A Local Trace Formula for the Local Gan-Gross-Prasad Conjecture for Special Orthogonal Groups
aa r X i v : . [ m a t h . R T ] S e p A LOCAL TRACE FORMULA FOR THE LOCALGAN-GROSS-PRASAD CONJECTURE FOR SPECIALORTHOGONAL GROUPS
ZHILIN LUO
Abstract.
Through combining the work of Jean-Loup Waldspurger ([Wal10] [Wal12b])and Raphal Beuzart-Plessis ([BP15]), we give a proof for the tempered part of thelocal Gan-Gross-Prasad conjecture ([GGP12]) for special orthogonal groups overany local fields of characteristic zero, which was already proved by Waldspurgerover p -adic fields. Contents
1. Introduction 22. The Gan-Gross-Prasad triples 72.1. Definitions 82.2. Spherical variety structure 102.3. Analytical estimations 123. Explicit tempered intertwining 164. The distributions 215. Spectral expansion for the distribution J J Lie ′ ′ ′ ( F ) /H ( F ) 366.6. Spectral expansion of J Lie m geom and m Liegeom
Introduction
The restriction and branching problems can be traced back to the theory of sphereharmonics. Let SO ( R ) ֒ → SO ( R ) be a pair of compact special orthogonal groupsassociated to 2, resp. 3 dimensional anisotropic quadratic spaces over R . For anyfinite dimensional irreducible representation π of SO ( R ), the spectral decompositionof the restriction of π to SO ( R ) can be detected by the spectral decomposition ofthe canonical action of SO ( R ) on L (SO ( R ) / SO ( R )) ≃ L ( S ) where S is the2-dimensional real sphere, which is essentially provided by the classical theory ofspherical harmonics.The local Gan-Gross-Prasad conjecture aims to generalize the phenomenon to non-compact classical groups over any local field F of characteristic zero. Let ( W, V ) bea pair of quadratic space defined over F with W a subspace of V and the orthogonalcomplement W ⊥ of W in V is odd dimensional and split. Let G = SO( W ) × SO( V )be the product of two special orthogonal groups, and H the subgroup of G defined bythe semi-direct product of SO( W ) with the unipotent subgroup N of SO( V ) definedby the full isotropic flag determined by W ⊥ . Fix a generic character ξ of N whichextends to H . For any irreducible smooth admissible representation π of G ( F ),consider the dimension of the following space m ( π ) = dim Hom H ( π, ξ ) . By [AGRS10] [GGP12] when F is p -adic, and [SZ12] [JSZ11] when F is archimedean, m ( π ) ≤ m ( π ). The pure inner inner forms of the special orthogonal group SO( W ) areparametrized by the set H ( F, SO( W )) ≃ H ( F, H ). For each α ∈ H ( F, H ), onecan associate a pair of quadratic space ( W α , V α ) that enjoys similar properties as LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 3 ( W, V ). Moreover, G α = SO( W α ) × SO( V α ) has the same Langlands dual group as G , which is denoted as L G . For any generic L -parameter ϕ : W F → L G where W F is the Weil-Deligne group of F , let Π G α ( ϕ ) be the associated L -packet of G α . Conjecture 1.0.1 ([GGP12]) . The following identity holds for any generic L -parameter ϕ : W F → L G . X α ∈ H ( F,H ) X π ∈ Π Gα ( ϕ ) m ( π ) = 1 . Moreover, there is a refinement determining when m ( π ) is not equal to zero. It canbe detected by the representation of the component group A ϕ associated to ϕ , whichin particular can be detected by the sign of the relevant symplectic root number.Over p -adic fields, in a series of papers ([Wal10], [Wal12b], [Wal12a]), Waldspurgerproved a multiplicity formula for any tempered representation of G ( F ) and twistedgeneral linear groups, from which he could deduce Conjecture 1.0.1 and the refine-ment for any tempered L -parameters of G . Later the work of M. Moeglin andWaldspurger ([MgW12]) established the full conjecture for any generic L -parametersof G .There is a similar conjecture for unitary groups associated to a pair of hermit-ian spaces with parallel descriptions. Over p -adic fields, following the approach ofWaldspurger, Beuzart-Plessis established Conjecture 1.0.1 and the refinement forany tempered L -parameters of G over p -adic fields ([BP16], [BP14]). Later, aftercombining the work of Beuzart-Plessis, Moeglin and Waldspurger, W. Gan and A.Ichino established the full conjecture for any generic L -parameters of G ([GI16]).The cases above are usually called the Bessel case . There are also parallel conjec-tures called the
Fourier-Jacobi case , which treats unitary groups (skew-hermitian)and symplectic-metaplectic groups. Over p -adic fields, they were established byGan and Ichino for unitary groups case (skew-hermitian) ([GI16]), and H. Atobe forsymplectic-metaplectic case ([Ato18]), using the techniques of theta correspondence.Over archimedean local fields, Beuzart-Plessis established Conjecture 1.0.1 fortempered L -parameters of unitary groups in the Bessel case ([BP15]). The methodworks uniformly for p -adic fields as well.When F = C , Conjecture 1.0.1 for generic L -parameters of special orthogonalgroups ([M¨ol17]) is proved using different methods.The goal of the present paper is to prove Conjecture 1.0.1 for tempered L -parametersof special orthogonal groups. In particular, the result is new only when F is archimedean.We are going to combine the work of Waldspurger ([Wal10] [Wal12b]) over p -adicfields and Beuzart-Plessis ([BP15]) over archimedean fields to prove the conjecture. The proof . In the following, we will give a brief introduction to the proof.
