Suppose that F is a local field of characteristic zero, and that ( G , e G )belongs to one of the four families of twisted spaces considered above, that is G = GSpin n +1 , Sp n × GL or G = GSpin α n with e G = G , or G = GL n × GL with e G = G ⋊ θ . Given an endoscopic datum e = ( H , H , s, ξ ) for e G , and a choiceof an extension L ξ : L H → L G of the embedding ξ , Kottwitz and Shelstad defined transfer factors in [KS99], that is a function on the set of matching pairs of stronglyregular semisimple G ( F )-conjugacy classes in e G ( F ) and regular semisimple stableconjugacy classes in H ( F ). In general such a function is only canonical up to C × ,but in all cases considered in this paper there is a Whittaker datum w = ( U , λ )of G fixed by an element of e G ( F ) and this provides [KS99, § e , L ξ, w ]. To be more precise we use thetransfer factors called ∆ D in [KS], corresponding to the normalisation of the localLanglands correspondence identifying uniformizers to geometric Frobenii. In allcases of ordinary endoscopy one can choose an arbitrary Whittaker datum of G .In the case that G = GSpin α n , there is an outer automorphism δ of G whichpreserves the Whittaker datum. This δ can be chosen to have order 2 and be inducedby an element of the orthogonal group having determinant −
1; if F is Archimedean,for simplicity we can and do choose the maximal compact subgroup K of G ( F ) tobe δ -stable.In this paper we are particularly interested in the case G = GSpin . ByHilbert’s theorem 90 the morphism GSpin n +1 ( F ) → SO n +1 ( F ) is surjective,so GSpin n +1 is of adjoint type and there is up to conjugation by GSpin n +1 ( F )only one Whittaker datum in this case.For e G = ( GL n × GL ) ⋊ θ we choose for U the subgroup of unipotent uppertriangular matrices in GL n and λ (( g i,j ) i,j ) = κ ( P n − i =1 g i,i +1 ) where κ : F → S isa non-trivial continuous character. This is the Whittaker datum associated to E and κ . This Whittaker datum is fixed by θ (this is the reason for the choice of thisparticular θ in its G ( F )-orbit). RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp Definition 2.3.2. If F is p -adic, then we let H ( e G ) denote the space of smoothcompactly supported distributions on e G ( F ) with C -coefficients. Then H ( e G ) =lim −→ K H ( e G ( F ) //K ) where the limit is over compact open subgroups of G ( F ) and H ( e G ( F ) //K ) is the subspace of bi- K -invariant distributions. If F is Archimedean,then we fix a maximal compact subgroup K of G ( F ), and write H ( e G ) for thealgebra of bi- K -finite smooth compactly supported distributions on e G ( F ) with C -coefficients.Under convolution, the space H ( e G ) is a bi- H ( G )-module, where H ( G ) is theusual (non-twisted) Hecke algebra for G .In the case that G = GSpin α n , we let e H ( G ) denote the subalgebra of H ( G )consisting of δ -stable distributions, and otherwise we set e H ( G ) = H ( G ) and δ = 1.An admissible twisted representation of e G is by definition a pair ( π, e π ) consistingof an admissible representation π of G ( F ) and a map e π from e G to the automorphismgroup of the underlying vector space of π , which satisfies e π ( gγg ′ ) = π ( g ) e π ( γ ) π ( g ′ )for all g, g ′ ∈ G ( F ), γ ∈ e G . (This is the special case ω = 1 of the notion of an ω -representation of a twisted space, which is defined in [MW16a].) If F = R or C there is an obvious notion of ( g , e K )-module where e K ⊂ G ( F ) is a torsor under K normalising K .We will consider (invariant) linear forms on e H ( e G ). In particular, for each ad-missible representation π of G ( F ), there is the linear formtr( π ( f ( g ) dg )) = tr Z G ( F ) f ( g ) π ( g ) dg ! . If F is Archimedean and π is an admissible ( g , e K )-module the action of e H ( e G ) is notobviously well-defined but it is so when π arises as the space of K -finite vectors ofan admissible Banach representation of e G ( F ), independently of the choice of thisrealisation (see [War72, p. 326, Theorem 4.5.5.2]). In this paper all ( g , e K )-moduleswill naturally arise in this way, even with “Hilbert” instead of “Banach”, althoughnot all of them will be unitary.We write I ( e G ) for the quotient of e H ( e G ) by the subspace of those distribu-tions f ( g ) dg with the property that for any semisimple strongly regular γ ∈ e G ( F ),the orbital integral O γ ( f ( g ) dg ) vanishes. There is a natural topology on I ( e G ): see[MW16a, I.5.2]. Similarly, we write SI ( e G ) for the quotient by the subspace forwhich the stable orbital integrals SO γ ( f ( g ) dg ) vanish. We say that a continuouslinear form on e H ( G ) is stable if it descends to a linear form on SI ( e G ).Given an endoscopic datum ( H , H , s, ξ ) for e G , and our choice of Whittakerdatum, there is a notion of transfer from I ( e G ) to SI ( H ) (see [KS99, § § I.2.4 and IV.3.4]); this transfer is defined by the property that it relates the valuesof orbital integrals on e G to stable orbital integrals on H , using the transfer factorsrecalled above. Most importantly, this transfer exists ([Wal97], [Ngˆo10], [She12]).Dually, we may transfer stable continuous linear forms on e H ( H ) to continuous linearforms on e H ( G ). In the twisted case where e G = ( GL N × GL ) ⋊ θ over a p -adic field F , the cho-sen Whittaker datum yields a hyperspecial maximal compact subgroup K of G ( F )(see [CS80]), which is stable under θ , so it is natural to consider the hyperspecialsubspace (see [MW16a, § I.6]) e K = K ⋊ θ of e G ( F ). For any unramified endoscopicdatum ( H , H , ˜ s, ξ ) for e G (also defined in [MW16a, § I.6]), with the above trivialisa-tion of e H , the associated H ad ( F )-orbit of hyperspecial subspaces of e H is simply theobvious one, that is the set of K ′ ⋊ θ where K ′ is a hyperspecial maximal compactsubgroup of H ( F ).By the existence of transfer and [LMW15], [LW15] ([Hal95] in the case of stan-dard endoscopy), the twisted fundamental lemma is now known for all elements ofthe unramified Hecke algebra, with no assumption on the residual characteristic.We formulate it in our situation, which is slightly simpler than the general case bythe above remarks. Theorem 2.3.3.
Let e G be a twisted group over a p -adic field F belonging to oneof the four families introduced at the beginning of this section. Assume that G is unramified. Let ( H , H , ˜ s, ξ ) be an unramified endoscopic datum for e G . Choosean unramified L-embedding L ξ : L H → L G extending ξ . Let e K be the hyperspe-cial subspace of e G ( F ) associated to the chosen Whittaker datum for G . Let e K be the characteristic function of e K multiplied by the G ( F ) -invariant measure on e G ( F ) such that e K has volume . Let b : H ( G ( F v ) //K v ) → H ( H ( F v ) //K ′ v ) be themorphism dual to (cid:16) b H ⋊ Frob (cid:17) ss / b H − conj → (cid:16) b G ⋊ Frob (cid:17) ss / b G − conj via the Satake isomorphisms ( see [Bor79, § . Then for any f ∈ H ( G ( F v ) //K ) , b ( f ) is a transfer of f ∗ e K . Remark 2.3.4.
In the above setting, there is a natural notion of unramified twistedrepresentation: extend an unramified representation ( π, V ) of G ( F ) which is iso-morphic to its twist by e G ( F ) to a twisted representation by imposing that ˜ K actstrivially on V K .2.4. Local parameters.
Let F be a local field of characteristic zero. Let Ψ + ( G )denote the set of b G -conjugacy classes of continuous morphisms ψ : WD F × SL ( C ) → L G such that • the composite with the projection L G → W F is the natural projectionWD F × SL ( C ) → W F , • for any w ∈ WD F , ψ ( w ) is semisimple, and • the restriction ψ | SL ( C ) is algebraic.We let Ψ( G ) ⊂ Ψ + ( G ) be the subset of bounded parameters. By a standardargument (see for example the proof of [GT11a, Lem. 6.1]), the { , b δ } -orbit of aparameter ψ is determined by the data of the conjugacy class of Std G ◦ ψ . Let e Ψ( G )and e Ψ + ( G ) be the set of { , b δ } -orbits of parameters as above.For ψ ∈ Ψ + ( G ) let ϕ ψ be the Langlands parameter associated to ψ , that is ψ composed with the embedding w ∈ WD F (cid:16) w, diag( | w | / , | w | − / ) (cid:17) ∈ WD F × SL ( C ) . RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp We write C ψ for the centraliser of ψ in b G , S ψ = Z ( b G ) C ψ , and S ψ = π ( S ψ /Z ( b G )) , an abelian 2-group. We let S ∨ ψ = Hom( S ψ , C × ) be the character group of S ψ .Write s ψ for the image in C ψ of − ∈ SL ( C ).We can now formulate the conjectures on local Arthur packets in terms of endo-scopic transfer relations. Conjecture 2.4.1.
Let G = GSpin n +1 , Sp n × GL or GSpin α n . Then thereis a unique way to associate to each ( ψ ) ∈ e Ψ( G ) a multi-set Π ψ of { , δ } -orbits ofirreducible smooth unitary representations of G ( F ) , together with a map Π ψ → S ∨ ψ ,which we will denote by π
7→ h· , π i , such that the following properties hold.(1) Let π GL ψ be the representation of GL N ( b G ) ( F ) × GL ( F ) associated to (Std G ◦ ϕ ψ ) by the local Langlands correspondence for GL N ( b G ) × GL , and let e π GL ψ beits extension to (cid:16) GL N ( b G ) ( F ) × GL ( F ) (cid:17) ⋊ θ recalled in Section 3.2. Then P π ∈ Π ψ h s ψ , π i tr π is stable and its transfer to GL N ( b G ) ( F ) × GL ( F ) ⋊ θ is tr e π GL ψ , i.e. for any f ∈ I ( (cid:16) GL N ( b G ) ( F ) × GL ( F ) (cid:17) ⋊ θ ) having transfer f ′ ∈ SI ( G ) we have tr e π GL ψ ( f ) = X π ∈ Π ψ h s ψ , π i tr π ( f ′ ) . (2) Consider a semisimple s ∈ C ψ with image ¯ s in S ψ . The pair ( ψ, s ) deter-mines an endoscopic datum ( H , H , s, ξ ) for G ( with H = Cent ( s, b G ) ψ (WD F )) ,and if we fix an L-embedding L ξ : L H → L G extending ξ we obtain ψ ′ : WD F × SL ( C ) → L H such that ψ = L ξ ◦ ψ ′ . Then for any f ∈ I ( G ) with transfer f ′ ∈ SI ( H ) , we have: X π ∈ Π ψ h ¯ ss ψ , π i tr π ( f ) = X π ′ ∈ Π ψ ′ h s ψ ′ , π ′ i tr π ′ ( f ′ ) . (3) If ψ | SL ( C ) = 1 , then the elements of Π ψ are tempered and Π ψ is multiplicityfree, and the map Π ψ → S ∨ ψ is injective; if F is non-Archimedean, then itis bijective. Every tempered irreducible representation of G ( F ) belongs toexactly one such Π ψ . Remark 2.4.2.
Note that the uniqueness of the classification is clear from prop-erties (1) and (2) and Proposition 2.4.3 below, as irreducible representations aredetermined by their traces.
Proposition 2.4.3 (Arthur) . In the situation of Conjecture 2.4.1, the transfermap I ( ^ GL N ( b G ) × GL ) → SI ( G ) δ is surjective.Proof. This is [Art13, Cor. 2.1.2] slightly generalised from ^ GL N to ^ GL N × GL .Note that the general version of [Art13, Prop. 2.1.1] was later proved in [MW16a, § I.4.11] (see § IV.3.4 loc. cit. to extend to the Archimedean case with K -finiteness). (cid:3) Remark 2.4.4.
Part (3) of this conjecture gives the local Langlands correspon-dence for tempered representations of G ( F ) (up to outer conjugacy in case G = GSpin α n ). It can be extended to give the local Langlands correspondence forall local parameters ψ ∈ Ψ + ( G ) with ψ | SL ( C ) = 1; indeed if Conjecture 2.4.1 isknown for all G , then a version can be deduced for Ψ + ( G ) using the Langlandsclassification (see [Lan89], [Sil78] and [SZ14]). Remark 2.4.5.
In the case where F is Archimedean and for an arbitrary reduc-tive group the local Langlands correspondence was established by Langlands andShelstad (see [She10], [She08]). Compatibility with twisted endoscopy was provedby Mezo [Mez16] (under a minor assumption, see (3.10) loc. cit., which is satisfiedin all cases considered in the present article) up to a constant which a priori mightdepend on the parameter (see [AMR15, Annexe C]). Remark 2.4.6. If F is p -adic and G is unramified over F , then there is a unique G ( F )-conjugacy class of hyperspecial maximal compact subgroups of G ( F ) whichis compatible with the Whittaker datum fixed above (in the sense of [CS80]), andwe will say that a representation of G ( F ) is unramified if it is unramified withrespect to a subgroup in this conjugacy class.If ψ ∈ e Ψ + ( G ) and ψ | WD F is unramified, then assuming the conjecture thepacket Ψ ψ contains a unique unramified (orbit of) representation. It has Satakeparameter ϕ ψ (up to outer conjugation if G = GSpin α n ) and corresponds to thetrivial character on S ψ . This follows from the fundamental lemma (Theorem 2.3.3). Remark 2.4.7.
By [Mœg11] if F is p -adic and the conjecture holds then the packetsΠ ψ are sets rather than multi-sets.2.5. Global parameters and the conjectural multiplicity formula.
Nowlet F be a number field, and fix a continuous unitary character χ : A × F /F × → C × .If π is a cuspidal automorphic representation of GL N /F such that π ∨ ⊗ χ ∼ = π , thenwe say that π is χ -self dual. Note that this implies that ω π = χ N (so in particularif N is odd, then χ = ( ω π χ (1 − N ) / ) is a square).If π is χ -self dual and S is a big enough set of places of F then precisely one ofthe L -functions L S ( s, χ − ⊗ V ( π )) and L S ( s, χ − ⊗ Sym ( π )) has a pole at s = 1,and this pole is simple (see [Sha97]). In the former case we say that ( π, χ ) is ofsymplectic type, and set sign( π, χ ) = −
1, and in the latter we say that it is oforthogonal type, and we set sign( π, χ ) = 1.We write Ψ( ^ GL N × GL , χ ) for the set of formal unordered sums ψ = ⊞ i π i [ d i ],where the π i are χ -self dual automorphic representations for GL N i /F and the d i ≥ ( C )), with the property that P i N i d i = N . We refer to sucha sum as a parameter , and say that it is discrete if the (isomorphism classes of)pairs ( π i , d i ) are pairwise distinct. Remark 2.5.1. (1) By the main result of [MW89], a discrete automorphic representation π of GL N /F with π ∨ ⊗ χ ∼ = π gives rise to an element of Ψ( ^ GL N × GL , χ ).Indeed, there is a natural bijection between such representations π and theelements of Ψ( ^ GL N × GL , χ ) of the form π [ d ] (that is, the elements wherethe formal sum consists of a single term). We will use this bijection withoutfurther comment below. RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp (2) The set of formal parameters Ψ( ^ GL N × GL , χ ) that we consider does notcontain all non-discrete χ -self-dual parameters, for example those contain-ing a summand of the form π ⊞ ( χ ⊗ π ∨ ) for a non- χ -self-dual cuspidalautomorphic representation π for GL m . Our ad hoc definition will turnout to be convenient when we will consider the discrete part of (the stabil-isation of) trace formulas. Definition 2.5.2.
Let G = GSpin n +1 , Sp n × GL or GSpin α n over F . We let e Ψ disc ( G , χ ) be the subset of e Ψ( GL N ( G ) , χ ) given by those ψ = ⊞ i π i [ d i ] with theproperties that • ψ is discrete, • for each i , we have sign( π i , χ ) = ( − d i − sign( G ), • if G = GSpin α n , then χ − n Q i ω d i π i is the quadratic character correspondingto the extension F α /F .(Conditions analogous to this last bullet point could be formulated for the othergroups G , but in fact they are conjecturally automatically satisfied.)If G = GSpin α n we also let Ψ disc ( G , χ ) = e Ψ disc ( G , χ ). The reason for writing e Ψ in the case of even
GSpin groups is that this set only sees orbits of (substitutesfor) Arthur-Langlands parameters under outer conjugation.As a particular case of the above definition, for π a cuspidal automorphic rep-resentation for GL N /F such that χ ⊗ π ∨ ≃ π there is a unique group G as abovesuch that N ( b G ) = N and π [1] ∈ e Ψ disc ( G ). Conjecture 2.5.3.
