Periods of Automorphic Forms Associated to Strongly Tempered Spherical Varieties
PPERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLYTEMPERED SPHERICAL VARIETIES
CHEN WAN AND LEI ZHANG
Abstract.
In this paper, we compute the local relative character for 10 strongly temperedspherical varieties in the unramified case. We also study the local multiplicity for thesemodels. By proving a multiplicity formula, we show that the summation of the multiplicitiesis always equal to 1 over each local tempered Vogan L -packet defined on the pure inner formsof the spherical varieties. Finally, we formulate the Ichino–Ikeda type conjecture on a relationbetween the period integrals and the central values of certain automorphic L -functions forthose strongly tempered spherical varieties. Contents
1. Introduction and main results 21.1. The local relative character 41.2. The local multiplicity 71.3. The Ichino–Ikeda type conjecture 71.4. Organization of the paper 81.5. Acknowledgments 92. The strategy 92.1. Notation 92.2. The reductive case: some reduction 102.3. The computation of S θ (cid:82) ∗ K Y θ,ξ ( k ) d k × GSp , (GSp × GSp ) ) 283.1. The model and some orbit computation 293.2. The computation 314. The model (GL × GL , GL × GL ) 345. The model (GL , GL (cid:110) U ) 376. The models (GU , GU (cid:110) U ) and (GU × GU , (GU × GU ) ) 406.1. The models 406.2. Two identities 416.3. The computation for (GU , GU (cid:110) U ) 426.4. The computation for (GU × GU , (GU × GU ) ) 467. The model ( E , PGL (cid:110) U ) 498. The remaining models 568.1. The model (GSp , GL (cid:110) U ) 56 Mathematics Subject Classification.
Primary 11F67; Secondary 22E50.
Key words and phrases.
Period integrals of automorphic forms, central values of automorphic L -functions,local multiplicity of spherical varieties, strongly tempered spherical varieties. a r X i v : . [ m a t h . N T ] F e b CHEN WAN AND LEI ZHANG × GL , GL (cid:110) U ) 598.3. The model (GSO , GL (cid:110) U ) 618.4. The model (GSO × GL , GL (cid:110) U ) 649. Local multiplicity 669.1. The character identity between inner forms 669.2. The reductive case 679.3. The non-reductive case 72References 731. Introduction and main results
Let k be a number field and A its ring of adeles. Let G be a reductive group defined over k , and H a closed reductive subgroup of G . Assume that X = H \ G is a spherical G -variety(i.e., a Borel subgroup of G has a dense orbit in X ). Let Z G be the center of G and let Z G,H = Z G ∩ H . For a cuspidal automorphic form φ on G ( A ) whose central character istrivial on Z G,H ( A ), we define the period integral P H ( φ ) to be P H ( φ ) := (cid:90) H ( k ) Z G,H ( A ) \ H ( A ) φ ( h ) d h. Besides the reductive cases, one can also study the case when the spherical pair (
G, H )is the Whittaker induction of a reductive spherical pair ( G , H ) (we refer the reader toDefinition 2.2 for the definition of Whittaker induction). In this case, we have H = H (cid:110) U where U is the unipotent radical of H and is also the unipotent radical of a parabolicsubgroup of G , and the period integral is defined to be P H ( φ ) := (cid:90) H ( k ) Z G,H ( A ) \ H ( A ) φ ( h ) ξ ( h ) − d h where ξ = Π v ξ v is a generic character on U ( k ) \ U ( A ), extended to H ( A ) trivially on thereductive part H ( A ). We refer the reader to Definition 2.1 for the definition of genericcharacters.Let π be a cuspidal automorphic representation of G ( A ) whose central character is trivialon Z G,H ( A ). One of the most fundamental problems in the relative Langlands programis to establish the relation between P H | π -the period integral restricted to the space of π ,and special values of some automorphic L -functions L ( s , π, ρ X ) of π . For example, if G =SO n +1 × SO n and H = SO n , then ( G, H ) is the famous Gross–Prasad model and its periodintegrals are related to the central value of the tensor L -function L (1 / , π × π ) (here π = π ⊗ π is a cuspidal automorphic representation of SO n +1 ( A ) × SO n ( A )). This point ofview was most systematically put forward by Sakellaridis [Sa12], and Sakellaridis-Venkatesh[SV17]. As in [SV17], the spherical varieties under the consideration in this paper have noType N spherical root and are wavefront. We refer the reader to Sections 2.1 and 3.1 of[SV17] for the definitions of wavefront and spherical roots.In general, in order to find the L -functions related to the period integral P H ( φ ) for φ = ⊗ v φ v ∈ ⊗ v π v , one needs to compute the local relative character I H v ( φ v ) for the sphericalpair ( G v , H v ) := ( G ( k v ) , H ( k v )) over unramified places v ∈ | k | . If the model ( G, H ) is In general if we allow φ to have nontrivial central character, then we can also put some character on H ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES3 strongly tempered (see Section 2.1 for the definition of strongly tempered) or is the Whittakerinduction of a strongly tempered pair ( G , H ), the local relative character I H v ( φ v ) is definedto be the integration of the matrix coefficients over H ( k v ), i.e.(1.1) I H v ( φ v ) = (cid:90) Z G,H ( k v ) \ H ( k v ) (cid:104) π v ( h ) φ v , φ v (cid:105) ξ v ( h ) − d h. Note that if (
G, H ) is the Whittaker induction of a strongly tempered pair, the integralabove needs to be regularized (see Section 2.4 for details). In general, if the model (
G, H )is not strongly tempered, the local relative character I H v ( φ v ) is defined via the Plancherelformula. For details, see Section 17.3 of [SV17].For each spherical pair ( G, H ), one expects that the local relative character I H v ( φ v ) equalsthe product of some local L -functions L ( s ,π v ,ρ X ) L (1 ,π v , Ad) with a product of certain special values oflocal zeta functions (denoted by ∆ X v ) over all the unramified places. For instance, for theorthogonal Gross–Prasad model (which is strongly tempered), the local relative characterwas computed by Ichino–Ikeda [II], equal to L ( , π ,v × π ,v ) L (1 , π v , Ad) · ∆ SO n +1 ,v (1) . Here for any reductive group G defined over k that is split over an unramified extension,we use ∆ G ( s ) = Π v ∈| k | ∆ G,v ( s ) to denote the L -function of the dual M ∨ to the motive M introduced by Gross in [G].In [Sa], Sakellaridis developed a general method to compute the local relative characterat unramified places under certain conditions. He showed that the L -function L ( s, π, ρ X ) isdetermined by the so-called “virtual colors” of the spericial variety X and the extra factor∆ X v is related to the volume of X ( O v ) ( O v is the ring of integers of k v ). He also explicitlycomputed the virtual colors of many spherical varieties and hence the L -functions L ( s, π, ρ X )(see Page 1379 of [Sa]).In this paper, following the method of Sakellaridis, we explicitly compute the local relativecharacters for all the strongly tempered reductive spherical varieties without the Type N spherical root. We also compute the local relative characters for 7 non-reductive sphericalvarieties that are the Whittaker inductions of the trilinear GL model (GL , GL ). Ourcomputation shows that the period integrals for these strongly tempered spherical varietiesare always related to the central value of some L -functions of symplectic type, i.e. s = and ρ X is a self-dual representation of ˆ G of symplectic type. Moreover, we show that theextra factors ∆ X v is equal to ∆ G,v (1) / ∆ H /Z G,H ,v (1) for all the models under consideration(we would like point out that this is only true in the strongly tempered case). Note if H isreductive we just let H = H and U = 1.In addition, we study the local multiplicities for all models (except for the E case). Byproving a multiplicity formula, we show that the summation of the multiplicities is alwaysequal to 1 over each local tempered Vogan L -packet defined on the pure inner forms of thesespherical varieties. In other words, our results indicate that all these strongly temperedspherical varieties enjoy the same local and global properties with the Gan–Gross–Prasadmodels.Finally, combining our formulas of the local relative characters and our results for the localmultiplicities, we are able to formulate the Ichino–Ikeda type conjectures for these models. CHEN WAN AND LEI ZHANG
The local relative character.
By the classification of split reductive spherical vari-eties in [BP], it is easy to show that a split strongly tempered reductive spherical pair iseither one of the following 4 cases(1.2)(GL n +1 × GL n , GL n ) , (SO n +1 × SO n , SO n ) , (GL × GL , GL × GL ) , (GSp × GSp , (GSp × GSp ) ) , or it is a split symmetric pair, (GL n , SO n ) and (Sp n , GL n ) for instance. Here (GSp × GSp ) = { ( g, h ) ∈ GSp × GSp | l ( g ) = l ( h ) } where l is the similitude character of GSp.We refer the reader to Section 3.1 for the explicit description of the embeddings. All the splitsymmetric pairs have Type N spherical root unless G only has one simple root. If G onlyhas one simple root, then split symmetric pair ( G, H ) is essentially the model (PGL , GL ).So we only need to consider the 4 models in (1.2). Remark 1.1.
For each model in (1.2) , we can always modify the groups up to some centralelements and some finite isogeny, which will give us some other models with the same rootsystems (this will also preserve the strongly tempered property). For example, the model (GSp × GSp , (GSp × GSp ) ) and the model (Sp × Sp , Sp × Sp ) have the same rootsystems. In this paper, we will always choose the spherical pairs ( G, H ) so that over thelocal field k v , there is only one open Borel orbit in G ( k v ) /H ( k v ) . For example, the model (GSp × GSp , (GSp × GSp ) ) we choose indeed has only one open Borel orbit (see Section3.1) while the model (Sp × Sp , Sp × Sp ) has | k × v / ( k × v ) | -many open Borel orbits. The first one (GL n +1 × GL n , GL n ) is the model for the Rankin-Selberg integral of GL n +1 × GL n . There is also an analogue of this model for unitary groups, which is call the unitaryGan–Gross–Prasad model. The local relative characters have been computed by R. NealHarris [H] for both the general linear case and the unitary case. The second one (SO n +1 × SO n , SO n ) is the Gross–Prasad model for special orthogonal groups and the local relativecharacter has been computed by Ichino and Ikeda [II]. For these three models, the periodintegrals are related to the central values of the tensor L -functions.In this paper, we give an explicit formula for the local relative characters over unramifiedplaces for the remaining two cases (GL × GL , GL × GL ) and (GSp × GSp , (GSp × GSp ) ), as well as the analogue of the model (GL × GL , GL × GL ) for unitary groups.We also computed 7 non-reductive cases that are the Whittaker inductions of the trilinearGL -model (GL , GL ) (which is strongly tempered). № G H ρ X ∆ X,v = ∆
G,v (1) / ∆ H /Z G,H ,v (1)1 GL × GL GL × GL ( ∧ ⊗ std ) ⊕ std ⊕ std ∨ ζ v (1) ζ v (3) ζ v (4)2 GU × GU (GU × GU ) ( ∧ ⊗ std ) ⊕ std ⊕ std ∨ ∗ × GSp (GSp × GSp ) Spin ⊗ Spin ζ v (1) ζ v (4) ζ v (6)4 GL GL (cid:110) U ∧ ζ v (1) ζ v (3) ζ v (4) ζ v (5) ζ v (6)5 GU GU (cid:110) U ∧ ∗∗ GL (cid:110) U Spin ζ v (1) ζ v (4) ζ v (6) ζ v (8) ζ v (10)7 GSp × GL GL (cid:110) U Spin ⊗ std ζ v (1) ζ v (2) ζ v (4) ζ v (6)8 GSO × GL GL (cid:110) U HSpin ⊗ std ζ v (1) ζ v (2) ζ v (4) ζ v (6)9 GSO GL (cid:110) U HSpin ζ v (1) ζ v (4) ζ v (6) ζ v (8) ζ v (10)10 E PGL (cid:110) U ω ζ v (6) ζ v (8) ζ v (10) ζ v (12) ζ v (14) ζ v (18) Table 1
ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES5
Here std n is the standard representation of GL n ( C ) and std ∨ n is its dual representation,Spin n +1 is the Spin representation of GSpin n +1 ( C ), HSpin n is a half-Spin representationof GSpin n ( C ), ω is the 56 dimensional representation of E , and ∗ = ζ v (1) ζ v (4) L (1 , η k (cid:48) v /k v ) L (3 , η k (cid:48) v /k v ) , ∗∗ = ζ v (1) ζ v (4) ζ v (6) L (1 , η k (cid:48) v /k v ) L (3 , η k (cid:48) v /k v ) L (5 , η k (cid:48) v /k v )where η k (cid:48) v /k v is the quadratic character for the quadratic extension k (cid:48) v /k v . We refer the readerto Section 6 for more details about the representation ρ X for Models 2 and 5. Theorem 1.2.
For all the spherical pairs in Table 1, assume that all the data are unramifiedover v . Then (1.3) I H v ( φ v ) = ∆ G,v (1)∆ H /Z G,H ,v (1) · L ( , π v , ρ X ) L (1 , π v , Ad) where ρ X is a self-dual symplectic representation of ˆ G given in Table 1. Remark 1.3. In (1.1) , we choose the local Haar measure d h such that vol( H ( O v ) , d h ) = 1 .If we replace it by Weil’s canonical measure d can h = ∆ H /Z G,H ,v (1)d h ( [Weil, Chapter 2] ),then the constant ∆ H /Z G,H ,v (1) in the above theorem will disappear. In Section 2, we will explain our strategies of the proof of this theorem. We will also givethe formulas of the Whittaker–Shintani functions of these 10 spherical pairs in Propositions2.13 and 2.29.For the rest of this subsection, we explain how we derive the non-reductive models inTable 1. Model 4 was introduced by Ginzburg–Rallis in [GR] and Model 5 is an analogueof Model 4 for similitude unitary groups. Model 9 and 10 are inspired by one row of theMagic Triangle introduced by Deligne and Gross in [DG] (which is a generalization of thethe Freudenthal’s Magic Square). We recall the following row in the Magic Triangle in [DG,Table 1], a series of algebraic groups of type: A ⊂ A := A × A × A ⊂ C ⊂ A ⊂ D ⊂ E . In this sequence, we observe the spherical pair of type ( A , A ) corresponding to the trilinearGL -model. And the algebraic groups G of types A , D and E have a parabolic subgroup P = LU such that the Levi subgroup L is of type A and the stabilizer H of the genericcharacters ξ of U is of type A . This gives us the Whittaker inductions of the trilinear GL -model for these 3 groups, which are the Models 4, 9, and 10 respectively. Meanwhile, thegroup of type C does not have a Levi subgroup of type A , but it can be fixed by consideringthe product C × A . This explains Model 7.In addition, these non-reductive models are also related to the degenerated Whittakermodels of smooth admissible representations (we refer the reader to [GZ] for more details.)For instance, consider the degenerated Whittaker model W h ξ ( π ) of an irreducible representa-tion π of GSO with respect to ( U, ξ ) in Model 9. Here (
U, ξ ) is arisen from a nilpotent orbitof partition [6 ,
6] in the Lie algebra of GSO and W h ξ ( π ) is considered as an H -module.(Note that the partition [6 ,
6] is used to label two distinct stable nilpotent orbits. However,the corresponding models have no essential differences as explained in Section 8.3.) Thedistinguished problem in Model 9 is equivalent to determine when the trivial representationis a quotient representation in
W h ξ ( π ). By using the theta correspondence, Gomez and Zhuin [GZ] showed that the H -module W h ξ ( π ) is isomorphic to the degenerated Whittakermodel of certain representations of GSp as an H -module, arisen from the nilpotent orbitof the partition [5 ,
5] in the Lie algebra of GSp . Hence, following [GZ], Model 6 and Model CHEN WAN AND LEI ZHANG in Model 9 whose Levi subgroup is isomorphic toGSO × GL such that the intersection of the Levi subgroup with the subgroup H of GSO in Model 9 is exactly the subgroup H of GSO × GL in Model 8. Under this point ofview, we can also view Model 7 as a reduced model of Model 6, view Model 4 as a reducedmodel of Model 9 and view Model 9 as a reduced model of Model 10. This explains all thenon-reductive models in Table 1. We summarize the relations among these models in thefollowing diagram: (GL , GL (cid:111) U ) outer auto. (cid:47) (cid:47) (cid:79) (cid:79) reduced (GU , GU (cid:111) U )( E , PGL (cid:111) U ) reduced (cid:47) (cid:47) (GSO , GL (cid:111) U ) reduced (cid:47) (cid:47) (cid:79) (cid:79) θ -correspondence (cid:15) (cid:15) (GSO × GL , GL (cid:111) U ) (cid:79) (cid:79) θ -correspondence (cid:15) (cid:15) (GSp , GL (cid:111) U ) reduced (cid:47) (cid:47) (GSp × GL , GL (cid:111) U ) Remark 1.4.
Besides the 7 non-reductive cases in the table above, there are another threenon-reductive spherical pairs that are the Whittaker induction of strongly tempered reductivespherical pairs without Type N spherical root:(1) The Whittaker models for quasi-split reductive groups.(2) The non-reductive Gan–Gross–Prasad models for the general linear groups, the uni-tary groups, or the orthogonal groups. They are the Whittaker inductions of thereductive Gan–Gross–Prasad models.(3) The model (GSO , (GL × GL ) (cid:110) U ) introduced by Ginzburg [Gi] in his study ofthe Spin L-function of GSO . This is the Whittaker induction of the model (GL × GL , GL ) .The local relative characters of the Whittaker models have been computed by Lapid-Mao in [LM] ; the local relative characters of the non-reductive Gan–Gross–Prasad models have beencomputed by Liu in [L] ; and the local relative character of (GSO , (GL × GL ) (cid:110) U ) hasbeen computed by Ginzburg in [Gi] . The local relative characters over unramified places forthese models are also of the form (1.3) as our models in Table 1. The representation ρ X is the tensor representation for the non-reductive Gan–Gross–Prasad models, and the Spinrepresentation of GSpin ( C ) for the model (GSO , (GL × GL ) (cid:110) U ) . For the Whittakermodel, the numerator L -function L ( , π, ρ X ) is just 1.In general, by a tedious case by case argument (i.e. we checked all the parabolic subgroupsof all the reductive groups) which we will not include in this paper, we believe that anyspherical pair that are the Whittaker induction of a strongly tempered spherical pair withoutType N spherical root must be one of the 10 cases above (7 in Table 1 and 3 in this remark).Hence the local relative character of a spherical pair that is either strongly tempered or theWhittaker induction of a strongly tempered spherical pair should always be the form (1.3) over unramified places. ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES7
The local multiplicity.
Let (
G, H ) be one of the models in Table 1. If H is reductive,take χ to be the trivial character of H ( k v ); if H = H (cid:110) U is non-reductive, take χ to be thecharacter 1 ⊗ ξ v of H ( k v ) = H ( k v ) (cid:110) U ( k v ) where ξ v is the generic character of U ( k v ). Let π v be an irreducible representation of G ( k v ), whose central character is trivial on Z G,H ( k v ).Define the multiplicity m ( π v ) := dim Hom H ( k v ) ( π v , χ v ) . In Section 9, for all the models in Table 1 except the E case, we will prove a multiplicityformula m ( π v ) = m geom ( π v ) for all the tempered representations over non-archimedean fieldsor complex field. Then using this multiplicity formula, together with the assumption thatthe conjectural character identity holds for the local L -packet of G , we proved that thesummation of the multiplicities is always equal to 1 over every local tempered Vogan L -packet. We can also prove the same results for Models 1–4 in the real case. Remark 1.5.
The local multiplicity of the some models in Table 1 has already been studied inour previous works. More specifically, Model 4 has been studied by the first author ( [Wan15] , [Wan16] , [Wan17] ), Model 5 has been studied in our previous paper [WZ] , and Model 1 hasbeen studied in [PWZ19] . Remark 1.6.
Like in the Gan–Gross–Prasad model case (Section 17 of [GGP] ), one canalso formulate an explicit conjecture about the distinguished element in the L-packet usingthe local epsilon factor (cid:15) ( s, π v , ρ X ) . We will discuss this in our future work. The Ichino–Ikeda type conjecture.
Combining the results in the previous two sub-sections, we can now formulate the Ichino–Ikeda type conjectures for all the models in Table1. Let (
G, H ) be one of these models. Following the definition in Section 16.5 of [SV17],the pure inner forms of the spherical varieties are parameterized by the set H ( H/Z
G,H , k ).For all the models in Table 1 except Model 2, there is a natural bijection between the set H ( H/Z
G,H , k ) and the set of quaternion algebras D over k . For each quaternion algebra D/k (or for each D ∈ H ( H/Z
G,H , k ) in the case of Model 2), we can define an analogueof the model (
G, H ) associated to D , which will be denoted by ( G D , H D ). We can alsodefine the period integral P H D ( φ D ) and the local relative character I H D,v ( φ D,v ) where φ D is acuspidal automorphic form on G D ( A ). We refer the reader to later sections for the detaileddescriptions of ( G D , H D ) for each spherical variety in Table 1. Remark that in our cases G D and H D are not the pure inner forms of G and H in general. But after module the centralpart Z G,H , they become pure inner forms of
G/Z
G,H and
H/Z
G,H , respectively.We fix a global tempered cuspidal L -packet Π φ = ∪ D Π φ ( G D ) of G ( A ), whose centralcharacter is trivial on Z G,H ( A ). For each π D ∈ Π φ ( G D ) in the L -packet, as in Section17.4 of [SV17], let ν : π D → A cusp ( G D ( A )) be an embedding such that the period integralis identically zero on the orthogonal complement of ν ( π D ) in the π D -isotypic component A cusp ( G D ( A )) π D . This embedding is not unique if the multiplicity of π D in A cusp ( G D ( A )) isgreater than 1, but it does not affect the global conjecture.We first consider all the models in Table 1 except the first one. For those models, thecenter of H/Z
G,H is anisotropic.
Conjecture 1.7.
Let
D/k be a quaternion algebra that maybe split (or D ∈ H ( H/Z
G,H , k ) if we are in the case of Model 2), π D ∈ Π φ ( G D ) and φ D ∈ ν ( π D ) . We have |P H D ( φ D ) | = 1 | S φ | · C H/Z
G,H ∆ H /Z G,H (1) S · lim s → ∆ G ( s ) S L (1 , Π φ , Ad ) S · L (1 / , Π φ , ρ X ) S · Π v ∈ S I H D,v ( φ D,v ) CHEN WAN AND LEI ZHANG where • S is a finite subset of | k | such that φ is unramified outside S , and ∆ H/Z
G,H (1) S , ∆ G ( s ) S ,L (1 / , Π φ , ρ X ) S , L (1 , Π φ , Ad ) S are the partial L-functions. • C H/Z
G,H is the Haar measure constant of
H/Z
G,H defined in Section 1 of [II] (see alsoSection 1 of [L] ), and the period integral P H D is defined by the Tamagawa measureon Z G D ,H D ( A ) \ H D ( A ) . • S φ is the conjectural global component group associated to the L-packet Π φ . We referthe reader to Section 3.2 of [LM] for details. Then we consider the first model (GL × GL , GL × GL ) in Table 1. In this case, wehave Z H /Z G,H ∼ = GL . Conjecture 1.8.
Under the above notation, we have |P H D ( φ D ) | = 1 | S φ | · C H/Z H ∆ H /Z H (1) S · lim s → ∆ G ( s ) S L (1 , Π φ , Ad ) S · L (1 / , Π φ , ρ X ) S · Π v ∈ S ζ v (1) I H D,v ( φ D,v ) . Note that we have the extra factor ζ v (1) due to Z H /Z G,H = GL . This point of view hasbeen discussed in Section 17.5 of [SV17].In particular, we have the following weak global conjecture, which is a direct consequenceof the conjectures above and the multiplicity-one theorems on the local Vogan packets. Conjecture 1.9.
The following are equivalent:(1) L ( , Π φ , ρ X ) (cid:54) = 0 ;(2) There exists a quaternion algebra D/k (or D ∈ H ( H/Z
G,H , k ) if we are in the caseof Model 2) such that the period integral P H D ( φ D ) is nonzero for some φ D ∈ ν ( π D ) and π D ∈ Π φ ( G D ) .Moreover, if the above conditions hold, there exist a unique D and a unique π D ∈ Π φ ( G D ) that satisfy Condition (2). When
D/k is split, one direction of Conjecture 1.9 has been proved for Models 1 and 4 injoint works of the first author with Pollack and Zydor ([PWZ18], [PWZ19]).Finally, similar to Gross–Prasad models as discussed in Section 27 of [GGP], one expectsthat the central value of L -functions in Models 2 and 3 of Table 1 are related to the arithmeticgeometry of the cycles of the certain Shimura varieties. In Model 2, GU × GU and (GU × GU ) ◦ can be associated with Shimura varieties of dimensions 5 and 2 (resp. 3 and 1). InModel 3, GSp × GSp and (GSp × GSp ) ◦ can be associated with Shimura varieties ofdimensions 9 and 4. Then predicted by Beilinson–Bloch Conjecture, the order of L ( s, π, ρ X )at s = 1 / L (cid:48) (1 / , π, ρ X ).1.4. Organization of the paper.
In Section 2, we explain the strategy of our computationof the local relative characters. In Sections 3 and 4, we compute the local relative charactersfor the two split reductive cases in Table 1. In Sections 5 and 7, we study the non-reductivecases for GL and E , respectively. In Section 6, we deal with the non-split models (GU × GU , (GU × GU ) ) and (GU , GL (cid:110) U ) in Table 1. In Section 8, we compute the formulasfor the remaining 4 models. Finally, in Section 9, we will study the local multiplicity for allthese models. ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES9
Acknowledgments.
We thank Yiannis Sakellaridis for the helpful comments on thefirst draft of this paper and for the helpful discussions about the virtual colors. We thankAaron Pollack and Michal Zydor for the many helpful discussions about magic triangle whichlead to the discovery of Model 9 and 10 in Table 1. We thank Wei Zhang for the helpfuldiscussions about Model 3 in Table 1. We thank Dihua Jiang and Yifeng Liu for the helpfulcomments on the first draft of this paper. The work of the first author is partially supportedby the NSF grant DMS-2000192 and DMS-2103720. The work of the second author ispartially supported by AcRF Tier 1 grants R-146-000-277-114 and R-146-000-313-144 ofNational University of Singapore. 2.
The strategy
In this section, we will explain the strategy of our computation. In the reductive cases,we closely follow the method developed by Sakellaridis in [Sa]. For the Whittaker inductioncases, due to the non-trivial unipotent radical of H , the local characters I H v ( φ v ) in (1.1)in these cases are not absolutely convergent. To overcome this convergent issue, we modifythe method by regularizing the unipotent integrals. Then for all cases, we can reduce thecomputation of local relative characters to evaluate the local integrals associated to eachsimple root of G and verify certain combinatoric identities. We refer the reader to thedetailed strategies in Section 2.3.1 for the reduction case and in Section 2.5.1 for the non-reductive case.More precisely, in Section 2.1 we discuss some notation and conventions of spherical vari-eties. Then we discuss the strategies for the reductive cases in Sections 2.2 and 2.3, and forthe non-reductive cases in Sections 2.4 and 2.5, respectively.In Sections 2–8, we only consider the non-archimedean places v such that all data areunramified. Denote by F = k v a p -adic field. Let O F be its ring of integers. Fix a uniformizer (cid:36) , and denote by F q the residue field of F with cardinality q and of characteristic p with p (cid:54) = 2. Fix a nontrivial unramified additive character ψ : F → C × of F .2.1. Notation.
Let G be a connected reductive group defined over F , and Z G be the centerof G . We fix a maximal open compact subgroup K of G ( F ) and let d g be the Haar measureon G ( F ) such that the volume of K is equal to 1. Denote by W G the Weyl group of G ( F ). Definition 2.1.
Let P = LU be a proper parabolic subgroup of G defined over F . For acharacter ξ : U ( F ) → C × of U ( F ) , denote by L ξ the neutral component of the stabilizer of ξ in L (under the adjoint action).A character ξ is called a generic character of U ( F ) if dim( L ξ ) is minimal, i.e. dim( L ξ ) ≤ dim( L ξ (cid:48) ) for any character ξ (cid:48) of U ( F ) . It is easy to see that if ξ is a generic character, sois l ξ for all l ∈ L ( F ) , where l ξ is the character of U ( F ) defined by l ξ ( n ) = ξ ( l − nl ) . Moreover, there are finitely many generic characters of U ( F ) up to L ( F )-conjugation,which is in bijection with the open L ( F )-orbits in u ( F ) / [ u ( F ) , u ( F )] induced by the adjointaction on the Lie algebra u ( F ) of U ( F ).Let H ⊂ G be a connected closed subgroup also defined over F . We say that H is aspherical subgroup if there exists a Borel subgroup B of G (not necessarily defined over F since G ( F ) may not be quasi-split) such that BH is Zariski open in G . Such a Borelsubgroup is unique up to H ( ¯ F )-conjugation. Then, ( G, H ) is called a spherical pair and X = G/H is the corresponding spherical variety of G . From now on, we assume that H is a spherical subgroup. We say the spherical pair ( G, H )is reductive if H is reductive. Definition 2.2.