ZHILIN LUO
Fix a tempered representation π of G ( F ). To study the multiplicity space Hom H ( π, ξ ),by Frobenius reciprocity, it is equivalent to study the spectral decomposition of L ( H \ G, ξ ) as a unitary representation R of G ( F ) via right translation, where L ( H \ G, ξ ) is the L -induction from the character ξ of H ( F ) to G ( F ). From the spiritof J. Arthur ([Art91]), one may use the convolution action of C ∞ c ( G ) on L ( H \ G, ξ ),and study the spectral and geometric expansion of the trace of the operator R ( f ).More precisely, fix f ∈ C ∞ c ( G ), x ∈ G ( F ) and ϕ ∈ L ( H \ G, ξ ),( R ( f ) ϕ )( x ) = Z G ( F ) f ( g ) ϕ ( xg ) d g = Z H ( F ) \ G ( F ) K f ( x, y ) ϕ ( y ) d y, where K f ( x, y ) = Z H ( F ) f ( x − hy ) ξ ( h ) d h, x, y ∈ G ( F ) . In other words, the operator R ( f ) has integral kernel K f ( x, y ).Formally, the trace of R ( f ) can be computed via integrating K f ( x, y ) along the di-agonal H ( F ) \ G ( F ). Unfortunately the integral is usually not absolutely convergent.To rectify the issue one works instead with the space of strongly cuspidal functions.Recall that a function f ∈ C ∞ c ( G ) is called strongly cuspidal if Z U ( F ) f ( mu ) d u = 0 , m ∈ M ( F )for any nontrivial parabolic subgroup P of G with Levi decomposition P = M U .Similarly, one can also introduce the notion of strongly cuspidal functions in the
Harish-Chandra Schwartz space of G , which is denoted by C scusp ( G ).It turns out that the following theorem holds . Theorem 1.0.2 (Section 4) . For any f ∈ C scusp ( G ) , the following integral is abso-lutely convergent J ( f ) = Z H ( F ) \ G ( F ) K f ( x, x ) d x. The next step is to establish the spectral and geometric expansions for J ( f ). Spectral expansion.
The spectral expansion is similar to the situation of unitarygroups ([BP15]), except mild structural and combinatorial difference between specialorthogonal groups and unitary groups.More precisely, let X ( G ) be the set of virtual tempered representations built fromArthur’s elliptic representations of G and its Levi subgroups (for details see [BP15,2.7]). For any π ∈ X ( G ), Arthur defined a weighted character f ∈ C ( G ) → J M ( π ) ( π, f ) LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 5 where M ( π ) is the Levi such that π is induced from an elliptic representation of M ( π ).For any f ∈ C scusp ( G ), define b θ f on X ( G ) via b θ f ( π ) = ( − a M ( π ) J M ( π ) ( π, f ) , π ∈ X ( G ) , where a M ( π ) is the dimension of the maximal central split sub-torus of M ( π ). Let D ( π ) be the discriminant coming from Arthur’s definition of elliptic representations.We have the following spectral expansion for J ( f ). Theorem 1.0.3 (Section 5) . For any f ∈ C scusp ( G ) , J ( f ) = Z X ( G ) D ( π ) b θ f ( π ) m ( π ) d π. Geometric expansion.
The geometric expansion turns out to be different from thesituation of unitary groups. We first give the statement of the geometric expansion.