For π and G as above and for each place v of F , the repre-sentation (rec( π v ) , rec( χ v )) factors through Std G : L G → GL N ( b G ) ( C ) × GL ( C ) , sothat we can regard ( π v , χ v ) as an element of e Ψ + ( G ( F v )) . Remark 2.5.4. (1) This conjecture is the analogue of [Art13, Theorem 1.4.1] (reformulatedusing Theorem 1.5.3 loc. cit.). In particular it holds for G = Sp n × GL .(2) Since we do not know the generalised Ramanujan conjecture for GL n , anddo not wish to assume it, we can at present only hope to establish that thelocal parameters ψ v are elements of e Ψ + ( G F v ); they are, however, expectedto be elements of e Ψ( G F v ).Given a global parameter ψ ∈ e Ψ disc ( G , χ ), we define groups C ψ , S ψ , S ψ as fol-lows. For each i , there is a unique group G i of the kind we are considering forwhich π i ∈ e Ψ disc ( G i , χ ). We let L ψ denote the fibre product of the L G i over W F .Then there is a map ˙ ψ : L ψ × SL ( C ) → L G such that Std G ◦ ˙ ψ is conjugateto ⊕ i Std G i ⊗ ν d i , where ν d i is the irreducible representation of SL ( C ) of dimen-sion d i . The map ˙ ψ is well-defined up to the action of Aut( L G ). We let C ψ be thecentraliser of ˙ ψ , and similarly define S ψ and S ψ .For each finite place v , under Conjecture 2.5.3 (applied to the π i ’s) we may forma local Arthur-Langlands parameter ψ v : WD F v × SL ( C ) → L ψ . Composing with˙ ψ , we obtain ψ v ∈ e Ψ + ( G F v ). The composition of ψ v with Std G is given by • χ v on the GL factor, • the direct sum of the representations ϕ π i ,v ⊗ ν d i on the GL N ( b G ) factor,where ϕ π i ,v = rec( π i,v ). Conjecture 2.5.6 below makes precise the expectation that the elements of thecorresponding multi-sets Π ψ v of Conjecture 2.4.1 are the local factors of the discreteautomorphic representations of G with multiplier χ . Before stating it, we need tointroduce some more notation and terminology.For each place v of F , write e H ( G v ) for the Hecke algebra defined after Definition2.3.2, and write e H ( G ) for the restricted tensor product of the e H ( G v ). AssumingConjecture 2.5.3, we have an obvious map S ψ → S ψ v for each v , and we canassociate to ψ a global packet (a multi-set) of representations of e H ( G ): e Π ψ := {⊗ ′ v π v : π v ∈ Π ψ v with π v unramified for all but finitely many v } . For each π ∈ e Π ψ , we have the associated character on S ψ , h x, π i := Y v h x v , π v i (note that by Remark 2.4.6, we have h· , π v i = 1 for all but finitely many v , so thisproduct makes sense).Associated to each ψ is a character ε ψ : S ψ → {± } which can be definedexplicitly in terms of symplectic ε -factors. In the case χ = 1 this is defined in [Art13,Theorem 1.5.2], and this definition can be extended to the case of general χ withoutdifficulty. Since we will only need the case G = GSpin in this paper, and in thiscase the characters ε ψ are given explicitly in [Art04] and are recalled below inRemark 6.1.4, we do not give the general definition here. Definition 2.5.5. e Π ψ ( ε ψ ) is the subset of e Π ψ consisting of those elements forwhich h· , π i = ε ψ .This is the correct definition only because the groups S ψ v are all abelian.Recall that we have fixed a maximal compact subgroup K ∞ of G ( F ⊗ Q R ) inSection 2.3. Let g = C ⊗ R Lie ( G ( F ⊗ Q R )). We write A ( G ( F ) \ G ( A F ) , χ ) forthe space of χ -equivariant (where the action of A × F /F × is via µ ) square integrableautomorphic forms on G ( F ) \ G ( A F ). It decomposes discretely under the actionof G ( A F,f ) × ( g , K ∞ ). Conjecture 2.5.6.
Assume that Conjectures 2.4.1 and 2.5.3 hold. Then there isan isomorphism of e H ( G ) -modules A ( G ( F ) \ G ( A F ) , χ ) ∼ = M ψ ∈ e Ψ disc ( G ,χ ) m ψ M π ∈ e Π ψ ( ε ψ ) π, where m ψ = 1 unless G = GSpin α n , in which case m ψ = 2 if and only if each N i is even. The results of [Art13] . As we have already remarked, the conjectures aboveare all proved in [Art13] in the case that χ = 1. As we now explain, the casethat χ is a square follows immediately by a twisting argument. The main resultsof this paper are a proof of Conjectures 2.4.1 (Theorem 3.1.1) and 2.5.6 (Theorem7.4.1) in the case that G = GSpin ∼ = GSp for general χ . Conjecture 2.5.3 for G = GSpin is a consequence of [GT11a], see Proposition 7.3.1. The case that χ isa square will be a key ingredient in our arguments, as if χ is not a square, then it iseasy to see that there are considerably fewer possibilities for the parameters ψ , andthis will reduce the number of ad hoc arguments that we need to make. Moreover RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp in the remaining cases, the statements pertaining to local tempered representationsare covered by [CG15]. Theorem 2.6.1 (Arthur) . If χ = η is a square, then Conjectures 2.4.1, 2.5.3and 2.5.6 hold.Proof. Given a χ -self dual cuspidal automorphic representation π , the twist π ⊗ η − is self dual. Similarly, we may twist the local parameters by the restriction to W F v ofthe character corresponding to η − , and we can also twist representations of G ( F )and G ( F v ) by η − . All of the conjectures are easily seen to be compatible with thesetwists, so we reduce to the case χ = 1. In this case, representations of GSpin n +1 ,(resp. GSpin α n , resp. Sp n × GL ) with trivial similitude factor (recall that thiswas defined in Section 2.1 as the composition of the central character with µ ) areequivalent to representations of SO n +1 , (resp. representations of SO α n , resp. pairsgiven by a representation of Sp n and a character of GL of order 1 or 2), so theconjectures are equivalent to the main results of [Art13]. (cid:3) In particular, since in the case G = Sp n × GL the character χ is always asquare, Theorem 2.6.1 always holds in this case.2.7. Low rank groups. If N ( b G ) ≤ N = 1 the results are tautological.(2) if N = 2 then G = GSpin or G = GSpin α . In the first case G ≃ GL and the results are also tautological. In the second case where G = GSpin α ≃ Res F ( √ α ) /F ( GL ) we are easily reduced to the well-known The-orem 2.7.1 below, the symplectic/orthogonal alternative for GL .(3) If N = 3 then G = Sp × GL and we are reduced to a special caseof Theorem 2.6.1. Note that the local Langlands correspondence and themultiplicity formula in this case go back to Labesse–Langlands [LL79] and[Ram00]. Theorem 2.7.1.
Let π be a χ -self dual cuspidal automorphic representation of GL . Then either(1) χ = ω π , and L S ( s, V ( π ) ⊗ χ − ) has a pole at s = 1 ; or(2) ω π χ − is the quadratic character given by some quadratic extension E/F , π is the automorphic induction of a character of A × E /E × which is not fixedby the non-trivial element of Gal(
E/F ) , and L S ( s, Sym ( π ) ⊗ χ − ) has apole at s = 1 .Proof. Certainly L S ( s, V ( π ) ⊗ χ − ) = L S ( s, ω π χ − ) has a pole at s = 1 if and onlyif χ = ω π . So if L S ( s, Sym ( π ) ⊗ χ − ) has a pole at s = 1, we see that ω π χ − is anon-trivial quadratic character corresponding to an extension E/F . Since we alwayshave π ∨ ⊗ ω π ∼ = π , this implies that π ∼ = π ⊗ ( ω π χ − ), and it follows (see [Lan80,end of § π is the automorphic induction of a character of A × E /E × which isnot fixed by the non-trivial element of Gal( E/F ). (cid:3) The local Langlands correspondence for GSp . Let F be a p -adic field.The local Langlands correspondence for GSp ( F ) was established in [GT11a], butwas characterised by relations with γ -factors, rather than endoscopic characterrelations. The necessary endoscopic character relations were then proved in [CG15].In particular, we have: Theorem 2.8.1 (Chan–Gan) . If F is a p -adic field then Conjecture 2.4.1 holdsfor GSpin and parameters ψ which are trivial on SL ( C ) , i.e. tempered Langlandsparameters.Proof. Parts (1) and (2) of Conjecture 2.4.1 are an immediate consequence of themain theorem of [CG15] (note that bounded parameters are automatically generic,in the sense that their adjoint L -functions are holomorphic at s = 1). Part (3) thenfollows from the main theorem of [GT11a]. (cid:3) Remark 2.8.2.
Recall from Remark 2.4.5 that over an Archimedean field the localLanglands correspondence and (ordinary) endoscopic character relations are knownin complete generality, and the twisted endoscopic character relations are knownup to a constant (which might depend on the parameter).If F is Archimedean and ψ is a tempered and non discrete Langlands parameterfor GSpin , then the twisted endoscopic character relation was verified in [CG15, § ψ .3. Construction of missing local Arthur packets for
GSpin Local packets.
Let F be a local field of characteristic zero. In this sectionwe complete the proof of the following theorem, which completes the proof of Con-jecture 2.4.1 for GSpin . Theorem 3.1.1.
Let ψ : WD F × SL → GSp be an element of Ψ( GSpin ) .Then there is a unique multi-set Π ψ of irreducible smooth unitary representationsof GSpin ( F ) , together with a map Π ψ → S ∨ ψ , which we will simply denote by π
7→ h· , π i , such that the following holds:(1) Let π Γ ψ be the representation of Γ ( F ) associated to Std
GSpin ◦ ϕ ψ by the lo-cal Langlands correspondence, and let π e Γ ψ be its extension to e Γ ( F ) ( Whittaker-normalised as explained in Section 3.2 ) . Then the linear form P π ∈ Π ψ h s ψ , π i tr π on I ( GSpin ( F )) is stable and its transfer to e Γ is tr π e Γ ψ .(2) Consider a semisimple s ∈ Cent ( ψ, GSp ) , and denote by ¯ s its image in S ψ .The pair ( ψ, s ) determines an endoscopic datum ( H , H , s, ξ ) for GSpin ,as well as ψ ′ : WD F × SL → b H such that ψ = ξ ◦ ψ ′ . Then for any f ∈ I ( GSpin ( F )) we have X π ∈ Π ψ h ¯ ss ψ , π i tr π ( f ) = X π ′ ∈ Π ψ ′ h s ψ ′ , π ′ i tr π ′ ( f ′ ) . Note that in the second point H is either GSpin or a quotient of a product ofgeneral linear groups by a split torus, and so Π ψ ′ is well-defined. In the latter caseit is a singleton and S ψ ′ is trivial.As we recalled above (Theorems 2.6.1, 2.8.1 and Remark 2.8.2) this theorem isalready known in the following cases: • if b µ ◦ ψ is a square, • if F is p -adic and ψ | SL = 1, • if F is Archimedean, ψ | SL and ψ is not discrete. RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp We will prove the case where F is Archimedean, ψ tempered discrete and χ nota square later in Proposition 7.2.1, since we will use a global argument using thestabilisation of the trace formula.This section is devoted to the proof of Theorem 3.1.1 in the remaining cases,where ψ | SL is not trivial and b µ ◦ ψ is not a square. It is easy to see thatStd GSpin ◦ ψ ≃ ( ϕ [2] , χ ), where ϕ : WD F → GL is χ -self-dual of orthogonaltype. Then ϕ factors through W F and det ϕ/ ( b µ ◦ ψ ) has order 1 or 2. There aretwo cases to consider.(1) If ϕ is irreducible then det ϕ/ ( b µ ◦ ψ ) has order 2. Let E/F be the correspond-ing quadratic extension and denote c the non-trivial element of Gal( E/F ).We have ϕ ≃ Ind
E/F µ for a character µ : E × → C × such that µ c = µ and µ | F × = χ . Then Cent ( ψ, GSp ) = Z (GSp ) and so we simply have toproduce Π ψ = { π } such that tr π transfers to the trace of π e Γ ψ .(2) If ϕ is reducible then ϕ = η ⊕ η with η η = χ and η = η . ThenCent ( ψ, GSp ) = { diag( u I , u I ) } and so we are led to define Π ψ = { Ind
GSpin L ((rec( η ) ◦ det) ⊗ rec( χ )) } where L ≃ GL × GSpin . Then thesecond point in Theorem 3.1.1 is automatically satisfied (see [CG15, § F is p -adic, real, or complex (in which case only the second case occurs). Beforedoing so, we recall some material on Whittaker normalisations.3.2. Whittaker normalisation for general linear groups.
In this section F denotes a local field of characteristic zero, G = GL n × GL over F and e G = G ⋊ θ . Following [MW06, § §
8] we briefly recall the Whittakernormalisation of extensions to e G ( F ) of irreducible representations of G ( F ) fixedby θ . Recall that we have fixed a θ -stable Whittaker datum ( U , λ ) for G . If F is Archimedean for simplicity we choose the maximal compact subgroup K to beO n ( F ) × {± } (resp. U( n ) × U(1)) if F is real (resp. complex), so that θ ( K ) = K .First consider the case of essentially tempered representations. Let π be anessentially tempered (in particular, essentially unitary) irreducible representationof G ( F ). By [Sha74] there exists a continuous Whittaker functional Ω for π . If F is p -adic this is just an element of the algebraic dual of the space π K of smoothvectors. If F is Archimedean this is a continuous functional on the space π ∞ ofsmooth vectors for the topology defined by seminorms as in [Sha74, p. 183]. Nowif π is fixed by θ , define ˜ π ( θ ) as the unique element A ∈ Isom( π, π θ ) such thatΩ ◦ A = Ω. This does not depend on the choice of Ω. So we have an extension ˜ π of π to a representation of e G ( F ), well-defined using the Whittaker datum ( U , λ ).Next consider representations parabolically induced from a θ -stable parabolicsubgroup. Fix the usual (diagonal) split maximal torus T of G , as well as theusual (upper triangular) Borel subgroup B = TU of G . Both are θ -stable. Let w G be the longest element of the Weyl group W ( T , G ). Let P = MN be a standardparabolic subgroup of G , with standard Levi subgroup M ⊃ T . Assume that P is θ -stable, which means that M = ( GL n × · · · × GL n r ) × GL (block diagonal)with n i = n r +1 − i for all i . Let σ be an irreducible admissible representation of M ( F ) fixed by θ , that is σ ≃ ( σ ⊗ · · · ⊗ σ r ) ⊗ χ with (det ◦ χ ) ⊗ σ ∨ i ≃ σ r +1 − i forall i . Let D M be the largest split torus which is a quotient of M , so that we have a canonical isogeny A M → D M . In the present case we have a natural identification D M ≃ GL r × GL via the determinants GL n i → GL . For ν ∈ X ∗ ( D M ) ⊗ C inducing a character of M ( F ), consider the parabolically induced (normalised)representation π ν := Ind G ( F ) P ( F ) σ ⊗ ν . We also assume that ν = ( ν , . . . , ν r , ν ) is fixedby θ , i.e. ν i + ν r +1 − i = ν for all i . Let w M be the longest element of W ( T , M ) (for B ∩ M ) and w = w G w M . Let P − = MN − be the parabolic subgroup of G oppositeto P with respect to M , and let P ′ = M ′ N ′ = w P − w − = w G P − w − G be thestandard parabolic subgroup conjugated to P − . Choose a lift ˜ w of w in N G ( F ) ( T ).Let λ ˜ w M : ( M ∩ U )( F ) → S be the generic character defined by λ ˜ w M ( u ) = λ ( ˜ wu ˜ w − ).Assume that the space Hom ( M ∩ U )( F ) ( σ, λ ˜ w M ) of Whittaker functionals for σ withrespect to λ ˜ w M is non-zero and thus one-dimensional, and fix a basis Ω σ of this line.In the p -adic case, according to a theorem of Rodier ([Rod73], [CS80], explained in[Sha10, § U ( F ) (Ind G ( F ) P ( F ) ( σ ⊗ ν ) , λ ) also has dimensionone. A basis Ω π ν can be made explicit: for f in the space of Ind GP σ ⊗ ν whosesupport is contained in the big cell P ( F ) w − U ( F ),(3.2.1) Ω π ν ( f ) := Z N ′ ( F ) Ω σ ( f ( ˜ w − n )) λ ( n ) − dn is well-defined (the integrand is smooth and compactly supported). For arbitrary f the same formula holds with N ′ ( F ) replaced by large enough open compactsubgroup which depends on f but not on ν (as usual realising the vector spaceunderlying Ind G ( F ) P ( F ) σ ⊗ ν independently of ν by restriction to K ), so that ν Ω π ν ( f ) is holomorphic.The Archimedean case is more subtle, since the notion of Whittaker functionalrequires a topology on the underlying space of the representation to be well-behaved(it is not defined directly on ( g , K )-modules). So in this case one considers thesmooth parabolically induced representation π ν := Ind GP ( σ ∞ ⊗ ν ), whose subspace π ν,K of K -finite vectors is naturally isomorphic to the ( g , K )-module algebraicallyinduced from σ M ( F ) ∩ K (see [BW00, § III.7]). Assume that the central character of σ is unitary. Then the integral (3.2.1) is absolutely convergent for ν ∈ X ∗ ( D M ) ⊗ C satisfying(3.2.2) ∀ α ∈ Φ( T , N ) , h α ∨ , ℜ ν i > , and extends analytically to X ∗ ( D M ) ⊗ C ([Sha10, Theorem 3.6.4]). The proof ofTheorem 3.6.7 in [Sha10] also shows uniqueness (up to a scalar) of a Whittakerfunctional for Ind GP ( σ ∞ ⊗ ν ) (note that the argument for uniqueness only involvesthe Jordan–H¨older factors of a principal series representation, and so one mayreplace P by another parabolic subgroup of G admitting M as a Levi factor andsuch that the opposite of (3.2.2) is satisfied, so that any generic subquotient ofInd GP ( σ ⊗ ν ) appears as a quotient).We can now treat the p -adic and Archimedean cases together. Assume that ν ischosen so that End G ( F ) ( π ν ) = C . This is the case if the central character of σ isunitary and ν satisfies (3.2.2) (this follows from the fact that π ν then has a uniqueirreducible quotient which occurs with multiplicity one in its composition series), orif − ν satisfies (3.2.2) ( π ν then has a unique irreducible subrepresentation). Thenone can define the action of θ on π ν to be the unique A θ ∈ End( π ν ) such that A θ ◦ π ν ( g ) = π ν ( θ ( g )) ◦ A θ for all g ∈ G ( F ) and Ω π ν ◦ A = Ω π ν . This can be made RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp more explicit in the case at hand, see [MW06, § A θ does notdepend on the choice of ˜ w made above.For this definition we followed [AMR15, § π ν coincides with the extension defined by Arthur in [Art13, § § π of G ( F ) (ad-missible ( g , K )-module in the Archimedean case). By the Langlands classification([Lan89, Lemmas 3.14 and 4.2], [Sil78], [BW00, Chapter IV]), π is the unique ir-reducible quotient of Ind GP ( σ ⊗ ν ) (resp. unique irreducible subrepresentation ofInd GP − ( σ ⊗ ν )) for ν ∈ X ∗ ( D M ) ⊗ C satisfying (3.2.2), with σ tempered (in partic-ular, with unitary central character) and the pair ( P , σ ⊗ ν ) is well-defined up toconjugation. These two realisations of π as quotient (resp. subrepresentation) of aparabolically induced representation give two canonical extensions of π to e G , by theabove. In fact these two canonical extensions coincide: consider the compositionInd GP ( σ ⊗ ν ) → π → Ind GP − ( σ ⊗ ν )which is clearly non-zero. From the properties of these induced representationsmentioned above it follows that dim Hom G ( F ) (Ind GP ( σ ⊗ ν ) , Ind GP − ( σ ⊗ ν )) ≤ ν ). But this operator varies an-alytically if we vary ν , and generically it is an isomorphism between irreducibleparabolically induced representations, thus generically it intertwines the two A θ ’s,and by continuity this also holds for the original ν .3.3. Proof of Theorem 3.1.1.