A spherical pair ( G, H ) is called the Whittaker induction of a reductivespherical pair ( G , H ) if there exist a parabolic subgroup P = LU of G , and a genericcharacter ξ : U ( F ) → C × such that H = H (cid:110) U where G ∼ = L and H ∼ = L ξ ⊂ L is theneutral component of the stabilizer of ξ in L . Alternatively, we say that (
G, H ) is the Whittaker induction of ( G , H , ξ ). For conve-nience, we also consider a reductive spherical pair ( G, H ) as the Whittaker induction of(
G, H,
Remark 2.3.
In general the stabilizer of a generic character is not necessarily a reductiveor spherical subgroup of L . For instance, if we take G = GL and a parabolic subgroupwith Levi subgroup L ∼ = GL × GL , then L ξ is isomorphic to the Borel subgroup of GL ,which is not reductive; if we take G = GL and a parabolic subgroup with Levi subgroup L ∼ = GL × GL × GL , then L ξ ∼ = GL is not a spherical subgroup of L . Finally, for a reductive spherical pair (
G, H ), we say it is strongly tempered if all thetempered matrix coefficients of G ( F ) is absolutely convergent on H ( F ) /Z G,H ( F ). If thespherical pair ( G, H ) is the Whittaker induction of a reductive spherical pair ( G , H ), wesay ( G, H ) is strongly tempered if ( G , H ) is strongly tempered.In the rest of this section, we assume that G is split (this is true for all the models inTable 1 except the GU × GU and GU cases). The computation for the quasi-split case isslightly different from the split case. We refer the reader to Section 6 for details.2.2. The reductive case: some reduction.
Let (
G, H ) be a reductive strongly temperedspherical pair with G ( F ) split. Assume that it does not have Type N spherical root. Let B = T N be a Borel subgroup of G defined over F , T the maximal split torus in B and N the unipotent radical of B , and ¯ B = T ¯ N be its opposite. There exists a unique open Borelorbit B ( F ) ηH ( F ) (note that for each root system, we already choose suitable representatives( G, H ) so that it has unique open Borel orbit, see Remark 1.1). For all the four models in(1.2), it is easy to verify H ( F ) ∩ η − B ( F ) η = Z G,H ( F ), i.e. the stabilizer of the open Borelorbit belongs to the center of G . Remark 2.4.
This is not true if the spherical pair has a Type N spherical root. For example,for the model (GL , SO ) , the stabilizer of the open orbit is isomorphic to Z / Z and does notbelong to the center of G . Our goal is to compute the local relative character I ( φ θ ) = (cid:90) H ( F ) /Z G,H ( F ) φ θ ( h ) d h where φ θ is the unramified matrix coefficient of I GB ( θ ) normalized by φ θ (1) = 1, θ is a unitaryunramified character of T ( F ), and I GB ( · ) is the normalized induced representation from theBorel subgroup B . The integral is absolutely convergent since ( G, H ) is strongly tempered.We follows the method in Sections 6-7 of [Sa].Let f θ be the unramified vector in I GB ( θ ) with f θ (1) = 1. Then the normalized unramifiedmatrix coefficient φ θ is given by φ θ ( g ) = (cid:82) K f θ ( kg ) d k. This implies that I ( φ θ ) = (cid:90) H ( F ) /Z G,H ( F ) φ θ ( h ) d h = (cid:90) H ( F ) /Z G,F ( F ) (cid:90) K f θ ( kh ) d k d h = (cid:90) K (cid:90) H ( F ) /Z G,F ( F ) f θ ( kh ) d h d k. ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES11
Note that since the integral is convergent if we replace θ by its absolute value (which changes f θ to f | θ | = | f θ | ), the above double integral is absolutely convergent. In particular, the integral(2.1) (cid:90) H ( F ) /Z G,F ( F ) f θ ( kh ) d h is absolutely convergent for almost all k ∈ K . As a function on k ∈ G , this integral is right H ( F )-invariant and left ( B ( F ) , δ / B θ )-invariant, where δ B is the modular character of B .Since B ( F ) ηH ( F ) is open in G ( F ), we have the integral (2.1) is absolutely convergent forall k ∈ B ( F ) ηH ( F ).On the other hand, consider the function Y θ on G ( F ) satisfying the following conditions:(1) Y θ is supported on the open orbit B ( F ) ηH ( F ) with Y θ ( η ) = 1;(2) Y θ is right H ( F )-invariant and left ( B ( F ) , θ − δ / B )-invariant.For g ∈ B ( F ) ηH ( F ), Y θ − ( g ) is proportional to (2.1) and then (cid:90) H ( F ) /Z G,F ( F ) f θ ( gh ) d h = (cid:90) H ( F ) /Z G,H ( F ) f θ ( ηh ) d h · Y θ − ( g ) . In consequence, since the complementary set of B ( F ) ηH ( F ) has measure zero, we have I ( φ θ ) = (cid:90) K (cid:90) H ( F ) /Z G,F ( F ) f θ ( kh ) d h d k = (cid:90) K Y θ − ( k ) d k × (cid:90) H ( F ) /Z G,H ( F ) f θ ( ηh ) d h. To obtain a formula of I ( φ θ ), it suffices to compute (cid:90) K Y θ − ( k ) d k and (cid:90) H ( F ) /Z G,H ( F ) f θ ( ηh ) d h. To evaluate the integral (cid:82) H ( F ) /Z G,H ( F ) f θ ( ηh ) d h , we need the following lemma. Lemma 2.5.
Under the above notation, for f ∈ C ∞ c ( G ( F )) , we have (cid:90) G f ( g ) d g = ∆ G (1)∆ H/Z
G,H (1) ζ (1) − rk ( G ) (cid:90) H ( F ) /Z G,H ( F ) (cid:90) B ( F ) f ( bηh ) d b d h, where rk ( G ) is the F -rank of G .Proof. Without loss of generality, it is sufficient to consider the case η = 1, that is, H ( F ) B ( F )is an open dense subset of G ( F ). Denote by d can g , d can b , and d can h the Weil’s canonicalmeasures on the smooth varieties G , B and H/Z
G,H , respectively. Since B ∩ H = Z G,H and BH is open dense in G , by [Weil, Chapter 2] we have (cid:90) G ( F ) f ( g ) d can g = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) B ( F ) f ( bh ) d can b d can h. By [Weil, Chapter 2], for a smooth variety X defined over F , we have vol ( X ( O F ) , d can x ) = | X ( F q ) | q dim X . This implies thatd can g = | G ( F q ) | q dim G d g, d can b = | B ( F q ) | q dim B d b, and d can h = | H/Z
G,H ( F q ) | q dim( H ) − dim( Z G,H ) d h. Since B ∩ H = Z G,H , we have (cid:90) G f ( g ) d g = | B ( F q ) | · | H/Z
G,H ( F q ) || G ( F q ) | (cid:90) H ( F ) /Z G,H ( F ) (cid:90) B ( F ) f ( bh ) dbdh. Now the lemma follows from the following equation which is a consequence of (3.1) and (5.1)of [G] | B ( F q ) | · | H/Z
G,H ( F q ) || G ( F q ) | = ∆ G (1)∆ H/Z
G,H (1) ζ (1) − rk ( G ) . (cid:3) By Lemma 2.5, (cid:82) K Y θ ( k ) d k is equal to (cid:90) G ( F ) K ( g ) Y θ ( g ) d g = ∆ G (1)∆ H/Z
G,H (1) ζ (1) − rk ( G ) (cid:90) H ( F ) /Z G,H ( F ) (cid:90) B ( F ) K ( bηh ) θ − δ ( b ) d b d h = ∆ G (1)∆ H/Z
G,H (1) ζ (1) − rk ( G ) (cid:90) H ( F ) /Z G,H ( F ) f θ ( ηh ) d h, where 1 K is the characteristic function on K . As a result, we have proved the followingproposition, which reduces to evaluate the integral (cid:82) K Y θ ( k ) d k . Proposition 2.6.
The local relative character I ( φ θ ) is equal to (cid:90) K Y θ − ( k ) dk × (cid:90) H ( F ) /Z G,H ( F ) f θ ( ηh ) d h = ∆ H/Z
G,H (1)∆ G (1) ζ (1) rk ( G ) (cid:90) K Y θ − ( k ) d k × (cid:90) K Y θ ( k ) d k. In the next subsection 2.2, we will explain how to compute the integral (cid:82) K Y θ ( k ) d k . Proposition 2.7.
Let Φ + be the set of positive roots of G . There is a decomposition of theweights of a representation ρ X of ˆ G , denoted by Θ = Θ + ∪ Θ − , such that (2.2) (cid:90) K Y θ ( k ) d k = ∆ G (1)∆ H/Z
G,H (1) ζ (1) − rk ( G ) · β ( θ ) , where β ( θ ) = (cid:81) α ∈ Φ + − q − e α ∨ (cid:81) γ ∨ ∈ Θ + − q − e γ ∨ ( θ ) . Moreover, we have (2.3) (cid:89) γ ∨ ∈ Θ + − q − e γ ∨ ( θ − ) = (cid:89) γ ∨ ∈ Θ − − q − e γ ∨ ( θ ) . Here for α ∈ Φ + , we use e α ∨ ( θ ) to denote θ ( e α ∨ ( (cid:36) )). For γ ∨ ∈ Θ + , we can identify it witha co-weight of G and we let e γ ∨ be the associated homomorphism from GL to T . We define e γ ∨ ( θ ) = θ ( e γ ∨ ( (cid:36) )). Remark 2.8.
For all the models in Table 1, the representation ρ X in the proposition above(or Proposition 2.23 for the non-reductive cases) is just the representation ρ X listed in Table1. In Theorem 7.2.1 of [Sa] , for general (not necessarily strongly tempered) spherical vari-eties, Sakellaridis proved the identities (2.2) and (2.3) for a W X -invariant set Θ of weights of ˆ G . Here W X ⊂ W is the little Weyl group of X and we have W = W X if the model is stronglytempered. Later in Corollary 7.3.3 of [SW] , Sakellaridis-Wang proved that in the case when ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES13 ( G, H ) is strongly tempered and H is reductive, Θ must be the set of weights of a represen-tation ρ X of ˆ G . Our computation in later sections shows that for all the non-reductive casesin Table 1, Θ is also the set of weights of a representation ρ X of ˆ G . Combining Propositions 2.6 and 2.7, we have I ( φ θ ) = ∆ H/Z
G,H (1)∆ G (1) ζ (1) rk ( G ) (cid:90) K Y θ − ( k ) d k × (cid:90) K Y θ ( k ) d k = ∆ G (1)∆ H/Z
G,H (1) ζ (1) − rk ( G ) · β ( θ ) · β ( θ − ) = ∆ G (1)∆ H/Z
G,H (1) · L (1 / , π, ρ X ) L (1 , π, Ad) . This finishes the computation. The L -functions L (1 / ,π,ρ X ) L (1 ,π, Ad) is just the L X , L -function of thespherical variety X = G/H , defined in Definition 7.2.3 of [Sa].2.3.
The computation of S θ . Set S θ ( g ) = (cid:90) K Y θ ( kg − ) d g for g ∈ G ( F ) , which is the Whittaker-Shintani function. (See [KMS03] for instance.) In this section, ourgoal is to prove Proposition 2.7, i.e. compute S θ (1).Let I = B ( O F ) ¯ N ( (cid:36) O F ) be the Iwahori subgroup of G ( F ). For all the strongly temperedmodels in the introduction, we can choose a representative η in the open double coset of B ( F ) \ G ( F ) /H ( F ) so that it satisfies the following lemma (we will check this lemma foreach model in the later sections). Lemma 2.9.
Then there exists a representative η for the open double coset of B ( F ) \ G ( F ) /H ( F ) such that η ∈ K and ¯ N ( (cid:36) O F ) η ⊂ T ( O F ) N ( (cid:36) O F ) ηH ( O F ) . For w ∈ W ( W is the Weyl group of G ), let Φ w = 1 I w I be the characteristic function of I w I . Then 1 K is equal to (cid:80) w ∈ W Φ w . Let α be a simple root and w α be the correspondingreflection in W . We would need to compute I α ( θ ) = vol ( I ) − (cid:90) G ( F ) Y θ ( xη )(Φ ( x ) + Φ w α ( x )) d x for all simple roots α .First, by Lemma 2.9, we have I η ⊂ B ( O F ) ηH ( O F ). Hence Y θ ( xη ) = 1 for all x ∈ I . Thisimplies that(2.4) vol ( I ) − (cid:90) G ( F ) Y θ ( xη )Φ ( x ) d x = 1 . For each root α ∈ Φ G of G , define (note that all the root spaces are one dimensional sincewe have assumed that G is split)(2.5) x α : a ∈ F (cid:55)→ u α ( a ) ∈ N ( F )where u α ( · ) is the one-parameter unipotent subgroup of G ( F ) associated to the root α . Lemma 2.10.
We have (2.6) I α ( θ ) = 1 + q (cid:90) O F ( θ − δ B )( e α ∨ ( a − )) Y θ ( x − α ( a − ) η ) d a, where δ B is the modular character of B . Proof.
It is sufficient to compute the integral vol ( I ) − (cid:90) I w α I Y θ ( xη ) d x. First, let us evaluate Y θ ( · ) on the set I w α I η . Since I w α I = B ( O F ) w α X α ( O F ) ¯ N ( (cid:36) O F ) and X α ( O F ) = { x α ( a ) | a ∈ O F } , by Lemma 2.9, we have I w α I η ⊂ B ( O F ) w α X α ( O F ) ηH ( O F ) . This implies that (cid:90) I w α I Y θ ( xη ) d x = vol ( I w α I ) (cid:90) O F Y θ ( w α x α ( a ) η ) d a = q · vol ( I ) (cid:90) O F Y θ ( w α x α ( a ) η ) d a. Then, since w α x α ( a ) = x α ( a − ) t · e α ∨ ( a − ) x − α ( a − ) for some t ∈ T ( O F ), we have Y θ ( w α x α ( a ) η ) = ( θ − δ B )( e α ∨ ( a − )) Y θ ( x − α ( a − ) η ) . This proves the lemma. (cid:3)
Then for each model, by an explicit matrix computation, we will show that there exists β ∨ α ∈ Θ such that − β ∨ α + α ∨ ∈ Θ (here we view α ∨ as a weight on the dual group) and(2.7) Y θ ( x − α ( a − ) η ) = θ ( e β ∨ α (1 + a − )) · | a − | − / . This implies that I α ( θ ) =1 + q (cid:90) O F ( θ − δ )( e α ∨ ( a − )) Y θ ( x − α ( a − ) η ) d a (2.8) =1 + q (cid:90) O F ( θ − δ )( e α ∨ ( a − )) θ ( e β ∨ α (1 + a − )) · | a − | − / d a =1 + q (cid:90) O F θ ( e α ∨ ( a )) · | a | − · θ ( e β ∨ α (1 + a − )) · | a − | − / d a =1 + q (cid:90) O F ( θ ( e β ∨ α ) · | | − / )(1 + a ) · ( θ ( e α ∨ − β ∨ α ) · | | − / )( a ) d a =( q − · − q − e α ∨ ( θ )(1 − q − / e β ∨ α ( θ ))(1 − q − / e − β ∨ α + α ∨ ( θ )) . Here we use the fact that for unitary unramified characters χ , χ of F × , the integral(2.9) q (cid:90) O F ( χ · | | − / )(1 + a ) · ( χ · | | − / )( a ) da is absolutely convergent and is equal to q − q − · q − / χ ( (cid:36) ) + q − / χ ( (cid:36) ) − q − χ χ ( (cid:36) )(1 − q − / χ ( (cid:36) ))(1 − q − / χ ( (cid:36) )) . Remark 2.11.
The set { β ∨ α , α ∨ − β ∨ α | α ∈ ∆( G ) } is the set of virtual weighted colorsof X = G/H defined in Section 7.1 of [Sa] . There is another way to compute the virtualweighted colors using the Luna diagram of X = G/H . In [Lu] , Luna computed the Lunadiagram for all the split reductive spherical varieties of Type A. The Luna diagram of all the
ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES15 split reductive spherical varieties was computed in [BP] . In Section 3, we will use the model (GSp × GSp , (GSp × GSp ) ) as an example to explain how to use the Luna diagram tocompute the virtual weighted colors. We refer the reader to Remark 3.6 for details. Definition 2.12.
Let Θ + be the smallest subset of Θ satisfying the following condition: • For every simple root α , we have Θ + − w α Θ + = { β ∨ α , α ∨ − β ∨ α } .Recall that for all the models in Table 1, Θ is the set of weights of the representation ρ X of ˆ G listed in Table 1. We define β ( θ ) = (cid:81) α ∈ Φ + − q − e α ∨ (cid:81) γ ∨ ∈ Θ + − q − e γ ∨ ( θ ) and c W S ( θ ) = (cid:81) γ ∨ ∈ Θ + − q − e γ ∨ (cid:81) α ∈ Φ + − e α ∨ ( θ ) . For a Weyl element w ∈ W , the intertwining operator T w : I GB ( θ ) → I GB ( wθ ) is defined by T w ( f )( g ) = (cid:90) N ( F ) ∩ wN ( F ) w − \ N ( F ) f ( w − ng ) d n, f ∈ I GB ( θ ) . It is absolutely convergent when θ is positive enough and admits a meromorphic continuation.By Theorem 1.2.1 of [Sa08], the space Hom H ( F ) ( I GB ( θ ) ,
1) is one-dimensional for almost all θ in the unitary line. In fact, for all the cases in Table 1, the little Weyl group W X of thespherical variety is equal to the Weyl group W . This implies that the factor ( N W ( δ − / A ∗ X ) : W X ) in loc. cit. is equal to 1. Moreover, since the spherical pairs ( G, H ) have the uniqueopen Borel orbits, the factor | H ( k, A X ) | in loc. cit. is also equal to 1. This implies thatthe Hom-space is one dimensional. In Section 9, we will prove a multiplicity formula of thedimension of this Hom space for all the tempered representations which will imply that theHom-space Hom H ( F ) ( I GB ( θ ) ,
1) is actually one dimensional for all unitary characters. But wedon’t need this result in our computation.By the definition of Y θ , we can define an element (cid:96) θ ∈ Hom H ( F ) ( I GB ( θ ) ,
1) to be(2.10) (cid:96) θ ( P θ ( f )) = (cid:90) G ( F ) f ( g ) Y θ ( g ) d g, for f ∈ C ∞ c ( G ) , where P θ ( f ) = (cid:82) B ( F ) ( θ − δ B )( b ) f ( bg ) d b is the canonical G ( F )-equivariant map from C ∞ c ( G ( F ))to I GB ( θ ). Since the Hom-space is one dimensional for almost all θ , for each simple reflection w α ∈ W associate to a simple root α , there exists a rational function b w α ( θ ) on θ such that(2.11) (cid:96) w α θ ◦ T w α = c α ( θ ) b w α ( θ ) (cid:96) θ . Here c α ( θ ) = − q − e α ∨ − e α ∨ ( θ ) is the c -function defined in [C80].Our goal is to obtain a formula of b w α ( θ ). Similar to the proof of Theorem 10.5 in [KMS03],to obtain b w α ( θ ), we evaluate both sides of equation (2.11) at P θ (Φ + Φ w α ). Note that T w α ( P θ (Φ + Φ w α )) = c α ( θ ) P θ (Φ + Φ w α ) . Then, under the choice (2.10) of (cid:96) θ , we have vol ( I ) I α ( θ ) = (cid:96) θ ( P θ ◦ R ( η )(Φ + Φ w α )) , where R is the right translation of G ( F ). On the other hand, (cid:96) w α θ ◦ T w α ( P θ ◦ R ( η )(Φ + Φ w α )) = c α ( θ ) (cid:96) w α θ ( P w α θ ◦ R ( η )(Φ + Φ w α )) = c α ( θ ) vol ( I ) I α ( w α θ ) . Following (2.11), we obtain(2.12) b w α ( θ ) = I α ( w α θ ) I α ( θ ) . Recall that S θ ( g ) = (cid:96) θ ( R ( g ) P θ (1 K )). Plugging T w α ( P w α θ (1 K )) = c α ( θ ) P θ (1 K ) into the lefthand side of (2.11), we have S w α θ ( g ) = (cid:96) w α θ ( R ( g ) P w α θ (1 K )) = c α ( θ ) − ( (cid:96) w α θ ◦ T w α )( R ( g ) P θ (1 K ) = b w α ( θ ) S θ ( g ) . Thus for all simple roots α , we have S w α θ ( g ) S θ ( g ) = b w α ( θ ) = I α ( w α θ ) I α ( θ ) = β ( αθ ) β ( θ ) , This implies that S θ ( g ) /β ( θ ) is W -invariant as a function of θ . Proposition 2.13.
Let T ( F ) + = { t ∈ T ( F ) | t − N ( O F ) t ⊂ N ( O F ) } be the positive chamberof T ( F ) . Then S θ ( η − t ) /β ( θ ) = q l ( W ) vol ( I ) (cid:88) w ∈ W c W S ( wθ )( wθ ) − δ B ( t − ) , for t ∈ T ( F ) + , where l ( W ) is the length of the longest Weyl element in W .Proof. First, we show that(2.13) S θ ( η − t ) = vol ( I t ( λ ) I ) − R (1 I t I ) S θ ( η − ) , where R is the right convolution defined by R (1 I t I ) S θ ( η − ) = (cid:90) G ( F ) I t I ( x ) S θ ( η − x ) d x = (cid:90) I t I S θ ( η − x ) d x. Now, it is enough to show that η − I t I ⊂ H ( O F ) η − t I ⊂ H ( O F ) η − tK, which follows from Lemma 2.9.Similar to Proposition 1.10 of [KMS03], there exists a basis { f w : w ∈ W } of I GB ( θ ) I suchthat(2.14) R (1 I t I ) f w = vol ( I t − I )( wθ ) − δ B ( t − ) f w , f = P θ (Φ ) , P θ (1 K ) = q l ( w (cid:96) ) (cid:88) w ∈ W c w ( θ ) f w for t ∈ T ( F ) + where c w ( θ ) = (cid:89) α> ,wα> c α ( θ ).Recall that S θ ( g ) = (cid:96) θ ( R ( g ) P θ (1 K )). Substituting (2.14) into S θ , we have S θ ( ηt ) /β ( θ ) = q l ( w (cid:96) ) β ( θ ) − (cid:88) w ∈ W c w ( θ ) (cid:96) θ ( R ( η ) f w ) · ( wθ ) − δ ( t − ) . By (2.4) and f = P θ (Φ ), we have the coefficient of ( wθ ) − δ ( t − ) for w = 1 in S θ ( ηt ) /β ( θ )is equal to q l ( w (cid:96) ) β ( θ ) − c ( α ) = q l ( w (cid:96) ) vol ( I ) c W S ( θ ) , ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES17 where c ( θ ) = (cid:81) α ∈ Φ + − q − e α ∨ − e α ∨ ( θ ). Since S θ ( ηt ) /β ( θ ) is W -invariant, by the linear indepen-dence of the characters ( wθ ) − δ ( t − ), we obtain S θ ( η − t ) /β ( θ ) = q l ( w (cid:96) ) vol ( I ) (cid:88) w ∈ W c W S ( wθ )( wθ ) − δ ( t − ) . (cid:3) Since η ∈ K , we have S θ (1) = S θ ( η − ). Combining with the proposition above, we have S θ (1) /β ( θ ) = q l ( W ) vol ( I ) (cid:88) w ∈ W c W S ( wθ ) . Since vol ( I ) = ∆ G (1) ζ (1) − rk ( G ) · q − l ( W ) , we have S θ (1) /β ( θ ) = ∆ G (1) ζ (1) − rk ( G ) (cid:88) w ∈ W c W S ( wθ ) . Hence in order to prove Proposition 2.7, it is enough to prove the following lemma.
Lemma 2.14.
The summation (cid:88) w ∈ W c W S ( wθ ) is independent of the choice of θ and is equalto H/ZG,H (1) .Proof.
Since the spherical varieties for the reductive cases are affine, the first part of thisstatement follows from Theorem 7.2.1 of [Sa]. For the second part, since the summation isindependent of the choice of θ , we can compute it by plug in some special θ . We will computeit for each of our models in later sections. (cid:3) Remark 2.15.
For all reductive cases, if we set θ = δ / B as in Lemma 4.2.3 of [Sa] (which isused in the proof of [Sa, Theorem 7.2.1] ), then the only nonvanishing term in the summationis the term corresponds to the longest Weyl element, which is equal to H/ZG,H (1) . We willshow this for all the reductive models in Table 1 in later sections.
The summary.
By the discussion in the previous two subsections, in order to computethe relative character in the reductive case, we just need to perform the following steps:(1) Show that the double cosets B ( F ) \ G ( F ) /H ( F ) have a unique open orbit B ( F ) ηG ( F )and the representative η can be chosen to satisfy Lemma 2.9.(2) Verify the identity (2.7) by expressing the product x − α ( a ) η in terms of the decom-position B ( F ) ηH ( F ). This gives us the set of virtual weighted colors of X .(3) Compute the subset Θ + of Θ and show that it satisfies (2.3).(4) Compute the constant (cid:80) w ∈ W c W S ( wθ ), i.e. Lemma 2.14. This computation is easyfor the reductive case, see Remark 2.15.2.3.2. The trilinear model.
To end this subsection, we use the trilinear GL model as anexample to explain the method. This example also appeared in Section 7.2.4 of [Sa] and wewill use it for the non-reductive cases in Table 1 , which are the Whittaker inductions of thetrilinear GL -model. Let G = GL × GL × GL and H = GL diagonally embedded into G . Let B be the upper triangular Borel subgroup of G and η = ( I , ( ) , ( ) ( )) . It iseasy to see that B ( F ) η H ( F ) is the unique open orbit and η satisfies Lemma 2.9. Let Θ be the set of weights of the tensor product representation of GL ( C ) × GL ( C ) × GL ( C ). We can write it as { e i + e (cid:48) j + e (cid:48)(cid:48) k | ≤ i, j, k ≤ } . Let α i (1 ≤ i ≤
3) be the simpleroot of the i -th copy of GL . We have x − α ( a ) η = ( (cid:18) − a
00 1 (cid:19) , (cid:18) − a − a − a (cid:19) , (cid:18) − a − a − a (cid:19) ) · η · (cid:18)(cid:18) − a a (cid:19)(cid:19) diag ,x − α ( b ) η = ( (cid:18) − b − b − b (cid:19) , (cid:18) − b
00 1 (cid:19) , (cid:18) − b
00 1 (cid:19) ) · η · (cid:18) b − b (cid:19) diag ,x − α ( c ) η = ( (cid:18) c (cid:19) , (cid:18) c
00 1 (cid:19) , (cid:18) c
00 1 (cid:19) ) · η · (cid:18) c (cid:19) diag . This proves (2.7) and implies that (note that the representation has trivial central character) β ∨ α = e + e (cid:48) + e (cid:48)(cid:48) , α ∨ − β ∨ α = e + e (cid:48) + e (cid:48)(cid:48) , β ∨ α = e + e (cid:48) + e (cid:48)(cid:48) ,α ∨ − β ∨ α = e + e (cid:48) + e (cid:48)(cid:48) , β ∨ α = e + e (cid:48) + e (cid:48)(cid:48) , α ∨ − β ∨ α = e + e (cid:48) + e (cid:48)(cid:48) . Then Θ + will be the smallest subset of Θ satisfying the following two conditions: • e + e (cid:48) + e (cid:48)(cid:48) , e + e (cid:48) + e (cid:48)(cid:48) , e + e (cid:48) + e (cid:48)(cid:48) ∈ Θ + . • Θ + − w α Θ + = { e + e (cid:48) + e (cid:48)(cid:48) , e + e (cid:48) + e (cid:48)(cid:48) } , Θ + − w α Θ + = { e + e (cid:48) + e (cid:48)(cid:48) , e + e (cid:48) + e (cid:48)(cid:48) } , Θ + − w α Θ + = { e + e (cid:48) + e (cid:48)(cid:48) , e + e (cid:48) + e (cid:48)(cid:48) } . As a result, we haveΘ + = { e + e (cid:48) + e (cid:48)(cid:48) , e + e (cid:48) + e (cid:48)(cid:48) , e + e (cid:48) + e (cid:48)(cid:48) , e + e (cid:48) + e (cid:48)(cid:48) } . It is easy to see that Θ + satisfies (2.3).Finally, if we let θ = δ / B , it is easy to see that for w ∈ W , c W S ( wθ ) = 0 unless w isthe longest Weyl element. If w is the longest Weyl element, we have c W S ( wθ ) = 1 − q − = ζ (2) − = H/ZG,H (1) . This proves Lemma 2.9. In conclusion, we have proved that the localrelative character in this case is equal to ζ (1) ζ (2) · L (1 / , π × π × π ) L (1 , π, Ad)where π = π ⊗ π ⊗ π is an unramified representation of G ( F ).2.4. The Whittaker induced case: some reductions.