We now prove Theorem 3.1.1 in the cases describedat the end of Section 3.1.
Proof in the first case for
F p -adic.
The proof is a very special case of the gener-alisation of [MW06, Th´eor`eme 4.7.1] to essentially self-dual representations. Seealso [Mœg06].Let ρ be the supercuspidal representation of GL ( F ) such that rec( ρ ) = ϕ . Then χ ⊗ ρ ∨ ≃ ρ . We will give an ad hoc definition of Π ψ , using special cases of resultsof [MW06] to check compatibility with twisted endoscopy for GL × GL . In[MW06] Mœglin and Waldspurger consider self-dual parameters, and we will arguethat their arguments extend to the case at hand without substantial modification,the essential input being compatibility of local Langlands for GSpin for twistedendoscopy (and the same for GSpin and GSpin , which is trivial).Let ∆ be the diagonal embedding SU(2) ֒ → SU(2) × SL ( C ), so that ψ ◦ ∆ isthe essentially tempered Langlands parameter obtained by tensoring ϕ with the 2-dimensional irreducible representation of the factor SU(2) of WD F . Then Cent ( ψ ◦ ∆ , GSp ) = Z (GSp ), and so Π ψ ◦ ∆ (as defined by Gan–Takeda in [GT11a]) consistsof a single irreducible discrete series representation π ψ ◦ ∆ of GSpin ( F ). Let P bethe standard parabolic subgroup of GSpin with Levi subgroup L ≃ GL × GSpin (conventions as in Section 2.2). Then Jac P ( π ψ ◦ ∆ ) = ρ | det | / ⊗ χ where Jac de-notes the normalised Jacquet module. We briefly recall the proof. Let π GL ψ ◦ ∆ bethe (discrete series) representation of GL ( F ) corresponding to pr Std ◦ ψ ◦ ∆ :WD F → GL ( C ). Denoting by P GL the upper block triangular parabolic subgroupof GL with Levi subgroup GL × GL , it is well-known that Jac P GL (cid:16) π GL ψ ◦ ∆ (cid:17) = ρ | det | / ⊗ ρ | det | − / . Let π e Γ ψ ◦ ∆ be the Whittaker-normalised (see Section 3.2 or[MW06, § π GL ψ ◦ ∆ ⊗ χ to e Γ ( F ). By (iii) in the main theorem of [CG15]we have that tr π e Γ ψ ◦ ∆ is a transfer of tr π ψ ◦ ∆ . The parabolic subgroup P GL × GL of Γ is stable under θ , write ˜ P = ( P GL × GL ) ⋊ θ . By (an obvious generalisationof) [MW06, Lemme 4.2.1], tr Jac ˜ P ( π e Γ ψ ◦ ∆ ) is a transfer of tr Jac P ( π ψ ◦ ∆ ), and thusJac P ( π ψ ◦ ∆ ) = ρ | det | / ⊗ χ . By Frobenius reciprocity, π ψ ◦ ∆ is naturally a subrep-resentation of Ind GSpin P (cid:0) ρ | det | / ⊗ χ (cid:1) . By [BZ77, Theorem 2.8] this parabolicinduction has length ≤ π ψ ◦ ∆ ֒ → Ind
GSpin P (cid:16) ρ | det | − / ⊗ χ (cid:17) is an irreducible Langlands quotient which we denote π ψ . We let Π ψ = { π ψ } . SinceCent ( ψ, GSp ( C )) = C × , we only have to check the twisted endoscopic characterrelation (Theorem 3.1.1 (1)). Following [MW06], this will be a consequence ofcomparing the short exact sequence(3.3.1) 0 → π ψ ◦ ∆ → Ind
GSpin P (cid:16) ρ | det | / ⊗ χ (cid:17) → π ψ → e Γ .We have a short exact sequence of representations of Γ ( F ) = GL ( F ) × GL ( F ):(3.3.2) 0 → π GL ψ ◦ ∆ ⊗ χ → E ( π GL ψ ◦ ∆ ) ⊗ χ → π GL ψ ⊗ χ → π GL ψ ◦ ∆ to get a resolution of π GL ψ ◦ ∆ by sums of standard modules except possibly for the lastterm, which is defined as a cokernel and shown to be irreducible with Langlandsparameter ( ψ ◦ ∆) ♯ = ψ (the general definition of ψ ♯ is given in [MW06, § E ( π GL ψ ◦ ∆ ) := Ind GL P GL (Jac P GL ( π ψ ◦ ∆ )) ≃ Ind GL P GL (cid:16) ρ | det | / ⊗ ρ | det | − / (cid:17) and in the present case Mœglin and Waldspurger’s resolution does not involve anynon-trivial “proj”, so that the resolution actually goes back to [Aub95], [SS97].Following Mœglin and Waldspurger one can extend π GL ψ ◦ ∆ ⊗ χ from Γ ( F ) to Γ + ( F )by choosing an action of θ (see [MW06, §§ θ MW . Theresolution (3.3.2) inherits an action of θ by functoriality (see [MW06, § π GL ψ ⊗ χ happens to coincide with θ MW (see[MW06, Lemma 3.2.2], in which we have j ( ψ ) = 1 and so β ( ψ ◦ ∆ , ρ, ≤ d ) = +1).Another way to choose an extension of π GL ψ ◦ ∆ ⊗ χ (resp. π GL ψ ⊗ χ ) to θ is to useWhittaker functionals and the Langlands classification as we recalled in Section 3.2.Denote the resulting actions of θ by θ W . In general θ W and θ MW differ by a sign,but here fortunately θ W = θ MW on both π GL ψ ◦ ∆ ⊗ χ and π GL ψ ⊗ χ (a special case of[MW06, Prop. 5.4.1]). Thus we have a well-defined extension(3.3.3) 0 → (cid:0) π GL ψ ◦ ∆ ⊗ χ (cid:1) + → (cid:0) E ( π GL ψ ◦ ∆ ) ⊗ χ (cid:1) + → (cid:0) π GL ψ ⊗ χ (cid:1) + → Γ + ( F ). The trace of the left term is known to be the transfer of tr π ψ ◦ ∆ .By compatibility of stable transfer with Jacquet modules [MW06, Lemme 4.2.1] andparabolic induction (a consequence of the explicit formula for parabolic induction([vD72], [Clo84], [Lem10, § RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp transfer of the middle term of (3.3.1). So we can conclude that tr (cid:16) π GL ψ ⊗ χ (cid:17) + isthe transfer of tr π ψ . (cid:3) Proof in the second case for p -adic F . This is similar to the previous case but now ϕ : W F → GL ( C ) is reducible and so it defines a principal series representationof GL ( F v ). Write ϕ ≃ rec( η ) ⊕ rec( η ), so that χ = η η . As explained abovewe can assume that η = η . Define π ψ = Ind GSpin P (( η ◦ det) ⊗ χ ) where thestandard parabolic subgroup P has Levi L ≃ GL × GSpin and Π ψ = { π ψ } . Therepresentation π ψ is certainly irreducible (see [Mœg11, § ψ = { π ψ } to meanthat Π ψ is the multi-set of constituents of π ψ .Consider the parabolic induction for GL × GL (3.3.4) π Γ ψ := Ind GL P GL (( η ◦ det) ⊗ ( η ◦ det)) ⊗ χ where P GL is the standard parabolic subgroup of GL with Levi GL × GL .The twisted representation π e Γ ψ of e Γ ( F ) obtained from (3.3.4) using the canonicalaction of θ (defined as in [MW06, § π ψ , by compatibility of parabolic induction with transfer. This is almostthe twisted endoscopic character relation, but again we need to be careful with thedefinition of Whittaker normalisation. The Whittaker-normalised action of θ on π Γ ψ is obtained by realising it as the Langlands quotient of(3.3.5) Ind GL B GL (cid:16) η | · | / ⊗ η | · | / ⊗ η | · | − / ⊗ η | · | − / (cid:17) ⊗ χ where B GL is the standard Borel subgroup of GL , which coincides with the canoni-cal action of θ on this parabolic induction by (the obvious generalisation of) [MW06,Lemme 5.2.1].Let us sketch the proof of the fact that these two actions of θ on π Γ ψ coincide.It will be convenient to denote σ × · · · × σ r for the parabolic induction (using thestandard parabolic) of an admissible representation σ ⊗ · · ·⊗ σ r of GL n ( F ) × · · ·× GL n r ( F ) to GL n + ··· + n r ( F ). Recall that for any s ∈ C the parabolic induction η | · | / s × η | · | − / − s is irreducible by [BZ76, Theorem 3], since the assumptionthat χ = η η is not a square implies that η | O × F = η | O × F . The intertwiningoperator I s : η | · | / s × η | · | − / − s −→ η | · | − / − s × η | · | / s defined by the usual integral formula for ℜ ( s ) ≫
0, is rational in q − s (where q isthe cardinality of the residue field of F ) by [Wal03, Th´eor`eme IV.1.1], and so thereis a polynomial r ( s ) in q − s such that r ( s ) I s is well-defined and non-zero for any s ,and therefore an isomorphism. It induces an isomorphism I s, norm : η |·| / × η |·| / s × η |·| − / − s × η |·| − / → η |·| / × η |·| − / − s × η |·| / s × η |·| − / . Denote π ,s (resp. π ,s ) the LHS (resp. RHS). Since η | · | − / = χ/ (cid:0) η | · | / (cid:1) and η | · | − / − s = χ/ (cid:0) η | · | / s (cid:1) , there is a canonical extension of π ,s ⊗ χ to Γ + ( F )(see [MW06, § θ this canonical action of θ on the space of π ,s ⊗ χ (one can easily check that it does not depend on s ), so that for s = 0 we recoverthe Whittaker normalisation on (3.3.5). The irreducible representation (cid:16)(cid:16) η | · | / × η | · | − / − s (cid:17) ⊗ (cid:16) η | · | / s × η | · | − / (cid:17)(cid:17) ⊗ χ of the θ -stable parabolic subgroup P × GL of Γ is also fixed by θ , and so π ,s ⊗ χ also admits a canonical extension to e Γ ( F ). Denote θ this canonical action of θ on the space of π ,s ⊗ χ , which for s = 0 recovers the canonical action on thequotient (3.3.4). An easy computation that we skip shows that for ℜ ( s ) ≫ I s, norm ◦ θ = θ ◦ I s, norm , and the case of an arbitrary s ∈ C follows by analyticcontinuation. (cid:3) Proof in the first case for F = R . This is similar to the first case for F a p -adicfield except we now follow arguments of [AMR15]. For a ∈ Z ≥ let I a be thetempered Langlands parameter W R → GL ( C ) obtained by inducing the character z ( z/ ¯ z ) a := ( z/ | z | ) a of C × . Up to twisting we can assume that ϕ = I a with a > χ equal to the sign character sign of W R . Let π GL ψ be theirreducible unitary representation of GL ( R ) associated to ϕ ψ . Let χ : GL ( R ) →{± } be the sign character, so that χ ⊗ ( π GL ψ ) ∨ ≃ π GL ψ . As in the p -adic case wehave the Whittaker-normalised extension π e Γ ψ of π Γ ψ := π GL ψ ⊗ χ .We have a (short) resolution from [Joh84] (see [AMR15, § GL n and parameters I w [ n ] for w ∈ Z > )0 → π GL ψ → π GL I a |·| − / × π GL I a |·| / → π GL I a +1 / × π GL I a − / → | · | is the norm character of W R (i.e. the square of the usual absolute valueon C × and | j | = 1) and we denoted parabolic induction for standard parabolicsubgroups of GL as in the p -adic case. In [AMR15, Lemme 9.9] only the firstcase occurs, so comparing normalisations (Whittaker and imposed by inductionin Johnson’s construction of the resolution) is particularly simple: we obtain theanalogue of [AMR15, Th´eor`eme 9.7] with A s = A + s . (cid:3) Proof in the second case for F = R or C . Up to twisting we can assume that ϕ ≃ ⊕ χ with χ = sign in the real case and χ ( z ) = ( z/ ¯ z ) a | z | it with a ∈ Z r Z and t ∈ R in the complex case. The proof is identical to the p -adic case and we do notrepeat the argument. Note that the complex case is the analogue of [MR15, Prop.6.5]. (cid:3) Stabilisation of the twisted trace formula
We now state the stabilisation of the twisted trace formula proved by Mœglinand Waldspurger in [MW16a], [MW16b] following the case of ordinary (i.e. non-twisted) endoscopy proved by Arthur in [Art02], [Art01], [Art03] (also following[Lan83], [Kot86], [Lab99], and of course [LW13]). We recall some of the definitionsneeded to state the stabilisation, and mention some simplifications occurring in thecases at hand.4.1.
The discrete part of the spectral side.
Consider a connected reductivegroup G over a number field F and an automorphism θ of G of finite order. Let e G = G ⋊ θ . Let A be a maximal split torus in G . We will only consider Levisubgroups of G which contain A . Let K = Q v K v be a good maximal compactsubgroup of G ( A F ) with respect to A as in [LW13, § P of G containing A .Following [MW16b, § X.5], let us recall the terms occurring in the discrete part ofthe spectral side of the twisted trace formula. To work with discrete automorphicspectra it is necessary to fix central characters (at least on a certain subgroup of
RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp the centre), and we follow [MW16b, § X.5.1]. We now elaborate on the notationfor the discrete automorphic spectrum introduced in Section 1.3.3. Recall that A G denotes the vector group A G ( R ) where A G is the biggest central split torus inRes F/ Q ( G ). Then G ( A F ) = G ( A F ) × A G , where G ( A F ) = (cid:8) g ∈ G ( A F ) (cid:12)(cid:12) ∀ β ∈ X ∗ ( G ) Gal F , | β ( g ) | = 1 (cid:9) , so that G ( F ) \ G ( A F ) has finite measure. Let A e G = A θ G . Then A G = (1 − θ )( A G ) × A e G .In the general definition of twisted endoscopy one considers a character ω of G ( A F );in all cases considered in this paper we have ω = 1. Mœglin and Waldspurger con-sider a character χ G of A G which is trivial on A e G and satisfies θ ( χ G ) = χ G ω | A G ;since we will always have ω = 1 in this paper, we will have χ G = 1.Let L be a Levi subgroup of G . Up to conjugating by G ( F ) we can assumethat L is the standard Levi subgroup of a standard parabolic subgroup P of G . There is a canonical splitting A L = A G × A GL (with A GL included in the de-rived subgroup of G ( F ⊗ Q R )), and we write χ G , L for the extension of χ G to A L such that χ G , L | A GL = 1. As remarked above in all cases considered in this pa-per we simply have χ G , L = 1. The space of square integrable automorphic forms A ( L ( F ) \ L ( A F ) , χ G , L ) decomposes discretely, i.e. it is canonically the direct sum,over the countable set Π disc ( L , χ G , L ) of discrete automorphic representations π L for L such that π L | A L = χ G , L , of isotypical components A ( L ( F ) \ L ( A F ) , χ G , L ) π L which have finite length. Denote by U P the unipotent radical of P . Recall [MW94, § I.2.17] the space A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) of smooth K -finite functions φ on U P ( A F ) L ( F ) \ G ( A F ) such that for any k ∈ K , x δ P ( x ) − / ( x ) φ ( xk )is an element of A ( L ( F ) \ L ( A F ) , χ G , L ). In other words, A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) = Ind G ( A F ) P ( A F ) (cid:0) A ( L ( F ) \ L ( A F ) , χ G , L ) (cid:1) K − fin . This space is endowed with the usual left action of H ( G ), which we will de-note by ρ GP . If π L is an irreducible admissible representation of L ( A F ) such that ω π L | A L = χ G , L , denote by A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) π L the sub- H ( G )-module of A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) consisting of functions φ such that for any k ∈ K , (cid:16) x δ P ( x ) − / ( x ) φ ( xk ) (cid:17) ∈ A ( L ( F ) \ L ( A F ) , χ G , L ) π L . Let W ( L , e G ) = N e G ( F ) ( L ) / L ( F ), where the action of e G ( F ) on G is the adjointaction coming from the definition of a twisted space [MW16a, § I.1.1]. For ˜ w ∈ W ( L , e G ) and f (˜ x ) d ˜ x ∈ H ( e G ), we have a map [MW16b, bottom of p. 1204] ρ e GP , ˜ w ( f ) : A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) −→ A ( U ˜ w ( P ) ( A F ) L ( F ) \ G ( A F ) , χ G , L ) ,φ g Z e G ( A F ) φ ( ˜ w − g ˜ x ) f (˜ x ) d ˜ x ! (4.1.1) and for f , f ∈ H ( G ) and f ∈ H ( e G ) we have ρ e GP , ˜ w ( f ∗ f ∗ f ) = ρ G ˜ w ( P ) ( f ) ◦ ρ e GP , ˜ w ( f ) ◦ ρ GP ( f ) . If π L is an irreducible admissible representation of L ( A F ) such that ω π L | A L = χ G , L , then for any f ∈ H ( e G ), ρ e GP , ˜ w ( f ) restricts to A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) π L −→ A ( U ˜ w ( P ) ( A F ) L ( F ) \ G ( A F ) , χ G , L ) ˜ w ( π L ) where ˜ w ( π L ) = π L ◦ Ad( ˜ w − ).By meromorphic continuation of the usual integral formula, there is an inter-twining operator M P | ˜ w ( P ) (0) : A (cid:0) U ˜ w ( P ) ( A F ) L ( F ) \ G ( A F ) , χ G , L (cid:1) → A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) . Since χ G , L is unitary, M P | ˜ w ( P ) is well-defined (i.e. holomorphic) at 0, and is infact unitary. Moreover for any irreducible admissible representation π L of L ( A F ), M P | ˜ w ( P ) (0) restricts to A ( U ˜ w ( P ) ( A F ) L ( F ) \ G ( A F ) , χ G , L ) π L −→ A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) π L . Therefore for f ∈ H ( e G ) the composition M P | ˜ w ( P ) (0) ◦ ρ e GP , ˜ w ( f ) maps A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L )to itself and restricts to A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) π L −→ A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) ˜ w ( π L ) . We can finally recall the contribution of L to the discrete part of the spectralside of the twisted trace formula for e G . For f ∈ H ( e G ), let I e G , L disc ( f ) = | W ( L , G ) | − X ˜ w ∈ W ( L , e G ) reg | det (cid:0) ˜ w − | A GL (cid:1) | − tr (cid:16) M P | ˜ w ( P ) (0) ◦ ρ e GP , ˜ w ( f ) (cid:17) where W ( L , e G ) reg is the set of ˜ w ∈ W ( L , e G ) such that ( a GL ) ˜ w = 0. As the no-tation suggests, I e G , L disc ( f ) only depends on f and the G ( F )-conjugacy class of L .In fact it depends on f only via its image in I ( e G ). The fact that the trace of M P | ˜ w ( P ) (0) ◦ ρ e GP , ˜ w ( f ) on A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) is well-defined and equalsthe absolutely convergent sum X π L ∈ Π disc ( L ,χ G , L )˜ w ( π L ) ≃ π L tr (cid:16) M P | ˜ w ( P ) (0) ◦ ρ e GP , ˜ w ( f ) (cid:12)(cid:12)(cid:12) A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) π L (cid:17) is a consequence of work of Finis, Lapid and M¨uller, as explained in [LW13, § § X.5.2 and X.5.3].The most interesting case is of course for L = G , since I e G , G disc ( f ) is simply thetrace of f on the discrete automorphic spectrum for G and χ G . We will recall belowthe refinement of discrete terms by infinitesimal character and Hecke eigenvaluesfollowing Arthur and Mœglin–Waldspurger, that allows one to forget about con-vergence issues and work with finite sums. But first we make explicit the condition˜ w ( π L ) ≃ π L in the cases at hand. RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp (1) For G = GL N × GL and a standard (i.e. block diagonal) Levi L ≃ (cid:16)Q k ≥ ( GL k ) n k (cid:17) × GL (where n k = 0 for almost all k and P k ≥ kn k = N ), there always exists an element of e G ( F ) normalising L (for example θ : g t g − ), and W ( L , G ) ≃ Q k ≥ S n k . For ˜ w = ( σ k ) k ≥ θ ∈ W ( L , e G ),˜ w is regular if and only if for every k ≥
1, the decomposition of σ k incycles only involves cycles of odd length. For such a regular ˜ w and if π = (cid:16)N k ≥ ( π k, ⊗ · · · ⊗ π k,n k ) (cid:17) ⊗ χ is an irreducible admissible representa-tion of L ( A F ), then ˜ w ( π ) ≃ π if and only if each π k,i satisfies π ∨ k,i ⊗ χ ≃ π k,i and for every k ≥
1, the isomorphism class of ( π k,i ) ≤ i ≤ n k is fixed by σ k .(2) In the non-twisted cases G = GSpin n +1 or GSpin α n , recall that in Sec-tion 2.2 we chose L ≃ Q k ≥ ( GL k ) n k × G m where m + P k ≥ kn k = n and G m is a GSpin group of the same type as G of absolute rank m . There isa natural embedding W ( L , G ) ֒ → Q k ≥ ( {± } n k ⋊ S n k ) which is surjectiveunless G = GSpin α n , m = 0, and there exists an odd k ≥ n k >
0, in which case it is of index two.An element w = (( ε k,i ) ≤ i ≤ n k ⋊ σ k ) k ≥ is regular if and only if for ev-ery k ≥ i . . . i r ) appearing in the decomposition of σ k , Q rj =1 ε k,i j = −
1. For such w ∈ W ( L , G ) reg and π L ≃ N k ≥ ( π k, ⊗ · · · ⊗ π k,n k ) ⊗ π G m an irreducible admissible representation of L ( A F ), we have w ( π L ) ≃ π L if and only(a) for every k ≥ ≤ i ≤ n k , π ∨ k,i ⊗ χ ≃ π k,i where χ : A × F → C × is π G m ◦ µ , and(b) for every k ≥ π k,i ) ≤ i ≤ n k is fixed by σ k .We now recall from [MW16b, p. 1212] the refinement of the discrete part of thespectral side of the twisted trace formula by infinitesimal characters (using Arthur’stheory of multipliers) and families of Satake parameters. Definition 4.1.2. (1) Let IC ( G ) be the set of semisimple conjugacy classes in the Lie algebra ofthe dual group (over C ) of Res F/ Q ( G ). This is the set where infinitesimalcharacters for irreducible representations of G ( F ⊗ Q R ) live. In the twistedcase let IC ( e G ) = IC ( G ) ˆ θ . For π ∞ an irreducible admissible representationof G ( F ⊗ Q R ), denote by ν ( π ∞ ) ∈ IC ( G ) its infinitesimal character.(2) Let S be a large enough (i.e. containing V ram as in [MW16b, § VI.1.1]) fi-nite set of places of F . Let F S S ( G ) = Q v S (cid:16) b G ⋊ Frob v (cid:17) ss / b G , and in thetwisted case let F S S ( e G ) = (cid:0) F S S ( G ) (cid:1) ˆ θ . Write also F S ( G ) = lim −→ S F S S ( G )and in the twisted case F S ( e G ) = lim −→ S F S S ( e G ). If π = ⊗ ′ v π v is an ir-reducible admissible representation of G ( A F ), we will write c ( π ) for theassociated element of F S ( G ) via the Satake isomorphisms.(3) For ν ∈ IC ( e G ), S as above, c S = ( c v ) v S ∈ F S S ( e G ), and L a Levisubgroup of G , let Π disc ( L , χ G , L ) ν,c S be the set of π L ∈ Π disc ( L , χ G , L ) suchthat the infinitesimal character of π L , ∞ maps to ν via Lie (cid:16) \ Res F/ Q ( L ) (cid:17) → Lie (cid:16) \ Res F/ Q ( G ) (cid:17) , and for every v S , π L ,v is unramified for K v and itsSatake parameter maps to c v via L L → L G . For f ∈ N v ∈ S H ( e G ( F v )), let I e G , L disc ,ν,c S ( f ) = | W ( L , G ) | − X ˜ w ∈ W ( L , e G ) reg det (cid:0) ˜ w − | A GL (cid:1) | − X π L ∈ Π disc ( L ,χ G , L ) ν,cS ˜ w ( π L ) ≃ π L tr π L ( f ) , where we writetr π L ( f ) = (cid:16) M P | ˜ w ( P ) (0) ◦ ρ e GP , ˜ w ( f ) (cid:12)(cid:12)(cid:12) A ( U P ( A F ) L ( F ) \ G ( A F ) , χ G , L ) π L (cid:17) . Finally let(4.1.3) I e G disc ,ν,c S ( f ) = X L I e G , L disc ,ν,c S ( f )where the sum is over G ( F )-conjugacy classes of Levi subgroups of G .Seeing this as a sum over triples ( L , ˜ w, π L ), all but finitely many terms vanish.Indeed, if we fix ν , S , c S and an idempotent e of N v ∈ Sv ∤ ∞ H ( G ( F v )), then there is afinite set Υ( ν, S, c S , e ) of triples ( L , ˜ w, π L ) such that for any f ∈ N v ∈ S H ( e G ( F v ))for which e ∗ f = f ∗ e = f , the terms corresponding to ( L , ˜ w, π L ) Υ( ν, S, c S , e )in the double sum defining I e G disc ,ν,c S ( f ) all vanish. Remark 4.1.4. (1) By [JS81] and [MW89], taking the image in
F S ( GL N )is injective on formal sums of elements of Π disc ( GL n i , χ ) (note that it isessential that all of the summands are χ self-dual for the same character χ ).For this reason we will often identify such formal sums and their image.(2) In [MW16b] Mœglin–Waldspurger multiply (4.1.3) by j ( e G ) − := | det(1 − θ |A G / A e G ) | − , but this factor is also present in ι ( e G , H ) with their defini-tion. Definition 4.1.5. (1) We will say that c S ∈ F S ( e G ) occurs in I e G , L disc if thereexists ν ∈ IC ( e G ) and f ∈ H ( e G ) such that up to enlarging S we have I e G , L disc ,ν,c S ( f ) = 0.(2) Let D be an induced central torus in G , so that there is a dual morphism L G → L D . For c S ∈ F S ( e G ) occurring in I e G , L disc we define the central charac-ter of c S to be the (unique by weak approximation for D [PR94, Proposition7.3]) character ω c : A G ( A F ) / A G ( F ) → C × such that for almost all places v of F , the Langlands parameter of ( ω c ) v equals the image of c v in L D .Note that in all cases considered in this paper the connected centre of G is splitand so one can take D to be the full connected centre. Lemma 4.1.6.
Let G = GL N × GL and e G = G ⋊ θ . If c ∈ F S ( e G ) occurs in I e G disc and χ is the central character of c , then there is a unique ψ ∈ Ψ( e G , χ ) suchthat c is associated to ψ .Proof. This simply follows from Remark 4.1.4 (1) and the above description inthe case at hand of the pairs ( ˜ w, π L ) with ˜ w ∈ W ( L , e G ) reg , π L ∈ Π disc ( L ) and π ˜ w L ≃ π L . (cid:3) Remark 4.1.7.
Let G = GL N × GL and e G = G ⋊ θ . RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp (1) For P a parabolic subgroup of G with Levi L and π L ∈ Π disc ( L , χ G , L ), theparabolically induced representation A ( U P ( A F ) L ( F ) \ G ( A F )) π L is irre-ducible by [MW89] (implying multiplicity one for the discrete automorphicspectrum for L ) and [Ber84, § c ∈ F S ( e G ) oc-curring in I e G , L disc , L is determined by c .(3) For S ⊂ S ′ , the linear form I e G disc ,ν,c S ′ on I ( e G S ′ ) is simply the tensor productof I e G disc ,ν,c S with the unramified linear form on I ( e G S ′ r S ) corresponding tothe Satake parameters ( c v ) v ∈ S ′ r S (see Remark 2.3.4). This is particular to GL n and is a direct consequence of strong multiplicity one. Also by strongmultiplicity one for a given c S ∈ F S S ( e G ) there is at most one ν ∈ IC ( e G )such that I disc ,ν,c S = 0. By these remarks, for c ∈ F S ( e G ) we have a well-defined linear form I e G disc ,c on I ( e G ), whose restriction to I ( e G S ) (for largeenough S ) is I e G disc ,ν,c S for the unique ν such that this is non-zero, or 0 if nosuch ν exists.4.2. Elliptic endoscopic groups.
Consider the split group Γ = GL × GL over F and its automorphism θ : ( g, x ) ( J t g − J − , x det g ), where J = −
10 0 1 00 − was chosen so that the usual pinning of GL × GL is stable under θ . Note thatif ( π, χ ) is a representation of Γ ( F v ) for some place v of F , then ( π, χ ) ◦ θ ≃ (˜ π ⊗ ( χ ◦ det) , χ ). The dual group b Γ is naturally identified with GL ( C ) × GL ( C ), and b θ ( g, x ) = ( b J t g − b J − x, x ), where b J = J (but with coefficients in a different field).Denote e Γ = Γ ⋊ θ (that is, the non-identity connected component of Γ ⋊ { , θ } ).We consider twisted endoscopy with ω = 1.Then the elliptic endoscopic data ( H , H , s, ξ ) for e Γ are easily seen to be of thefollowing form.(1) H = GSpin , dual b H = GSp , for s = 1: The first projection identifies ξ ( \ GSpin ) = b Γ b θ with the general symplectic group defined by b J , and the“similitude factor” morphism \ GSpin → GL equals pr ◦ ξ | \ GSpin . Both Γ and GSpin are split, so there is an obvious choice for L ξ : L GSpin → L Γ .(2) GSpin α , with α ∈ F × /F × , , dual \ GSpin α = GSO with action of Gal( E/F )if α is not a square, where E = F ( √ α ). Pick s = diag( − , − , , b Γ Ad( s ) ◦ b θ = GO for the Gram matrix − − . If α = 1 the group GSpin is split and we choose the obvious L ξ . Otherwiselet c be the non-trivial element of Gal( E/F ), and define L ξ by mapping 1 ⋊ c to , . (3) R α := ( GSpin α × GSpin ) / { ( z, z − ) | z ∈ GL } , for non-trivial α . Thedual c R α is the subgroup of GSO × GSp of pairs of elements with equalsimilitude factors, and Gal( E/F ) acts on the first factor. Let s = diag( − , , , ξ α R ( c R α ) = { diag( x , A, x ) | A ∈ GL , x x = det A } . Define L ξ by mapping 1 ⋊ c to , . We also need to consider the elliptic endoscopic groups for
GSpin and GSpin .Let H be the unique non-trivial elliptic endoscopic group for GSpin , so that H ≃ GL × GL / { ( zI , z − I ) } . Then b H is the subgroup of GSp ( C ) × GSp ( C )of pairs of elements with equal similitude factors, so we have an obvious embeddingof dual groups b H → \ GSpin = GSp ( C ), inducing an embedding of L -groups L ξ ′ : L H → L GSpin .Let H α be the elliptic endoscopic group for GSpin associated to α ∈ F × /F × , , α = 1, so that H α ≃ GSpin α × GSpin α / { ( z, z − ) | z ∈ GL } . Recall that GSpin α is naturally isomorphic to Res F ( √ α ) /F ( GL ). Then d H α is the subgroup ofGSO ( C ) × GSO ( C ) consisting of pairs of elements with equal similitude factors,so we again have an obvious embedding of dual groups d H α → \ GSpin = GSO ( C ).If α = 1 then this trivially extends to an embedding of L -groups, while if α = 1,writing Gal( F ( √ α ) /F ) = { , c } , define L ξ ′ : L H α → L GSpin by mapping 1 ⋊ c to , . Stabilisation of the trace formula.
We will need to use the stabilisationof the (twisted) trace formula for e Γ and its elliptic endoscopic groups. Considerthe latter first: let ( H , H , s, ξ ) be an elliptic endoscopic datum for ( Γ , e Γ ). Thestabilisation of the trace formula for H is as follows. Fix ν ∈ IC ( H ), S a bigenough set of places, and c ∈ F S S ( H ). Choose representatives ( H ′ , H ′ , s, ξ ) for theisomorphism classes of elliptic endoscopic data for H , and for each representativechoose L ξ ′ : L H ′ → L H extending ξ (for example as in the previous section). Itinduces maps L ξ ′ : F S ( H ′ ) → F S ( H ) and L ξ ′ : IC ( H ′ ) → IC ( H ). Inductively RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp define a linear form on I ( H ( F S )) by(4.3.1) S H disc ,ν,c ( f ) := I H disc ,ν,c ( f ) − X e ′ =( H ′ , H ′ ,s ′ ,ξ ′ ) H ′ = H ι ( e ′ ) X c ′ cν ′ ν S H ′ disc ,ν ′ ,c ′ ( f H ′ )where the sum is over equivalence classes of nontrivial elliptic endoscopic data for H , f H ′ is a transfer of f (see Section 2.3), and the constants ι ( e ′ ) are recalled after thefollowing theorem. Theorem 4.3.2 ([Art02, Global Theorems 2 and 2’ and Lemma 7.3(b)]) . Thelinear form S H disc ,ν,c is stable, i.e. factors through SI ( H ( F S )) . Note that in general (4.3.1) is only well-defined thanks to Theorem 4.3.2 appliedto H ′ . However, for the groups H considered here, and for any non-trivial endo-scopic group H ′ , the only elliptic endoscopic group for H ′ is H ′ , and so S H ′ disc = I H ′ disc .Let us recall the definition of ι ( e ′ ), both for ordinary endoscopy and for twistedendoscopy. Assume that e G is a twisted space and e = ( H , H , s, ξ ) is an ellipticendoscopic datum. Let ι ( e ) = τ ( G ) τ ( H ) (cid:12)(cid:12)(cid:12) π (cid:16) Z ( b G ) Gal F , ∩ b T ˆ θ, (cid:17)(cid:12)(cid:12)(cid:12) | π (Aut( e )) | where τ is the Tamagawa number and the superscript 0 denotes the identity com-ponent. We have not included the factor | det(1 − θ | . . . ) | − from [MW16b, VI.5.1]because of Remark 4.1.4 (2); compare with the definition on p. 109 of [KS99] using[KS99, Lem. 6.4.B]. Recall [MW16b, p. 693] that there is a short exact sequence1 → (cid:16) Z ( b G ) /Z ( b G ) ∩ b T ˆ θ, (cid:17) Gal F → Aut( e ) / b H → Out ( e ) → . In the ordinary (non-twisted) case we have b T ˆ θ, = b T ⊃ Z ( b G ) and thus ι ( e ) = τ ( b G ) τ ( b H ) − | Out ( e ) | − . The only twisted case that we need in this paper is thecase of e Γ , when b T ˆ θ, = { (( t , . . . , t ) , x ) |∀ i, t i = t − − i x } and so Z ( b G ) ∩ b T ˆ θ, ≃ C × .Similarly it is easy to see that Z ( b G ) /Z ( b G ) ∩ b T ˆ θ, ≃ C × with trivial action of Gal F ,so we can conclude that ι ( e ) = τ ( Γ ) τ ( H ) − | Out ( e ) | − for any elliptic endoscopicdatum e = ( H , H , s, ξ ) of e Γ .Let us make the constant ι ( e ) explicit in the only two cases where it will beneeded in this paper:(1) For the elliptic endoscopic group H of GSpin , ι ( e ) = 1 / GSpin of e Γ , ι ( e ) = 1.We can finally state the stabilisation of the twisted trace formula for ( Γ , e Γ ).As in the case of ordinary endoscopy we fix representatives e = ( H , H , s, ξ ) ofisomorphism classes of elliptic endoscopic data for e Γ and for each e we also choosean L -embedding L ξ : L H → L G extending ξ (for example the ones defined in theprevious section). Theorem 4.3.3 ([MW16b, X.8.1]) . For any ν and c we have I e Γ disc ,ν,c ( f ) = X e =( H , H ,s,ξ ) ι ( e ) X ν ′ νc ′ c S H disc ,ν ′ ,c ′ ( f H ) where the first sum is over equivalence classes of elliptic endoscopic data for e Γ . Restriction of automorphic representations
Restriction for general groups.