In this subsection, we considerthe Whittaker induced case. Let (
G, H ) be a Whittaker induction of a strongly temperedmodel ( G , H ). In other words, there exists a parabolic subgroup P = LU of G and a genericcharacter ξ of U ( F ) such that G (cid:39) L and H is the neutral component of the stabilizer of ξ in M . Note that for all the cases we considered in Table 1, the model ( G , H ) is essentiallythe trilinear GL -model.Let B = T N be a Borel subgroup of G , ¯ B = T ¯ N be its opposite, and ¯ P = L ¯ U be theopposite parabolic subgroup of P . Let N = N U and ¯ N = ¯ N ¯ U . Then B = T N is a Borelsubgroup of G and ¯ B = T ¯ N is its opposite.For all of our cases (as well as all the other Whittaker inducted cases in Remark 1.4),there exists a Weyl element w such that the w -conjugation map • induces an isomorphism between U and ¯ U , • stabilizes L and fixes H ⊂ L . ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES19
Also there exists a homomorphism λ : U ( F ) → F such that ξ ( u ) = ψ ( λ ( u )) for all u ∈ U ( F ).We extend λ to H ( F ) by making it trivial on H ( F ). We also have a map a : GL → Z L such that(2.15) w − a ( t ) w = a ( t ) − , and λ ( a ( t ) ua ( t ) − ) = tλ ( u ) , for t ∈ F × , u ∈ U ( F ) . Let B ( F ) η H ( F ) be the unique open Borel orbit of the model ( G , H ), and let η = η w .Then B ( F ) ηH ( F ) is the unique open Borel orbit of the model ( G, H ) and the stabilizer ofthis orbit is Z G,H = H ∩ Z G . Note that we always assume ( G , H ) does not have Type N spherical root. The equation (2.15) implies that η − a ( t ) η = a ( t ) − .We want to compute the local relative character(2.16) I ( φ θ ) = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) φ θ ( hu ) ξ ( u ) − d u d h where φ θ is the unramified matrix coefficient of I GB ( θ ) with φ θ (1) = 1, and θ is a unitaryunramified character of T ( F ). The general idea of the computation is the same as thereductive case, the only difference is some convergent issue. Unlike the reductive case, theintegral above is not absolutely convergent because of the extra unipotent integral. Hencewe need to regularize the unipotent integral.There are three (equivalent) ways to regularize the unipotent integral. The first one isusing the fact the the unipotent integral is stable, i.e. there exists a compact open subgroup U of U ( F ) such that (cid:90) U (cid:48) −U φ θ ( hu ) ξ ( u ) − d u = 0for all compact open subgroup U (cid:48) of U ( F ) with U ⊂ U (cid:48) . Hence we can replace the integralover U ( F ) by an integral over the compact subgroup U . This regularization has been usedby Lapid–Mao [LM] in their computation for the Whittaker model, and used by Liu [L]in his computation of the non-reductive Gan–Gross–Prasad models. The advantage of thisregularization is that it works for general matrix coefficients, not just the tempered ones.(Of course, in order for the integral of H ( F ) to be convergent, we still need the character θ to be close to the unitary line.)The rest two regularizations are only for the tempered case. It uses a critical fact thatalthough the integral (2.16) is not absolutely convergent, the integral will become absolutelyconvergent if we replace U ( F ) by U (cid:48) ( F ) = { u ∈ U ( F ) | λ ( u ) = 0 } . (For all the modelsconsidered in this paper, this can be proved by the same argument as Lemma 4.3.1 of[Wan17b].) As a result, we only need to regularize the integral over λ ( u ) ∈ F . Remark 2.16.
In particular, the integral (cid:90) H ( F ) /Z G,H ( F ) φ θ ( h )Φ( λ ( h )) d h is absolutely convergent for all Φ ∈ C ∞ c ( F ) . There are two ways to regularize the integral over λ ( u ). The first way is to replace thethe integral over U ( F ) by the integral over U n ( F ) = { u ∈ U ( F ) | | λ ( u ) | ≤ q n } . Then one can show that for every matrix coefficient φ of I GB ( θ ), there exists N > φ , i.e. the open compact subgroup K (cid:48) ⊂ G ( F ) where φ is bi- K (cid:48) -invariant) such that the integral I n ( φ θ ) = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) φ θ ( hu ) ξ ( u ) − d u d h is independent of n for n > N , i.e. the unipotent integral is stable on the sequence U n ( F )(the difference between this regularization and the previous one is that the unipotent groups U n ( F ) we used here is not compact). Hence we can just replace the integral over U ( F ) in thedefinition of I ( φ θ ) by the integral over U n ( F ) for some large n (in fact, as φ θ is unramified,one can easily show that we can just replace U ( F ) by U ( F )). This regularization has beenused by Waldspurger in Lemma 5.1 of [W12] for the orthogonal Gan–Gross–Prasad model.The same arguments work for all the Whittaker induction cases in this paper.Another way is to replace the character ξ ( u ) − = ψ ( λ ( u )) − by some Schwartz function ϕ n ( λ ( u )) ( n ≥
0) of λ ( u ) where ϕ = ϕ = 1 O F − q − · (cid:36) − O × F is the Fourier transform of vol ( O × F ) · O × F and ϕ n is the Fourier transform of the function vol (1+ (cid:36) n O × F ) (cid:36) n O × F for n ≥
1. One can show that for every matrix coefficient φ of I GB ( θ ),there exists N > φ ) such that the integral I n ( φ θ ) = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) φ θ ( hu ) ξ ( u ) − d u d h is independent of n for n > N .This regularization has been used by Beuzart-Plessis (Proposition 7.1.1 of [B15]) for theunitary Gan–Gross–Prasad model (note that the group 1 + (cid:36) n O × F is just the group K a in loc.cit.) and by the first author (Proposition 5.1 of [Wan16]) for the Ginzburg–Rallis model. Thesame argument works for all the Whittaker induction cases in this paper. In the unramifiedcase, we may just take n = 0 and the regularized integral is given by the formula I ( φ θ ) = (cid:90) H ( F ) /Z G,H ( F ) φ θ ( h ) ϕ ( λ ( h )) d h. In order to compute this regularized integral, we need another two regularized integrals.
Lemma 2.17.
For f ∈ I GB ( θ ) , we have the integrals (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( ηhu ) ξ ( u ) − d u d h and (cid:90) H ( F ) f ( ηh ) ϕ n ( λ ( h )) d h are absolutely convergent for all n .Moreover, there exists N ≥ (depends on the level of f , i.e. the open compact subgroup K (cid:48) ⊂ G ( F ) where f is right K (cid:48) -invariant) such that both integrals are equal to each otherand are independent of n for n > N . Thus, we may use (cid:90) ∗ H ( F ) /Z G,H ( F ) f ( ηh ) ξ ( h ) − d h = lim n →∞ (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( ηhu ) ξ ( u ) − d u d h = lim n →∞ (cid:90) H ( F ) /Z G,H ( F ) f ( ηh ) ϕ n ( λ ( h )) d h ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES21 to denote this regularized integral.
Proof.
We first prove the convergence. By replacing f by | f | and θ by | θ | we may assumethat f is a non-negative real valued function. Let f θ be the unramified vector in I GB ( θ ) with f θ (1) = 1. Then the matrix coefficient of f and f θ is given by φ f,f θ ( g ) = (cid:90) K f ( kg ) d g. By the discussion above, we know that the integral (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) φ f,f θ ( hu ) ξ ( u ) − d u d h is absolutely convergent for all n . This implies that (note that f is a non-negative real valuedfunction) the triple integral (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) (cid:90) K f ( khu ) ξ ( u ) − d k d u d h is absolutely convergent. In particular, the integral (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( khu ) ξ ( u ) − d u d h is absolutely convergent for almost all k ∈ K . But as a function on k ∈ K , this integral isleft B ( O F ) and right H ( O F ) invariant. Combining with the fact that η ∈ K and BηH isZariski open in G , we know that the integral (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( ηhu ) ξ ( u ) − d u d h is absolutely convergent for all n . This also implies that the integral (cid:90) H ( F ) f ( ηh ) ϕ n ( λ ( h )) d h is absolutely convergent for all n since ϕ n is a compactly supported function.Now we prove the second part of the theorem. We first prove the following statement(1) there exists N ≥ n > N , we have (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f ( ηhu u ) ξ ( u ) − d u d h = 0for all u ∈ U ( F ) − U n ( F ).In fact, for n large, we have the function f ( g ) is right a ( t )-invariant for all t ∈ (cid:36) n O × F .It is also left a ( t )-invariant for all t ∈ (cid:36) n O × F since θ is an unramified character. Thenfor u ∈ U ( F ) − U n ( F ), there exists t ∈ (cid:36) n O × F such that ( t − λ ( u ) ∈ (cid:36) − O × F (inparticular, ψ (( t − λ ( u )) (cid:54) = 1). This implies that (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f ( ηhu u ) ξ ( u ) − d u d h = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f ( a ( t − ) ηhu u ) ξ ( u ) − d u d h = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f ( ηhu ( u − a ( t ) u ) ua ( t ) − ) ξ ( u ) − d u d h = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f ( ηhu ( u − a ( t ) u a ( t ) − ) u ) ξ ( u ) − d u d h. Here we use the fact that since ψ is unramified, ξ ( u ) = ξ ( a ( t ) ua ( t ) − ) for u ∈ U ( F ) and t ∈ (cid:36) n O F . By our choice of u and t , we know that u − a ( t ) u a ( t ) − ∈ U ( F ), thenan easy change of variable shows that (cid:82) H ( F ) /Z G,H ( F ) (cid:82) U ( F ) f ( ηhu u ) ξ ( u ) − d u d h is equal to ξ ( u − a ( t ) u a ( t ) − ) (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f ( ηhu u ) ξ ( u ) − d u d h. Since ξ ( u − a ( t ) u a ( t ) − ) = ψ (( t − λ ( u )) (cid:54) = 1, we have (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f ( ηhu u ) ξ ( u ) − d u d h = 0 . This proves (1).Now we are ready to prove the theorem. By (1), we know that for all n > N , we have (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n +1 ( F ) − U n ( F ) f ( ηhu ) ξ ( u ) − d u d h = 0 . In particular, the integral (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( ηhu ) ξ ( u ) − d u d h is independent of n for n > N .For the second integral, choose n large so that the function f ( g ) is right a ( t )-invariant forall t ∈ (cid:36) n O × F . Then we have (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( ηhu ) ξ ( u ) − d u d h = (cid:90) (cid:36) n O × F (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( a ( t ) ηhua ( t )) ξ ( u ) − d u d h d t = (cid:90) (cid:36) n O × F (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( ηha ( t ) − ua ( t )) ξ ( u ) − d u d h d t = (cid:90) (cid:36) n O × F (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( ηhu ) ψ ( tλ ( u )) − d u d h d t = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) (cid:90) (cid:36) n O × F f ( ηhu ) ψ ( tλ ( u )) − (cid:36) − n O F ( λ ( u )) d t d u d h. Here the measure d t on 1 + (cid:36) n O × F is chosen so that the total volume is equal to 1. Thefunction x (cid:55)→ (cid:90) (cid:36) n O × F ψ ( tx ) − (cid:36) − n O F ( x ) d t = 1 (cid:36) − n O F ( x ) · (cid:90) F vol (1 + (cid:36) n O × F ) 1 (cid:36) n O × F ( t ) ψ ( tx ) − d y ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES23 is just 1 (cid:36) − n O F · ϕ n (recall that ϕ n is the Fourier transform of the function vol (1+ (cid:36) n O × F ) (cid:36) n O × F ).A direct computation shows that the function ϕ n is supported on (cid:36) − n O F , hence 1 (cid:36) − n O F · ϕ n = ϕ n . As a result, we have (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f ( ηhu ) ξ ( u ) − d u d h = (cid:90) H ( F ) f ( ηh ) ϕ n ( λ ( h )) d h. This proves the lemma. (cid:3)
Remark 2.18.
As long as f is right T ( O F ) -invariant (for example when f is unramifiedor when f is an Iwahori fixed vector), we can just take N = 0 . We can also show that theintegral (cid:90) H ( F ) f ( ηh ) ϕ n ( λ ( h )) d h is independent of n for n ≥ . Let Y θ,ξ (resp. Y θ,ξ,n , Y nθ,ξ ) be the function on G ( F ) satisfying the following conditions: • Y θ,ξ (resp. Y θ,ξ,n , Y nθ,ξ ) is supported on the open orbit B ( F ) ηH ( F ). • Y θ,ξ ( bηh ) = θ − δ / B ( b ) ξ ( h ) (resp. Y θ,ξ,n ( bηh ) = θ − δ / B ( b ) ξ ( h )1 (cid:36) − n O F ( λ ( h )), Y nθ,ξ ( bηh ) = θ − δ / B ( b ) ϕ n ( λ ( h ))) for b ∈ B ( F ) , h ∈ H ( F ). Lemma 2.19.
For Φ ∈ C ∞ c ( G ( F )) , the integrals (cid:90) G ( F ) Φ( g ) Y θ,ξ,n ( g ) d g and (cid:90) G ( F ) Φ( g ) Y nθ,ξ ( g ) d g are absolutely convergent for all n . Moreover, there exists N ≥ (depends on the level of f , i.e. the open compact subgroup K (cid:48) ⊂ G ( F ) where f is right K (cid:48) -invariant) such that bothintegrals are equal to each other and are independent of n for n > N . Then we may use (cid:90) ∗ G ( F ) Φ( g ) Y θ,ξ ( g ) d g = lim n →∞ (cid:90) G ( F ) Φ( g ) Y θ,ξ,n ( g ) d g = lim n →∞ (cid:90) G ( F ) Φ( g ) Y nθ,ξ ( g ) d g to denote this regularized integral. Proof.
By the same arguments as Lemma 2.5, we can prove the following statement • For Φ ∈ C ∞ c ( G ( F )), we have(2.17) (cid:90) G ( F ) Φ( g ) d g = ∆ G (1)∆ H /Z G,H (1) ζ (1) − rk ( G ) (cid:90) H ( F ) /Z G,H ( F ) (cid:90) B ( F ) Φ( bηh ) d b d h. This implies that the integral (cid:82) G ( F ) Φ( g ) Y θ,ξ,n ( g ) d g is equal to∆ G (1)∆ H /Z G,H (1) ζ (1) − rk ( G ) (cid:90) H ( F ) Z G,H ( F ) (cid:90) U n ( F ) (cid:90) B ( F ) Φ( bηhu ) θ − δ / ( b ) ξ ( u ) − d b d u d h = ∆ G (1)∆ H /Z G,H (1) ζ (1) − rk ( G ) (cid:90) H ( F ) Z G,H ( F ) (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U n ( F ) f Φ ( ηhu ) ξ ( u ) − d u d h and the integral (cid:82) G ( F ) Φ( g ) Y nθ,ξ ( g ) d g is equal to∆ G (1)∆ H /Z G,H (1) ζ (1) − rk ( G ) (cid:90) H ( F ) /Z G,H ( F ) (cid:90) B ( F ) Φ( bηh ) θ − δ / ( b ) ϕ n ( λ ( h )) d b d h = ∆ G (1)∆ H /Z G,H (1) ζ (1) − rk ( G ) (cid:90) H ( F ) Z G,H ( F ) (cid:90) H ( F ) /Z G,H ( F ) f Φ ( ηh ) ϕ n ( λ ( h )) d h. Then the lemma just follows from the lemma above. (cid:3)
Remark 2.20. If Φ is right T ( O F ) -invariant, then we can just take N = 0 . We can alsoshow that the integral (cid:90) G ( F ) Φ( g ) Y nθ,ξ ( g ) d g is independent of n for n ≥ . For any open compact subset K (cid:48) ⊂ G ( F ) , we let (cid:90) ∗ K (cid:48) Y θ,ξ ( k ) d k := (cid:90) ∗ G ( F ) K (cid:48) ( g ) Y θ,ξ ( g ) d g. Recall that f θ is the unramified vector in I GB ( θ ) with f θ (1) = 1. We have φ θ ( g ) = (cid:90) K f θ ( kg ) d k, I ( φ θ ) = (cid:90) H ( F ) /Z G,H ( F ) φ θ ( h ) ϕ ( λ ( h )) d h = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) φ θ ( hu ) ξ ( u ) − d u. This implies that(2.18) I ( φ θ ) = (cid:90) K (cid:90) H ( F ) /Z G,H ( F ) f θ ( kh ) ϕ ( λ ( h )) d h d k = (cid:90) K (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f θ ( khu ) ξ ( u ) − d u d h d k. The next lemma follows from the proof of Lemma 2.17.
Lemma 2.21.
For u ∈ U ( F ) − U ( F ) , we have (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f θ ( ηu hu ) ξ ( u ) − d u d h = 0 . Corollary 2.22.
We have I ( φ θ ) = ∆ H /Z G,H (1)∆ G (1) ζ (1) rk ( G ) · (cid:90) ∗ K Y θ − ,ξ ( k ) d k · (cid:90) ∗ K Y θ,ξ − ( k ) d k. Proof.
We have I ( φ θ ) = (cid:90) K ∩ B ( F ) ηH ( F ) U ( F ) (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f θ ( khu ) ξ ( u ) − d u d h d k. The function k (cid:55)→ (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f θ ( khu ) ξ ( u ) − d u d h on K ∩ B ( F ) ηH ( F ) U ( F ) is a scalar of the restriction of the function Y θ − ,ξ to K ∩ B ( F ) ηH ( F ) U ( F ) and the scalar is equal to (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f θ ( ηhu ) ξ ( u ) − d u d h. ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES25
This implies that I ( φ θ ) = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f θ ( ηhu ) ξ ( u ) − d u d h · (cid:90) K ∩ B ( F ) ηH ( F ) U ( F ) Y θ − ,ξ ( k ) d k. By Lemma 2.19 and Remark 2.20, we have (cid:90) K ∩ B ( F ) ηH ( F ) U ( F ) Y θ − ,ξ ( k ) d k = (cid:90) ∗ K Y θ − ,ξ ( k ) d k. Hence it remains to show that(2.19) ∆ H /Z G,H (1)∆ G (1) ζ (1) rk ( G ) · (cid:90) ∗ K Y θ,ξ − ( k ) d k = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f θ ( ηhu ) ξ ( u ) − d u d h. By Lemma 2.19, Remark 2.20 and (2.17), we have∆ H /Z G,H (1)∆ G (1) ζ (1) rk ( G ) · (cid:90) ∗ K Y θ,ξ − ( k ) d k = ∆ H /Z G,H (1)∆ G (1) ζ (1) rk ( G ) · (cid:90) K ∩ B ( F ) ηH ( F ) U ( F ) Y θ,ξ − ( k ) d k = (cid:90) H ( F ) (cid:90) U ( F ) (cid:90) B ( F ) K ( bηhu ) θ − δ / B ( b ) ξ ( u ) − du = (cid:90) H ( F ) /Z G,H ( F ) (cid:90) U ( F ) f θ ( ηhu ) ξ ( u ) − d u d h. This proves (2.19) and finishes the proof of the lemma. (cid:3)
In the next subsection, we will explain how to compute the regularized integral (cid:82) ∗ K Y θ,ξ ( k ) d k .The result is summarized in the proposition below. Proposition 2.23.
Let Φ + be the set of positive roots of G . There is a decomposition ofweights of a representation ρ X of ˆ G (denoted by Θ = Θ + ∪ Θ − ) such that (cid:90) ∗ K Y θ,ξ ( k ) d k = ∆ G (1)∆ H /Z G,H (1) ζ (1) − rk ( G ) · β ( θ ) where β ( θ ) = (cid:81) α ∈ Φ + − q − e α ∨ (cid:81) γ ∨ ∈ Θ + − q − e γ ∨ ( θ ) . Also (2.3) still holds.
The proposition above implies that I ( φ θ ) = ∆ G (1)∆ H /Z G,H (1) ζ (1) − rk ( G ) · β ( θ ) · β ( θ − ) = ∆ G (1)∆ H /Z G,H (1) · L (1 / , π, ρ X ) L (1 , π, Ad) . This finishes the computation.2.5.
The computation of (cid:82) ∗ K Y θ,ξ ( k ) d k . Let S θ ( g ) := (cid:90) ∗ K Y θ,ξ ( kg − ) d k = (cid:90) ∗ G ( F ) K ( g (cid:48) g ) Y θ,ξ ( g (cid:48) ) d g (cid:48) . Our goal is to prove Proposition 2.23, i.e. compute S θ (1) = (cid:90) ∗ K Y θ,ξ ( k ) d k = (cid:90) K Y θ,ξ ( k ) d k. Let I = B ( O F ) ¯ N ( (cid:36) O F ) (resp. I = B ( O F ) ¯ N ( (cid:36) O F )) be the Iwahori subgroup of G ( F )(resp. G ( F ) = L ( F )). As in Lemma 2.9, we can choose η so that(2.20) ¯ N ( (cid:36) O F ) η ⊂ T ( O F ) N ( (cid:36) O F ) η H ( O F ) . Lemma 2.24.
We have η ∈ K and ¯ N ( (cid:36) O F ) η ⊂ T ( O F ) N ( (cid:36) O F ) ηH ( O F ) U ( (cid:36) O F ) . Proof.
Since η , w ∈ K , we have η = η w ∈ K . For ¯ n ∈ ¯ N ( (cid:36) O F ), we write it as ¯ n (cid:48) ¯ u with ¯ n (cid:48) ∈ ¯ N ( (cid:36) O F ) and ¯ u ∈ ¯ U ( (cid:36) O F ). Since η = η w and η ∈ L ( O F ), we have η − ¯ uη ∈ U ( (cid:36) O F ). Hence it is enough to consider the case when ¯ n ∈ ¯ N ( (cid:36) O F ). Then the lemmafollows from (2.20) and the fact that H commutes with w . (cid:3) For w ∈ W , let Φ w = 1 I w I . Let α be a simple root and w α be the corresponding elementin W . As in the reductive case, we would need to compute I α ( θ ) = vol ( I ) − (cid:90) ∗ G ( F ) Y θ,ξ ( x )(Φ ( xη − )+Φ w α ( xη − )) d x = vol ( I ) − (cid:90) G ( F ) Y θ,ξ ( xη )(Φ ( x )+Φ w α ( x )) d x. First, by the lemma above, we have I η ⊂ B ( O F ) ηH ( O F ) U ( (cid:36) O F ). Hence Y θ,ξ ( xη ) = 1for all x ∈ I . This implies that vol ( I ) − (cid:90) G ( F ) Y θ,ξ ( xη )Φ ( x ) d x = 1 . The next lemma follows from the same arguments as in the reductive case.
Lemma 2.25.
Let x α : F → N ( F ) be the homomorphism whose image is the root space of α (the root space is one dimensional since we assume that the group is split). Then I α ( θ ) = 1 + q (cid:90) O F ( θ − δ )( e α ∨ ( a − )) Y θ,ξ ( x − α ( a − ) η ) d a, Then for each model, by an explicit matrix computation, we will show that for α ∈ ∆( G ),there exists β ∨ α ∈ Θ such that − β ∨ α + α ∨ ∈ Θ and(2.21) Y θ,ξ ( x − α ( a − ) ηh − ) = ϕ ( λ ( h )) · θ ( e β ∨ α (1 + a − )) · | a − | − / . As in the reductive case, this will imply that(2.22) α = ( q − · − q − e α ∨ ( θ )(1 − q − / e β ∨ α ( θ ))(1 − q − / e − β ∨ α + α ∨ ( θ )) . Remark 2.26.
In fact, by our choice of η = η w , we only need to verify the identity forthe reductive model ( G , H ) and we know that β ∨ a is just the color associated to α for thereductive model ( G , H ) . For all of our cases in Table 1, since it is induced from the trilinear GL -model, we can just use the computations in Section 2.3.2. On the other hand, if α ∈ ∆( G ) − ∆( G ), we will show that(2.23) Y θ,ξ ( x − α ( a − ) η ) = ϕ ( a − ) . This implies that(2.24) I α ( θ ) = 1 + q (cid:90) O F θ ( e α ∨ ( a )) · | a | − · ϕ ( a − ) da = q (1 − q − e α ∨ ( θ )) . ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES27
Remark 2.27.
Note that if we don’t regularize the unipotent integral, the integral we gethere will be I α ( θ ) = 1 + q (cid:90) O F θ ( e α ∨ ( a )) · | a | − · ψ ( a − ) d a. This is not absolutely convergent (which is also the reason why the original unipotent integralis not absolutely convergent). There are two ways to regularize this integral which correspondto the two ways to regularize the unipotent integral.The first way is to use the fact that (cid:82) (cid:36) n O × F θ ( e α ∨ ( a )) · | a | − · ψ ( a − ) d a = 0 for n > , andregularize the integral as I α ( θ ) = 1 + q (cid:90) (cid:36) − O F θ ( e α ∨ ( a )) · | a | − · ψ ( a − ) d a, which is equal to q (1 − q − e α ∨ ( θ )) . The second way is to replace ψ ( a − ) by ϕ ( a − ) as wedid above which gives the same answer. Definition 2.28.
Let Θ + be the smallest subset of Θ satisfying the following two conditions: • For every simple root α ∈ ∆ G , we have Θ + − w α Θ + = { β ∨ α , α ∨ − β ∨ α } ; • For every simple root α ∈ ∆ G − ∆ G , Θ + is stable under w α .We then define β ( θ ) = (cid:81) α ∈ Φ + − q − e α ∨ (cid:81) γ ∨ ∈ Θ + − q − e γ ∨ ( θ ) and c W S ( θ ) = (cid:81) γ ∨ ∈ Θ + − q − e γ ∨ (cid:81) α ∈ Φ + − e α ∨ ( θ ) . Now by the exactly same arguments as in the reductive case (the only difference is that forthe definition of l θ in (2.10), we replace the integral (cid:82) G ( F ) by the regularized integral (cid:82) ∗ G ( F ) ),we can prove the following proposition. Proposition 2.29.
Let T ( F ) + = { t ∈ T ( F ) | t − N ( O F ) t ⊂ N ( O F ) } be the positive chamberof T ( F ) . Then S θ ( η − t ) /β ( θ ) = q l ( W ) vol ( I ) (cid:88) w ∈ W c W S ( wθ )( wθ ) − δ ( t − ) , for t ∈ T ( F ) + , where l ( W ) is the length of the longest Weyl element in W . Since η ∈ K , we have S θ (1) = S θ ( η − ). Combining with the proposition above, we have S θ (1) /β ( θ ) = q l ( W ) vol ( I ) (cid:88) w ∈ W c W S ( wθ ) . Since vol ( I ) = ∆ G (1) ζ (1) − rk ( G ) · q − l ( W ) , we have S θ (1) /β ( θ ) = ∆ G (1) ζ (1) − rk ( G ) (cid:88) w ∈ W c W S ( wθ ) . Hence in order to prove Proposition 2.23, it is enough to prove the following lemma.
Lemma 2.30.
The summation (cid:80) w ∈ W c W S ( wθ ) is independent of the choice of θ and is equalto H /ZG,H (1) . This lemma is much more difficult than the reductive case for two reasons. First, Theorem7.2.1 of [Sa] only works for the reductive case, so we can not use it to imply that thesummation is a constant. Secondly, in the reductive case, if we set θ = δ / B , then the onlynonvanishing term in the summation is the term corresponding to the longest Weyl element,and it will be equal to H /ZG,H (1) . But for all the non-reductive cases in Table 1, this is nottrue and actually it is impossible to choose a θ so that all the terms in the summation areequal to 0 except one. We believe that this is related to the fact that for the models in Table1, the Type T spherical roots and Type ( U, ψ ) spherical roots will sometimes interlace eachother. For example, for the model (GL , GL (cid:110) U ), the roots α i = e i − e i +1 is of Type T when i = 1 , , U, ψ ) when i = 2 , , GL (cid:110) U ),we will prove the identity by proving another identity which allows us to reduce to theidentity for the model (GL , GL (cid:110) U ); for the model (GL , GL (cid:110) U ), we will prove theidentity by proving another identity which allows us to reduce to an identity related to thegroup GL × GL .2.5.1. The summary.