Let us recall a consequence of [HS12, §
4] that we will need. Since in all cases considered in this paper the assump-tion of [Che18, Proposition 1 (iii)] will be satisfied, one can use the more pre-cise result of [Che18] (which can be formally generalised from cuspidal to square-integrable forms) instead. Consider an injective morphism G ֒ → G ′ between con-nected reductive groups over a number field F such that G is normal in G ′ and G ′ / G is a torus. Choose a maximal compact subgroup K ′∞ of G ′ ( F ⊗ Q R ); then K ∞ := G ( F ⊗ Q R ) ∩ K ′∞ is a maximal compact subgroup of G ( F ⊗ Q R ). Notethat if π ′ is an irreducible unitary admissible ( g ′ , K ′∞ ) × G ′ ( A F,f )-module thenRes G ′ G ( π ′ ) is a unitary admissible ( g , K ∞ ) × G ( A F,f )-module, but it has infinitelength in general. We have a ( g , K ∞ ) × G ( A F,f )-equivariant mapres G ′ G : A ( A G ′ G ′ ( F ) \ G ′ ( A F )) → A ( A G G ( F ) \ G ( A F ))obtained by restricting automorphic forms. The fact that res G ′ G takes values in A ( A G G ( F ) \ G ( A F )) is a routine verification, except for square-integrability whichfollows from the proof of [HS12, Lemma 4.19] (see also Remark 4.20 op. cit. ). If π ′ ∈ Π disc ( G ′ ) and ι : π ′ ֒ → A ( A G ′ G ′ ( F ) \ G ′ ( A F )), then res G ′ G ( ι ( π ′ )) is naturallyidentified with a quotient of Res G ′ G ( π ′ ). This quotient can be proper and of infinitelength, but in any case it is non-zero . In particular there exists an irreducibleconstituent π of Res G ′ G ( π ′ ) such that π ∈ Π disc ( G ). In this situation we will saythat π is an automorphic restriction of π ′ . Unsurprisingly, this notion of restrictionis compatible with the Satake isomorphism at almost all places: Lemma 5.1.1 (Satake) . Suppose that π ≃ ⊗ ′ v π v ∈ Π disc ( G ) is an automorphicrestriction of π ′ ≃ ⊗ ′ v π ′ v ∈ Π disc ( G ′ ) , then for almost all places v of F the Satakeparameter c ( π v ) of π v is the image of c ( π ′ v ) under the natural map (cid:16)c G ′ ⋊ Frob v (cid:17) ss / c G ′ − conj −→ (cid:16) b G ⋊ Frob v (cid:17) ss / b G − conj . Proof.
For almost all places v , π v is the unique unramified direct summand inRes G ′ ( F v ) G ( F v ) ( π ′ v ). The result follows from [Sat63, § G × T → G ′ , where T is any central torus in G isogenous to G ′ / G , and the translation in terms of dualgroups [Bor79, Prop. 6.7]. (cid:3) Let us now formulate a direct consequence of [HS12, Theorem 4.14], ignoringmultiplicities.
Theorem 5.1.2 (Hiraga–Saito) . The map res G ′ G is surjective, and so any discreteautomorphic representation for G is an automorphic restriction of a discrete auto-morphic representation for G ′ . In other words, there exists a surjective map ext G ′ G : Π disc ( G ) −→ Π disc ( G ′ ) / ( G ′ ( A F ) / G ( A F ) G ( F ) A G ′ ) ∨ such that for any π ′ ∈ ext G ′ G ( π ) , π is a subrepresentation of Res G ′ G ( π ′ ) . In general this map ext G ′ G is not uniquely determined.We will mainly use this result for Sp ֒ → GSpin . This will be fruitful thanksto exterior square functoriality for GL [Kim03] and the commutativity of the RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp following commutative diagram of dual groups:(5.1.3) \ GSpin = GSp d Sp = SO GL × GL SL L ξ Std ⊕ f where f := V (pr ) ⊗ pr − .6. Global Arthur–Langlands parameters for
GSpin Classification of global parameters.
Let χ : A × F /F × → C × be a continu-ous unitary character. Recall the set Ψ( e Γ , χ ) of formal global parameters definedin Section 2.5.In this section we will denote the functorial transfer GL × GL → GL by( π , π ) π ⊠ π (we will only need this in the weak sense, i.e. compatibility withSatake parameters at all but finitely many places). This transfer exists for cuspidalrepresentations by [Ram00], and is easily extended to discrete representations: • if π = η [2] for some character η and π is cuspidal, then π ⊠ π = η ⊗ π [2]. • if π = η [2] and π = η [2], then π ⊠ π = η η ⊞ η η [3].Recall that in Section 4.2 we fixed a representative ( H , H , s, ξ ) for each equiv-alence class of elliptic endoscopic data for e Γ , and in each case an L -embedding L ξ : L H → L Γ = b Γ × W F . We also fixed, for each H as above, a representative( H ′ , H ′ , s ′ , ξ ′ ) for each equivalence class of elliptic endoscopic data for H , as wellas an L -embedding L ξ ′ : L H ′ → L H . We use this generic notation in the followingProposition, which shows that we may associate a parameter in the set Ψ( e Γ , χ )to each discrete automorphic representation of GSpin or GSpin with centralcharacter χ ; we will refine this in Propositions 6.1.5 and 6.1.6 to show that theseparameters are in fact contained in the subsets e Ψ disc ( GSpin , χ ), Ψ disc ( GSpin , χ )respectively. Proposition 6.1.1. (1) For L a proper Levi subgroup of GSpin , any c ∈ F S ( GSpin ) occurringin I GSpin , L disc such that b µ ( c ) = c ( χ ) satisfies L ξ ( c ) ∈ Ψ( e Γ , χ ) and is notdiscrete.(2) Let H = ( GL × GL ) / { ( zI , z − I | z ∈ GL } be the unique non-trivialelliptic endoscopic group for GSpin . Then any c ∈ F S ( H ) occurring in I H disc = S H disc and such that b µ ( c ) = c ( χ ) satisfies ( L ξ ◦ L ξ ′ )( c ) ∈ Ψ( e Γ , χ ) .(3) Let H ′ be a non-trivial elliptic endoscopic group for GSpin . Then any c ∈ F S ( H ′ ) occurring in I H ′ disc = S H ′ disc and such that b µ ( c ) = c ( χ ) satisfies ( L ξ ◦ L ξ ′ )( c ) ∈ Ψ( e Γ , χ ) .(4) For L a Levi subgroup of GSpin , any c ∈ F S ( GSpin ) occurring in I GSpin , L disc and such that b µ ( c ) = c ( χ ) satisfies L ξ ( c ) ∈ Ψ( e Γ , χ ) . If L = GSpin then L ξ ( c ) is not discrete.(5) Any c ∈ F S ( GSpin ) occurring in S GSpin disc and such that b µ ( c ) = c ( χ ) satisfies L ξ ( c ) ∈ Ψ( e Γ , χ ) .(6) Any c ∈ F S ( GSpin ) occurring in S GSpin disc and such that b µ ( c ) = c ( χ ) satisfies L ξ ( c ) ∈ Ψ( e Γ , χ ) . (7) Any c ∈ F S ( GSpin ) associated to a discrete automorphic representationfor GSpin with central character χ satisfies L ξ ( c ) ∈ Ψ( e Γ , χ ) .Proof. We use repeatedly the description of I e G , L disc explained in Section 4.1, namelythat if c ∈ F S ( e G ) occurs in I e G , L disc , then there is a regular element ˜ w ∈ W ( L , e G ),and π L ∈ Π disc ( L ) such that π ˜ w ≃ π and c ( π L ) maps to c via L L → L G .(1) The possible proper Levi subgroups L and the embeddings L L → L GSpin are listed in Section 2.2. In the case at hand, the possibilities are(a) GL × GSpin ∼ = GL × GL ,(b) GL × GSpin ∼ = GL × GL , and(c) GL × GL × GSpin ∼ = GL × GL × GL .In the first case we find that the corresponding parameter is of the form η ⊞ π ⊞ η , where π is a unitary discrete automorphic representation of GL ( A F )with ω π = χ and η = χ ; in the second case, that the parameter is ofthe form π ⊞ π , where π is a unitary discrete automorphic representationof GL ( A F ) such that π ∨ ⊗ χ ≃ π ; and in the third case that the parameteris of the form η ⊞ η ⊞ η ⊞ η with η = η = χ .(2) By the description of H as a quotient, c corresponds to a pair ( π , π )with each π i either an element of Π disc ( GL ) with ω π i = χ or η ⊞ η , with η = χ . It is easy to check that ( L ξ ◦ L ξ ′ )( c ) = ( c ( π ) ⊕ c ( π ) , c ( χ )), so thatthe corresponding parameter is π ⊞ π .(3) This is similar to the previous two parts. Write H ′ = H α as in Section 4.2,so that an element of Π disc ( H ′ ) is given by a pair of automorphic represen-tations ρ , ρ for the torus GSpin α ≃ Res
E/F ( GL ) (here E = F ( √ α ))whose restrictions to GL are equal to χ . Then via the natural embed-ding L GSpin α = GSO ⋊ Gal(
E/F )) → GL , we have ( L ξ ◦ L ξ ′ )( c ) =( c ( π ) ⊕ c ( π ) , c ( χ )) where π and π are the cuspidal automorphic repre-sentations for GL with central character χ automorphically induced (for E/F ) from ρ and ρ seen as unitary characters of A × E /E × .(4) Recall that GSpin is isomorphic to the subgroup of elements of GL × GL such that the determinants of the two elements are equal. Accordingly, if c is discrete automorphic, i.e. it occurs in I GSpin , GSpin disc , then by The-orem 5.1.2 it comes from the automorphic restriction of some ( π , π ) ∈ Π disc ( GL × GL ), with c ( ω π ) c ( ω π ) = c ( χ ) and so ω π ω π = χ . Then L ξ ( c ) = ( c ( π ) ⊗ c ( π ) , c ( χ )), and the corresponding parameter is π ⊠ π .Otherwise c occurs in I GSpin , L disc for some proper Levi subgroup. By thedescription given in Section 2.2, we see that L is isomorphic to GL × GSpin ∼ = GL × GL or to GL × GL × GSpin ∼ = GL × GL × GL .In either case we can compute explicitly as in (1), and we find that weobtain parameters of the form π ⊞ π , where π is a discrete automorphicrepresentation of GL ( A F ) such that π ∨ ⊗ χ ≃ π , and parameters of theform η ⊞ η ⊞ η ⊞ η with η = η = χ .(5) This follows immediately from the stable trace formula (4.3.1) for GSpin and the two previous points.(6) This follows from the stable twisted trace formula of Theorem 4.3.3. Ob-serve that we can associate an element of Ψ( e Γ , χ ) to any family of Satakeparameters occurring in S GSpin disc or to I e Γ disc ; in the former case this is thecontent of (5), and in the latter case it follows from Lemma 4.1.6. RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp (7) This follows as in (6), this time using the stable trace formula for GSpin ,and applying parts (2) and (6). (cid:3) We can now prove the symplectic/orthogonal alternative for GL . This is wellknown, and can also be proved using the theta correspondence or converse theo-rems; indeed, [AS14, Thm. 4.26] proves a corresponding result for GSpin groups ofarbitrary rank, showing that a χ -self dual cuspidal automorphic representation π of GL n arises as the transfer of a globally generic representation of a GSpin groupwhich is uniquely determined by the data of which of the corresponding symmetricand alternating square L -functions has a pole, together with the central characterof π .However, our emphasis here is slightly different (we wish to determine which rep-resentations have Satake parameters which occur in the discrete spectrum of GSpin ),and in any case we find it instructive to show how this follows from the trace formulatogether with Kim’s exterior square transfer [Kim03].The following remark will help us to distinguish parameters coming from differentendoscopic subgroups. Remark 6.1.2.
The sets (cid:0) L ξ ( F S ( GSpin )) ∪ L ξ ( F S ( GSpin )) (cid:1) and (cid:0) L ξ ( F S ( GSpin α )) ∪ L ξ ( F S ( R α )) (cid:1) (where α ∈ F × /F × , is non-trivial) are pairwise disjoint, because we can recover α as follows (by the definition of L ξ ): for H = GSpin α or H = R α , c S ∈ F S ( H ) and( g S , x S ) = L ξ ( c S ), for any v S , then v splits in F ( √ α ) if and only if det g v = x v .On the other hand if H = GSpin or H = GSpin then we always have det g v = x v . Proposition 6.1.3.
Let π be a χ -self dual cuspidal automorphic representation for GL , and let S be a finite set of places of F containing all Archimedean places andall non-Archimedean places where π is ramified.(1) If there are cuspidal automorphic representations π i for GL such that ω π ω π = χ and π ≃ π ⊠ π , then L S ( s, Sym ( π ) ⊗ χ − ) has a pole at s = 1 , and there exists c ′ ∈ F S ( GSpin ) occurring in S GSpin disc and suchthat L ξ ( c ′ ) = ( c ( π ) , c ( χ )) .(2) If ω π = χ , then π is an Asai transfer from a cuspidal automorphic repre-sentation of GL /E , for E = F ( √ α ) the quadratic extension of F corre-sponding to the quadratic character χ /ω π , and L S ( s, Sym ( π ) ⊗ χ − ) hasa pole at s = 1 . Furthermore there exists c ′ ∈ F S ( GSpin α ) occurring in S GSpin α disc and such that L ξ ( c ′ ) = ( c ( π ) , c ( χ )) .(3) Otherwise ( i.e. if ω π = χ and π does not come from a pair of automorphiccuspidal representations for GL as in (1) ) L S ( s, V ( π ) ⊗ χ − ) has a poleat s = 1 , c := L ξ − ( c ( π ) , c ( χ )) ∈ F S ( GSpin ) occurs in S GSpin disc , and forany large enough S and any ν ∈ IC ( GSpin ) S GSpin disc ,ν,c S = I GSpin disc ,ν,c S = I GSpin , GSpin disc ,ν,c S . Proof. (1) It suffices to note that L S ( s, ^ ( π ⊠ π ) ⊗ ( ω π ω π ) − ) = L S ( s, ad ( π )) L S ( s, ad ( π )) is holomorphic at s = 1 since for each i = 1 , ( π i ) defined in [GJ78] is either(a) a self-dual cuspidal automorphic representation for GL ([GJ78, The-orem 9.3]),(b) η ⊞ σ where η is a character of order two and σ is a self-dual cuspidalautomorphic representation for GL such that η ⊗ σ ≃ σ ([GJ78,Remark 9.9] with (Ω / Ω ′ ) = 1),(c) η ⊞ η ⊞ η η where η and η are distinct characters of order two([GJ78, Remark 9.9] with Ω / Ω ′ of order two).As in the proof of Proposition 6.1.1 (4) we see that the element c ′ ∈ F S ( GSpin ) which is the image of ( c ( π ) , c ( π )) via either of the two ten-sor product morphisms GL × GL → GSO = \ GSpin occurs in I GSpin disc and S GSpin disc , and satisfies L ξ ( c ′ ) = ( c ( π ) , c ( χ )).(2) By Remark 4.1.7 (2) we know that ( c ( π ) , c ( χ )) does not occur in I e Γ , L disc forany proper Levi subgroup L of Γ . Since ( π, χ ) occurs with multiplicity onein the discrete automorphic spectrum for Γ , the automorphic extension e π of π to e Γ (provided by (4.1.1) for L = G , with ˜ w = θ ) has non-vanishing trace(see [Lem10, Proposition A.5] for the p -adic case, the Archimedean case isproved similarly) and so ( c ( π ) , c ( χ )) occurs in I e Γ disc . In the stabilisation ofthe twisted trace formula (Theorem 4.3.3) this contribution comes from atleast one elliptic endoscopic datum, i.e. there is an elliptic endoscopic group H and c ′ ∈ F S ( H ) occurring in S H disc such that L ξ ( c ′ ) = ( c ( π ) , c ( χ )).The character ω π /χ corresponds to some quadratic extension E = F ( √ α ), and by Remark 6.1.2, in the stabilisation of the twisted trace for-mula for e Γ this contribution must come from S GSpin α disc or S R α disc (a priorinon-exclusively). In the latter case, we see that π has the same Satake pa-rameters as π ⊞ π , where π is either a discrete automorphic representationfor GL with central character χ , or π = η ⊞ η with η = χ , and π is acuspidal χ -self-dual automorphic representation for GL , corresponding tothe extension E/F ; but either possibility contradicts [JS81].Thus ( c ( π ) , c ( χ )) comes from S GSpin α disc . • If it comes from S H disc = I H disc for some elliptic endoscopic group H = GSpin α for GSpin α then H ≃ GSpin β × GSpin γ / { ( z, z − ) | z ∈ GL } for some β, γ ∈ F × /F × , r { } satisfying βγ = α . Recall that GSpin β ≃ Res F ( √ β ) /F GL . Then we see that π = π ⊞ π where π (resp. π ) is the automorphic induction of a character of A × F ( √ β ) /F ( √ β ) × (resp. A × F ( √ γ ) /F ( √ γ ) × ) and this contradicts the cuspidality of π . • If ( c ( π ) , c ( χ )) comes from I GSpin α , L disc for the proper Levi subgroup L ≃ GL × GSpin α of GSpin α then π = η ⊞ π ⊞ η where η = χ and π is the automorphic induction of a character of A × E /E × and we alsoget a contradiction with the cuspidality of π .Therefore ( c ( π ) , c ( χ )) comes from a discrete automorphic representationfor GSpin α . As explained in [AR11, § GSpin α ֒ → Res
E/F GL ), this is equivalent to π being the Asai transfer of RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp a cuspidal automorphic representation π E of GL ( A E ). Then L S ( s, V π ⊗ χ − ) = L S ( s, Ind FE (Sym π E ⊗ ω ′ π E ) ⊗ χ − ), where ω ′ π E is the Gal( E/F )-conjugate of ω π E .If π E is not dihedral then V π is cuspidal by [AR11, Prop. 3.2], so itis enough to consider the case that π E is dihedral, induced from a charac-ter χ E ′ of A × E ′ / ( E ′ ) × , where E ′ /E is a quadratic extension. Then Sym π E =Ind EE ′ χ E ′ ⊞ χ E ′ | A × E , and it is easy to verify explicitly that the isobaric repre-sentation Ind FE (Sym π E ⊗ ω ′ π E ) ⊗ χ − cannot contain the trivial character.(3) As in the previous case, ( c ( π ) , c ( χ )) occurs in I e Γ disc . By Remark 6.1.2,in the stabilisation of the twisted trace formula for e Γ this contributioncomes from H = GSpin or H = GSpin on the right-hand side. Inthe former case, as GSpin embeds into GL × GL , we would be in thesituation of part (1); so it must occur in S GSpin disc . Moreover, it cannotcome from S H , disc or I GSpin , L disc for a proper Levi L , as by (the proof of)Proposition 6.1.1 this would contradict strong multiplicity one, so we canconclude that I GSpin , GSpin disc ,c = S GSpin disc ,c is not identically zero.In particular there is a discrete automorphic representation Π for GSpin such that L ξ ( c (Π)) = ( c ( π ) , c ( χ )). Let Π ′ be an automorphic restriction(in the sense of Section 5) of Π to Sp . Then Π ′ is a discrete automor-phic representation for Sp , and Arthur associates a discrete parameter ψ ′ ∈ Ψ disc ( Sp ) to Π ′ (see Theorem 2.6.1). Now V ( c ( π )) ⊗ c ( χ ) − =1 ⊕ c ( ψ ′ ) (see the commutative diagram (5.1.3)) and so L S ( s, V ( π ) ⊗ χ − ) = ζ SF ( s ) L S ( s, ψ ′ ). Moreover by [Kim03, Thm. 5.3.1], 1 ⊕ c ( ψ ′ ) is associated toa (unique) isobaric sum of unitary cuspidal representations, and by [JS81,Thm. 4.4] the same holds for ψ ′ . This implies that L S ( s, ψ ′ ) does not vanishon the line ℜ ( s ) = 1, by the main result of [JS77]. (cid:3) Remark 6.1.4.
By Theorem 2.7.1 and Proposition 6.1.3, we see that Ψ disc ( GSpin )is the subset of Ψ( e Γ , χ ) consisting of pairs ( ψ, χ ) with ψ of the following kinds. (Wehave labelled them in the same way as in [Art04]. The groups S ψ are easy to com-pute; for the values of ε ψ , see [Art13, (1.5.6)].)(a) cuspidal automorphic representations π of GL such that π ∨ ⊗ χ ≃ π and L S ( s, χ − ⊗ V π ) has a pole at s = 1. (General type, S ψ = 1, ε ψ = 1.)(b) π ⊞ π where π i are cuspidal automorphic representations of GL , ω π = ω π = χ and π π . (Yoshida type, S ψ = Z / Z , ε ψ = 1.)(c) π [2] for π a cuspidal automorphic representation for GL such that ω π /χ has order 2 (i.e. ( π, χ ) is of orthogonal type, which means that π is auto-morphically induced from a character η : A × E /E × → C × for the quadraticextension E/F corresponding to ω π /χ , such that η c = η and η | A × F /F × = χ ).(Soudry type, S ψ = 1, ε ψ = 1.)(d) π ⊞ η [2] with π cuspidal for GL and ω π = η = χ . (Saito–Kurokawa type, S ψ = Z / Z , ε ψ = sgn if ε (1 / , π ⊗ η − ) = −
1, and ε ψ = 1 otherwise.)(e) η [2] ⊞ η [2] with η = η = χ and η = η . (Howe–Piatetski-Shapiro type, S ψ = Z / Z , ε ψ = 1.)(f) η [4] with η = χ . (One dimensional type, S ψ = 1, ε ψ = 1.) Proposition 6.1.5.
For c ∈ F S ( GSpin ) associated to a discrete automorphicrepresentation Π of GSpin with central character χ , the associated element of Ψ( e Γ , χ ) (by Proposition 6.1.1) belongs to the subset Ψ disc ( GSpin ) .Proof. As in the proof of Proposition 6.1.3 3, we use an automorphic restriction Π ′ of Π to ( GSpin ) der ≃ Sp , and the associated parameter ψ ′ , which we know tobe discrete. We know that 1 ⊕ c ( ψ ′ ) = V ( c ( ψ )) ⊗ c ( χ ) − .By Theorem 2.6.1, we can and do assume that χ is not a square. In particular,this implies that ψ does not have a summand of the form η , η [2] or η [4] (as thecondition that η is χ -self dual forces η = χ ). In addition, if ψ = ψ ⊞ ψ , then c ( ψ ′ ) = (cid:16)V ( c ( ψ )) ⊗ c ( χ ) − (cid:17) ⊕ ⊕ ad ( c ( ψ )), which contradicts the discretenessof ψ ′ . Thus we have the following possibilities for ψ .(1) ψ = ψ ⊞ ψ where ψ i is a cuspidal automorphic representation for GL such that ψ ∨ i ⊗ χ ≃ ψ i and ψ ψ . We need to show that ω π i = χ ,i.e. that ( π i , χ ) is of symplectic type. Suppose not. We have ω π i = χ ,and by Remark 6.1.2 we also have ω π ω π = χ and so ω π = ω π . Thenwe find that V ( ψ ) ⊗ χ − = ( ω π /χ ) ⊞ ( ω π /χ ) ⊞ ( χ − π ⊠ π ). Since ω π /χ = ω π /χ is a non-trivial quadratic character, this cannot be writtenas 1 ⊞ ψ ′ with ψ ′ discrete, a contradiction.(2) ψ = π [2], where π is a cuspidal automorphic representation for GL suchthat π ∨ ⊗ χ ≃ π . In this case we need to check that ω π /χ has order 2,i.e. is non-trivial. But if χ = ω π then ψ ′ = V ( π [2]) ⊗ ω − π = ad ( π ) ⊞ [3],which cannot be written as an isobaric sum of 1 and discrete automorphicrepresentations for general linear groups, a contradiction.(3) ψ = π [1] where π is a cuspidal automorphic representation for GL suchthat π ∨ ⊗ χ ≃ π . In this case we need to check that ( π, χ ) is of symplectictype, i.e. that L S ( s, V ( π ) ⊗ χ − ) has a pole at s = 1. Exactly as in theproof of Proposition 6.1.3 (3), we have L S ( s, V ( π ) ⊗ χ − ) = ζ SF ( s ) L S ( s, ψ ′ ),and L S ( s, ψ ′ ) does not vanish on the line ℜ ( s ) = 1, as required. (cid:3) Proposition 6.1.6. (1) For c ∈ F S ( GSpin ) associated to a discrete automorphic representation π for GSpin having central character χ , the element L ξ ( c ) ∈ Ψ( e Γ , χ ) associated to c by Proposition 6.1.1 belongs to e Ψ disc ( GSpin , χ ) .(2) For c ∈ F S ( GSpin ) , occurring in S GSpin disc , such that b µ ( c ) = c ( χ ) andsuch that L ξ ( c ) is discrete, we have that L ξ ( c ) ∈ e Ψ disc ( GSpin , χ ) .Proof. (1) By Theorem 2.6.1, we can and do assume that χ is not a square. Asexplained in the proof of Proposition 6.1.1 (4), the parameter of π is of theform π ⊠ π , where π , π are discrete automorphic representations of GL with ω π ω π = χ . If neither π , π were cuspidal, then χ would be a square,so we may assume that π is cuspidal. If π = η [2] then π ⊠ π = π [2]where π = η ⊗ π , so ω π = χ , and it follows from Theorem 2.7.1 that thisparameter belongs to e Ψ disc ( GSpin , χ ).It remains to consider the case that π , π are both cuspidal. If π ⊠ π is cuspidal, then the parameter belongs to e Ψ disc ( GSpin , χ ) by Proposi-tion 6.1.3. If π ⊠ π is not cuspidal, then since ω π ω π = χ is not a square, π cannot be a twist of π , and it follows from Theorem A of the appendix RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp to [Kri12] that π , π are both automorphic inductions of characters from acommon quadratic extension E/F . In this case π ⊠ π is the isobaric directsum π ⊞ π where π , π are distinct automorphic inductions of charactersfrom E/F (see [Kri12, (A.2.2)]), so it again follows from Theorem 2.7.1that this parameter belongs to e Ψ disc ( GSpin , χ ), as required.(2) By Proposition 6.1.1 (4) and the stabilisation of the trace formula for GSpin , either c is associated to a discrete automorphic representation for GSpin , or there exists α ∈ F × /F × , r { } and c ′ ∈ F S ( H α ) occurring in S H α disc = I H α disc such that c = L ξ ′ ( c ′ ). In the first case we conclude by the pre-vious point, so we are left to consider the second case. Denote E = F ( √ α )and Gal( E/F ) = { , σ } . By the description of H α in Section 4.2 we obtainthat c = c ( π ) ⊕ c ( π ) where each π i is a cuspidal automorphic represen-tation automorphically induced for E/F from a character χ i of A × E /E × such that χ i | A × F = χ and χ , χ σ , χ , χ σ are pairwise distinct, and again weconclude by Theorem 2.7.1. (cid:3) Multiplicity formula
In this section we prove the multiplicity theorem for
GSpin (Theorem 7.4.1),which describes the discrete automorphic spectrum in terms of the packets Π ψ ( ε ψ )defined in Definition 2.5.5. We begin with some preliminaries.7.1. Canonical global normalisation versus Whittaker normalisation.
Re-call from Remark 4.1.7 that for G = GL N × GL and e G = G ⋊ θ , for a Levisubgroup L of G and π L ∈ Π disc ( L ) the parabolically induced representation A ( U P ( A F ) L ( F ) \ G ( A F )) π L is irreducible. For ˜ w ∈ W ( L , e G ) we have a canon-ical (“automorphic”) extension of this representation of G ( A F ) to e G , denoted M P | ˜ w ( P ) (0) ◦ ρ GP , ˜ w in Section 4. We have another canonical normalisation of thisextension, namely the Whittaker normalisation recalled in Section 3.2. Lemma 7.1.1 (Arthur) . These two extensions coincide.Proof.
The proof of [Art13, Lemma 4.2.3] readily extends to the case at hand. (cid:3)
The twisted endoscopic character relation for real discrete temperedparameters.Proposition 7.2.1.
Let ϕ : W R → GSp be a discrete parameter. Then the twistedendoscopic character relation holds for Π ϕ ( as defined by Langlands in [Lan89]) ,i.e. part 1 of Theorem 3.1.1 holds. Recall that for ϕ such that b µ ◦ ϕ is a square, this twisted endoscopic characterrelation is a direct consequence of [Mez16] and [AMR15, Annexe C]. Proof.
We use a global argument similar to (but simpler than) [AMR15, AnnexeC]. Up to twisting we can assume that Std
GSpin ◦ ϕ ≃ ( I a ⊕ I a , sign a ), where a , a ∈ Z > are such that a − a ∈ Z > (and as before, I a = Ind W R C × ( z ( z/ ¯ z ) a )).Fix a continuous character χ : A × / R > Q × → C × such that χ | R × = sign a . Thereare cuspidal automorphic representations π , π for GL / Q with central characters ω π = ω π = χ and such that rec( π i, ∞ ) = I a i (apply [Ser97, Proposition 4] with n = 1, k = 2 a i +1 fixed and N of the form ℓ cond( χ ) where cond( χ ) is the conductorof χ and ℓ → + ∞ prime). Let ψ = π ⊞ π ∈ Ψ disc ( GSpin , χ ), so that ψ ∞ = ϕ . By [Mez16] there is z ( ϕ ) ∈ C × such that for any f ∞ ∈ I ( e Γ R ) we havetr π e Γ ϕ ( f ∞ ) = z ( ϕ ) (cid:0) tr π + ∞ ( f ′ ) + tr π −∞ ( f ′∞ ) (cid:1) where π + ∞ (resp. π −∞ ) is the generic (resp. non-generic) element of Π ϕ , i.e. h· , π + ∞ i (resp. h· , π −∞ i ) is the trivial (resp. non-trivial) character of S ϕ . We need to showthat z ( ϕ ) = 1. Recall that for any finite prime p the twisted endoscopic characterrelation tr π e Γ ψ p ( f p ) = X π p ∈ Π ψp tr π p ( f ′ p )holds by the main theorem of [CG15].In the discrete part of the trace formula for e Γ , the contribution I e Γ disc ,c ( ψ ) of c ( ψ )only comes from L = GL × GL and ˜ w = θ , using notation as in the discussionpreceding Definition 4.1.2. By Lemma 7.1.1 and since det( ˜ w − | A ΓL ) = 2 thiscontribution is (on I ( e Γ S ) for S containing ∞ and all places where π or π ramify) Y v ∈ S h v Y v tr π e Γ ψ v ( h v )where π e Γ ψ v is the Whittaker-normalised extension to e Γ ( F v ) of the irreducible parabol-ically induced representation π ,v × π ,v . Thus we get for h = Q v ∈ S h v ∈ I ( e Γ S )(7.2.2) I e Γ disc ,c ( ψ ) ( h ) = z ( ϕ )2 Y v ∈ S X π v ∈ Π ψv tr π v ( h GSpin v ) . By the stabilisation of the twisted trace formula (Theorem 4.3.3) and Remark 6.1.2and Proposition 6.1.6 (2) which imply that the endoscopic groups
GSpin α and R α have vanishing contributions corresponding to c ( ψ ) S , (7.2.2) equals S disc ,ν ( ϕ ) ,c ( ψ ) S ( h GSpin ) . By surjectivity of the transfer map h h GSpin (Proposition 2.4.3), this determinesthe stable linear form S GSpin disc ,ν ( ψ ) ,c ( ψ ) S . Let H = ( GL × GL ) / { ( zI , z − I | z ∈ GL } be the unique non-trivial elliptic endoscopic group for GSpin . The ( ν ( ψ ) , c ( ψ ) S )-part of the stabilisation of the trace formula (Theorem 4.3.2) for GSpin now reads,for f = Q v ∈ S f v ∈ I ( GSpin ), I GSpin disc ,ν ( ψ ) ,c ( ψ ) S ( f ) = z ( ϕ )2 Y v ∈ S X π v ∈ Π ψv tr π v ( f v ) + 14 X ν ′ ν ( ψ ) c ′ S c ( ψ ) S S H disc ,ν ′ ,c ′ S ( f H ) . Now S H disc ,ν ′ ,c ′ S = I H disc ,ν ′ ,c ′ S is non-vanishing if and only if ( ν ′ , c ′ S ) is associated to( π , π ) or to ( π , π ), in which case it equals tr ( π ⊗ π ) or tr ( π ⊗ π ). By theendoscopic character relations, in either case we have S H disc ,ν ′ ,c ′ S ( f H ) = Y v ∈ S X π v ∈ Π ψv h s, π v i tr π v ( f v ) , RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp where s is the non-trivial element of S ψ . Thus we obtain I GSpin disc ,ν ( ψ ) ,c ( ψ ) S ( f ) = X ( π v ) v ∈ Q v ∈ S Π ψv z ( ϕ ) + Q v ∈ S h s, π v i Y v ∈ S tr π v ( f v ) . By Proposition 6.1.1 (1) the left-hand side simply equals the trace of f in the( ν ( ψ ) , c ( ψ ) S )-part of the discrete automorphic spectrum for GSpin . Varying S ,the above equality means that the multiplicity of π = ⊗ ′ v π v ∈ Π ψ in A ( GSpin )equals ( z ( ϕ ) + h s, π i ) /
2. Comparing with [CG15, Theorem 3.1] (which relies on thetheta correspondence and not trace formulas) for any π we finally obtain z ( ϕ ) =1. (cid:3) Remark 7.2.3.