By the discussion in the previous two subsections, in order to computethe local relative character in the non-reductive case, we just need to do the following steps:(1) Define the Weyl element w so that the w -conjugation map • induces an isomorphism between U and ¯ U , • stabilizes L and fixes H ⊂ L .(2) Define the map a : GL → Z L so that it satisfies (2.15).(3) Show that the double coset B ( F ) \ G ( F ) /H ( F ) has a unique open Borel B ( F ) η G ( F )and the representative η can be chosen to satisfy (2.20). Since all the cases in Table1 are Whittaker inductions of the trilinear GL -model, this step has already beenverified in Section 2.3.2.(4) Verify the identity (2.21) and (2.23) by expressing the product x − α ( a ) η in termsof the decomposition B ( F ) ηH ( F ). Since all the cases in Table 1 are Whittakerinduction of the trilinear GL -model, the identity (2.21) and the colors have alreadybeen computed in Section 2.3.2 (see Remark 2.26), so we only need to verify (2.23).(5) Compute the subset Θ + of Θ and show that it satisfies (2.3).(6) Compute the constant (cid:80) w ∈ W c W S ( wθ ), i.e. Lemma 2.30. This is the most technicalpart of the computation. A final remark for the spherical roots.
In Table 1, if a model is reductive, then allthe simple roots of the spherical variety are of Type T , and our computation of I α ( θ ) in (2.8)confirms Statement 6.3.1 of [Sa]; If a model is non-reductive, the Whittaker induction of thetrilinear model ( G , H ), then for a simple root α of the spherical variety, α is of Type T if α is a simple root of G (recall that G is embedded as the Levi subgroup of G ) and theremaining simple roots are of Type ( U, ψ ). In such case, our computation of I α ( θ ) in (2.22)and (2.24) also confirms Statement 6.3.1 of [Sa].3. The model (GSp × GSp , (GSp × GSp ) )In this section, we compute the local relative character for the model (GSp × GSp , (GSp × GSp ) ). We closely follow the four steps in Section 2.3.1. In Section 3.1, we will define this ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES29 model and verify Step (1), i.e. there is only one open orbit under the action of the Borel sub-group. Then in Section 3.2, we will first study the matrix identities of the product x − α ( x ) η to get the set of virtual weighted colors (Step (2)). Then we will compute the set Θ + (Step(3)) and finally we will compute the constant (cid:80) w ∈ W c W S ( wθ ) (Step (4)).3.1. The model and some orbit computation.
Define the split symplectic similitudegroupGSp n = { g ∈ GL n | g t J n g = l ( g ) J n } , where J n = (cid:18) − w n w n (cid:19) and w n = (cid:18) ... (cid:19) . Here l is the similitude character. Let B n be the Borel subgroup of GSp n consisting of allupper triangular matrices. Set G = GSp × GSp and H = (GSp × GSp ) := { ( h , h ) ∈ GSp × GSp | l ( h ) = l ( h ) } embeds into G via the map h × ( a bc d ) ∈ (GSp × GSp ) = H (cid:55)→ ( (cid:16) a b h c d (cid:17) , h ) ∈ GSp × GSp = G, ( a bc d ) ∈ GSp . For the non-split version of this model, let
D/F be a quaternion algebra. LetGSp n ( D ) = { g ∈ GL n ( D ) | ¯ g t w n g = l ( g ) w n } where ¯ g is the conjugation map on GL n ( D ) induced by the conjugation map on D . Let G D ( F ) = GSp ( D ) × GSp ( D ) and H D ( F ) = (GSp ( D ) × GSp ( D )) = { ( h , h ) ∈ GSp ( D ) × GSp ( D ) | l ( h ) = l ( h ) } embeds into G D ( F ) via the map h × h ∈ (GSp ( D ) × GSp ( D )) (cid:55)→ ( (cid:16) a b h c d (cid:17) , h ) ∈ GSp ( D ) × GSp ( D ) , h = ( a bc d ) ∈ GSp ( D ) . Set η = − − − − − and η − = (cid:32) (cid:33) . Proposition 3.1.
The double cosets B ( F ) \ G ( F ) /H ( F ) contain a unique open orbit B ( F )( η, I ) H ( F ) .Here B ( F ) = B ( F ) × B ( F ) . Let H (cid:48) ( F ) = { h × h ∈ (GSp × GSp ) = H | h ∈ B ( F ) } be a subgroup of GSp ( F ).Let X ( F ) = GSp ( F ) /B ( F ) be the flag variety associated to GSp ( F ). We have a naturalaction of GSp ( F ) on X ( F ) which induces an action of H (cid:48) ( F ) on X ( F ).Let W = Span { w, w , w , w ⊥ , w ⊥ , w ⊥ } be the six dimensional symplectic space defin-ing GSp where { w, w , w , w ⊥ , w ⊥ , w ⊥ } is the standard basis induced by B , i.e. w =(1 , , , , , T , w = (0 , , , , , T , · · · . Then X ( F ) is characterized by X (cid:48) ( F ) = { ( v , v , v ) | (cid:104) v i , v j (cid:105) = 0 , v , v , v are linearly independent } . More specifically, X ( F ) = { Span { v } , Span { v , v } , Span { v , v , v } | ( v , v , v ) ∈ X (cid:48) ( F ) } .The GSp ( F )-action is just g · ( v , v , v ) = ( gv , gv , gv ) . Note that η − · ( w, w , w ) = ( w + w ⊥ , w + w ⊥ + w ⊥ , w + w ⊥ + w ⊥ ) . Hence in order to prove the proposition, it is enough to prove the following lemma.
Lemma 3.2.
The H (cid:48) ( F ) -action on X ( F ) contains a unique open orbit represented by ( w + w ⊥ , w + w ⊥ + w ⊥ , w + w ⊥ + w ⊥ ) .Proof. First, we assume that ( v , v , v ) belongs to the Zariski open subset such that • v has nonzero projection to the subspaces Span { w, w ⊥ } and Span { w ⊥ } ; • The projections of v and v to Span { w, w ⊥ } and Span { w ⊥ , w ⊥ } are linearly inde-pendent; • The projections of v , v and v to Span { w , w ⊥ , w ⊥ } are linearly independent; • The projections of v , v and v to Span { w, w ⊥ , w ⊥ } are linearly independent.Up to the H (cid:48) ( F )-action and because of the first condition we may assume that v = w + w ⊥ .Then by the second condition and since (cid:104) v , v (cid:105) = 0, we may assume v = w + w ⊥ + aw ⊥ + bw + cw ⊥ with a, b, c ∈ F and a (cid:54) = 0. Up to the action of an element diag ( I , (cid:18) t x t − (cid:19) , I ) ∈ H (cid:48) ( F ) , we may assume that v = w + w ⊥ + w ⊥ + cw ⊥ . Note that such an element fixes v . Nowlet h be the element in H (cid:48) ( F ) that fixes w, w , w , w ⊥ , w ⊥ and maps w ⊥ to w ⊥ + cw . Then hv = v , hv = cv + ( w + w ⊥ + w ⊥ ) . Hence we may assume that v = w + w ⊥ + w ⊥ .Finally, because ( v , v ) = ( w + w ⊥ , w + w ⊥ + w ⊥ ) and by the third condition, we mayassume that v = w + aw ⊥ + bw + cw ⊥ . Since (cid:104) v , v (cid:105) = (cid:104) v , v (cid:105) = 0, we have a = 1 , b = 0.Hence v = w + w ⊥ + cw ⊥ . By the fourth condition, we know that c (cid:54) = 0.Now consider the element h = diag (1 , c − , , c − , , c − ) ∈ H (cid:48) ( F ). We have h v = v , h v = c − v , h v = w + w ⊥ + w ⊥ . This proves the lemma. (cid:3)
Now if we let ¯ N (resp. ¯ N ) be the lower triangular unipotent subgroup of GSp (resp.GSp ) and we embed ¯ N into ¯ N via the embedding of GSp to GSp . We also let T (resp. T ) be the diagonal torus of GSp (resp. GSp ). The following lemma is a direct consequenceof the proof of the previous lemma. Lemma 3.3.
For all n ∈ ¯ N ( (cid:36) O F ) and n (cid:48) ∈ ¯ N ( (cid:36) O F ) , we have nηn (cid:48) ∈ B ( F ) ηN ( (cid:36) O F ) T ( O F ) N (cid:48) ( (cid:36) O F ) where N (cid:48) is the lower triangular unipotent subgroup of GSp via as a subgroup of GSp . We need a stronger result.
Lemma 3.4.
For all n ∈ ¯ N ( (cid:36) O F ) and n (cid:48) ∈ ¯ N ( (cid:36) O F ) , we have nηn (cid:48) ∈ T ( O F ) N ( (cid:36) O F ) ηN ( (cid:36) O F ) T ( O F ) N (cid:48) ( (cid:36) O F ) . Proof.
The above lemma implies that nηn (cid:48) ∈ T ( O F ) N ( O F ) ηN ( (cid:36) O F ) T ( O F ) N (cid:48) ( (cid:36) O F ) . We just need to prove the element in N ( O F ) actually belongs to N ( (cid:36) O F ), which is equiv-alent to show that this element preserves the sets V , V , V where V i = { ( a , a , a , a , a , a ) T | a i ∈ O × F , a j ∈ (cid:36) O F for all j (cid:54) = i } . ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES31
But this just follows from the fact that these three sets are fixed by η, η − , T ( O F ) , ¯ N ( (cid:36) O F ) ¯ N ( (cid:36) O F ) , N ( (cid:36) O F ) , N (cid:48) ( (cid:36) O F ) . This proves the lemma. (cid:3)
The above lemma implies the following proposition which is Lemma 2.9 for the current case.
Proposition 3.5.
For all n ∈ ¯ N ( (cid:36) O F ) and n (cid:48) ∈ ¯ N ( (cid:36) O F ) , we have ( n, n (cid:48) )( η, I ) ∈ T ( O F ) N ( (cid:36) O F )( η, I ) H ( O F ) with B = T N . The computation.
Let α = ε − ε , α = ε − ε , α = 2 ε be the simple roots ofGSp and α (cid:48) = ε (cid:48) − ε (cid:48) , α (cid:48) = 2 ε (cid:48) be the simple roots of GSp . We want to compute thevirtual weighted colors associated to these simple roots. Set x − α ( x ) = x − x , x − α ( x ) = x − x , x − α ( x ) = (cid:32) x (cid:33) . We also let Θ the weights of the 32-dimensional representation Spin ⊗ Spin of GSpin ( C ) × GSpin ( C ). We can write it asΘ = { ± e ± e ± e } + { ± e (cid:48) ± e (cid:48) } . For α , we have(3.1) ( x − α ( x ) η, I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F )where g = x − x x x x
00 0 1 0 0 00 0 0 x x x
00 0 0 0 1 0 x x x , h = (cid:32) x − x x x x x x x (cid:33) , b = x +1 0 0 − x
00 0 1 0 0 00 0 0 x +1 0 00 0 0 0 1 00 0 0 0 0 x +1 . This implies that β ∨ α = e − e + e + − e (cid:48) + e (cid:48) and α ∨ − β ∨ α = e − e − e + e (cid:48) − e (cid:48) (note that therepresentation has trivial central character).For α , we have have(3.2) ( x − α ( x ) η, I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F )where g = − x − x − x x − x − x − x
00 0 0 0 0 1 , h = x − x
00 0 − x
00 0 0 − x , b = − x x x − x − x − x − x . This implies that β ∨ α = − e + e − e + e (cid:48) + e (cid:48) and α ∨ − β ∨ α = e + e − e + − e (cid:48) − e (cid:48) .For α , we have(3.3) ( x − α ( x ) η, I ) = ( g, h ) · ( η, I ) · ( g − , h − ) , ( g, h ) ∈ B ( F ) ∩ H ( F )where h = diag (1 − x, , − x,
1) and g = diag (1 , − x, , − x, , − x ). This implies that β ∨ α = e − e + e + − e (cid:48) + e (cid:48) and α ∨ − β ∨ α = − e + e + e + e (cid:48) − e (cid:48) . For α (cid:48) , we can reduce to the root α (because the open orbit is represented by the element( η, I )) but we need to change x − α ( x ) η to ηx − α ( x ) − = ηx − α ( − x ). We have(3.4) ( ηx − α ( − x ) , I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F )where g = − x − x − x − x − x − x − x
00 0 0 0 0 1 , h = (cid:32) − x − x − x
00 0 1 00 0 0 − x (cid:33) , b = − x x x x − x − x − x . This implies that β ∨ α (cid:48) = − e + e + e + e (cid:48) − e (cid:48) and α (cid:48)∨ − β ∨ α (cid:48) = e − e − e + e (cid:48) − e (cid:48) .For α (cid:48) , we can reduce to the root α but we need to change x − α ( x ) η to ηx − α ( x ) − = ηx − α ( − x ). We have(3.5) ( ηx − α ( − x ) , I ) = ( g, h ) · ( η, I ) · ( g − , h − ) , ( g, h ) ∈ B ( F ) ∩ H ( F )where h = diag (1 + x, , x,
1) and g = diag (1 , x, , x, , x ). This implies that β ∨ α (cid:48) = e − e + e + − e (cid:48) + e (cid:48) and α (cid:48)∨ − β ∨ α (cid:48) = − e + e − e + e (cid:48) + e (cid:48) . Remark 3.6.
In this remark, we explain how to use the Luna diagram to compute the virtualcolors. We recall the following Luna diagram of the model (GSp × GSp , (GSp × GSp ) ) in Case (48) of [BP] :The middle row is the Dynkin diagram of G , from left to right we have the simple roots α (cid:48) , α (cid:48) , α , α , α . For each simple root, there are two colors associated to it (represented bythe two ◦ above and below the simple root). There is a line connecting two colors if and onlyif they are equal to each other. For α = α , α , α (cid:48) (resp. α = α , α (cid:48) ), we use β ∨ α to denotethe color above (resp. below) α in the Luna diagram and we use α ∨ − β ∨ α to denote the colorbelow (resp. above) α in the Luna diagram. The Luna diagram above implies that β ∨ α = β ∨ α = β ∨ α (cid:48) , β ∨ α (cid:48) = α ∨ − β ∨ α , α (cid:48)∨ − β ∨ α (cid:48) = α ∨ − β ∨ α , α (cid:48)∨ − β ∨ α (cid:48) = β ∨ α . Combining the first three equations, we have α (cid:48)∨ = β ∨ α (cid:48) + ( α (cid:48)∨ − β ∨ α (cid:48) ) = ( α ∨ − β α ) + ( α ∨ − β ∨ α ) = α ∨ + α ∨ − β ∨ α . This implies that β ∨ α = β ∨ α = β ∨ α (cid:48) = α ∨ + α ∨ − α (cid:48)∨ = e − e + e + − e (cid:48) + e (cid:48) . Combining with the lastthree equations, we have β ∨ α (cid:48) = α ∨ − β ∨ α = − e + e + e e (cid:48) − e (cid:48) , α (cid:48)∨ − β ∨ α (cid:48) = α ∨ − β ∨ α = e − e − e e (cid:48) − e (cid:48) ,α (cid:48)∨ − β ∨ α (cid:48) = β ∨ α = − e + e − e e (cid:48) + e (cid:48) , α ∨ − β ∨ α = e + e − e − e (cid:48) − e (cid:48) . This recovers the above computation of colors using matrix identities.
Proposition 3.7. Θ + is consisting of the following 16 elements: (3.6) e + e ± e ± e (cid:48) ± e (cid:48) , e − e + e e (cid:48) ± e (cid:48) , e − e + e − e (cid:48) + e (cid:48) , ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES33 ± ( e − e − e )2 + e (cid:48) ± e (cid:48) , − e + e − e e (cid:48) + e (cid:48) . Proof.
By the computations of the virtual weighted colors above, we know that Θ + is thesmallest subset of Θ satisfies the following 6 conditions:(1) { e + e − e + − e (cid:48) − e (cid:48) e − e + e + − e (cid:48) + e (cid:48) , − e + e + e + e (cid:48) − e (cid:48) , e − e − e + e (cid:48) − e (cid:48) , − e + e − e + e (cid:48) + e (cid:48) } ⊂ Θ + .(2) Θ + − (Θ + ∩ w α Θ + ) = { e − e + e + − e (cid:48) + e (cid:48) , e − e − e + e (cid:48) − e (cid:48) } .(3) Θ + − (Θ + ∩ w α Θ + ) = { e + e − e + − e (cid:48) − e (cid:48) , − e + e − e + e (cid:48) + e (cid:48) } .(4) Θ + − (Θ + ∩ w α Θ + ) = { e − e + e + − e (cid:48) + e (cid:48) , − e + e + e + e (cid:48) − e (cid:48) } .(5) Θ + − (Θ + ∩ w α (cid:48) Θ + ) = { − e + e + e + e (cid:48) − e (cid:48) , e − e − e + e (cid:48) − e (cid:48) } .(6) Θ + − (Θ + ∩ w α (cid:48) Θ + ) = { e − e + e + − e (cid:48) + e (cid:48) , − e + e − e + e (cid:48) + e (cid:48) } .It is clear that the set in the statement satisfies these conditions. So we just need to showthat the set is the unique subset of Θ satisfying these 6 conditions. Let Θ (cid:48) + be another subsetof Θ satisfies these 6 conditions. Then the set Θ + ∩ Θ (cid:48) + also satisfies these 6 conditions.This implies that Θ + − (Θ + ∩ Θ (cid:48) + ) and Θ (cid:48) + − (Θ + ∩ Θ (cid:48) + ) are W -invariant subsets of Θ. Butthe only W -invariant subsets of Θ are Θ and the empty set. Hence we must have Θ + = Θ (cid:48) + .This proves the proposition. (cid:3) It is clear that Θ + satisfies (2.3). So the last thing remains to prove Lemma 2.14 for thecurrent case. Lemma 3.8.
With the notation above, we have (cid:88) w ∈ W c W S ( wθ ) = 1∆ H/Z
G,H (1) = 1 ζ (2) ζ (4) = (1 − q − ) (1 − q − ) . Recall that c W S ( θ ) = (cid:81) γ ∨∈ Θ+ − q − e γ ∨ (cid:81) α ∈ Φ+ − e α ∨ ( θ ) .Proof. Since the summation is independent of θ , we set θ = δ / B . The lemma follows fromthe following two claims:(1) c W S ( wθ ) is zero unless w is the longest Weyl element.(2) If w is the longest Weyl element, we have c W S ( wθ ) = (1 − q − ) (1 − q − ).The second claim is easy to prove so we will focus on the first one. Let w = ( s, s (cid:48) ) ∈ W so that c W S ( wθ ) is nonzero. Here s is a Weyl element of GSp and s (cid:48) is a Weyl element ofGSp . By abuse of language, we can also view w = ( s, s (cid:48) ) as a Weyl element of the dualgroup GSpin ( C ) × GSpin ( C ). Then(3.7) e γ ∨ ( wθ ) = e w − γ ∨ ( θ ) (cid:54) = q / , γ ∨ ∈ Θ + . The values of the modular character δ / B on the weights e + e + e , e + e − e , e − e + e , − e + e + e , e − e − e , − e + e − e , − e − e + e , − e − e − e q , q , q, , , q − , q − , q − . The values of the modular character δ / B on the weights e (cid:48) + e (cid:48) , e (cid:48) − e (cid:48) , − e (cid:48) + e (cid:48) , − e (cid:48) − e (cid:48) q / , q / , q − / , q − / . Apply (3.7) to the first eight weights in Θ + , we know that s − ( e + e ± e ∈ { e + e + e , − e − e ± e } . Since s is a Weyl element, we must have s − ( e + e ± e ) = { − e − e ± e } . This implies that { s − ( e ) , s − ( e ) } = {− e , − e } , s ( e ) ∈ {± e } . If s − ( e ) = − e , s − ( e ) = − e and s − ( e ) = e , then s − fixes e − e + e and ± ( e − e − e )2 .Combining with (3.7) and the fact that Θ + contains the following 7 elements e − e + e e (cid:48) ± e (cid:48) , e − e + e − e (cid:48) + e (cid:48) , ± ( e − e − e )2 + e (cid:48) ± e (cid:48) , we know that s (cid:48)− { e (cid:48) ± e (cid:48) , − e (cid:48) + e (cid:48) } = { e (cid:48) ± e (cid:48) , − e (cid:48) − e (cid:48) } , s (cid:48)− ( e (cid:48) ± e (cid:48) ∈ { ± e (cid:48) + e (cid:48) , − e (cid:48) − e (cid:48) } . It is easy to see that such an s (cid:48) does not exist, so we get a contradiction. Similarly we canalso get a contradiction when s − ( e ) = − e , s − ( e ) = − e , s − ( e ) = − e or s − ( e ) = − e , s − ( e ) = − e , s − ( e ) = e .Now the only case left is when s − ( e ) = − e , s − ( e ) = − e and s − ( e ) = − e . In thiscase, we have s − ( α ∨ ) = − α ∨ for all α ∨ ∈ { ± e ± e ± e } . By the same argument as in theprevious cases, we know that s (cid:48)− { e (cid:48) ± e (cid:48) , − e (cid:48) + e (cid:48) } = { − e (cid:48) ± e (cid:48) , e (cid:48) − e (cid:48) } , s (cid:48)− ( e (cid:48) ± e (cid:48) ∈ { ± e (cid:48) + e (cid:48) , − e (cid:48) − e (cid:48) } . This implies that s (cid:48)− ( e (cid:48) ) = − e (cid:48) and s (cid:48)− ( e (cid:48) ) = − e (cid:48) . In particular w = ( s, s (cid:48) ) is the longestWeyl element. This proves the lemma. (cid:3) To sum up, we have proved that the local relative character is equal to ζ (1) ζ (4) ζ (6) L (1 / , π, Spin ⊗ Spin ) L (1 , π, Ad)where π = π ⊗ π is an unramified representation of GSp ( F ) × GSp ( F ).4. The model (GL × GL , GL × GL )In this section, we compute the local relative character for the model (GL × GL , GL × GL ). We again closely follow four steps in Section 2.3.1.Let G = GL × GL and H = GL × GL embed into G via the map ( a, b ) (cid:55)→ ( diag ( a, b ) , b ).Similarly, we can also define the quaternion version of this model. Let D/F be a quaternionalgebra, and let G D ( F ) = GL ( D ) × GL ( D ) and H D ( F ) = GL ( D ) × GL ( D ) embed into G D via the map ( a, b ) (cid:55)→ ( diag ( a, b ) , b ). ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES35
We let Θ = Θ ∪ Θ ∪ Θ with Θ being the weights of the representation ∧ ⊗ std ofGL ( C ) × GL ( C ), Θ being the weights of the standard representation of GL ( C ) and Θ being the weights of the dual of the standard representation of GL ( C ). We can write Θ i asΘ = { e i + e j + e (cid:48) k | ≤ i < j ≤ , ≤ k ≤ } , Θ = { e i | ≤ i ≤ } , Θ = {− e i | ≤ i ≤ } . Set η = (cid:18) − − − (cid:19) and η − = (cid:18) (cid:19) . The proofs of the following two lemmas aresimilar to the (GSp × GSp , (GSp × GSp ) ) case, and we will skip them here. Lemma 4.1.
The double cosets B ( F ) \ G ( F ) /H ( F ) contain a unique open orbit B ( F )( η, I ) H ( F ) .Here B ( F ) = B ( F ) × B ( F ) is the upper triangular Borel subgroup. Lemma 4.2.
For all n ∈ ¯ N ( (cid:36) O F ) and n (cid:48) ∈ ¯ N ( (cid:36) O F ) , we have ( n, n (cid:48) )( η, I ) ∈ T ( O F ) N ( (cid:36) O F )( η, I ) H ( O F ) with B = T N . Then we compute the colors for this case. Let α = ε − ε , α = ε − ε , α = ε − ε bethe simple roots of GL and α (cid:48) = ε (cid:48) − ε (cid:48) be the simple root of GL . Set x − α ( x ) = (cid:18) x (cid:19) , x − α ( x ) = (cid:18) x (cid:19) , x − α ( x ) = (cid:18) x (cid:19) . We first study α , we have(4.1) ( x − α ( x ) η, I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F )where g = (cid:32) xx +1 1 x +1 x +1 xx +1 (cid:33) , h = (cid:18) x +1 xx +1 (cid:19) , b = (cid:18) x +1 0 00 0 1 − x x +1 (cid:19) . This implies that β ∨ α = e + e + e (cid:48) and α ∨ − β ∨ α = e + e + e (cid:48) (note that the representationhas trivial central character).For α , we have(4.2) ( x − α ( x ) η, I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F )where g = − x − x − x − x
00 0 0 − x , h = (cid:18) − x − x (cid:19) , b = (cid:18) − x x − x
00 0 0 1 − x (cid:19) . This implies that β ∨ α = e and α ∨ − β ∨ α = − e .For α , we have(4.3) ( x − α ( x ) η, I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F )where g = (cid:32) − x − x (cid:33) , h = (cid:18) − x (cid:19) , b = (cid:18) − x − x (cid:19) . This implies that β ∨ α = e + e + e (cid:48) and α ∨ − β ∨ α = e + e + e (cid:48) .For the root α (cid:48) on GL , we can reduce to the root α on GL but we need to change x − α ( x ) η to ηx − α ( − x ). We have(4.4) ( ηx − α ( − x ) , I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F ) where g = (cid:32) x x x x (cid:33) , h = (cid:18) x (cid:19) , b = (cid:18) x − x x (cid:19) . This implies that β ∨ α (cid:48) = e + e + e (cid:48) and α (cid:48)∨ − β ∨ α (cid:48) = e + e + e (cid:48) . Proposition 4.3. Θ + is consisting of the following 10 elements: e + e i + e (cid:48) j , ≤ i ≤ , ≤ j ≤ e + e + e (cid:48) , e + e + e (cid:48) ; e , e , − e , − e . Proof.
By the computation of colors above, we know that Θ + is the smallest subset of Θsatisfying the following 5 conditions:(1) e + e + e (cid:48) , e + e + e (cid:48) , e + e + e (cid:48) , e , − e ∈ Θ + .(2) Θ + − (Θ + ∩ w α Θ + ) = { e + e + e (cid:48) , e + e + e (cid:48) } .(3) Θ + − (Θ + ∩ w α Θ + ) = { e , − e } .(4) Θ + − (Θ + ∩ w α Θ + ) = { e + e + e (cid:48) , e + e + e (cid:48) } .(5) Θ + − (Θ + ∩ w α (cid:48) Θ + ) = { e + e + e (cid:48) , e + e + e (cid:48) } .It is clear that the set in the statement satisfies these conditions. So we just need to showthat the set is the unique subset of Θ satisfying these conditions. The argument is exactlythe same as the case (GSp × GSp , (GSp × GSp ) ) in Proposition 3.7. We will skip ithere. (cid:3) It is clear that Θ + satisfies (2.3). The last thing remains to prove Lemma 2.14 for thecurrent case. Lemma 4.4.
With the notation above, we have (cid:88) w ∈ W c W S ( wθ ) = 1∆ H/Z
G,H (1) = 1 ζ (1) ζ (2) = (1 − q − )(1 − q − ) . Proof.