Arguing as in Lemma C.1 of [AMR15] one could certainly prove theProposition without using [CG15, Theorem 3.1], since | z ( ψ ) | = 1 and ( z ( ψ ) − / ∈ Z ≥ imply z ( ψ ) = 1 (consider the multiplicity of π −∞ ⊗ N ′ p π p where h s, π p i = +1for all p ).7.3. Local parameters.
In this section we obtain Arthur’s multiplicity formula for
GSpin , by formally using the stable twisted trace formula and twisted endoscopiccharacter relations to get the desired expression for S GSpin disc ,c for c corresponding to ψ ∈ Ψ disc ( GSpin ), and then the stable trace formula for GSpin .We begin with the following important point, which is Conjecture 2.5.3 for G = GSpin . Proposition 7.3.1. If π is a χ -self dual cuspidal automorphic representationof GL ( A F ) of symplectic type, then for any place v of F , the pair (rec( π v ) , rec( χ v )) is of symplectic type, i.e. factors through GSp ( C ) .Proof. This follows from [GT11a, Thm. 12.1], which shows that π arises as thetransfer of a (globally generic) automorphic representation Π of GSp ( A F ), andthat at each place v , the pair (rec( π v ) , rec( χ v )) is obtained from the L -parameterassociated to Π v by the main theorem of [GT11a]. (cid:3) Remark 7.3.2.
There are at least two alternative ways of proving Proposition 7.3.1.One is to use the main results of [Kim03] and [Hen09], which imply in particularthat for each place v the representation V rec( π v ) ⊗ rec( χ v ) − contains the trivialrepresentation, together with a case by case analysis. The other is to follow theargument of [Art13, § The global multiplicity formula.
Given Proposition 6.1.5, the multiplicityformula is morally equivalent to the following formula for any ψ ∈ Ψ disc ( GSpin ), f ∈ H ( GSpin ) and S large enough: S GSpin disc ,ν,c ( ψ ) S = ( ε ψ ( s ψ ) |S ψ | P π ∈ Π ψ h s ψ , π i tr π if ν = ν ( ψ )0 otherwise.This is the simplification (for discrete parameters) of the general stable multiplicityformula (see [Art13, Theorem 4.1.2]).We now prove the multiplicity formula; the following theorem is Conjecture 2.5.6,specialised to the case G = GSpin . We write Π ψ ( ε ψ ) for the set of representationsdefined in 2.5.5 (with no tilde, since we are working with GSpin ). Theorem 7.4.1.
There is an isomorphism of H ( GSpin ) -modules (7.4.2) A ( GSpin ) ∼ = M χ : A × F /F × R > → C × ψ ∈ Ψ disc ( GSpin ,χ ) π ∈ Π ψ ( ε ψ ) π where χ runs over the continuous (automatically unitary) characters.Proof. Fix a continuous character χ : A × F /F × R > → C × , and write A ( GSpin , χ )for the space of χ -equivariant square-integrable automorphic forms on which A × F /F × acts via χ . For any ν ∈ IC ( G ) and c ∈ F S ( G ), write A ( GSpin , χ ) ν,c := lim −→ S A ( GSpin , χ ) ν,c s . Then we have A ( GSpin , χ ) = M ν ∈ IC ( G ) c ∈ F S ( G ) A ( GSpin , χ ) ν,c = M ν ∈ IC ( G ) ψ ∈ Ψ disc ( GSpin ,χ ) A ( GSpin ) ν,c ( ψ ) . Indeed, it follows from Proposition 6.1.5 that for any c with A ( GSpin , χ ) c = 0,there is some ψ ∈ Ψ disc ( GSpin , χ ) such L ξ ( c ( π )) = c ( ψ ). It follows that we arereduced to showing that for each ψ ∈ Ψ disc ( GSpin , χ ), we have(7.4.3) A ( GSpin ) ν,c ( ψ ) ∼ = (L π ∈ Π ψ ( ε ψ ) π if ν = ν ( ψ )0 if ν = ν ( ψ ) . Fix ν ∈ IC ( G ) and ψ ∈ Ψ disc ( GSpin , χ ). If χ is a square, then we are doneby Theorem 2.6.1 (that is, by reducing to SO , already proved by Arthur). So weonly have to consider the following cases:(1) Cuspidal π for GL such that π ∨ ⊗ χ ≃ π and ( π, χ ) is of symplectic type.(2) π ⊞ π where the π i ’s are distinct cuspidal automorphic representations for GL with ω π i = χ (Yoshida type).(3) π [2] where π is a cuspidal automorphic representation for GL such that ω π /χ is a quadratic character, i.e. π ∨ ⊗ χ ≃ π and ( π, χ ) is of orthogonaltype (Soudry type).In case (2), the multiplicity formula is a special case of [CG15, Theorem 3.1], provedusing the global theta correspondence. So we can and do assume that we are incase (1) or case (3), so that in particular S ψ = 1 and ε ψ = 1. Furthermore, ineither case we know that for any place v , the parameter ψ v is of symplectic type,i.e. factors through GSp (in case (1) this is Proposition 7.3.1, and in case (3) itfollows from Theorem 2.7.1).We will prove (7.4.3) by computing I GSpin , GSpin disc ,ν,c ( ψ ) ( f ) for each f ∈ H ( GSpin ),which by definition is the trace of f on the left hand side of (7.4.3) (note that thisis well-defined, and equal to I GSpin , GSpin disc ,ν,c ( ψ ) S ( f ) for any sufficiently large S ). To thisend, note firstly that by Proposition 6.1.1 (1), we know that for any proper Levi L of GSpin , and for any c ∈ F S ( GSpin ) occurring in I GSpin , L disc ,ν , with central RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp character χ , we have L ξ ( c ) ∈ Ψ( e Γ , χ ) r Ψ disc ( GSpin , χ ). Consequently, we seethat for any ψ ∈ Ψ disc ( GSpin , χ ), we have(7.4.4) I GSpin disc ,ν,c ( ψ ) = I GSpin , GSpin disc ,ν,c ( ψ ) . Denoting as usual the unique non-trivial elliptic endoscopic group of
GSpin by H , we have that S H disc ,ν ′ ,c ′ vanishes identically for any ν ′ ∈ IC ( H ) and any c ′ ∈ F S ( H ) such that L ξ ′ ( c ′ ) = c ( ψ ) (because the proof of Proposition 6.1.1 (2)shows that any c ′ occurring in S H disc is such that L ξ ◦ L ξ ′ ( c ′ ) is a sum of at least twodiscrete automorphic representations of general linear groups). It follows that wehave(7.4.5) I GSpin disc ,ν,c ( ψ ) = S GSpin disc ,ν,c ( ψ ) . By Proposition 6.1.6 (2), for any c ′ occurring in S GSpin disc we have L ξ ( c ′ ) = c ( ψ ),so that (using also Remark 6.1.2) the contribution of ψ to the stabilisation of thetwisted trace formula for e Γ simply reads(7.4.6) I e Γ disc ,ν,c ( ψ ) ( h ) = S GSpin disc ,ν,c ( ψ ) ( h GSpin )where on the right-hand side c ( ψ ) denotes the unique element of F S ( GSpin )which is the preimage of c ( ψ ) ∈ F S ( e Γ ) by L ξ , and similarly for ν seen as anelement of IC ( GSpin ). By surjectivity of h h GSpin (see Proposition 2.4.3),and Remark 4.1.7, this implies that S GSpin disc ,ν,c ( ψ ) vanishes identically if ν = ν ( ψ ). Inthe definition of I e Γ disc ,ν,c ( ψ ) as a sum over Levi subgroups, the only non-vanishingsummand corresponds to L = GL . By Lemma 7.1.1 we have for h = Q v h v ∈ I ( e Γ ) I e Γ disc ,ν ( ψ ) ,c ( ψ ) ( h ) = Y v tr π e Γ ψ v ( h v ) . Applying Theorem 3.1.1 (1) (or rather its extension to parameters in Ψ + ( GSpin )via parabolic induction; see [Art13, § S GSpin disc ,ν ( ψ ) ,c ( ψ ) ( Y v f v ) = Y v X π v ∈ Π ψv tr π v ( f v ) . Combining this with (7.4.4) and (7.4.5), we conclude that I GSpin , GSpin disc ,ν,c ( ψ ) ( Y v f v ) = (Q v P π v ∈ Π ψv tr π v ( f v ) if ν = ν ( ψ )0 if ν = ν ( ψ )Recalling that S ψ = 1 and ε ψ = 1, this is equivalent to 7.4.3, so we are done. (cid:3) Remark 7.4.7.
A consequence of the multiplicity formula and [AS14] is that forany discrete automorphic representation π for GSpin which is formally tempered(i.e. of general or Yoshida type), there exists a globally generic discrete automorphicrepresentation π ′ for GSpin such that for any place v of F , π v and π ′ v have thesame Langlands parameter. Indeed letting ψ ∈ Ψ disc ( GSpin , χ ) be the parameterof π (well-defined by the multiplicity formula), Shahidi’s conjecture (proved in[GT11a]) implies that there is a unique representation in Π ψ which is generic ateach place. In fact the multiplicity formula asserts that it is automorphic withmultiplicity one. By (the converse part of) [AS14, Theorem 4.26] there exists aglobally generic discrete (even cuspidal) automorphic representation π ′ for GSpin
52 TOBY GEE AND OLIVIER TA¨IBI such that π ′ v ≃ π v for almost all v . In particular π ′ has parameter ψ , and for anyplace v of F , π ′ v is generic.Note that in the case χ = 1, Arthur used the the analogue of [AS14] in orderto prove Shahidi’s conjecture: see [Art13, Proposition 8.3.2]. More precisely, heused the descent theorem of Ginzburg, Rallis and Soudry (and thus indirectly theconverse theorem of Cogdell, Kim, Piatestski-Shapiro and Shahidi). Remark 7.4.8.
Let G be an inner form of GSpin over a number field F . Us-ing the stabilisation of the trace formula for G qualitatively (i.e. only consideringfamilies of Satake parameters), we see that for any π ∈ Π disc ( G , χ ), there is a well-defined ψ ∈ Ψ( e Γ , χ ) such that c ( π ) = ( c ( ψ ) , c ( χ )). Moreover if ψ is discrete then ψ ∈ Ψ disc ( GSpin , χ ). If ψ ∈ Ψ disc ( GSpin , χ ) is tempered (i.e. either of generaltype or of Yoshida type) then using the stabilisation of the trace formula quanti-tatively and the endoscopic character relations proved in [CG15] for inner forms aswell, one could certainly prove the multiplicity formula for the part of the discreteautomorphic spectrum for G corresponding to ( c ( ψ ) , c ( χ )) ∈ F S ( G ). The proofwould be similar to those of Proposition 7.2.1 and Theorem 7.4.1. Note howeverthat to even state the multiplicity formula, one has to fix a normalisation of localtransfer factors satisfying a product formula. This normalisation was achieved in[Kal] and used in [Ta¨ı] to prove the multiplicity formula for certain inner forms ofsymplectic groups. It would thus be necessary to compare Kaletha’s normalisationof local transfer factors for the non-split inner form of GSp realised as a rigidinner twist with Chan–Gan’s ad hoc normalisation [CG15, § Compatibility of the local Langlands correspondences for Sp and GSpin In this section, we study the compatibility of the local Langlands correspondencewith restriction from
GSp ( F ) ≃ GSpin ( F ) to Sp ( F ), where F is a p -adic field.We do not consider the case of Archimedean places, which could certainly be doneby a careful examination of the Langlands–Shelstad correspondence.8.1. Compatibility with restriction.
Let F be a p -adic field. The proof ofthe existence of the local Langlands correspondence for GSp ( F ) ≃ GSpin ( F )in [GT11a] used the theta correspondence, and its compatibility with the correspon-dence stated in [Art04] (characterised by (twisted) endoscopic character relations)was proved in [CG15]. In the paper [GT10], a local Langlands correspondencefor Sp ( F ) was deduced from the correspondence for GSp ( F ) by restriction. Thiscorrespondence is uniquely characterised by the commutativity of the diagram(8.1.1) Π( GSpin ) Φ( GSpin )Π( Sp ) Φ( Sp ) pr where Π( GSpin ) (resp. Π( Sp )) is the set of equivalence classes of irreducible ad-missible representations of GSpin ( F ) (resp. Sp ( F )), Φ( GSpin ) (resp. Φ( Sp ))is the set of equivalence classes of continuous semisimple representations of WD F valued in GSp ( C ) (resp. SO ( C )), the horizontal arrows are the local Langlandscorrespondences, and pr is the projection GSp ( C ) → PGSp ( C ) ∼ = SO ( C ). The RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp left hand vertical arrow is not in fact a map at all, but a correspondence, given bytaking any restriction of an element of Π( GSpin ) to Sp ( F ).Of course, [Art13] gives another definition of the local Langlands correspondencefor Sp , which is characterised by twisted endoscopy for ( GL , g t g − ). It isnot obvious that this correspondence agrees with that of [GT10]; this amountsto proving the commutativity of the diagram (8.1.1), where now the horizontalarrows are the correspondences characterised by twisted endoscopy. Proving thisis the main point of this section; we will also prove a refinement, describing theconstituents of the restrictions of representations of GSpin ( F ) to Sp ( F ) in termsof the parameterisation of L -packets.We begin by recalling some results about restriction of admissible representa-tions, most of which go back to [GK82], and are explained in the context of GSp n in [GT10]. They are also proved in [Xu16], which we refer to as a self-contained ref-erence. If e π is an irreducible admissible representation of GSpin ( F ), then e π | Sp ( F ) is a direct sum of finitely many irreducible representations of Sp ( F ) ([Xu16, Lem.6.1]), and these representations are pairwise non-isomorphic ([AP06, Thm. 1.4]).Furthermore if π is an irreducible admissible representation of Sp ( F ), then thereexists an irreducible representation e π of GSpin ( F ) whose restriction to Sp ( F )contains π , and e π is unique up to twisting by characters ([Xu16, Cor. 6.3, 6.4]).There is also an analogue of these statements for L -parameters, which is that L -parameters for Sp may be lifted to GSpin , and such lifts are unique up to twist;see [GT10, Prop. 2.8] (see also [Lab85, Th´eor`eme 7.1] for a more general liftingresult).In particular, it follows that if π ∈ Π( Sp ), and e π lifts π , with L -parameter ϕ e π ,then pr ◦ ϕ e π depends only on π (because ϕ e π is well-defined up to twist, as e π itselfis); we need to show that it is equal to the L -parameter of π defined by the localLanglands correspondence of [Art13]. Theorem 8.1.2.