Since the summation is independent of θ , we set θ = δ / B . The lemma follows fromthe following two claims:(1) c W S ( wθ ) is zero unless w is the longest Weyl element.(2) If w is the longest Weyl element, we have c W S ( wθ ) = (1 − q − )(1 − q − ) .The second claim is easy to prove so we will focus on the first one. For w = ( s, s (cid:48) ) ∈ W = S × S , we know that c W S ( wθ ) is nonzero if and only if(4.5) 1 − q − / θ s ( i ) , − q − / θ s (1) θ s ( j ) θ (cid:48) s (cid:48) ( k ) , − q − / θ s (1) θ s (4) θ (cid:48) s (cid:48) (1) , − q − / θ s (2) θ s (3) θ (cid:48) s (cid:48) (1) are nonzero for 1 ≤ i ≤ , ≤ j ≤ , ≤ k ≤ θ = q / , θ = θ (cid:48) = q / , θ = θ (cid:48) = q − / , θ = q − / .Using the four terms 1 − q − / θ s ( i ) in (4.5), we have s (1) , s (2) ∈ { , , } , s (3) , s (4) ∈{ , , } . This implies that { s (1) , s (2) } is equal to { , } or { , } . If it is equal to { , } , then θ s (1) θ s (2) = q . Hence θ s (1) θ s (2) θ (cid:48) s (cid:48) (1) or θ s (1) θ s (2) θ (cid:48) s (cid:48) (2) is equal to q / . This is a contradiction.So we must have { s (1) , s (2) } = { , } .If s (1) = 3, then θ s (1) θ s (3) is equal to 1 or q (depends on whether s (3) = 2 or s (3) = 1). Inboth cases, we have θ s (1) θ s (3) θ (cid:48) s (cid:48) (1) or θ s (1) θ s (3) θ (cid:48) s (cid:48) (2) is equal to q / . This is a contradiction.So we must have s (1) = 4 and s (2) = 3.Now if s (3) = 1, then θ s (1) θ s (3) = 1, which implies that θ s (1) θ s (3) θ (cid:48) s (cid:48) (1) or θ s (1) θ s (3) θ (cid:48) s (cid:48) (2) isequal to q / . This is a contradiction. So we must have s (3) = 2 and s (4) = 1. ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES37
Finally, using the fact that 1 − q − / θ s (1) θ s (4) θ (cid:48) s (cid:48) (1) (cid:54) = 0 we know that s (cid:48) (1) = 2 and s (cid:48) (2) = 1.Hence w is the longest Weyl element. This proves the lemma. (cid:3) To sum up, we have proved that the local relative character is equal to ζ (1) ζ (3) ζ (4) L (1 / , π, ∧ ⊗ std ) L (1 / , π , std ) L (1 / , π , std ∨ ) L (1 , π, Ad)where π = π ⊗ π is an unramified representation of GL ( F ) × GL ( F ).5. The model (GL , GL (cid:110) U )In this section, we compute the local relative character for the model (GL , GL (cid:110) U ). Weclosely follow the six steps in Section 2.5.1.Let G = GL , H = H (cid:110) U with H = { diag ( h, h, h ) | h ∈ GL } , U = { u ( X, Y, X ) = (cid:16) I X Z I Y I (cid:17) | X, Y, Z ∈ M at × } . Let P = LU be the parabolic subgroup of G with L = { ( h , h , h ) | h i ∈ GL } . We define ageneric character ξ on U ( F ) to be ξ ( u ( X, Y, Z )) = ψ ( λ ( u ( X, Y, Z ))) where λ ( u ( X, Y, Z )) =tr( X ) + tr( Y ). It is easy to see that H is the stabilizer of this character and ( G, H ) is theWhittaker induction of the trilinear GL model ( L, H , ξ ). The model ( G, H, ξ ) is the socalled Ginzburg–Rallis model introduced by Ginzburg and Rallis in [GR].We can also define the quaternion version of this model. Let
D/F be a quaternion algebra,and let G D ( F ) = GL ( D ), H D = H ,D (cid:110) U D with H ,D ( F ) = { diag ( h, h, h ) | h ∈ GL ( D ) } , U D ( F ) = { (cid:16) X Z Y (cid:17) | X, Y, Z ∈ D } . Like the split case, we can define the character ξ D on U D ( F ) by replacing the trace map of M at × by the trace map of D .Let w = (cid:16) I I I (cid:17) be the Weyl element that sends U to its opposite. It is clear that the w -conjugation map stabilizes L and fixes H . We define the map a : GL → Z L to be a ( t ) = ( tI , I , t − I ) . This clearly satisfies (2.15). For the open Borel orbit, let η = diag ( I , (cid:18) (cid:19) , (cid:18) (cid:19) (cid:18) (cid:19) )be the representative of the open Borel orbit for the model ( L, H ) as in Section 2.3.2, and η = η w . The relation (2.20) has already been verified in Section 2.3.2. This finishes thefirst three steps in Section 2.5.1.Now we compute the set of colors and also the set Θ + . Let Θ be the weights of the exteriorcube representation of GL ( C ). We can write it asΘ = { e i + e j + e k | ≤ i < j < k ≤ } . Let α i = ε i − ε i +1 be the simple roots for 1 ≤ i ≤
5. By the computation of the trilinearGL -model in Section 2.3.2 and the discussion in Section 2.5 (in particular, Remark 2.26),we get the set of colors for this case: β ∨ α = e + e + e , α ∨ − β ∨ α = e + e + e , β ∨ α = e + e + e , α ∨ − β ∨ α = e + e + e ,β ∨ α = e + e + e , α ∨ − β ∨ α = e + e + e . Then we verify (2.23) for α and α .For α , let x − α ( a ) = ( x ij ) ≤ i,j ≤ with x ii = 1 , x = a and x ij = 0 for all the other ( i, j ).We have x − α ( a ) η = η (cid:16) I I X I (cid:17) , X = ( a ) . This proves (2.23) for α .For α , let x − α ( a ) = ( x ij ) ≤ i,j ≤ with x ii = 1 , x = a and x ij = 0 for all the other ( i, j ).We have x − α ( a ) η = η (cid:16) I X I
00 0 I (cid:17) , X = ( − a a ) . This proves (2.23) for α . Next, we compute the set Θ + . Proposition 5.1. Θ + is consisting of the following 10 elements: e + e + e i , e + e + e j , e + e + e , e + e + e , e + e + e , ≤ i ≤ , ≤ j ≤ . Proof.
By the computations above, we know that Θ + is the smallest subset of Θ satisfyingthe following 5 conditions:(1) e + e + e , e + e + e , e + e + e ∈ Θ + .(2) Θ + − (Θ + ∩ w α Θ + ) = { e + e + e , e + e + e } .(3) Θ + − (Θ + ∩ w α Θ + ) = { e + e + e , e + e + e } .(4) Θ + − (Θ + ∩ w α Θ + ) = { e + e + e , e + e + e } .(5) Θ + is stable under w α and w α .It is clear that the set in the statement satisfies these conditions. So we just need to showthat the set is the unique subset of Θ satisfying these conditions. The argument is exactlythe same as the case (GSp × GSp , (GSp × GSp ) ) in Proposition 3.7. We will skip ithere. (cid:3) It is clear that Θ + satisfies (2.3). The last thing remains to prove Lemma 2.30 for thecurrent case. Lemma 5.2.
With the notation above, we have (cid:88) w ∈ W c W S ( wθ ) = 1∆ H /Z G,H (1) = 1 ζ (2) = (1 − q − ) . Proof.
Recall that W = S is the permutation group of 6 variables. The goal is to show that (cid:88) s ∈ S Π e i + e j + e k ∈ Θ + (1 − q − / θ s ( i ) θ s ( j ) θ s ( k ) )Π ≤ i We embed S × S into S by letting S act on the first four elements and S acts on the last two elements. Then (cid:88) s ∈ S × S Π e i + e j + e k ∈ Θ +0 (1 − q − / θ s ( i ) θ s ( j ) θ s ( k ) )Π { ( i,j ) | ≤ i Proof. By the identity for the triple product in Section 2.3.2, we have (here we embed S × S into S by letting the first S -copy act on the first two elements and the second S -copy acton the last two elements) (cid:88) s ∈ S × S × S Π e i + e j + e k ∈ Θ +1 (1 − q − / θ s ( i ) θ s ( j ) θ s ( k ) )(1 − θ s (1) /θ s (2) )(1 − θ s (3) /θ s (4) )(1 − θ s (5) /θ s (6) ) = 1 − q − , where Θ +1 = { e + e + e , e + e + e , e + e + e , e + e + e } . Hence in order to prove thelemma, it is enough to show that (cid:88) s ∈ S /S × S (1 − q − / θ s (1) θ s (2) θ )(1 − q − / θ s (1) θ s (2) θ )Π ≤ i ≤ , ≤ j ≤ (1 − θ s ( i ) /θ s ( j ) ) = 1 . This follows from an easy computation. (cid:3) By the lemma above, we have (cid:80) s ∈ S Π ei + ej + ek ∈ Θ+ (1 − q − / θ s ( i ) θ s ( j ) θ s ( k ) )Π ≤ i Let E = F ( √ (cid:15) ) be a quadratic extension of F , η E/F be the quadraticcharacter associated to E , N E/F (resp. tr E/F ) be the norm map (resp. trace map), and x → ¯ x be the Galois action on E . Denote w n to be the longest Weyl element of GL n . Definethe quasi-split even unitary similitude group GU n,n ( F ) to be(6.1) GU n,n ( F ) = { g ∈ GL n ( E ) : t ¯ gw n g = l ( g ) w n } where l ( g ) ∈ F × is the similitude factor of g .We first define the model (GU , GU (cid:110) U ). Let G = GU , , and P = LU be the standardparabolic subgroup of G with L ( F ) = { m ( g, h ) = (cid:16) g h l ( h ) g ∗ (cid:17) | g ∈ GL ( E ) , g ∗ = w t ¯ g − w , h ∈ GU , ( F ) } ,U ( F ) = { u ( X, Y ) = (cid:16) I X YI X (cid:48) I (cid:17) | X, Y ∈ M at × ( E ) , X (cid:48) = − w t Xw , w Y + t Y w + t X (cid:48) w X (cid:48) = 0 } . Let ξ be a generic character of U ( F ) given by ξ ( u ( X, Y )) = ψ ( λ ( u ( X, Y ))) , λ ( u ( X, Y )) = tr E/F (tr( X )) . Then the stabilizer of ξ under the adjoint action of L ( F ) is H ( F ) := { m ( h, h ) | h ∈ GU , ( F ) } = { diag ( h, h, h ) | h ∈ GU , ( F ) } . Let H = H (cid:110) U and we extend the character ξ to H by making it trivial on H . Themodel ( G, H, ξ ) is the analogue of the Ginzburg–Rallis model in the previous section forunitary similitude group. We can also define the quaternion (non quasi-split) version of thismodel by letting G D be the non quasi-split unitary similitude group (in the archimedeancase G D = GU , ). ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES41 Now we define the model (GU × GU , (GU × GU ) ). Let G = GU , × GU , and H = (GU , × GU , ) = { ( h , h ) ∈ GU , × GU , | l ( h ) = l ( h ) } . We can embed H into G via the map ( h , h ) ∈ H (cid:55)→ ( (cid:16) a b h c d (cid:17) , h ) ∈ G, h = ( a bc d ) . For the pure inner forms of this model, we use GU , = GU , to denote the non quasi-splitunitary similitude group of rank 2, and we use GU , to denote the non quasi-split unitarysimilitude group of rank 4 and split rank 1 (we use these notation in order to be compatiblewith the standard notation in the archimedean case). In the p -adic case, the pure innerforms are (GU , × GU , , (GU , × GU , ) ) , (GU , × GU , , (GU , × GU , ) ) , (GU , × GU , , (GU , × GU , ) ). In the archimedean case, there is an extra compact pure innerform (GU , × GU , , (GU , × GU , ) ).The goal of this section is to compute the local relative character I ( φ θ ) for these twomodels. As we mentioned in Section 2, the difference between these models and all theother models is that since G is not split, the root space maybe two-dimensional. In the nextsubsection, we will prove two identities that will be used in our computation. Then we willcompute the relative character in the last two subsections. From now on, we assume that E/F is unramified and (cid:15) ∈ O × F .6.2. Two identities.Lemma 6.1. We have (6.2) 1 + q (cid:90) O F σ ( x − (cid:15)y − x ) η ( x + y √ (cid:15) ) d x d y = q (1 − q − )(1 − q − σ η ( (cid:36) ))(1 − q − σ ( (cid:36) ))(1 − q − ση ( (cid:36) )) . This integral is an analogy of (2.9). We need this identity when the root space is twodimensional. To compute it, we need the following lemma. Lemma 6.2. The equation x − (cid:15)y − x = 0 has q nonzero solutions in F q × F q .Proof. The equation is equivalent to (2 x − − ε (2 y ) = 1. So the number of solutions isequal to | U ( F q ) | = q + 1. In particular, there are q nonzero solutions. (cid:3) Now we prove Lemma 6.1. Set X = O F × O × F ∪ O × F × O F . The left hand side of (6.2) isequal to 1 + (cid:88) k ≥ q (cid:90) (cid:36) k X σ ( x − (cid:15)y − x ) η ( x + y √ (cid:15) ) d x d y = 1 + (cid:88) k ≥ q − k (cid:90) X σ ( (cid:36) k x − (cid:36) k (cid:15)y − (cid:36) k x ) η k ( (cid:36) ) d x d y = 1 + (cid:88) k ≥ q − k η k σ k ( (cid:36) ) (cid:90) X σ ( (cid:36) k x − (cid:36) k (cid:15)y − x ) d x d y. Now we study the integral (cid:82) X σ ( (cid:36) k x − (cid:36) k (cid:15)y − x ) d x d y . When k > 0, by Theorem 10.2.1in [I00], the integral is equal to q − ((( q − − ( q − q − 1) (1 − q − ) σ ( (cid:36) )1 − q − σ ( (cid:36) ) ) = q − (( q − q ) + ( q − 1) (1 − q − ) σ ( (cid:36) )1 − q − σ ( (cid:36) ) ) . When k = 0, by Theorem 10.2.1 in [I00] and the lemma above, the integral is equal to q − ((( q − − ( q ))+ q (1 − q − ) σ ( (cid:36) )1 − q − σ ( (cid:36) ) ) = q − (( q − q )+( q − 1) (1 − q − ) σ ( (cid:36) )1 − q − σ ( (cid:36) ) ) − q − + q − (1 − q − ) σ ( (cid:36) )1 − q − σ ( (cid:36) ) . This implies that the left hand side of (6.2) is equal to(1 − q − ) σ ( (cid:36) )1 − q − σ ( (cid:36) ) + (cid:88) k ≥ η k σ k ( (cid:36) ) q − k · (( q − q ) + ( q − 1) (1 − q − ) σ ( (cid:36) )1 − q − σ ( (cid:36) ) )= (1 − q − ) σ ( (cid:36) )1 − q − σ ( (cid:36) ) + 11 − ησ ( (cid:36) ) q − (( q − q ) + ( q − 1) (1 − q − ) σ ( (cid:36) )1 − q − σ ( (cid:36) ) )= q (1 − q − )(1 − q − σ η ( (cid:36) ))(1 − q − σ ( (cid:36) ))(1 − q − ση ( (cid:36) )) . This proves Lemma 6.1. We also need the following identity. Lemma 6.3. We have (recall that ϕ = ϕ = 1 O F − q − · (cid:36) − O × F ) q (cid:90) O F η ( x + √ εy ) · | x − εy | − · ϕ ( 2 xx − εy ) d x d y = q (1 − q − η ( (cid:36) )) . Proof. A direct computation shows that (cid:90) X η ( x + √ εy ) · | x − εy | − · ϕ ( 2 xx − εy ) d x d y = q − q , (cid:90) (cid:36)X η ( x + √ εy ) · | x − εy | − · ϕ ( 2 xx − εy ) d x d y = − η ( (cid:36) ) q , (cid:90) (cid:36) k X η ( x + √ εy ) · | x − εy | − · ϕ ( 2 xx − εy ) d x d y = 0 , k ≥ . This proves the lemma. (cid:3) The computation for (GU , GU (cid:110) U ) . In this subsection, we compute the localrelative character for the model (GU , GU (cid:110) U ). First, all the arguments in Section 2.4 stillwork for the current case, the only exception is that the equation (2.17) will become(6.3) (cid:90) G ( F ) Φ( g ) d g = ∆ G (1)∆ H /Z G,H (1) ζ E (1) − ζ (1) − (cid:90) H ( F ) /Z G,H ( F ) (cid:90) B ( F ) Φ( bηh ) d b d h. This is because in the split case, | T ( F q ) | = ( q − dim( T ) ; for our current model, | T ( F q ) | =( q − ( q − I ( φ θ ) = ∆ H /Z G,H (1)∆ G (1) ζ E (1) ζ (1) · (cid:90) ∗ K Y θ − ,ξ ( k ) d k · (cid:90) ∗ K Y θ,ξ − ( k ) d k. Now we compute the integral (cid:82) ∗ K Y θ,ξ ( k ) d k . Let α = ε − ε , α = ε − ε and α = 2 ε be the simple roots of G ( F ). Note that the root spaces of α and α are two dimensionaland the root space of α is one dimensional. Let w = (cid:16) I I I (cid:17) be the Weyl element thatsends U to its opposite. It is clear that the w -conjugation map stabilizes L and fixes H .We define the map a : GL → Z L to be a ( t ) = ( tI , I , t − I ) . This clearly satisfies (2.15).For the open Borel orbit, let η = diag ( (cid:18) (cid:19) (cid:18) (cid:19) , I , (cid:18) − (cid:19) (cid:18) (cid:19) )be the representative of the open Borel orbit for the model ( L, H ), and η = η w . Therelation (2.20) can be easily verified as in the trilinear GL -model case in Section 2.3.2. ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES43 Now we compute the colors. Let Θ be the weights of the exterior cube representation ofˆ G ( C ). We can write it as Θ = { e i , − e i , ± e ± e ± e | ≤ i ≤ } . The weight spaces of e i , − e i are two dimensional and the weight spaces of ± e ± e ± e are onedimensional. More precisely, the exterior representation of the L -group L GU of GU isexplicated in Section 3.1 [Z]. More details on the exterior cubic L -function of GU are alsogiven there.For α , as in the split case, we let I α ( θ ) = vol ( I ) − (cid:82) G ( F ) Y θ,ξ ( xη )(Φ ( x ) + Φ w α ( x )) d x. Since the root space is two dimensional, the same argument in the split case implies that I α ( θ ) = 1 + q (cid:90) O F ( θ − δ / )( e α ∨ (( x + y √ ε ) − )) Y θ,ξ ( x − α (( x + y √ ε ) − ) η ) d x d y. Here x − α ( a ) = diag ( (cid:18) a (cid:19) , I , (cid:18) − ¯ a (cid:19) ). Meanwhile, a direct computation shows that x − α ( x + y √ (cid:15) ) η is equal to diag ( (cid:18) x +1 00 1 (cid:19) , (cid:18) − y √ (cid:15)x +1 x +1 (cid:19) , (cid:18) x +1 00 1 (cid:19) ) · η · diag ( (cid:18) y √ (cid:15) x + 1 (cid:19) , (cid:18) y √ (cid:15) x + 1 (cid:19) , (cid:18) y √ (cid:15) x + 1 (cid:19) ) . Since x + y √ (cid:15) = xx − y (cid:15) − y √ (cid:15)x − y (cid:15) and 1 + xx − y (cid:15) = x + x − y (cid:15)x − y (cid:15) , we have I α ( θ ) = 1 + q (cid:90) O F σ ( x − εy − x ) η ( x + y √ ε ) d x d y where η = θ ( e α ∨ ) · ( | | − · σ − ) ◦ N E/F and σ = θ ( e β α ∨ ) | F × · | | − / with β ∨ α = e − e − e .Combing with Lemma 6.1, we have I α ( θ ) = q (1 − q − ) · − q − e α ∨ ( θ )(1 − q − / e β ∨ α ( θ ))(1 − q − / e α ∨ − β ∨ α ( θ ))with β ∨ α = e − e − e , α ∨ − β ∨ α = e − e + e . For α , as in the Ginzburg–Rallis model case, it is easy to see that Y θ,ξ ( x − α ( a ) η ) = ϕ ( a + ¯ a ) , x − α ( a ) = diag (1 , (cid:18) a (cid:19) , (cid:18) − ¯ a (cid:19) , . Then we have (note that the root space in this case is also 2-dimensional) I α ( θ ) = 1 + q (cid:90) O F θ ( e α ∨ ( x + √ εy )) · | x − εy | − · ϕ ( 2 xx − εy ) d x d y. By Lemma 6.3, we know that I α ( θ ) = q · (1 − q − e α ∨ ( θ )) . For α , the root space is one dimensional, so we have the identity I α ( θ ) = 1 + q (cid:90) O F ( θ − δ / )( e α ∨ ( a − )) Y θ,ξ ( x − α ( a − ) η ) d a where x − α ( x ) = diag ( I , (cid:18) x √ (cid:15) (cid:19) , I ). On the other hand, we have x − α ( x ) η = diag ( (cid:18) x √ (cid:15) − x √ (cid:15) − x √ (cid:15) (cid:19) , (cid:18) − x √ (cid:15) − x (cid:15) (cid:19) , (cid:18) − x √ (cid:15) − x √ (cid:15) x √ (cid:15) (cid:19) ) · η × diag ( (cid:32) − x (cid:15) x √ (cid:15) − x (cid:15)x √ (cid:15) − x (cid:15) − x (cid:15) (cid:33) , (cid:32) − x (cid:15) x √ (cid:15) − x (cid:15)x √ (cid:15) − x (cid:15) − x (cid:15) (cid:33) , (cid:32) − x (cid:15) x √ (cid:15) − x (cid:15)x √ (cid:15) − x (cid:15) − x (cid:15) (cid:33) ) . This implies that (note that all the characters are unramified and hence their values at a + √ ε, a − √ ε are equal to 1 for all a ∈ O F ) I α ( θ ) = q + 1 = ( q + 1) · − q − e α ∨ ( θ )1 − q − e β ∨ α ( θ ) , with β ∨ α = α ∨ − β ∨ α = e . Then we compute the set Θ + . Lemma 6.4. Let W = S (cid:110) ( Z / Z ) be the Weyl group of G and let Θ + be the smallestsubset of Θ satisfying the following two conditions:(1) e − e ± e , e ∈ Θ + .(2) Θ + − (Θ + ∩ w α Θ + ) = { e − e ± e } , Θ + = w α Θ + , Θ + − (Θ + ∩ w α Θ + ) = { e } .Then we have Θ + = { e , e , e , e ± e ± e } .Proof. It is clear that the set { e , e , e , e ± e ± e } satisfies both conditions. So we just needto show that the set is the unique subset of Θ satisfying these conditions. The argument isexactly the same as the case (GSp × GSp , (GSp × GSp ) ) in Proposition 3.7. We willskip it here. (cid:3) Now we decompose Θ as Θ ∪ Θ and Φ = Φ ∪ Φ where Θ i , Φ i contain the weights/rootswhose weight spaces/root spaces are i dimensional. More specifically,Φ = {± e i } , Φ = {± e i ± e j } , Θ = { ± e ± e ± e } , Θ = {± e i } , ≤ i, j ≤ , i (cid:54) = j. Similarly, we can define Φ + i and Θ + i for i = 1 , 2. Set β ( θ ) = Π i ∈{ , } Π α ∈ Φ + i − q − i e α ∨ Π i ∈{ , } Π γ ∨ ∈ Θ + i − q − i/ e γ ∨ . Then it is clear that ζ (1) − ζ − E (1) β ( θ ) β ( θ − ) = L (1 / , π, ∧ ) L (1 , π, Ad) . The next lemma is an analogue of Lemma 2.30 for the current case. Lemma 6.5. Set c W S ( θ ) = Π i ∈{ , } Π γ ∨∈ Θ+ i − q − i/ e γ ∨ Π i ∈{ , } Π α ∈ Φ+ i − e α ∨ ( θ ) . Then (cid:80) w ∈ W c W S ( wθ ) is independentof θ and is equal to H /ZG,H (1) = ζ (2) − = 1 − q − .Proof. Our goal is to show that ( θ i are arbitrary variables) (cid:88) w ∈ W w ( Π ε ,ε ∈{± } (1 − q − / (cid:112) θ θ ε θ ε ) · Π i =1 (1 − q − θ i )(1 − θ θ )(1 − θ θ )(1 − θ θ )(1 − θ θ )(1 − θ θ )(1 − θ θ )(1 − θ )(1 − θ )(1 − θ ) ) ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES45 is equal to ζ (2) − = 1 − q − . Multiplying both the denominator and the numerator by θ − / θ − / θ − / , the denominator will be ( W, sgn)-invariant. Hence it is enough to show that (cid:88) w ∈ W sgn( w ) · w ( θ − / θ − / θ − / · Π ε ,ε ∈{± } (1 − q − / (cid:112) θ θ ε θ ε ) · Π i =1 (1 − q − θ i ))is equal to 1 − q − times(6.4) θ − / θ − / θ − / (1 − θ θ )(1 − θ θ )(1 − θ θ )(1 − θ θ )(1 − θ θ )(1 − θ θ )(1 − θ )(1 − θ )(1 − θ ) . We need to study the q − k/ -coefficients (0 ≤ q ≤ 10) of θ − / θ − / θ − / · Π ε ,ε ∈{± } (1 − q − / (cid:112) θ θ ε θ ε ) · Π i =1 (1 − q − θ i ) . For k = 1 , , , , 9, the q − k/ -coefficients are combinations of θ a θ a θ a , a , a ∈ { , − , − } , a ∈ { , , − } . For any such triple ( a , a , a ), we either have a i = ± a j for some i (cid:54) = j or we have a i = 0for some i . Hence the ( W, sgn)-summation of the q − k/ -coefficients are all equal to 0 for k = 1 , , , , q -coefficient is equal to θ − / θ − / θ − / , and the ( W, sgn)-summation of it is equal tothe denominator (6.4).The q − -coefficient is equal to − θ θ θ , and the ( W, sgn)-summation of it is equal to zerosince the powers of θ and θ are equal.The q − -coefficient is equal to θ − / θ − / θ − / + θ − / θ − / θ / + θ − / θ − / θ − / + θ − / θ − / θ − / − θ − / θ − / θ − / + θ − / θ − / θ − / − θ − / θ − / θ / . The ( W, sgn)-summation of all the terms except the last two are equal to zero because eithertwo of the powers are equal to other or two of the powers are opposite to each other. The( W, sgn)-summation of θ − / θ − / θ − / and θ − / θ − / θ / are both equal to − W, sgn)-summation of the q − -coefficient is equal to 0.The q − -coefficient is equal to θ / θ − / θ − / + θ / θ − / θ / − θ − / θ − / θ / − θ − / θ − / θ − / − θ − / θ − / θ / − θ − / θ − / θ / − θ − / θ / θ / . The ( W, sgn)-summation of all the terms is equal to zero because either two of the powersare equal to other or two of the powers are opposite to each other. As a result, the ( W, sgn)-summation of the q − -coefficient is equal to 0.The q − -coefficient is equal to − θ − / θ − / θ − / − θ − / θ − / θ / − θ − / θ / θ − / − θ − / θ − / θ / − θ − / θ − / θ / − θ − / θ − / θ − / − θ − / θ − / θ / + θ − / θ − / θ / − θ − / θ − / θ − / − θ − / θ − / θ − / − θ − / θ − / θ − / − θ − / θ − / θ − / − θ − / θ − / θ − / − θ − / θ − / θ / . The ( W, sgn)-summation of all the terms except the last term is equal to zero because eithertwo of the powers are equal to other or two of the powers are opposite to each other. The( W, sgn)-summation of θ − / θ − / θ / is equal to the denominator (6.4). The q − -coefficient is equal to θ − / θ − / θ / + θ − / θ − / θ − / + θ − / θ − / θ / + θ − / θ − / θ − / + 2 θ − / θ − / θ − / + θ − / θ − / θ / + θ − / θ − / θ − / − θ / θ − / θ − / + θ − / θ − / θ / + 2 θ − / θ − / θ / + θ − / θ / θ / + θ − / θ − / θ / + θ − / θ / θ − / + θ − / θ − / θ / . The ( W, sgn)-summation of all the terms is equal to zero because either two of the powers areequal to other or two of the powers are opposite to each other. Hence the ( W, sgn)-summationof the q − -coefficient is equal to 0.This finishes the proof of the lemma. (cid:3) Now by a very similar argument as in Section 2.3, our computation of the colors and thelemma above implies that (cid:90) ∗ K Y θ ( k ) d k = ∆ G (1)∆ H /Z G,H ζ (1) − ζ E (1) − · β ( θ ) . There are only two differences • The c -function function for GU is defined to be c α ( θ ) = − q − e α ∨ − e α ∨ ( θ ) if the root spaceof α is one dimensional and is defined to be c α ( θ ) = − q − e α ∨ − e α ∨ ( θ ) if the root space of α is two dimensional. This matches our definition of β ( θ ) and c W S ( θ ) for this case. • The volume of Iwahori subgroup of GU is equal to ∆ G (1) ζ (1) − ζ E (1) − · q − l ( W ) . Thisis why we get ζ (1) − ζ E (1) − instead of ζ (1) − rk ( G ) for this case.This implies that I ( φ θ ) = ∆ G (1)∆ H /Z G,H (1) ζ (1) − ζ E (1) − · β ( θ ) · β ( θ − ) = ∆ G (1)∆ H /Z G,H (1) · L (1 / , π, ∧ ) L (1 , π, Ad) . The computation for (GU × GU , (GU × GU ) ) . In this subsection, we compute thelocal relative character for the model (GU × GU , (GU × GU ) ). We first study the openBorel orbit. Let B n be the upper triangular Borel subgroup of GU n,n and B = B × B bea Borel subgroup of G . We write B = T N and let ¯ B = T ¯ N be the opposite Borel subgroup.Set η − = (cid:18) − − (cid:19) and η = (cid:18) − − (cid:19) . The proofs of the following two lemmas aresimilar to the (GSp × GSp , (GSp × GSp ) ) case, and we will skip them here. Lemma 6.6. The double cosets B ( F ) \ G ( F ) /H ( F ) contain a unique open orbit B ( F )( η, I ) H ( F ) . Lemma 6.7. For all n ∈ ¯ N ( (cid:36) O F ) , we have n ( η, I ) ∈ T ( O F ) N ( (cid:36) O F )( η, I ) H ( O F ) . Now all the arguments in Section 2.2 still work for the current case, the only exception isthat the equation in Lemma 2.5 will become(6.5) (cid:90) G ( F ) Φ( g ) d g = ∆ G (1)∆ H/Z G,H (1) ζ E (1) − ζ (1) − (cid:90) H ( F ) /Z G,H ( F ) (cid:90) B ( F ) Φ( bηh ) d b d h. This implies that I ( φ θ ) = ∆ H/Z G,H (1)∆ G (1) ζ E (1) ζ (1) · (cid:90) K Y θ − ( k ) d k · (cid:90) K Y θ ( k ) d k. ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES47 Next we compute the colors. Let Θ be the weights of the representation ∧ ⊗ std ⊕ std ⊕ std ∨ . We can write it asΘ = { ± e ± e ± e (cid:48) , ± e (cid:48) , ± e i | ≤ i ≤ } . The weight spaces of ± e (cid:48) , ± e i are two dimensional and the weight spaces of ± e ± e ± e (cid:48) are onedimensional. For this model, since the representation π of G ( F ) is of trivial central character,the associated L -parameter factors through an endoscopic subgroup GL ( C ) × GL ( C ) × GL ( C ) of L GU . ∧ ⊗ std ⊕ std ⊕ std ∨ is a representation of GL ( C ) × GL ( C ) × GL ( C )of dimension 20, which is the restriction of ∧ of L GU into the endoscopic subgroup. Thenthis restriction can be naturally extended to an irreducible representation of the L -group L (GU × GU ), which is also of dimension 20. By abuse of notation, ∧ ⊗ std ⊕ std ⊕ std ∨ is considered as a representation of L (GU × GU ), which is ρ X for this model.Let α = ε − ε , α = 2 ε and α (cid:48) = 2 ε (cid:48) be the simple roots of G . We can define I α , I α and I α (cid:48) as in the previous case. For α , the root space is two dimensional and we have thematrix identity(6.6) ( x − α ( x + y √ ε ) η, I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F )where x − α ( x + y √ ε ) = (cid:32) x + y √ ε − x + y √ ε (cid:33) , g = (cid:32) x y √ ε 00 0 1+ x − y √ ε (cid:33) , h = (cid:18) y √ ε x (cid:19) , b = x − y √ ε x 00 0 x 00 0 0 1 . By the same argument as in the α case in the previous subsection, we have I α ( θ ) = q (1 − q − ) · − q − e α ∨ ( θ )(1 − q − / e β ∨ α ( θ ))(1 − q − / e α ∨ − β ∨ α ( θ ))with β ∨ α = e − e − e (cid:48) and α ∨ − β ∨ α = e − e + e (cid:48) .For α , the root space is one dimensional and we have the matrix identity(6.7) ( x − α ( x ) η, I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F )where x − α ( x ) = (cid:18) x √ ε (cid:19) , g = (cid:32) x √ ε x √ ε x √ ε x √ ε (cid:33) , h = (1 + x √ ε ) I ,b = 11 − x ε (cid:32) − x √ ε x √ ε − x √ ε − x √ ε − x √ ε − x √ ε − x ε − x ε + x √ ε − x √ ε (cid:33) . By the same argument as in the α case in the previous subsection (use the fact that all theunramified characters have value 1 at a ± √ ε for a ∈ O F ), we have I α ( θ ) = q + 1 = ( q + 1) · − q − e α ∨ ( θ )1 − q − e β ∨ α ( θ ) , with β ∨ α = α ∨ − β ∨ α = e . For α (cid:48) , the root space is one dimensional and it can be reduced to α but we need tochange x − α ( x ) η to ηx − α ( − x ). We have the matrix identity(6.8) ( ηx − α ( − x ) , I ) = ( b, h − ) · ( η, I ) · ( g, h ) , ( b, h − ) ∈ B ( F ) , ( g, h ) ∈ H ( F ) where g = − x √ ε − x √ ε − x √ ε x √ ε x √ ε − x √ ε 00 0 − x √ ε − x √ ε − x √ ε − x √ ε , h = (cid:32) − x √ ε − x √ ε x √ ε x √ ε (cid:33) , b = − x √ ε x √ ε x √ ε x √ ε x √ ε x √ ε − x √ ε x √ ε x √ ε x √ ε − x √ ε x √ ε − x √ ε x √ ε . By the same argument as in the α case in the previous subsection, we have I α (cid:48) ( θ ) = q + 1 = ( q + 1) · − q − e α (cid:48)∨ ( θ )1 − q − e β ∨ α (cid:48) ( θ ) , with β ∨ α (cid:48) = α (cid:48)∨ − β ∨ α (cid:48) = e (cid:48) . Then we compute the set Θ + . Lemma 6.8. Let W = ( S (cid:110) ( Z / Z ) ) × ( Z / Z ) be the Weyl group of G and let Θ + be thesmallest subset of Θ satisfying the following two conditions:(1) e − e ± e (cid:48) , e , e (cid:48) ∈ Θ + .(2) Θ + − (Θ + ∩ w α Θ + ) = { e − e ± e (cid:48) } , Θ + − (Θ + ∩ w α Θ + ) = { e } , Θ + − (Θ + ∩ w α (cid:48) Θ + ) = { e (cid:48) } .Then we have Θ + = { e , e , e (cid:48) , e ± e ± e (cid:48) } .Proof. It is clear that the set { e , e , e (cid:48) , e ± e ± e (cid:48) } satisfies the two conditions. So we just needto show that the set is the unique subset of Θ satisfying these conditions. The argument isexactly the same as the case (GSp × GSp , (GSp × GSp ) ) in Proposition 3.7. We willskip it here. (cid:3) Now as in the previous case, we decompose Θ as Θ ∪ Θ and Φ = Φ ∪ Φ where Θ i , Φ i contain the weights/roots whose weight spaces/root spaces are i dimensional:Φ = {± e i , ± e (cid:48) } , Φ = {± e ± e } , Θ = { ± e ± e ± e (cid:48) } , Θ = {± e i , ± e (cid:48) } , ≤ i ≤ . Similarly, we can define Φ + i and Θ + i for i = 1 , 2. Set β ( θ ) = Π i ∈{ , } Π α ∈ Φ + i − q − i e α ∨ Π i ∈{ , } Π γ ∨ ∈ Θ + i − q − i/ e γ ∨ . Then it is clear that ζ (1) − ζ − E (1) β ( θ ) β ( θ − ) = L (1 / , π, ρ X ) L (1 , π, Ad) . The next lemma is an analogue of Lemma 2.14 for the current case. Lemma 6.9. Set c W S ( θ ) = Π i ∈{ , } Π γ ∨∈ Θ+ i − q − i/ e γ ∨ Π i ∈{ , } Π α ∈ Φ+ i − e α ∨ ( θ ) . Then (cid:80) w ∈ W c W S ( wθ ) is independentof θ and is equal to H/ZG,H (1) = ζ (2) − L (1 , η E/F ) − = (1 − q − ) (1 + q − ) .Proof. Since H is reductive, Theorem 7.2.1 of [Sa] implies that the summation is independentof θ . Now we let θ = δ / B . The lemma follows from the following two claims:(1) c W S ( wθ ) is zero unless w is the longest Weyl element.(2) If w is the longest Weyl element, we have c W S ( wθ ) = (1 − q − ) (1 + q − ). ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES49 The second claim is easy to prove so we will focus on the first one. Let w = ( s, s (cid:48) ) ∈ W with s ∈ S (cid:110) ( Z / Z ) and s (cid:48) ∈ Z / Z so that c W S ( wθ ) is nonzero.The factor 1 − q − e e (cid:48) ( wθ ) in the numerator of c W S ( wθ ) forces s (cid:48) to be the longest Weylelement of GU , . The factors 1 − q − e e i ( wθ ) , i = 1 , s ( e ) , s ( e ) ∈{± e , − e } . Hence there are four possibilities of s : s ( e ) = ± e , s ( e ) = − e or s ( e ) = ± e , s ( e ) = − e . If s ( e ) = ± e , s ( e ) = − e or s ( e ) = e , s ( e ) = − e , one of thefactors 1 − q − / e e ± e + e (cid:48) ( wθ ) in the numerator is equal to 0. Hence we must have s ( e ) = − e , s ( e ) = − e , i.e. w is the longest Weyl element. This proves the lemma. (cid:3) As in the previous case, our computation of the colors and the lemma above imply that (cid:90) K Y θ ( k ) d k = ∆ G (1)∆ H/Z G,H ζ (1) − ζ E (1) − · β ( θ ) . This implies that I ( φ θ ) = ∆ G (1)∆ H/Z G,H (1) ζ (1) − ζ E (1) − · β ( θ ) · β ( θ − ) = ∆ G (1)∆ H/Z G,H (1) · L (1 / , π, ρ X ) L (1 , π, Ad) . The model ( E , PGL (cid:110) U )In this section, we compute the local relative character of the model ( E , PGL (cid:110) U ). Weclosely follow the six steps in Section 2.5.1.To define this model, we recall a description of the adjoint group of type E , followingnotation in [P20]. Let H ( H ) be the degree three central simple Jordan algebra over k . Here H is a quaternion algebra over k and denote by N its norm map, tr the trace, and x (cid:55)→ x ∗ itsconjugation. More precisely, one may realize H ( H ) as the vector space of all 3 × H , which are of form(7.1) J = (cid:16) a z y ∗ z ∗ b xy x ∗ c (cid:17) , where x, y, z ∈ H and a, b, c ∈ k . The Jordan algebra on H ( H ) is defined by the composition J ◦ J := ( J J + J J ) for J , J ∈ H ( H ), where J J and J J are under the matrixmultiplications. The cubic norm det on H ( H ) is defined by(7.2) det( J ) := abc − aN ( x ) − bN ( y ) − cN ( z ) + tr( xyz ) , and the adjoint map (cid:93) is J (cid:93) := (cid:18) bc − N ( x ) y ∗ x ∗ − cz zx − by ∗ xy − cz ∗ ac − N ( y ) z ∗ y ∗ − axx ∗ z ∗ − by yz − ax ∗ ab − N ( z ) (cid:19) . Denote by ( · , · , · ) the symmetric trilinear form corresponding to the cubic norm det with( A, A, A ) = det( A ) for A ∈ H ( H ).In [R97], Rumelhart constructed the Lie algebra g ( H ( H )) through a Z -grading. (Herewe following the notation in [P20, Section 4.2].) More precisely, define(7.3) g = sl ⊕ m ⊕ V ⊗ H ( H ) ⊕ V ∨ ⊗ H ( H ) ∨ where V and V ∨ are the standard representation of sl and its dual representation, respec-tively. Here let m be the Lie algebra consisting of all linear transformations φ on H ( H )such that( φ ( z ) , z , z ) + ( z , φ ( z ) , z ) + ( z , z , φ ( z )) = 0 for all z , z , z ∈ H ( H ) . And we refer the reader to Section 4.2.1 in [P20] for the description of the Lie bracket on g ( H ( H )).Now, let us consider the identity component of the automorphism group Aut( g ( H ( H ))),which is the quaternionic adjoint group of type E . In particular, if H is split, then it is thesplit adjoint group of E , denoted by G . If H is not split, then we denote it by G D , whichis of type E , and of k -rank 4.Next, let us explicate this model for the split case. In this case, the quaternion H is splitand take H = M × ( F ) with x ∗ = (cid:18) d − b − c a (cid:19) , tr( x ) = a + d, N ( x ) = det( x ) = ad − bc, for x = ( a bc d ) ∈ H . We may identify H ( H ) to { A ∈ M × ( F ) : A = Γ A t Γ − } as follows (cid:16) a z y ∗ z ∗ b xy x ∗ c (cid:17) (cid:55)→ (cid:18) aI z y ∗ z ∗ bI xy x ∗ cI (cid:19) ∈ M × ( F ) , where Γ = diag { ( − ) , ( − ) , ( − ) } . Then the cubic norm det in (7.2) on H ( H ) is givenby det( A ) = Pf(Γ A ) where Pf is the Pfaffian of the skew-symmetric matrices.The Lie algebra m ( F ) is isomorphic to sl ( F ) via the action of sl ( F ) on H ( H ) given by A · X := AX + XA ∗ for X ∈ H ( H ) where A ∗ = Γ t A Γ − . Consider V and V ∨ in (7.3) asthe 3-dimensional vector spaces of column vectors. The action of sl ( F ) on V and V ∨ aregiven by: for v ∈ V , δ ∈ V ∨ , and φ ∈ sl ( F ),(7.4) φ ( v ) = φv and φ ( δ ) = − t φδ where the products in the right hand sides are the matrix multiplications.For A ∈ GL ( F ), define Φ A ∈ End ( H ( H )) byΦ A ( X ) := AXA ∗ and Φ ∨ A ( X ) := ( A ∗ ) − XA − . Write (GL × GL × GL ) ◦ = { ( a, g, λ ) ∈ GL × GL × GL | λ Det( A )Det( a ) = 1 } where Det is the usual determinant of GL n . Define the map ι from (GL × GL × GL ) ◦ toGL( g ) as ι : φ (cid:55)→ aφa − A (cid:55)→ gAg − v ⊗ X (cid:55)→ ( av ) ⊗ λ Φ A ( X ) δ ⊗ γ (cid:55)→ ( a t ) − δ ⊗ λ − Φ ∨ A ( γ ) . By a straightforward computation, we have the image of ι lies in G . Moreover, the kernel of ι is ker ι = { ( wI , zI , ( wz ) − ) | w, z ∈ F × } ∼ = F × × F × .We take the unipotent subgroup U of Lie algebra u consisting of elements { (cid:16) v v v (cid:17) ∈ sl } ⊕ { (cid:18) × x y × z × (cid:19) ∈ sl } ⊕ F w ⊗ H ( H ) ⊕ F w ⊗ H ( H ) ⊕ F w ⊗ H ( H ) ∨ , where { w , w , w } is the standard basis of F . Then its corresponding Levi subgroup L isgiven by the image ι ( { ( (cid:16) a b c (cid:17) , (cid:16) g g g (cid:17) , λ ) | Det( g )Det( g )Det( g ) abc = λ − } ) . ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES51 For u ∈ u , define the character ξ of U by ξ (exp( u )) = ψ ( v + Tr( x ) + Tr( z ) + e )where e is the entry corresponding to the simple root α = e − e , i.e. the coefficient of w ⊗ ( E , + E , ). ( E i,j are the elementary matrices in M × ( F ).) The stabilizer H of ξ isgiven by the image ι ( { ( (cid:16) a a c (cid:17) , (cid:16) g g g (cid:17) , λ ) | aλ Det( g ) = 1 , Det( g ) a c = λ − } ) , which is isomorphic to PGL ( F ). Let H = H (cid:110) U and we extend the character ξ to H by making it trivial on H . The model ( G, H, ξ ) is the Whittaker induction of the trilinearGL model ( L, H , ξ ). We can also define the quaternion (non-split) version of this model byletting G D be of type E , . In the non-split case, L D (cid:111) U D is a minimal parabolic subgroupof G D defined over F and ξ D is a generic character of U D . Then the stabilizer H ,D of ξ D in L D is isomorphic to P D × . Thus we obtain the quaternion (non-split) version ( G D , H D , ξ D )with H D = H ,D (cid:111) U D .Define the Weyl element w of E by w : φ ∈ sl (cid:55)→ − φ t ∈ sl A ∈ sl (cid:55)→ − A ∗ ∈ sl v ⊗ X ∈ V ⊗ H (cid:55)→ v ⊗ X ∈ V ∨ ⊗ H ∨ δ ⊗ γ ∈ V ∨ ⊗ H ∨ (cid:55)→ δ ⊗ γ ∈ V ⊗ H . Then w = 1 and w sends U to its opposite. It is clear that the w -conjugation mapstabilizes L and fixes H . We define the map a : GL → Z L to be a ( t ) = ι ( (cid:16) t t − (cid:17) , (cid:18) tI I t − I (cid:19) , t ) . This clearly satisfies (2.15). For the open Borel orbit, let η = w (cid:48) γ , where w (cid:48) = ι ( I , (cid:32) I (cid:33) , 1) and γ = ι ( I , (cid:16) I (cid:17) , , be the representative of the open Borel orbit for the model ( L, H ) as in Section 2.3.2, and η = η w . The relation (2.20) has already been verified in Section 2.3.2. This finishes thefirst three steps in Section 2.5.1.Now we compute the set of colors and also the set Θ + . Following the notation in [B02],let α = ( ε + ε ) − (cid:80) i =2 ε i , α = ε + ε and α i +1 = ε i − ε i − for 3 ≤ i ≤ E ( C ),corresponding to the 7-th fundamental weight ω , where ω = e + ( e − e ). We can writeit asΘ = {± e i ± 12 ( e − e ) | ≤ i ≤ } ∪ { (cid:88) i =1 a i e i | { i : a i = 1 } is even and a i = ± } . By the computation of the trilinear GL -model in Section 2.3.2 and the discussion inSection 2.5 (in particular, Remark 2.26), we get the set of colors for this case: β ∨ α = e + e + e − e − e + e , α ∨ − β ∨ α = − e − e − e + e − e + e , β ∨ α = e + e − e + e + e − e , α ∨ − β ∨ α = − e − e − e + e − e + e ,β ∨ α = e + e − e + e + e − e , α ∨ − β ∨ α = e + e + e − e − e + e . Then we verify (2.23) for α , α , α and α . Let x − α ( a ) = ι ( (cid:16) a (cid:17) , I , , x − α ( a ) = Id + a · ad w ⊗ ( E , + E , ) ∨ ,x − α ( a ) = ι ( I , I + aE , , , x − α ( a ) = ι ( I , I + aE , , . We have the following 4 identities x − α ( a ) η = ηx α ( a ) , x − α ( a ) η = ηx α ( a ) ,x − α ( a ) η = ηx α ( − a ) x e + e ( − a ) , x − α ( a ) η = ηx e − e ( − a ) . This proves (2.23) for α , α , α and α . In addition, we label the type of each simple rootin the following weighted Dynkin Diagram:0 α , T α , ( U, ψ ) 0 α , T α , ( U, ψ ) 2 α , ( U, ψ ) 2 α , ( U, ψ )0 α , T Figure 1. Weighted Dynkin Diagram of E Note that this weighted Dynkin Diagram is associated to the special nilpotent stable orbitof Balar-Carter label E . Its corresponding unipotent stable orbit is the maximal unipotentorbit with a non-empty intersection with the unipotent subgroup U .Next, we compute the set Θ + . Proposition 7.1. Θ + is consisting of the following 28 elements: e + e + e + e + e + e − e i − e i , − e − e − e − e − e − e + 2 e i (cid:48) + 2 e j (cid:48) , (7.5) e + e + e + e + e + e , − e + e + e + e + e + e − e k , for ≤ k ≤ , (7.6) ± e m + e − e , for ≤ m ≤ , (7.7) where ( i, j ) ∈ { (23) , (24) , (34) , (25) , (35) , (45) , (26) , (36) } and ( i (cid:48) , j (cid:48) ) ∈ { (56) , (46) } .Proof. By the computation of the colors, we know that Θ + is the smallest subset of Θsatisfying the following 5 conditions:(1) ( e + e + e − e − e + e ) , ( − e − e − e + e − e + e ) , ( e + e − e + e + e − e ) ∈ Θ + .(2) Θ + − (Θ + ∩ w α Θ + ) = { ( e + e + e − e − e + e ) , ( − e − e − e + e − e + e ) } .(3) Θ + − (Θ + ∩ w α Θ + ) = { ( e + e − e + e + e − e ) , ( − e − e − e + e − e + e ) } .(4) Θ + − (Θ + ∩ w α Θ + ) = { ( e + e − e + e + e − e ) , ( e + e + e − e − e + e ) } .(5) Θ + is stable under w α , w α , w α and w α . ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES53 It is clear that the set in the statement satisfies these conditions. So we just need to showthat the set is the unique subset of Θ satisfying these conditions. The argument is exactlythe same as the case (GSp × GSp , (GSp × GSp ) ) in Proposition 3.7. We will skip ithere. (cid:3) It is clear that Θ + satisfies (2.3). The last thing remains is to prove Lemma 2.30 for thecurrent case. Denote by Θ +1 the subset of Θ + consisting of the 12 weights in (7.7) and Θ +2 the complement of Θ +1 in Θ + , that is, consisting of the 16 weights in (7.5) and (7.6). ThenΘ +2 corresponds to the weights of the GSO model in Proposition 8.4. We also decomposethe set of positive roots Φ + as Φ +1 ∪ Φ +2 where Φ +2 = { e j ± e i | ≤ i < j ≤ } is the set ofthe roots contained in GSO , and Φ +1 consists of the remaining positive roots, that is, e − e , 12 ( e − e + (cid:88) i =1 ( − a i e i ) with (cid:88) i =1 a i odd.Denote by W ( D ) the Wely group of the Levi subgroup of type D , generated by the simplereflections w α i for i (cid:54) = 1. We embed W ( D ) into the Weyl group W . Lemma 7.2. With the notation above, we have (cid:88) w ∈ W c W S ( wθ ) = 1∆ H (1) = 1 ζ (2) = (1 − q − ) . Proof. By the identity for the GSO model case proved in Lemma 8.5, we have (cid:88) w ∈ W c W S ( wθ ) = (cid:88) w ∈ W (cid:81) γ ∨ ∈ Θ + − q − e γ ∨ (cid:81) α ∈ Φ + − e α ∨ ( wθ ) = (1 − q − ) · (cid:88) w ∈ W/W ( D ) (cid:81) γ ∨ ∈ Θ +1 − q − e γ ∨ (cid:81) α ∈ Φ +1 − e α ∨ ( wθ ) . Hence it is enough to show that (cid:88) w ∈ W/W ( D ) (cid:81) γ ∨ ∈ Θ +1 − q − e γ ∨ (cid:81) α ∈ Φ +1 − e α ∨ ( wθ ) = 1 . It is easy to see that the constant coefficient of the above summation is equal to 1, so it isenough to show that all the q − i/ -coefficients are equal to 0 for 1 ≤ i ≤ 12. We can replacethe summation on W/W ( D ) by the summation on W and rewrite the function inside thesummation ( θ i are arbitrary variables): (cid:81) γ ∨ ∈ Θ +1 − q − e γ ∨ (cid:81) α ∈ Φ +1 − e α ∨ ( wθ ) = e − ρ ∨ (cid:81) α ∈ Φ +2 − e α ∨ · (cid:81) γ ∨ ∈ Θ +1 − q − e γ ∨ e − ρ ∨ (cid:81) α ∈ Φ + − e α ∨ ( wθ )= w ( θ − (cid:89) i =2 θ − i +1 i (cid:89) ≤ i 12. By the Weyl Denominator formula of type D , (cid:81) i =2 θ − i +1 i (1 − θ j θ − i )(1 + θ j θ i ) = (cid:80) w ∈ W ( D ) sgn ( w ) w ( (cid:81) i =2 θ i − i ) consists of the form (cid:89) i =1 θ a i i , {| a | , | a | , | a | , | a | , | a | , | a |} = { , , , , , } . Then any term (cid:81) i =1 θ b i i appearing in the q − / -coefficient of (7.8) satisfies b = − • b i = ± b j for some 1 ≤ i (cid:54) = j ≤ • {| b | , | b | , | b | , | b | , | b | , | b |} = { , , , , , } .In the second case, up to a Weyl element w action, w ( (cid:81) i =1 θ b i i ) = θ − θ (cid:81) i =2 θ i − i = e − e − e )+ (cid:80) i =2 ( i − e i +6 e ( θ ). However by changing variable θ − to θ , the weight 8( e − e ) + (cid:80) i =2 ( i − e i + 6 e is orthogonal to α ∨ . This implies that the ( W, sgn)-summation of the q − / -coefficient is equal to 0.For the q − -coefficient, any term Π i =1 θ b i i appearing in it satisfies b = − and one of thefollowing two conditions • b i = ± b j for some 1 ≤ i (cid:54) = j ≤ • {| b | , | b | , | b | , | b | , | b | , | b |} = { , , , , , } or { , , , , , } .In the second case, up to a Weyl element w action, w ( (cid:81) i =1 θ b i i ) = e − ( e − e )+ (cid:80) i =2 ( i − e i ( θ )or e − ( e − e )+ (cid:80) i =2 ( i − e i +5 θ +6 e ( θ ). By changing variable θ − → θ , the weight ( e − e ) + (cid:80) i =2 ( i − e i is orthogonal to α ∨ . And the weight w α ( ( e − e )+ (cid:80) i =2 ( i − e i +5 θ +6 e ) =8( e − e ) + ( e + e + 3 e + 5 e + 9 e + 11 e ) is orthogonal to α ∨ . This implies that the( W, sgn)-summation of the q − -coefficient is equal to 0.For the q − / -coefficient, any term Π i =1 θ b i i appearing in it satisfies b = − • b i = ± b j for some 1 ≤ i (cid:54) = j ≤ • {| b | , | b | , | b | , | b | , | b | , | b |} = { , , , , , } or { , , , , , } .In the second case, up to a Weyl element w action, w ( (cid:81) i =1 θ b i i ) = e − e − e )+ (cid:80) i =2 ( i − e i +6 e ( θ )or e − e − e )+ (cid:80) i =2 ( i − e i + (cid:80) i =4 ie i ( θ ). By changing variable θ − to θ , the weight w α (7( e − e )+ (cid:80) i =2 ( i − e i + 6 e ) = ( e − e ) + ( e + e + 3 e + 5 e + 7 e + 11 e ) is orthogonal to e − e ;the weight w α w α (7( e − e ) + (cid:80) i =2 ( i − e i + (cid:80) i =4 ie i ) = 8( e − e ) + e + e + 3 e + 4 e + 5 e is orthogonal to e − e . This implies that the ( W, sgn)-summation of the q − / -coefficient isequal to 0.For the q − -coefficient, any term Π i =1 θ b i i appearing in it satisfies b = − and one of thefollowing two conditions • b i = ± b j for some 1 ≤ i (cid:54) = j ≤ • {| b | , | b | , | b | , | b | , | b | , | b |} = { , , , , , } , { , , , , , } or { , , , , , } . ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES55 In the second case, up to a Weyl element w action, w ( (cid:81) i =1 θ b i i ) = e − ( e − e )+ (cid:80) i =2 ( i − e i ( θ ), e − ( e − e )+ (cid:80) i =2 ( i − e i + (cid:80) i =5 ie i ( θ ), or e − ( e − e )+ e + (cid:80) i =3 ie i ( θ ). By changing variable θ − to θ ,the weight w α ( − ( e − e ) + (cid:80) i =2 ( i − e i ) = 7( e − e ) + ( e + e + 3 e + 5 e + 7 e + 9 e ) isorthogonal to e − e ; the weight w α w α ( ( e − e )+ (cid:80) i =2 ( i − e i + (cid:80) i =5 ie i ) = ( e − e )+ e + e +2 e +4 e +5 e is orthogonal to e − e ; the weight w α w α ( ( e − e )+ e + (cid:80) i =3 ie i ) =8( e − e ) + ( − e + 3 e + 3 e + 5 e + 7 e + 9 e is orthogonal to e − e . This implies thatthe ( W, sgn)-summation of the q − -coefficient is equal to 0.For the q − / -coefficient, any term Π i =1 θ b i i appearing in it satisfies b = − • b i = ± b j for some 1 ≤ i (cid:54) = j ≤ • {| b | , | b | , | b | , | b | , | b | , | b |} = { , , , , , } , { , , , , , } , { , , , , , } .In the second case, up to a Weyl element w action, w ( (cid:81) i =1 θ b i i ) = e − e − e )+ (cid:80) i =2 ( i − e i +6 e ( θ ), e − e − e )+ (cid:80) i =2 ( i − e i + (cid:80) i =4 ie i ( θ ) or e − e − e )+ (cid:80) i =2 ie i ( θ ). By changing variable θ − to θ , underthe action of w α , we have w α (0 , , , , , , − , 6) = (1 , , , , , , − , w α (0 , , , , , , − , 6) = ( 32 , − , , , , , − , 152 ) w α (0 , , , , , , − , 6) = (2 , , , , , , − , . Here ( b , b , . . . , b ) corresponds the weight (cid:80) i =1 b i e i . After the action of w , we have b i = ± b j for some 1 ≤ i (cid:54) = j ≤ 6. This implies that the ( W, sgn)-summation of the q − / -coefficient isequal to 0.For the q − -coefficient, any term Π i =1 θ b i i appearing in it satisfies b = − and one of thefollowing two conditions • b i = ± b j for some 1 ≤ i (cid:54) = j ≤ • {| b | , | b | , | b | , | b | , | b | , | b |} is equal to { , , , , , } , { , , , , , } , { , , , , , } , or { , , , , , } . In the second case, up to a Weyl element w action, w ( (cid:81) i =1 θ b i i ) = e − ( e − e )+ (cid:80) i =2 ( i − e i ( θ ), e − ( e − e )+ (cid:80) i =2 ( i − e i + (cid:80) i =5 ie i ( θ ), e − ( e − e )+ e + (cid:80) i =3 ie i ( θ ), or e − ( e − e )+ (cid:80) i =2 ie i ( θ ). By chang-ing variable θ − to θ , under the action of w α , we have w α (0 , , , , , , − , 112 ) = (1 , , , , , , − , 132 ) w α (0 , , , , , , − , 112 ) = ( 32 , − , , , , , − , w α (0 , , , , , , − , 112 ) = (2 , − , , , , , − , 152 ) w α (1 , , , , , , − , 112 ) = (3 , , , , , , − , 152 ) . After the action of w α , we have b i = ± b j for some 1 ≤ i (cid:54) = j ≤ 6. This implies that the( W, sgn)-summation of the q − -coefficient is equal to 0.Due to symmetry, the remaining q − i/ -coefficients for 7 ≤ i ≤ 12 are vanishing by similararguments and we omit the details here. This finishes the proof of the lemma. (cid:3) To sum up, we have proved that the local relative character is equal to ζ (6) ζ (8) ζ (10) ζ (12) ζ (14) ζ (18) L (1 / , π, ω ) L (1 , π, Ad)where π is an unramified representation of E ( F ).8. The remaining models In this section, we will compute the local relative characters for the remaining 4 modelsin Table 1. The computations are very similar to the cases in the previous sections.8.1. The model (GSp , GL (cid:110) U ) . In this subsection, we compute the local relative char-acter for the model (GSp , GL (cid:110) U ). For simplicity, defineGSp n = { g ∈ GL n | t gJ (cid:48) n g = l ( g ) J (cid:48) n } , where J (cid:48) n = (cid:18) J (cid:48) n − J (cid:19) . Note that the skew-symmetric matrix J (cid:48) n is different with J n when n > J = J (cid:48) . We use J (cid:48) n here to simplify the definition and computation. Let G = GSp , H = H (cid:110) U with H = { diag ( h, h, h, det( h ) h ∗ , det( h ) h ∗ ) | h ∈ GL , h ∗ = J (cid:48) t h − ( J (cid:48) ) − } = { diag ( h, h, h, h, h ) | h ∈ GL } and U be the unipotent radical of the standard parabolic subgroup P = LU of G where L = { ( h , h , h , det( h ) h ∗ , det( h ) h ∗ ) | h i ∈ GL } . We define a generic character ξ on U ( F ) to be ξ ( u ) = ψ ( λ ( u )) where λ ( u ) = tr( X ) + tr( Y ) , u = (cid:32) I X ∗ ∗ ∗ I Y ∗ ∗ I ∗ ∗ I ∗ I (cid:33) . It is easy to see that H is the stabilizer of this character and ( G, H ) is the Whittakerinduction of the trilinear GL -model ( L, H , ξ ).We can also define the quaternion version of this model. Let D/F be a quaternion algebra,and let G D ( F ) = GSp ( D ) (the group GSp n ( D ) has been defined in Section 3.1), H D = H ,D (cid:110) U D with H ,D ( F ) = { diag ( h, h, h, N D/F ( h ) h ∗ , N D/F ( h ) h ∗ ) | h ∈ GL ( D ) , h ∗ = ¯ h − } and U D is the unipotent radical of the standard parabolic subgroup P D = L D U D of G D where L D ( F ) = { ( h , h , h , N D/F ( h ) h ∗ , N D/F ( h ) h ∗ ) | h i ∈ GL ( D ) } . Here N D/F : GL ( D ) → F × is the norm map and x → ¯ x is the conjugation map on thequaternion algebra. Like the split case, we can define the character ξ D on U D ( F ) by replacingthe trace map of M at × by the trace map of D .Let w = (cid:32) I I 00 0 I I I (cid:33) be the Weyl element that sends U to its opposite. It is clear thatthe w -conjugation map stabilizes L and fixes H . We define the map a : GL → Z L to be a ( t ) = diag ( t I , tI , I , t − I , t − I ) . ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES57 This clearly satisfies the equation (2.15). For the open Borel orbit, let η = diag ( I , (cid:18) (cid:19) , (cid:18) (cid:19) (cid:18) (cid:19) , (cid:18) (cid:19) , − I )be the representative of the open Borel orbit for the model ( L, H ) as in Section 2.3.2, and η = η w . The relation (2.20) has already been verified in Section 2.3.2. This finishes thefirst three steps in Section 2.5.1.Now we compute the set of colors and also the set Θ + . Let Θ be the weights of the32-dimensional representation Spin of GSpin ( C ). We can write it asΘ = { ± e ± e ± e ± e ± e } . Let α i = ε i − ε i +1 , 1 ≤ i ≤ α = 2 ε be the simple roots of GSp . By thecomputation of the trilinear GL -model in Section 2.3.2 and the discussion in Section 2.5 (inparticular, Remark 2.26), we have β ∨ α = e − e − e + e + e , α ∨ − β ∨ α = e − e + e − e − e , β ∨ α = − e + e + e − e + e ,α ∨ − β ∨ α = e − e + e − e − e , β ∨ α = − e + e + e − e + e , α ∨ − β ∨ α = e − e − e + e + e . By a similar argument as in the Ginzburg–Rallis model case in Section 5, we can also verify(2.23) for the roots α and α . Next, we compute the set Θ + . Proposition 8.1. Θ + is consisting of the following 16 elements: e + e ± e ± e ± e , e − e + e ± e ± e , e − e − e + e + e , − e + e + e + e ± e , − e + e + e − e + e . Proof. By the computation of the colors, we know that Θ + is the smallest subset of Θsatisfying the following 5 conditions:(1) e − e − e + e + e , e − e + e − e − e , − e + e + e − e + e ∈ Θ + .(2) Θ + − (Θ + ∩ w α Θ + ) = { e − e − e + e + e , e − e + e − e − e } .(3) Θ + − (Θ + ∩ w α Θ + ) = { e − e + e − e − e , − e + e + e − e + e } .(4) Θ + − (Θ + ∩ w α Θ + ) = { e − e − e + e + e , − e + e + e − e + e } .(5) Θ + is stable under w α and w α .It is clear that the set in the statement satisfies these conditions. So we just need to showthat the set is the unique subset of Θ satisfying these conditions. The argument is exactlythe same as the case (GSp × GSp , (GSp × GSp ) ) in Proposition 3.7. We will skip ithere. (cid:3) It is clear that Θ + satisfies (2.3). The last thing remains is to prove Lemma 2.30 for thecurrent case. For i = 1 , 2, we decompose Θ + as Θ +1 ∪ Θ +2 with Θ +1 consisting of the following10 elements: e + e + e ± e ± e , e + e + e + e ± e − e i , ≤ i ≤ +2 consisting of the remaining 6 elements. Then Θ +2 corresponds to the weights inLemma 5.3 (here we view GL × GL (cid:39) GL × GSp as a standard Levi subgroup of GSp ).Decompose the set of positive roots Φ + as Φ +1 ∪ Φ +2 where Φ +2 = { e i − e j , e | ≤ i < j ≤ } is the set of the positive roots contained in GL × GSp and Φ +1 contains the remainingpositive roots. We also embed the Weyl group S × S of GL × GL into W . Lemma 8.2. With the notation above, we have (cid:88) w ∈ W c W S ( wθ ) = 1∆ H /Z G,H (1) = 1 ζ (2) = (1 − q − ) . Proof. By the identity in Lemma 5.3, we have (cid:88) w ∈ W c W S ( wθ ) = (cid:88) w ∈ W (cid:81) γ ∨ ∈ Θ + − q − e γ ∨ (cid:81) α ∈ Φ + − e α ∨ ( wθ ) = (1 − q − ) · (cid:88) w ∈ W/S × S (cid:81) γ ∨ ∈ Θ +1 − q − e γ ∨ (cid:81) α ∈ Φ +1 − e α ∨ ( wθ ) . Hence it is enough to show that (cid:88) w ∈ W/S × S (cid:81) γ ∨ ∈ Θ +1 − q − e γ ∨ (cid:81) α ∈ Φ +1 − e α ∨ ( wθ ) = 1 . It is easy to see that the constant coefficient of the above summation is equal to 1, so itis enough to show that all the q − i/ -coefficients are equal to 0 for 1 ≤ i ≤ 10. Like in theprevious cases, we can replace the summation on W/S × S by the summation on W . Wealso need to rewrite the function inside the summation (cid:81) γ ∨∈ Θ+1 − q − e γ ∨ (cid:81) α ∈ Φ+1 − e α ∨ ( wθ ) as (here θ i arearbitrary variables): w ( (1 − q − / · √ θ θ θ θ θ ) · Π i =1 (1 − q − / · √ θ θ θ θ θ θ i ) · Π ≤ i ≤ (1 − q − / · √ θ θ θ θ θ θ i θ )Π ≤ i 10. The product ∗ consists of terms of the formΠ i =1 θ a i i , { a , a , a , a } = {− / , − / , − / , − / } , a = ± / . Then the q − -coefficients consisting of terms of the formΠ i =1 θ b i i , { b , b , b , b } = {− / , − / , / , / } , b = ± / . The ( W, sgn)-summation of these terms is equal to 0 since b i = ± b j for some i (cid:54) = j .For the q − / -coefficient, any term Π i =1 θ b i i appearing in it must satisfy one of the followingtwo conditions • b i = b j for some 1 ≤ i < j ≤ • { b , b , b , b } = {− , − , − , − } or {− , − , − , − } and b ∈ {− , , } . ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES59 In either case, we have b i = ± b j for some i (cid:54) = j or b = 0. This implies that the ( W, sgn)-summation of the q − / -coefficient is equal to 0. Similarly, we can also show that the ( W, sgn)-summation of the q − / -coefficient is equal to 0.For the q − -coefficient, any term Π i =1 θ b i i appearing in it must satisfy one of the followingtwo conditions • b i = b j for some 1 ≤ i < j ≤ • { b , b , b , b } = {− / , − / , − / , − / } , {− / , − / , − / , − / } , {− / , − / , − / , − / } or {− / , − / , − / , − / } and b ∈ {± / , ± / } .In either case, we have b i = ± b j for some i (cid:54) = j . This implies that the ( W, sgn)-summationof the q − -coefficient is equal to 0. Similarly, we can also show that the ( W, sgn)-summationof the q − -coefficient is equal to 0.For the q − / -coefficient, any term Π i =1 θ b i i appearing in it must satisfy one of the followingtwo conditions • b i = b j for some 1 ≤ i < j ≤ • b i = 0 for some 1 ≤ i ≤ W, sgn)-summation of the q − / -coefficient is equal to 0. Similarly, wecan also show that the ( W, sgn)-summation of the q − / -coefficient is equal to 0.For the q − -coefficient, any term Π i =1 θ b i i appearing in it must satisfy one of the followingtwo conditions • b i = ± b j for some 1 ≤ i < j ≤ • { b , b , b , b } = {− / , − / , − / , − / } , {− / , − / , − / , / } , or {− / , − / , − / , / } and b ∈ {± / , ± / , ± / } .In either case, we have b i = ± b j for some i (cid:54) = j . This implies that the ( W, sgn)-summationof the q − -coefficient is equal to 0. Similarly, we can also show that the ( W, sgn)-summationof the q − -coefficient is equal to 0.For the q − / -coefficient, any term Π i =1 θ b i i appearing in it must satisfy one of the followingtwo conditions • b i = ± b j for some 1 ≤ i < j ≤ • b i = 0 for some 1 ≤ i ≤ W, sgn)-summation of the q − / -coefficient is equal to 0. This finishesthe proof of the lemma. (cid:3) To sum up, we have proved that the local relative character is equal to ζ (1) ζ (4) ζ (6) ζ (8) ζ (10) L (1 / , π, Spin ) L (1 , π, Ad)where π is an unramified representation of GSp ( F ).8.2. The model (GSp × GL , GL (cid:110) U ) . In this subsection, we compute the local relativecharacter for the model (GSp × GL , GL (cid:110) U ). Let G = GSp × GL , H = H (cid:110) U with H = { diag ( h, h, h ) × h | h ∈ GL } and U be the unipotent radical of the standard parabolic subgroup P = LU of GSp em-bedded into G via the map u (cid:55)→ ( u, I ) where L = { ( h , h , det( h ) h ∗ ) | h i ∈ GL } . We define a generic character ξ on U ( F ) to be ξ ( u ) = ψ ( λ ( u )) where λ ( u ) = tr( X ) , u = (cid:16) I X ∗ I ∗ I (cid:17) . The model ( G, H ) is the Whittaker induction of the trilinear GL -model ( L × GL , H , ξ ).As in the previous case, we can also define the quaternion version of this model.Let w = (cid:16) I I I (cid:17) be the Weyl element that sends U to its opposite. It is clear that the w -conjugation map stabilizes L and fixes H . We define the map a : GL → Z L to be a ( t ) = diag ( tI , I , t − I ) × I . This clearly satisfies the equation (2.15). For the open Borel orbit, let η = diag ( I , (cid:18) (cid:19) , I ) × (cid:18) (cid:19) (cid:18) (cid:19) be the representative of the open Borel orbit for the model ( L × GL , H ) as in Section 2.3.2,and η = η w . The relation (2.20) has already been verified in Section 2.3.2. This finishesthe first three steps in Section 2.5.1.Let Θ be the weights of the 16-dimensional representation Spin × std of GSpin ( C ) × GL ( C ). We can write it asΘ = { ± e ± e ± e e (cid:48) i | ≤ i ≤ } . Let α i = ε i − ε i +1 , ≤ i ≤ α = 2 ε be the simple roots of GSp and α (cid:48) = ε (cid:48) − ε (cid:48) be the simple root of GL . By the computation of the trilinear GL -model in Section 2.3.2and the discussion in Section 2.5 (in particular, Remark 2.26), we have β ∨ α = e − e − e e (cid:48) , α ∨ − β ∨ α = e − e + e e (cid:48) , β ∨ α = − e + e + e e (cid:48) ,α ∨ − β ∨ α = e − e + e e (cid:48) , β ∨ α (cid:48) = − e + e + e e (cid:48) , α (cid:48)∨ − β ∨ α (cid:48) = e − e − e e (cid:48) . By a similar argument as in the Ginzburg–Rallis model case in Section 5, we can also verify(2.23) for the root α . The proof of the following proposition follows from a similar but easierargument as the model (GSp , GL (cid:110) U ) in the previous subsection. The only differenceis to replace the identity in Lemma 5.3 by the identity in Section 2.3.2 for the trilinearGL -model. We will skip it here. Proposition 8.3. Θ + is consisting of the following 8 elements: e + e ± e e (cid:48) i , e − e + e e (cid:48) i , ± ( e − e − e )2 + e (cid:48) , ≤ i ≤ . The set Θ + satisfies (2.3) . Moreover, we have (cid:88) w ∈ W c W S ( wθ ) = 1∆ H /Z G,H (1) = 1 ζ (2) = (1 − q − ) . To sum up, we have proved that the local relative character is equal to ζ (1) ζ (2) ζ (4) ζ (6) L (1 / , π, Spin × std ) L (1 , π, Ad)where π is an unramified representation of GSp ( F ) × GL ( F ). ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES61 The model (GSO , GL (cid:110) U ) . In this subsection, we compute the local relative char-acter for the model (GSO , GL (cid:110) U ). There are two models in this case (correspondingto the two Siegel parabolic subgroups) and they are differed by the outer automorphism ofGSO . Each of them corresponds to one of the Half-Spin L -function of GSpin ( C ). Wewill only compute the local relative character of one of the models, the other one can becomputed just by applying the outer automorphism to the first one. Let J (cid:48) = (cid:18) − 11 0 (cid:19) .Set L = (cid:18) J (cid:48) − J (cid:48) (cid:19) and L n = (cid:18) J (cid:48) L n − − J (cid:48) (cid:19) . DefineGSO n = { g ∈ GL n | g t L n g = l ( g ) L n } . Let G = GSO , H = H (cid:110) U with H = { diag ( h, h, h, h, det( h ) h ∗ , det( h ) h ∗ ) = diag ( h, h, h, h, h, h ) | h ∈ GL , h ∗ = J (cid:48) t h − ( J (cid:48) ) − } and U be the unipotent radical of the standard parabolic subgroup P = LU of G where L = { diag ( h , h , h , th ∗ , th ∗ , th ∗ ) | h i ∈ GL , t ∈ GL } . We define a generic character ξ on U ( F ) to be ξ ( u ) = ψ ( λ ( u )) where λ ( u ) = tr( X ) + tr( Y ) + tr( Z ) , u = I X ∗ ∗ ∗ ∗ I Y ∗ ∗ ∗ I Z ∗ ∗ I ∗ ∗ I ∗ I . It is easy to see that H is the stabilizer of this character and ( G, H ) is the Whittakerinduction of the trilinear GL -model ( L, H , ξ ).We can also define the quaternion version of this model. Let D/F be a quaternion algebra,and let GSO n ( D ) = { g ∈ GL n ( D ) | t ¯ gJ n (cid:48) g = l ( g ) J (cid:48) n } . Let G D ( F ) = GSO ( D ), H D = H ,D (cid:110) U D with H ,D ( F ) = { diag ( h, h, h, h, h, h ) | h ∈ GL ( D ) } and U D be the unipotent radical of the standard parabolic subgroup P D = L D U D of G D where L D ( F ) = { ( h , h , h , th ∗ , th ∗ , th ∗ ) | h i ∈ GL ( D ) , t ∈ GL ( F ) , h ∗ = ¯ h − } . Here N D/F : GL ( D ) → F × is the norm map and x → ¯ x is the conjugation map on thequaternion algebra. Like the split case, we can define the character ξ D on U D ( F ) by replacingthe trace map of M at × by the trace map of D .Let w = I I 00 0 0 I I I I be the Weyl element that sends U to its opposite. It is clearthat the w -conjugation map stabilizes L and fixes H . We define the map a : GL → Z L tobe a ( t ) = diag ( t I , t I , tI , I , t − I , t − I ) . This clearly satisfies the second identity of the equation (2.15). Although it does not satisfythe first equation of (2.15), but the difference between a ( t ) − and w − a ( t ) w belongs to the center so all the arguments in Section 2.4 still work (because all the characters areunramified). For the open Borel orbit, let η = diag ( I , (cid:18) (cid:19) , (cid:18) (cid:19) (cid:18) (cid:19) , (cid:18) − − − (cid:19) , − (cid:18) (cid:19) , I )be the representative of the open Borel orbit for the model ( L, H ) as in Section 2.3.2, and η = η w . The relation (2.20) has already been verified in Section 2.3.2. This finishes thefirst three steps in Section 2.5.1.Now we compute the set of colors and also the set Θ + . Let Θ be the weights of the32-dimensional Half-Spin representation HSpin of GSpin ( C ) given byΘ = { ± e ± e ± e ± e ± e ± e | − appears odd times } . Let α i = ε i − ε i +1 , ≤ i ≤ α = ε + ε be the simple roots of GSO . By thecomputation of the trilinear GL -model in Section 2.3.2 and the discussion in Section 2.5 (inparticular, Remark 2.26), we have β ∨ α = e − e − e + e + e − e , α ∨ − β ∨ α = e − e + e − e − e + e ,β ∨ α = − e + e + e − e + e − e , α ∨ − β ∨ α = e − e + e − e − e + e ,β ∨ α = − e + e + e − e + e − e , α ∨ − β ∨ α = e − e − e + e + e − e . By a similar argument as in the Ginzburg–Rallis model case in Section 5, we can alsoverify (2.23) for the roots α , α and α . Next, we compute the set Θ + . Proposition 8.4. Θ + is consisting of the following 16 elements: e + e + e + e + e + e − e l , − e − e − e − e − e − e + 2 e i + 2 e j + 2 e k with ≤ l ≤ and ( i, j, k ) ∈ { (123) , (124) , (125) , (126) , (134) , (135) , (136) , (145) , (234) , (235) } .Proof. By the computation of the colors, we know that Θ + is the smallest subset of Θsatisfying the following 5 conditions:(1) e − e − e + e + e − e , − e + e + e − e + e − e , e − e + e − e − e + e ∈ Θ + .(2) Θ + − (Θ + ∩ w α Θ + ) = { e − e − e + e + e − e , e − e + e − e − e + e } .(3) Θ + − (Θ + ∩ w α Θ + ) = { e − e + e − e − e + e , − e + e + e − e + e − e } .(4) Θ + − (Θ + ∩ w α Θ + ) = { e − e − e + e + e − e , − e + e + e − e + e − e } .(5) Θ + is stable under w α , w α and w α .It is clear that the set in the proposition satisfies these conditions. So we just need to showthat the set is the unique subset of Θ satisfying these conditions. The argument is exactlythe same as the case (GSp × GSp , (GSp × GSp ) ) in Proposition 3.7. We will skip ithere. (cid:3) It is clear that Θ + satisfies (2.3). The last thing remains to prove Lemma 2.30 forthe current case. Let Θ +1 (resp. Θ +2 ) be the subset of Θ + consisting of elements of theform e + e + e + e + e + e − e l (resp. e + e + e + e + e + e − e i − e j − e k ). Then Θ +2 corresponds to theweights of the Ginzburg–Rallis model discussed in Section 5. We also decompose the set ofpositive roots Φ + as Φ +1 ∪ Φ +2 where Φ +2 = { e i − e j | ≤ i < j ≤ } is the set of the roots ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES63 contained in GL and Φ +1 contains the remaining positive roots. We also embed the Weylgroup S of GL into the Weyl group W . Lemma 8.5. With the notation above, we have (cid:88) w ∈ W c W S ( wθ ) = 1∆ H /Z G,H (1) = 1 ζ (2) = (1 − q − ) . Proof. By the identity for the Ginzburg–Rallis model case proved in Lemma 5.2, we have (cid:88) w ∈ W c W S ( wθ ) = (cid:88) w ∈ W (cid:81) γ ∨ ∈ Θ + − q − e γ ∨ (cid:81) α ∈ Φ + − e α ∨ ( wθ ) = (1 − q − ) · (cid:88) w ∈ W/S (cid:81) γ ∨ ∈ Θ +1 − q − e γ ∨ (cid:81) α ∈ Φ +1 − e α ∨ ( wθ ) . Hence it is enough to show that (cid:88) w ∈ W/S (cid:81) γ ∨ ∈ Θ +1 − q − e γ ∨ (cid:81) α ∈ Φ +1 − e α ∨ ( wθ ) = 1 . It is easy to see that the constant coefficient of the above summation is equal to 1, so itis enough to show that all the q − i/ -coefficients are equal to 0 for 1 ≤ i ≤ 6. Like in theprevious cases, we can replace the summation on W/S by the summation on W . We alsoneed to rewrite the function inside the summation ( θ i are arbitrary variables): (cid:81) γ ∨ ∈ Θ +1 − q − e γ ∨ (cid:81) α ∈ Φ +1 − e α ∨ ( wθ ) = Π i =1 (1 − q − / · w ( √ θ θ θ θ θ θ θ i ))Π ≤ i 6. The product Π ≤ i We define a generic character ξ on U ( F ) to be ξ ( u ) = ψ ( λ ( u )) where λ ( u ) = tr( X ) + tr( Y ) , u = (cid:18) I X ∗ ∗ I Y ∗ I ∗ I (cid:19) . The model ( G, H ) is the Whittaker induction of the trilinear GL -model ( L × GL , H , ξ ).Similarly we can also define the quaternion algebra version of this model.Let w = (cid:18) I I I I (cid:19) × I be the Weyl element that sends U to its opposite. It is clearthat the w -conjugation map stabilizes L and fixes H . We define the map a : GL → Z L as a ( t ) = diag ( t I , tI , I , t − I ) × I . This clearly satisfies the second identity of the equation (2.15). Although it does not satisfythe first equation of (2.15), but the difference between a ( t ) − and w − a ( t ) w belongs to thecenter so all the arguments in Section 2.4 still work. For the open Borel orbit, let η = diag ( I , (cid:18) (cid:19) , − (cid:18) (cid:19) , I ) × (cid:18) (cid:19) (cid:18) (cid:19) be the representative of the open Borel orbit for the model ( L, H ) as in Section 2.3.2, and η = η w . The relation (2.20) has already been verified in Section 2.3.2. This finishes thefirst three steps in Section 2.5.1.Now we compute the set of colors and also the set Θ + . Let Θ be the weights of the16-dimensional Half-Spin representation HSpin of GSpin ( C ) given byΘ = { ± e ± e ± e ± e e (cid:48) i | − appears even times , i ∈ { , }} . Let α i = ε i − ε i +1 , ≤ i ≤ α = ε + ε be the simple roots of GSO and α (cid:48) = ε (cid:48) − ε (cid:48) be the simple root of GL . By the computation of the trilinear GL -model in Section 2.3.2and the discussion in Section 2.5 (in particular, Remark 2.26), we have β ∨ α = e − e − e + e e (cid:48) , α ∨ − β ∨ α = e − e + e − e e (cid:48) , β ∨ α = − e + e + e − e e (cid:48) ,α ∨ − β ∨ α = e − e + e − e e (cid:48) , β ∨ α (cid:48) = − e + e + e − e e (cid:48) , α (cid:48)∨ − β ∨ α (cid:48) = e − e − e + e e (cid:48) . By a similar argument as in the Ginzburg–Rallis model case in Section 5, we can also verify(2.23) for the roots α and α . The next proposition computes the set Θ + and proves Lemma2.30 for the current case. Proposition 8.6. Θ + is consisting of the following 8 elements: e + e ± ( e + e )2 + e (cid:48) i , e − e + e − e e (cid:48) i , ± ( e − e − e + e )2 + e (cid:48) , ≤ i ≤ . Moreover, we have (cid:88) w ∈ W c W S ( wθ ) = 1∆ H /Z G,H (1) = 1 ζ (2) = (1 − q − ) . Proof. The proof follows from a similar but easier argument as the (GSO , GL (cid:110) U ) modelcase in the previous subsection. The only difference is that we need to use Lemma 5.3 insteadof Lemma 5.2. We will skip the details here. (cid:3) To sum up, we have proved that the local relative character is equal to ζ (1) ζ (2) ζ (4) ζ (6) L (1 / , π, HSpin × std ) L (1 , π, Ad )where π is an unramified representation of GSO ( F ) × GL ( F ).9. Local multiplicity In this section we will study the multiplicity for the models in Table 1. Let F be a localfield of characteristic 0, ( G, H ) be one of the models in Table 1, and ξ be the character of H ( F ) defined in the previous sections (note that ξ is trivial in the reductive case). Let π bean irreducible admissible representation of G ( F ) with trivial central character. Recall thatthe multiplicity is defined by m ( π ) = dim Hom H ( F ) ( π, ξ ) . Similarly, if F (cid:54) = C , let D/F be the unique quaternion algebra (or D ∈ H ( F, H/Z G,H )if we are in the case of Model 2 of Table 1), and let ( G D , H D , ξ D ) be the quaternion ver-sion of the model ( G, H, ξ ) defined in the previous sections. Let π D be an irreducible rep-resentation of G D ( F ) with trivial central character. We can also define the multiplicity m ( π D ) = dim(Hom H D ( F ) ( π D , ξ D )) . In this section, we will prove a local multiplicity formula of m ( π ) and m ( π D ) in termsof the Harish-Chandra character. Then using the multiplicity formula, together with theassumption that the conjectural character identity holds for the local L -packet of G , we willshow that the summation of the multiplicities is equal to 1 over all the tempered L -packets.The proof of all the results in this section is very similar to the Gan–Gross–Prasad modelcase ([W10], [W12], [B15]) and the Ginzburg–Rallis model case ([Wan15], [Wan16], [Wan17],[WZ]) since similar to the Gan–Gross–Prasad model and the Ginburg–Rallis model, all themodels in Table 1 are strongly tempered without Type N root.In Section 9.1, we will recall the conjectural character identity for the local L -packets of G . In Section 9.2 we will study the reductive models in Table 1 and in Section 9.3 we willstudy the non-reductive models.9.1. The character identity between inner forms. Let G = GL n , GSp n or GSO n (resp. GU n,n ), and G D be the corresponding quaternion version of G (resp. the non quasi-split unitary similitude group GU n +1 ,n − of rank 2 n + 1 in the p -adic case, the unitarysimilitude group GU m, n − m in the archimedean case). We will discuss the conjectural char-acter identity between G and G D . For all these cases, G is quasi-split and G D /Z G D is a pureinner form of G/Z G (note that we always assume that the central character is trivial).Let φ be a tempered L -parameter of G and Π φ ( G ) , Π φ ( G D ) be the conjectural local L -packets. Let θ Π φ ( G ) = (cid:88) π θ π and θ Π φ ( G D ) = (cid:88) π θ π D be the associated characters of the L -packets where θ π is the Harish-Chandra character of π . According to the local Langlands conjecture, the characters θ Π φ ( G ) and θ Π φ ( G D ) are stableand we have the identity θ Π φ ( G ) ( γ ) = ( − a ( G ) − a ( G D ) θ Π φ ( G D ) ( γ D ) ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES67 for all regular semisimple elements γ ∈ G ( F ) , γ D ∈ G D ( F ) with γ ↔ γ D . Here a ( G ) (resp. a ( G D )) is the dimension of the maximal split torus of G (resp. G D ), and we say γ ↔ γ D ifthey have the same characteristic polynomial. Remark 9.1. When G = GL n or GSO n , a ( G ) − a ( G D ) = n . When G = GSp n , a ( G ) − a ( G D ) = [ n +12 ] . When G = GU n,n , a ( G ) − a ( G D ) = | n − m | . When F is archimedean, the L -packet was construct by Langlands [Lang] and the characteridentity was proved by Shelstad [S]. In the p -adic case, when G = GL n , the L -packetΠ φ ( G ) was constructed by Harris–Taylor [HT], the L -packet Π φ ( G D ) was constructed byDeligne–Kazhdan–Vigneras [DKV], and the character identity was also proved in [DKV].When G = GSp n and GSO n , the L -packet Π φ ( G ) was constructed by Bin Xu [X], whilethe construction of the L -packet Π φ ( G D ) and the character identity are still open. When G = GU n,n , the construction of L-packets and the character identity remain open. Theyshould follows from the method in [X] for lifting together with the results in [KMSW], [M]for unitary groups. For all these cases, we also know that the L -packet Π φ ( G ) contains aunique generic element.9.2. The reductive case. In this subsection we assume that H is reductive. The model(GL × GL , GL × GL ) has already been considered in the previous paper [PWZ19], so wewill focus on the models ( G, H ) = (GSp × GSp , (GSp × GSp ) ) and ( G, H ) = (GU × GU , (GU × GU ) ).Let π be an irreducible representation of G ( F ) with trivial central character and θ π be itsHarish-Chandra character. For a semisimple element x ∈ G ( F ), we let c π ( x ) be the averageof the regular germs of θ π at x . We refer the reader to Section 4.5 of [B15] for the definitionof regular germs. In fact, for all the cases we considered in the paper, g x ( F ) contains atmost one regular nilpotent orbit, so c π ( x ) is just the regular germ of θ π at x . We want toemphasize that c π ( x ) is zero if the centralizer G x is not quasi-split. We also let T ell ( G ) be aset of representatives of maximal elliptic tori of G ( F ).9.2.1. The model (GSp × GSp , (GSp × GSp ) ) . We first consider the case ( G, H ) =(GSp × GSp , (GSp × GSp ) ). For T ∈ T ell (GSp ), let T n, ◦ = { ( t , · · · , t n ) ∈ T n | det( t i ) = det( t j ) for all 1 ≤ i, j ≤ n } . We use ι n to denote the diagonal embedding from T to T n, ◦ . We can view T n, ◦ as a maximalelliptic torus of GSp n . Moreover, up to GSp n -conjugation, there are 2 n − -many differentembeddings from T n, ◦ to GSp n .When n = 2, there are two embeddings ν , ν (cid:48) from T , ◦ to GSp and the centralizerof the image of ν ◦ ι (resp. ν (cid:48) ◦ ι ) in GSp is the quasi-split (resp. non quasi-split)unitary similitude group of rank 3. Meanwhile, there are four embeddings from T , ◦ to(GSp × GSp ) and there are two of them whose projection to GSp coincide with ν .Compose with the embedding from (GSp × GSp ) to GSp , we get two embeddings ν , ν from T , ◦ to GSp . The centralizers of the image of ν i ◦ ι ( i = 1 , 2) in GSp are thetwo unitary similitude groups of rank 4 (both of them are quasi-split). We use ν T,i =( ν i ◦ ι ) × ( ν ◦ ι ) to denote the two embeddings from T to G (both factor through H ).Meanwhile, let ι , be the embedding from T , ◦ to T , ◦ given by ( t , t ) (cid:55)→ ( t , t , t ).Among the four embeddings from T , ◦ to GSp , there are two of them (denoted by ν , ν (cid:48) )such that the centralizers in GSp of the image of ν ◦ ι , and ν (cid:48) ◦ ι , are quasi-split (thecentralizer is the quasi-split unitary similitude group of rank 3 times an abelian group). Up to conjugation we may assume that ν , ν (cid:48) factor through (GSp × GSp ) and the projectionto GSp of ν ◦ ι , (resp. ν (cid:48) ◦ ι , ) is equal to ν (resp. ν (cid:48) ). We use ν T , ◦ , = ( ν ◦ ι , ) × ν and ν T , ◦ , = ( ν (cid:48) ◦ ι , ) × ν (cid:48) to denote the two embeddings from T , ◦ to G . Both of themfactor through H .Finally, for T , T ∈ T ell (GSp ) with T (cid:54) = T (this will not happen in the archimedeancase), let ( T × T ) ◦ = { ( t , t ) ∈ T × T | det( t ) = det( t ) } . Similarly, we can define ( T × T × T ) ◦ . Up to conjugation, there is only one embedding from( T × T ) ◦ to GSp and there are two embeddings from ( T × T × T ) ◦ to GSp . The twoembeddings induce two embeddings from ( T × T ) ◦ to GSp (we first map T diagonally into( T × T ) ◦ ). We let ν be the embedding such that the centralizer of its image is quasi-split(the centralizer of the other embedding is not quasi-split). Up to conjugation we may assumethat ν factors through (GSp × GSp ) and its projection to GSp is equal to the embeddingfrom ( T × T ) ◦ to GSp . This gives us an embedding ν T ,T from ( T × T ) ◦ to G that factorsthrough H .Define the geometric multiplicity to be ( D H ( · ) is the Weyl determinant) m geom ( π ) = c π (1) + (cid:88) T ∈T ell ( H ) | W ( H, T ) | − (cid:90) ∗ T ( F ) /Z G,H ( F ) D H ( t ) θ π ( t ) d t + 12 (cid:88) T ∈T ell (GSp ) ,i ∈{ , } (cid:0) (cid:90) ∗ T ( F ) /Z GL2 ( F ) D H ( ν T,i ( t )) c π ( ν T,i ( t )) d t + (cid:90) ∗ T , ◦ ( F ) /Z GL2 ( F ) D H ( ν T , ◦ ,i ( t )) c π ( ν T , ◦ ,i ( t )) d t (cid:1) + 14 (cid:88) T ,T ∈T ell (GSp ) ,T (cid:54) = T (cid:90) ∗ ( T × T ) ◦ ( F ) /Z GL2 ( F ) diag D H ( ν T ,T ( t )) c π ( ν T ,T ( t )) d t. Here 1 always stands for the identity element of G ( F ), W ( H, T ) is the Weyl group, all theHaar measures are chosen so that the total volume is equal to 1 (note that all the integraldomains are compact), and the factors , come from the cardinality of the Weyl groups W (GSp , T ) , W (GSp , T i ). Note that if F = C , then T ell ( H ) and T ell (GL ) are empty. Hencewe have m geom ( π ) = c π (1). When F = R , T ell (GSp ) only contains one element and the term (cid:80) T ,T ∈T ell (GSp ) ,T (cid:54) = T · · · will not appear. We leave it as an excise for the reader to check thatour definition of m geom ( π ) matches the definition in [Wan] for general spherical varieties. Remark 9.2. Like the unitary Gan–Gross–Prasad model case (Proposition 11.2.1 of [B15] ),the integrals defining the geometric mulitplicity are not necessarily absolutely convergentand they need to be regularized. The regularization is the same as the unitary Gan–Gross–Prasad model case. To be specific, one replace the Weyl determinant D H in the integrandby ( D G ) / · ( ( D H ) D G ) s − / . By a very similar argument as in Proposition 11.2.1 of [B15] , weknow that the integral is absolutely convergent when s > and has a limit as s → + . Thenwe can define the regularized integral to be this limit. This remark also applies to the nonquasi-split case below, and the model (GU × GU , (GU × GU) ) in the next subsection. Similarly, if F (cid:54) = C , for the quaternion version of the model, we can also define the embed-dings ν T D ,i , ν T , ◦ D ,i , ν T ,D ,T ,D for T D , T ,D , T ,D ∈ T ell (GSp ( D )) = T ell (GSp ) with T ,D (cid:54) = T ,D .We can define the geometric multiplicity to be m geom ( π D ) = (cid:88) T D ∈T ell ( H D ) (cid:90) ∗ T D ( F ) /Z GD,HD ( F ) D H D ( t ) θ π D ( t ) d t + 12 (cid:88) T D ∈T ell (GSp ( D )) ,i ∈{ , } ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES69 (cid:0) (cid:90) ∗ T D ( F ) /Z GL1( D ) ( F ) D H D ( ν T D ,i ( t )) c π D ( ν T D ,i ( t )) d t + (cid:90) ∗ T , ◦ D ( F ) /Z GL1( D ) ( F ) D H D ( ν T , ◦ D ,i ( t )) c π D ( ν T , ◦ D ,i ( t )) d t (cid:1) + 14 (cid:88) T ,D ,T ,D ∈T ell (GSp ( D )) ,T ,D (cid:54) = T ,D (cid:90) ∗ ( T ,D × T ,D ) ◦ ( F ) /Z GL1( D ) ( F ) diag D H D ( ν T ,D ,T ,D ( t )) c π D ( ν T ,D ,T ,D ( t )) d t. The only difference between m geom ( π ) and m geom ( π D ) is that m geom ( π ) contains the germat 1 (since G ( F ) is quasi-split) while m geom ( π D ) dose not. The following theorem gives amultiplicity formula for the model. Theorem 9.3. For all tempered representations π of G ( F ) (resp. π D of G D ( F ) ) with trivialcentral character, we have m ( π ) = m geom ( π ) , m ( π D ) = m geom ( π D ) . Proof. This follows from a similar but easier argument as in the Gan–Gross–Prasad modelcase ([W10], [W12], [B15]) and the Ginzburg–Rallis model case ([Wan15], [Wan16], [Wan17]).The argument is easier for this model because it is reductive and hence there is no need toregularize the integral over H . The only difference is that the proofs in the above papersused the Gelfand pair condition (i.e. m ( π ) ≤ π ) which is not known for this model.But this can be solved by the same argument as the unitary Ginzburg–Rallis model case inour previous paper (Section 6 and Appendix A of [WZ]). We will skip the proof. (cid:3) If F = C , then any tempered representation of G ( F ) is generic and we have m geom ( π ) = c π (1) = 1 by the result for Whittaker model in [Mat]. Hence the above theorem implies that m ( π ) = 1 for all tempered representation π of G ( F ) with trivial central character (note thatthe L -packet only contains one element in complex case).If F (cid:54) = C , let Π φ = Π φ ( G ) ∪ Π φ ( G D ) be a tempered local L -packet with trivial centralcharacter. We also let θ Π φ ( G ) = (cid:88) π ∈ Π φ ( G ) θ π , θ Π φ ( G D ) = (cid:88) π D ∈ Π φ ( G D ) θ π D be the corresponding stable characters. We assume that conjectural character identity be-tween G and G D holds (note that a ( G ) − a ( G D ) is odd in this case by Remark 9.1) θ Π φ ( G ) ( g ) = − θ Π φ ( G D ) ( g D ) , ∀ g ∈ G reg ( F ) , g D ∈ G D ( F ) , g ↔ g D . This implies that c θ Π φ ( G ) ( ν T,i ( t )) = − c θ Π φ ( GD ) ( ν T D ,i ( t D )) , ∀ t ∈ T ( F ) ↔ t D ∈ T D ( F ); c θ Π φ ( G ) ( ν T , ◦ ,i ( t )) = − c θ Π φ ( GD ) ( ν T , ◦ D ,i ( t D )) , ∀ t ∈ T , ◦ ( F ) ↔ t D ∈ T , ◦ D ( F ); c θ Π φ ( G ) ( ν T ,T ( t )) = − c θ Π φ ( GD ) ( ν T ,D ,T ,D ( t D )) , ∀ t ∈ ( T × T ) ( F ) ↔ t D ∈ ( T ,D × T ,D ) ◦ ( F ) . Here t ↔ t D means that they have the same characteristic polynomial. Together with themultiplicity formula, we have (cid:88) π ∈ Π φ ( G ) m ( π ) + (cid:88) π D ∈ Π φ ( G D ) m ( π D ) = m geom ( θ Π φ ( G ) ) + m geom ( θ Π φ ( G D ) ) = c θ Π φ ( G ) (1) = 1where the last equality follows from the results for Whittaker model in [Rod81], [Mat] and thefact that there is a unique generic element in the L -packet. In particular, we have proved thatthe summation of the multiplicities is equal to 1 over every tempered local Vogan L -packet. The model (GU × GU , (GU × GU ) ◦ ) . Let ( G, H ) = (GU , × GU , , (GU , × GU , ) ◦ ), and( G , H ) = (GU , × GU , , (GU , × GU , ) ◦ ) , ( G , H ) = (GU , × GU , , (GU , × GU , ) ◦ ) , ( G , H ) = (GU , × GU , , (GU , × GU , ) ◦ ) , ( G , H ) = (GU , × GU , , (GU , × GU , ) ◦ )be the pure inner forms (the pair ( G , H ) only appears in the archimedean case).Let T be the unique element in T ell (GU , ) = T ell (GU , ) that is isomorphic to E , ◦ := { ( a, b ) ∈ E × × E × | a ¯ a = b ¯ b } ⊂ E × × E × . For T ∈ T ell (GU , ) = T ell (GU , ) with T (cid:54) = T , let( T × T ) ◦ = { ( t , t ) ∈ T × T | λ ( t ) = λ ( t ) } . Up to conjugation, there is a unique embeddingfrom ( T × T ) ◦ to (GU , × GU , ) ◦ (resp. (GU , × GU , ) ◦ ). Combining with the diagonalembedding from T to ( T × T ) ◦ , we get an embedding (denoted by ν T ) from T to G (resp. G ( F )) that factors through H (resp. H ), and we will denote this embedding by ν T (resp. ν ,T ).For T , there are two embeddings from ( T × T ) ◦ to (GU , × GU , ) ◦ (resp. (GU , × GU , ) ◦ ). Combining with the diagonal embedding from T to ( T × T ) ◦ , we get two em-beddings from T to G (resp. G ). The centralizer of the image of one of the embedding isquasi-split (it is isomorphic to (GU , × GU , ) ◦ times a torus), we will denote this embed-ding by ν T (resp. ν ,T ), while the centralizer of the image of the other embedding is notquasi-split. In the archimedean case, there is only one embedding into (GU , × GU , ) ◦ andthe centralizer is quasi-split. Remark 9.4. For T (cid:54) = T , it is easy to see that the centralizer of the image of ν T (resp. ν ,T )in G (resp. G ) is quasi-split (it is isomorphic to a torus times Res F (cid:48) /F GU , where F (cid:48) /F is the quadratic extension associated to the the torus T ). In general, for T ∈ T ell (GU , ) ,we can also define the embeddings to G and G (also G in the archimedean case), but thecentralizer of the images will not be quasi-split. Meanwhile, consider the following two subgroups of ( T × T ) ◦ (we identify T with E , ◦ := { ( a, b ) ∈ E × × E × | a ¯ a = b ¯ b } ): T (cid:48) = { (1 , × (1 , a ) ∈ ( T × T ) ◦ | a ∈ E } , T (cid:48)(cid:48) = { (1 , a ) × (1 , b ) ∈ ( T × T ) ◦ | a, b ∈ E } . The two embeddings from ( T × T ) ◦ to (GU , × GU , ) ◦ (resp. (GU , × GU , ) ◦ ) inducetwo embeddings from T (cid:48) to G (resp. G ), and we will denote them by ν T (cid:48) ,i (resp. ν ,T ,i )with i = 1 , 2. Note that the projection of these embeddings to the GU , -factor is identity.The centralizers of the image of these embeddings are quasi-split (they are isomorphic toGU × GU , × U ). Remark 9.5. We can also define embeddings from T (cid:48) to G and G (also G in the archimedeancase), but the centralizer of the images will not be quasi-split. On the other hand, the two embeddings from ( T × T ) ◦ to (GU , × GU , ) ◦ induce twoembeddings from T (cid:48)(cid:48) to G . The centralizer of the image of one of the embedding is quasi-split (isomorphic to GU , times some torus, we will denote this embedding by ν T (cid:48)(cid:48) ) and thecentralizer of the image of the other embedding is not quasi-split. Similarly, we can alsodefine the embeddings ν i,T (cid:48)(cid:48) from T (cid:48)(cid:48) to G i for 1 ≤ i ≤ Remark 9.6. We can also define the embedding from T (cid:48)(cid:48) to G in the archimedean case butthe centralizer of the images will not be quasi-split. ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES71 Now we are ready to define the geometric multiplicity. Let π (resp. π i ) be an irreduciblerepresentation of G ( F ) (resp. G i ( F )) with trivial central character. For T ∈ T ell (GU , ) = T ell (GU , ), we use T ∗ ( F ) to denote T ( F ) /Z GU , ( F ) = T ( F ) /Z GU , ( F ). Define m geom ( π ) = c π (1) + (cid:88) T ∈T ell ( H ) | W ( H, T ) | − (cid:90) ∗ T ( F ) /Z G,H ( F ) D H ( t ) θ π ( t ) d t + 12 (cid:88) T ∈T ell (GU , ) (cid:90) ∗ T ∗ ( F ) D H ( ν T ( t )) c π ( ν T ( t )) d t + 12 (cid:88) i =1 (cid:90) ∗ T (cid:48) ( F ) D H ( ν T (cid:48) ,i ( t )) c π ( ν T (cid:48) ,i ( t )) d t + (cid:90) ∗ T (cid:48)(cid:48) ( F ) D H ( ν T (cid:48)(cid:48) ( t )) c π ( ν T (cid:48)(cid:48) ( t )) d t,m geom ( π ) = (cid:88) T ∈T ell ( H ) | W ( H , T ) | − (cid:90) ∗ T ( F ) /Z G ,H ( F ) D H ( t ) θ π ( t ) d t + 12 (cid:88) T ∈T ell (GU , ) (cid:90) ∗ T ∗ ( F ) D H ( ν ,T ( t )) c π ( ν ,T ( t )) d t + (cid:90) ∗ T (cid:48)(cid:48) ( F ) D H ( ν ,T (cid:48)(cid:48) ( t )) c π ( ν ,T (cid:48)(cid:48) ( t )) d t,m geom ( π ) = (cid:88) T ∈T ell ( H ) | W ( H , T ) | − (cid:90) ∗ T ( F ) /Z G ,H ( F ) D H ( t ) θ π ( t ) d t + 12 (cid:88) i =1 (cid:90) ∗ T (cid:48) ( F ) D H ( ν ,T (cid:48) ,i ( t )) c π ( ν ,T (cid:48) ,i ( t )) d t + (cid:90) ∗ T (cid:48)(cid:48) ( F ) D H ( ν ,T (cid:48)(cid:48) ( t )) c π ( ν ,T (cid:48)(cid:48) ( t )) d t,m geom ( π ) = (cid:88) T ∈T ell ( H ) | W ( H , T ) | − (cid:90) ∗ T ( F ) /Z G ,H ( F ) D H ( t ) θ π ( t ) d t + (cid:90) ∗ T (cid:48)(cid:48) ( F ) D H ( ν ,T (cid:48)(cid:48) ( t )) c π ( ν ,T (cid:48)(cid:48) ( t )) d t. If we are in the archimedean case, we also define m geom ( π ) = (cid:88) T ∈T ell ( H ) | W ( H , T ) | − (cid:90) ∗ T ( F ) /Z G ,H ( F ) D H ( t ) θ π ( t ) d t. Like in the previous case, we always choose the Haar measure so that the total volume isequal to 1 and the extra factor comes from the cardinality of the Weyl group of GU . Alsothe integrals in the geometric mulitplicity may not be absolutely convergent and they needto be regularized (see Remark 9.2). We leave it as an excise for the reader to check thatour definition of m geom ( π ) matches the definition in [Wan] for general spherical varieties.Like the previous case, by a similar but easier argument as in the Gan–Gross–Prasad modelcase ([W10], [W12], [B15]) and the Ginzburg–Rallis model case ([Wan15], [Wan16], [Wan17],[WZ]), we can prove the following theorem. Theorem 9.7. For all tempered representations π of G ( F ) (resp. π i of G i ( F ) ) with trivialcentral character, we have m ( π ) = m geom ( π ) , m ( π i ) = m geom ( π i ) . Now let Π φ = Π φ ( G ) ∪ Π φ ( G i ) be a tempered local L -packet with trivial central character(1 ≤ i ≤ p -adic case and 1 ≤ i ≤ θ Π φ ( G ) and θ Π φ ( G i ) as before. We assume that conjectural character identity between G and G i holds (note that a ( G ) − a ( G ) is even and a ( G ) − a ( G ) , a ( G ) − a ( G ) , a ( G ) − a ( G )are odd by Remark 9.1). Then we have (cid:88) π ∈ Π φ ( G ) m ( π ) + (cid:88) ≤ i ≤ k,π i ∈ Π φ ( G i ) m ( π i ) = m geom ( θ Π φ ( G ) ) + k (cid:88) i =1 m geom ( θ Π φ ( G i ) ) = c θ Π φ ( G ) (1) = 1where i = 3 in the p -adic case and i = 4 in the archimedean case. Here the last equalityfollows from the results for Whittaker model in [Mat], [MW] and the fact that there is aunique generic element in the L -packet. For the identity m geom ( θ Π φ ( G ) ) + k (cid:88) i =1 m geom ( θ Π φ ( G i ) ) = c θ Π φ ( G ) (1) , we just need to apply the following cancellations • The term (cid:80) T ∈T ell ( H ) in m geom ( θ Π φ ( G ) ) plus the term (cid:80) T ∈T ell ( H ) in m geom ( θ Π φ ( G ) ) canbe cancelled with the term (cid:80) T ∈T ell ( H ) in m geom ( θ Π φ ( G ) ) plus the term (cid:80) T ∈T ell ( H ) in m geom ( θ Π φ ( G ) ) (and also plus the term (cid:80) T ∈T ell ( H ) in m geom ( θ Π φ ( G ) ) if we are in thearchimedean case). • The term (cid:80) T ∈T ell (GU , ) in m geom ( θ Π φ ( G ) ) can be cancelled with the term (cid:80) T ∈T ell (GU , ) in m geom ( θ Π φ ( G ) ). • The term about T (cid:48) in m geom ( θ Π φ ( G ) ) can be cancelled with the term about T (cid:48) in m geom ( θ Π φ ( G ) ). • The terms about T (cid:48)(cid:48) in m geom ( θ Π φ ( G ) ) and m geom ( θ Π φ ( G ) ) can be cancelled with theterms about T (cid:48)(cid:48) in m geom ( θ Π φ ( G ) ) and m geom ( θ Π φ ( G ) ).In particular, we have proved that the summation of the multiplicities is equal to 1 overevery tempered local Vogan L -packet.9.3. The non-reductive case. In this subsection we consider the non-reductive cases. Let( G, H ) = ( G, H (cid:110) U ) be one of the non-reductive models in Table 1. For all the cases, H ( F ) is essentially GL ( F ) (up to the center). If F (cid:54) = C , we let ( G D , H ,D (cid:110) U D ) be thequaternion version of the model.Let T ell ( H ) (resp. T ell ( H ,D )) be a set of representatives of maximal elliptic tori of H ( F )(resp. H ,D ( F )). Define m geom ( π ) = c π (1) + (cid:88) T ∈T ell ( H ) | W ( H , T ) | − (cid:90) T ( F ) /Z G,H ( F ) D H ( t ) c π ( t ) d t,m geom ( π D ) = (cid:88) T D ∈T ell ( H ,D ) | W ( H ,D , T D ) | − (cid:90) T D ( F ) /Z GD,HD ( F ) D H D ( t ) c π D ( t ) d t where π (resp. π D ) is an irreducible representation of G ( F ) (resp. G D ( F )) with trivialcentral character, W ( H , T ) , W ( H ,D , T D ) are the Weyl group, and all the Haar measure arechosen so that the total volume is equal to 1. Again we leave it as an excise for the reader tocheck that our definition of m geom ( π ) matches the definition in [Wan] for general sphericalvarieties. ERIODS OF AUTOMORPHIC FORMS ASSOCIATED TO STRONGLY TEMPERED SPHERICAL VARIETIES73 Theorem 9.8. Assume that F (cid:54) = R , and ( G, H ) is not the last model ( E , PGL (cid:110) U ) inTable 1. For all tempered representations π of G ( F ) (resp. π D of G D ( F ) ) with trivial centralcharacter, we have m ( π ) = m geom ( π ) , m ( π D ) = m geom ( π D ) . Proof. The multiplicity formula for Model 4 in Table 1 has been proved in the previouspapers of the first author ([Wan15], [Wan16], [Wan17]), and the multiplicity formula for theModel 5 has been proved in our previous paper [WZ]. The argument for the remaining 4models is very similar to the Ginzburg–Rallis model case ([Wan15], [Wan16], [Wan17]), wewill skip it here. Like the reductive case, the Gelfand pair condition is not known for thesemodels, but it can be solved by the same argument as the unitary Ginzburg-–Rallis modelcase in our previous paper (Section 6 and Appendix A of [WZ]).The reason we need to assume that F (cid:54) = R is that in the case when F = R , we don’t knowhow to prove the nonvanishing property of certain explicit intertwining operator is invariantunder the parabolic induction because the operator is defined by a normalized integral in thenon-reductive case and it is not clear how to study it under the parabolic induction in thereal case. In Gan–Gross–Prasad case (Section 7.4 of [B15]), this can be solved by passing toa reductive model of a larger group (e.g. instead of studying ( U n +2 k +1 × U n , U n (cid:110) N ) one canjust study ( U n +2 k +1 , U n +2 k )). But for all the cases in Table 1, we cannot pass it to a reductivemodel of a larger group simply because such a module does not exist. For Model 4 in Table1, we solved this issue by using a special property that all the tempered representations ofGL ( R ) (resp. GL ( R )) are the parabolic induction of some tempered representations ofGL ( R ) × GL ( R ) × GL ( R ) (resp. GL ( R ) × GL ( R )), see Section 5.4 of [Wan16] for details.But this is not true for Models 5–10 of Table 1 (although it is still true in the complex casewhich why we can prove the multiplicity formula in the complex case). In general if one canprove that the nonvanishing property of the explicit intertwining operator is invariant underthe parabolic induction, then we can also prove the multiplicity formula in the real case.On the other hand, the reason we exclude the model ( E , PGL (cid:110) U ) is that in the proofof the geometric side of the trace formula, we need to study the slice representation, i.e.the conjugation action of H ( F ) on the tangent space. We need to show that the regularorbit coincides with the stable conjugacy classes of G ( F ). For all the other cases, this canbe down by computing the characteristic polynomials as the Gan–Gross–Prasad model case(Section 9 of [W10] and Section 10 of [B15]) and the Ginzburg–Rallis model case (Section 8of [Wan15]). But this is not possible for the E case since the matrix presentation of E isvery complicated. If one can prove this result for the model ( E , PGL (cid:110) U ), then we canalso prove the multiplicity formula for this case. (cid:3) As in the reductive cases, the multiplicity formulas will imply that (assume that characteridentity between G and G D holds, note that a ( G ) − a ( G D ) is odd for all these cases and hencethe term (cid:80) T ∈T ell ( H ) can be cancelled with the term (cid:80) T D ∈T ell ( H ,D ) ) and the summation ofthe multiplicities is equal to 1 over every tempered local L -packet. 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With appendices by M. Demazure and Takashi Ono. Progressin Mathematics, 23. Birkh¨auser, Boston, Mass., 1982.[WZ] C. Wan, L. Zhang, The Multiplicity Problems for the Unitary Ginzburg-Rallis Models. Priprint,arXiv:1808.02203.[X] B. Xu, Endoscopic classification of representations of GSp(2N) and GSO(2N) , PhD Thesis,arXiv:1503.04897.[Z] L. Zhang, The Exterior Cubic L -function of GU(6) and Unitary Automorphic Induction ,arXiv:1903.04322v1 Department of Mathematics & Computer Science, Rutgers University – Newark, Newark,NJ 07102, USA Email address : [email protected] Department of Mathematics, National University of Singapore, Singapore 119076 Email address ::