The diagram (8.1.1) commutes, where the horizontal arrows arethe correspondences of [Art13, Art04] determined by twisted endoscopy; that is, thelocal Langlands correspondences for Sp of [GT10] and [Art13] coincide.Proof. By the preceding discussion, it suffices to show that for each irreducibleadmissible representation π , there is some lift e π of π such that ϕ π = pr ◦ ϕ e π .We begin with the case that π is a discrete series representation. Then by [Clo86,Thm. 1B] and Krasner’s lemma, we can find a totally real number field K , a finiteplace v of K , and a discrete automorphic representation Π of Sp ( A K ), such that:(1) K v ∼ = F (so we identify K v with F from now on).(2) Π v ≃ π .(3) at each infinite place w of K , Π w is a discrete series representation.(4) for some finite place w of K , Π w is a discrete series representation whoseparameter is irreducible when composed with Std Sp : SO → GL (forexample the parameter which is trivial on W K w and the “principal SL ” onSU(2)).By Theorem 5.1.2, there is a discrete automorphic representation e Π of
GSpin ( A K )such that e Π | Sp ( A K ) contains Π. We can and do assume that the infinitesimal char-acter of Π is sufficiently regular, so that in particular the parameter ψ of Π istempered. By (4) above, ψ is just a self-dual representation for GL /K with triv-ial central character. Write e ψ for the parameter of e Π. As in the proof of Proposition 6.1.3 3 (i.e. comparing at the unramified placesusing (5.1.3)), we see that 1 ⊞ ψ = V ( e ψ ) ⊗ ω − e ψ . Given the possibilities in Remark6.1.4 we see (using [GJ78] to rule out the case e ψ = π [2], see the proof of Proposition6.1.3 (1)) that e ψ is necessarily tempered. If e ψ = π ⊞ π was of Yoshida type thenwe would have ψ = 1 ⊞ ( π ⊠ π ∨ ), a contradiction. Therefore e ψ is of general type, i.e.a χ -self-dual cuspidal automorphic representation for GL /K of symplectic typefor some character χ of A × K /K × . By the main results of [Kim03] and [Hen09], theLanglands parameter of 1 ⊞ ψ at v equals V (rec( e ψ v )) ⊗ rec( ω e ψ ) − , which impliesthat ϕ Π v = pr ◦ ϕ e Π v . Taking e π = e Π v , we are done in this case.We now treat the case that the parameter ϕ π is not discrete, but is boundedmodulo centre. Recall that a minimal Levi subgroup L M of L Sp such that ϕ π (WD F ) ⊂ L M is unique up to conjugation by Cent ( ϕ π , d Sp ) [Bor79, Proposition3.6]. Then ϕ π factors through a well-defined discrete parameter ϕ M : WD F → L M .Since Sp is quasi-split we have a natural identification of L M with the L -group ofa Levi subgroup M of GSp (well-defined up to conjugation by normalisers in Sp ,resp. d Sp ). Since ϕ π is assumed to be non-discrete we have L M = L Sp . It followsfrom the construction in [Art13] (see the proof of Proposition 2.4.3 loc. cit., inparticular (2.4.13)) that π is a constituent of the parabolic induction Ind G ( F ) P ( F ) π M ,where P is any parabolic subgroup of Sp with Levi M , and π M is in the L -packetof ϕ M . Recall that this L -packet is defined via the natural identification M witha product of copies of GL groups with Sp a for some 0 ≤ a <
2, using rec for the GL factors and Arthur’s local Langlands correspondence for the Sp factor.Write M = f M ∩ Sp where f M is a Levi subgroup of GSp , and similarly P = e P ∩ Sp . Let g π M be an essentially discrete series representation of f M ( F )whose restriction to M ( F ) contains π M . Then there is an irreducible constituent e π of the (semisimple) parabolic induction Ind GSpin ( F ) e P ( F ) g π M such that π is a restrictionof e π . We will prove that ϕ π = pr ◦ ϕ e π . Note that for non-discrete parameters, thelocal Langlands correspondence for GSpin ( F ) of [GT11a] is also compatible withparabolic induction (see [CG15, § e π is ϕ g π M (the Langlands parameter of g π M ) composed with L f M ⊂ L GSpin .Note that f M is isomorphic to a product of GL and for such a group the (bijec-tive) local Langlands correspondence is well-defined, i.e. it does not depend on thechoice of an isomorphism. This follows from compatibility of rec with twisting,central characters and duality. The same argument shows that any morphism withnormal image between two such groups is also compatible with the local Langlandscorrespondence. We have a commutative diagram L f M L GSpin L M L Sp so that to conclude that ϕ π = pr ◦ ϕ e π it is enough to show that ϕ M = pr ◦ ϕ g π M ,which is simply a compatibility of local Langlands correspondences for M and f M .There are three cases to consider. We write the standard parabolic subgroups of RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp GSpin and Sp as in Section 2.2. We do not justify the embedding M ֒ → f M , asthis is a simple but tedious exercise in root data. • f M = GL × GSpin , M = GL , the embedding M ֒ → f M is ( x , x ) ( x x , x /x , x − ). This case is trivial. • f M = GL × GSpin , M = Sp × GL , the embedding M ֒ → f M is( g, x ) ( gx , x − ). This case is not formal as for the factor Sp the localLanglands correspondence that is used is Arthur’s from [Art13] and it is notobvious that it is compatible with rec for GL , in other words that Arthur’slocal Langlands correspondence for Sp ≃ SL (characterised by twistedendoscopy for GL ) coincides with Labesse-Langlands [LL79]. FortunatelyArthur verified this compatibility in [Art13, Lemma 6.6.2]. • f M = GL × GSpin , M = GL × GL , the embedding M ֒ → f M is g (det g, g/ det g ) where we have identified GSpin with GL . This case alsofollows from the above remark about the local Langlands correspondencefor groups isomorphic to a product of GL .Finally, we must treat the case that ϕ is not bounded modulo centre. The de-scription of the L -packets in this case is again in terms of parabolic inductions fromLevi subgroups (“Langlands classification”). This is well-known and completelygeneral (see [Sil78], [SZ14]). The argument is similar to the above reduction, ex-cept that P and e P are uniquely determined by a positivity condition and that π and e π are unique quotients of standard modules and not arbitrary constituents. Wedo not repeat the argument. (cid:3) We now examine the restriction from
GSpin ( F ) to Sp ( F ) more closely, prov-ing a slight refinement of the results of [GT10]. In [GT10, App. A], a detailedqualitative description of the constituents of e π | Sp ( F ) is given, which is obtainedby examining the local Langlands correspondence (see [GT10, §
5, 6] for the cor-responding calculations with L -parameters). However, since the local Langlandscorrespondence of [GT10] is not characterised in terms of twisted and ordinary en-doscopic character relations, they cannot describe precisely which elements of the L -packets for Sp ( F ) arise as the restrictions of given elements of the L -packetsfor GSpin ( F ).Theorem 8.3.2 below answers this question. In its proof, we need to make use ofthe results of Section 5 for SO ֒ → H where H = ( GL × GL ) / { ( zI , z − I | z ∈ GL } is the non-trivial elliptic endoscopic group of GSpin . Here SO is identified withthe subgroup of pairs ( a, b ) with (det a )(det b ) = 1. Indeed, H may be identifiedwith the subgroup GSO of GO given by the elements for which det = ν , where ν is the similitude factor.Note that SO is an elliptic endoscopic group for Sp and that we have thefollowing commutative diagram:(8.1.3) b H d SO = SO \ GSpin = GSp d Sp = SO L ξ ′ L ξ ′ Multiplicity one.
In studying restriction from H to SO we will make useof the following variant of the results of [AP06]. In fact, we could prove the specialcase that we need in a simpler but more ad-hoc fashion by using the descriptionof H in terms of GL , but it seems worthwhile to prove this more general result. Proposition 8.2.1.
Let n ≥ , and let V be a vector space of dimension n over F endowed with a non-degenerate quadratic form q . Let π be an irreducible admissiblerepresentation of GSO(
V, q ) =
GSO ( V, q )( F ) . Then the irreducible constituents ofthe restriction π | SO(
V,q ) are pairwise non-isomorphic.Proof. By [AP06, Theorem 2.3], it suffices to show that there is an algebraicanti-involution τ of GSO ( V, q ) which preserves SO ( V, q ) and takes each SO(
V, q )-conjugacy class in GSO(
V, q ) to itself. To define τ , we set τ ( g ) = ν ( g ) δ n g − δ − n where δ ∈ O( V, q ) is an involution with det δ = −
1. This obviously preserves SO ( V, q ),so we need only check that it also preserves SO(
V, q )-conjugacy classes in GSO(
V, q ).To see this, we claim that it is enough to show that we can write g = xy with x ∈ O( V, q ), y ∈ GO(
V, q ) (so ν ( y ) = ν ( g )) satisfying x = 1, det( x ) = ( − n , y = ν ( y ). Indeed, we then have τ ( g ) = ν ( g ) δ n g − δ − n = δ n ν ( y ) y − x − δ − n = δ n yxδ − n = δ n x − ( xy ) xδ − n = ( xδ − n ) − g ( xδ − n ) , as required. The result then follows from Lemma 8.2.2 below, which is a slightrefinement of [RV16, Thm. A]. (cid:3) Lemma 8.2.2.
Let n ≥ , let K be a field of characteristic , and let V be a vectorspace of dimension n over K endowed with a non-degenerate quadratic form q . If g ∈ GSO(
V, q ) then we can write g = xy with x ∈ O( V, q ) , y ∈ GO(
V, q ) satisfying x = 1 , det( x ) = ( − n , y = ν ( y ) .Proof. We argue by induction on n , the case n = 0 being trivial. Suppose now that n >
0. By [RV16, Thm. A], we can write g = xy with x ∈ O( V, q ), y ∈ GO(
V, q )satisfying x = 1, y = ν ( y ) = ν ( g ). If det( x ) = ( − n then we are done, sosuppose that det( x ) = ( − n +1 and so det( y ) = ( − n +1 ν ( y ) n .Since y = ν ( y ), any eigenvalue (in an extension of K ) of y is a square rootof ν ( y ). Since det( y ) = ( − n +1 ν ( y ) n , we see that the two eigenspaces of y do nothave equal dimension. It follows that ν ( y ) is a square, as otherwise the characteristicpolynomial of y would be a power of the irreducible polynomial X − ν ( y ). So theeigenvalues of y are in K , and up to dividing g and y by one of these eigenvalueswe can assume that g ∈ SO(
V, q ) and y ∈ O( V, q ) with det( y ) = ( − n +1 . Then y has an eigenspace (for an eigenvalue ±
1) of dimension at least n + 1. The sameanalysis applies to x , and it follows that there is a subspace W (the intersection ofthese eigenspaces for x and y ) of dimension at least 2 of V on which g acts by ascalar which is ± g by − g and y by − y , we can assume that ker( g −
1) has dimensionat least 2. We have a canonical g -stable decomposition of V as the direct sum ofker(( g − n ) and its orthogonal complement, and they both have even dimensionover K since g ∈ SO(
V, q ) with dim K V even. If g is not unipotent, we concludeusing the induction hypothesis for the restriction of g to ker(( g − n ) and to itsorthogonal complement.Suppose for the rest of the proof that g is unipotent. If n = 1 the conclusionis trivial, so assume that n >
1, so that SO ( V, q ) is semisimple. By Jacobson–Morozov (see for example [Bou05, Ch. VIII § RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp SL → SO ( V, q ) mapping (cid:18) (cid:19) to g , unique up to conjugation by the centraliserof g in the subgroup Aut e ( so ( V, q )) of SO(
V, q ) / {± } where Aut e is the subgroup ofautomorphisms of the Lie algebra generated by exponentials of nilpotent elements.For d ≥ U d of SL of dimension d as well asa non-degenerate ( − d − -symmetric SL -invariant pairing B d on U d . We have acanonical decomposition V = M d ≥ U d ⊗ V d where V d = ( V ⊗ K U ∗ d ) SL . The quadratic form q corresponds to an element of(Sym V ∗ ) SL = M d ≥ Sym ( V ∗ d ) ⊕ M d ≥ ^ V ∗ d and non-degeneracy of q is equivalent to non-degeneracy of each factor. Writingeach V d for d odd (resp. even) as an orthogonal direct sum of quadratic lines (resp.planes endowed with a non-degenerate alternate form), we are left to prove a de-composition g ′ = x ′ y ′ in the following cases.(1) V ′ has odd dimension 2 m + 1 and is endowed with a non-degenerate qua-dratic form q ′ and a unipotent automorphism g ′ . Applying [RV16, Thm.A] we obtain g ′ = x ′ y ′ with x ′ , y ′ involutions in O( V, q ). Up to replacing( x ′ , y ′ ) by ( − x ′ , − y ′ ) we can assume that det( x ′ ) is ± V ′ = U m ⊗ V ′′′ where V ′′′ is 2-dimensional and endowed with a non-degenerate alternating form B ′′′ , and g ′ = g ′′ ⊗ Id V ′′′ ∈ SO( V ′ , q ′ ) for q ′ the quadratic form corresponding to the symmetric bilinear form B ′ = B m ⊗ B ′′′ and g ′′ a unipotent element of Sp( U m , B m ). Applying [RV16,Thm. A] again we can write g ′′ = x ′′ y ′′ where x ′′ , y ′′ are involutions inGSp( U m , B m ) having similitude factor −
1. Similarly write Id V ′′′ = x ′′′ y ′′′ where x ′′′ , y ′′′ are involutions in GSp( V ′′′ , B ′′′ ) having similitude factor − g ′ = ( x ′′ ⊗ x ′′′ )( y ′′ ⊗ y ′′′ ) is the desired decomposition as a productof involutions in SO( V ′ , q ′ ). (cid:3) Restriction of local Arthur packets.
We now give our description of therestriction of representations of
GSpin ( F ). Recall that if ϕ : WD F → GSp is abounded parameter, then the corresponding component group S ϕ is either trivial oris Z / Z = { , s } . In the former case, the L -packet Π ϕ associated to ϕ is a singleton,and in the latter case it is a pair { π + , π − } , where π ± is characterised by the factthat tr π + − tr π − is the transfer to GSpin ( F ) of tr π ϕ H where ϕ H ∈ Φ( H ) is theparameter mapping to ( ϕ, s ) via L ξ ′ . In either case, if we write ϕ ′ = pr ◦ ϕ , thenby [GT10, Prop. 2.8], we have(8.3.1) M π ∈ Π ϕ π | Sp ( F ) ∼ = M π ′ ∈ Π ϕ ′ π ′ . (Indeed, this follows from Theorem 8.1.2, the fact that lifts of representationsof Sp ( F ) to GSp ( F ) are unique up to twist, and the fact that the restrictionsof representations of GSp ( F ) to Sp ( F ) are semisimple and multiplicity free.)The following theorem improves on this result by giving a precise description of therestrictions of the individual elements of Π ϕ . Theorem 8.3.2.
Let ϕ be a bounded L -parameter, and write ϕ ′ = pr ◦ ϕ , so that S ϕ ֒ → S ϕ ′ . Write Π ϕ and Π ϕ ′ for the respective L -packets. If S ϕ is trivial,and Π ϕ = { π } , then π | Sp ( F ) ∼ = M π ′ ∈ Π ϕ ′ π ′ . If S ϕ = Z / Z = { , s } , and Π ϕ = { π + , π − } as above, then π ± | Sp ( F ) ∼ = M π ′ ∈ Π ϕ ′ h s,π ′ i = ± π ′ . Proof.
In the case that S ϕ is trivial, this is (8.3.1), so we may suppose that S ϕ is non-trivial, so that ϕ is endoscopic. We can write ϕ = ϕ ⊕ ϕ where ϕ , ϕ : WD F → GL are bounded with same determinant; that is, ϕ = L ξ ′ ◦ ϕ H , where ϕ H = ϕ × ϕ : WD F × SL ( C ) → b H . Via L ξ ′ we can see s as the non-trivial element of Z ( b H ) /Z ( \ GSpin ), i.e. the image of (1 , − ∈ b H ⊂ GL × GL . Then by Conjec-ture 2.4.1 (2) for GSpin (i.e. the main theorem of [CG15]), we have an equalityof traces tr π + ( f ) − tr π − ( f ) = X π H ∈ Π ϕ H tr π H ( f H ) . Applying Conjecture 2.4.1 (2) (or rather Theorem 2.6.1) for Sp , and writing ϕ ′ H for the composite of ϕ H and the natural map b H → d SO , we also have an equalityof traces X π ′ ∈ Π ϕ ′ h s,π ′ i =1 tr π ′ ( f ) − X π ′ ∈ Π ϕ ′ h s,π ′ i = − tr π ′ ( f ) = X π ′ SO ∈ Π ϕ ′ H tr π ′ SO ( f ′ ) . The result now follows from (8.3.1) and Theorem 8.3.3 below. (cid:3)
We end with a result on the restriction of representations from H ≃ GSO to SO that we used in the course of the proof of Theorem 8.3.2. The argumentsare very similar to those for GSpin , but are rather simpler, as H has no non-trivialelliptic endoscopic groups. Since H is isomorphic to the quotient of GL × GL by a split torus, the local Langlands correspondence for H , and the correspond-ing endoscopic character identities, are easily deduced from those for GL . Thecorrespondence and endoscopic character identities for SO are of course provedin [Art13] (up to the outer automorphism δ ).By Proposition 8.2.1, if π is an irreducible admissible representation of H ( F ),then π | SO ( F ) is a direct sum of representations occurring with multiplicity one.The proof of [GT10, Lem. 2.6] goes through unchanged and shows that π | SO ( F ) , π | SO ( F ) have a common constituent if and only if π , π differ by a twist bya character. By [GT10, Lem. 2.7], the analogous statement is also true for L -parameters: every L -parameter ϕ ′ : WD F → d SO ( C ) arises from some ϕ : WD F → b H ( C ), which is unique up to twist. Theorem 8.3.3.
Let ϕ : WD F → b H ( C ) be a bounded L -parameter, and let ϕ ′ :WD F → d SO ( C ) be the parameter obtained from (8.1.3) . Let π be the tempered RTHUR’S MULTIPLICITY FORMULA FOR
GSp AND RESTRICTION TO Sp irreducible representation of H associated to ϕ . Then π | e H ( SO ( F )) ∼ = M π ′ ∈ Π ϕ ′ π ′ . Proof.
By the preceding discussion, we need to show that for each bounded L -parameter ϕ ′ : WD F → d SO ( C ) (up to outer conjugacy), and each π ′ ∈ Π ϕ ′ , thereis some π lifting π ′ (or π ′ δ ) whose L -parameter ϕ lifts ϕ ′ .Suppose firstly that ϕ ′ is discrete. As in the proof of Theorem 8.1.2, by Krasner’slemma and [Clo86, Thm. 1B], we can find a totally real number field K , a finiteplace v of K , and a discrete automorphic representation Π ′ of SO ( A K ), such that: • K v ∼ = F (so we identify K v with F from now on). • Π ′ v = π ′ . • at each infinite place w of F , Π ′ w is a discrete series representation.By Theorem 5.1.2, there is a discrete automorphic representation Π of H ( A K )such that Π | SO ( A K ) contains Π ′ . Then Π corresponds to a pair π , π of discreteautomorphic representations of GL ( A K ) with equal central characters. The con-dition that Π ′ w is a discrete series representation at an infinite place w of K impliesthat π and π are cuspidal.We now consider the following commutative diagram of dual groups:(8.3.4) b H d SO = SO GL × GL GL where the vertical arrows are the natural inclusions, and the lower horizontal arrowis given by ( g, h ) (det g ) − g ⊗ h . Since the functorial transfer from GL × GL to GL exists (as we recalled at the beginning of Section 6), we may compare at theunramified places and then use strong multiplicity one to compare at the ramifiedplaces, and we obtain that the composite WD F ϕ ′ → b H → GL × GL → GL is givenby ϕ ,v ⊗ ϕ ∨ ,v , where ϕ ,v , ϕ ,v are the L -parameters of π ,v and π ,v respectively.Since the L -parameter of Π v is ϕ ,v ⊕ ϕ ,v , we can take π = Π v , so we are done inthe case that ϕ ′ is discrete.Suppose now that ϕ ′ is not discrete. Then one can argue as in the proof of 8.1.2,since both local Langlands correspondences for H and SO are compatible withparabolic induction. In fact the proof is simpler since all proper Levi subgroups aresimply products of GL , and we do not repeat the argument. (cid:3) Remark 8.3.5.
Theorem 8.3.2 (or rather its straightforward extension from tem-pered to generic parameters) gives the complete spectral description of the automor-phic restriction map of Section 5 for Sp ⊂ GSpin for formally tempered global pa-rameters . This is the analogue of the results of Labesse–Langlands [LL79] (ignoringinner forms) and the multiplicity one theorem of Ramakrishnan for SL [Ram00].It would perhaps be interesting to extend this to parameters which are not formallytempered, but in the interests of brevity we do not consider this question here. References [AMR15] Nicol´as Arancibia, Colette Mœglin, and David Renard,
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E-mail address : [email protected] Department of Mathematics, Imperial College London, London SW7 2AZ, UK
E-mail address : [email protected]@ens-lyon.fr