Theorem 1.0.4 (Section 7.3) . For any f ∈ C scusp ( G ) , J ( f ) = lim s → + Z Γ( G,H ) c f ( x ) D G ( x ) / ∆( x ) s − / d x. Here Γ(
G, H ) is a subset in the set of semi-simple conjugacy classes of H , equippedwith topology (Section 7.1).The definition of c f , which originally appears in [Wal10], is the key ingredientdifferent from the unitary group case appearing in [BP15]. We explicate in moredetail below.Recall that Arthur introduced the notion of weighted orbital integrals for any Levisubgroup M of G and x ∈ M ( F ) ∩ G rss ( F ) f ∈ C ( G ) → J M ( x, f ) . For f ∈ C scusp ( G ), define ([BP15, 5.2]) θ f ( x ) = ( − a G − a M ( x ) ν ( G x ) − D G ( x ) − / J M ( x ) ( x, f ) . The term θ f ( x ) is conjugation invariant, and it is a quasi-character ([BP15, § c f ( x ) is equal to the regular germ associated to a particularregular nilpotent orbit in g ( F ). The detailed construction will be specified in Section7.1.A key difference between the special orthogonal groups and unitary groups is that,for unitary groups, the regular nilpotent orbits can be permuted by scaling, which isnot the case for special orthogonal groups. Hence, for unitary groups, the term c f istaken to be the average of all regular nilpotent germs, which turns out to be not the ZHILIN LUO case for special orthogonal groups. We need to choose a particular regular nilpotentorbit (Section 7.1).In order to determine which regular nilpotent orbit shows up in the geometric ex-pansion, we follow the idea of Waldspurger. After localization (Proposition 7.3.5 andProposition 7.3.6), the problem can be reduced to Lie algebra. To adapt the strategyof Waldspurger ([Wal10, § § p -adic fields ([She89]).Then in appendix B, we review the work of Walspurger ([Wal01]) computing thetransfer factors over p -adic fields, which pass without hard effort to archimedeanlocal fields. Finally we are able to prove the geometric expansion following [Wal10, § § J ( f ),following [BP15, Section 11], we obtain the geometric multiplicity formula m geom ( π )for any tempered representation π of G (Section 7.3). Then following the strategy of[BP15, Section 12], Conjecture 1.0.1 for tempered L -parameters is established. Animportant step is Theorem 8.1.1, which shows that for a stable quasi-character θ on G ( F ), c θ defined in Section 7.1 is actually the same as the one introduced in [BP15,Section 5].When the ground field is C , Conjecture 1.0.1 for tempered L -parameters followseasily from the invariance of the multiplicity under parabolic induction (Corollary3.0.7), which reduce the conjecture to trivial case. Therefore when working withtrace formula, we will assume that the ground field is not C . More precisely, startingfrom Section 5, the ground field is assumed to be either p -adic or real.To save the length of the paper, many technical details similar to [Wal10], [Wal12b],[BP15] are omitted. The reader is referred to the upcoming thesis of the author[Luo21] for full detail. Notation and conventions . We will freely use notation from the first five sectionsin [BP15]. For convenience we list them below. • F is a local field of characteristic zero with fixed norm | · | ; • For a connected reductive algebraic group G defined over F , Temp( G ) isdenoted as the set of tempered representations of G ([BP15, 2.2]); • Gothic letters are used to denote the Lie algebras of the associated algebraicgroups; • For a locally compact Hausdorff totally disconnected topological space (resp.real smooth manifold) M , C ∞ c ( M ) is the space of smooth functions on M ,and C c ( M ) is the space of continuous functions on M ([BP15, 1.4]); LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 7 • C ( G ) is the space of Harish-Chandra Schwartz functions on G ( F ), Ξ G isthe associated Harish-Chandra Ξ-function ([BP15, 1.5]), and C scusp ( G ) isthe space of strongly cuspidal Harish-Chandra Schwartz functions on G ( F )([BP15, 5.1]); • S ( g ) = S ( g ( F )) is the space of Schwartz-Bruhat functions on g ( F ) ([BP15,1.4]), and S scusp ( g ) is the space of strongly cuspidal Schwartz-Bruhat functionson g ( F ) ([BP15, 5.1]); • Γ( G ) (resp. Γ( g )) is the set of semi-simple conjugacy classes in G ( F ) (resp. g ( F )) equipped with topologies ([BP15, 1.7]), and subscript ell (resp. rss, ss)denotes the elliptic (resp. regular semi-simple, semi-simple) elements ([BP15,1.7]); • For x ∈ G ss ( F ) (resp. X ∈ g ( F )), Z G ( x ) (resp. Z G ( X )) is the centralizer of x (resp. X ) in G ( F ), and G x is the identity connected component of Z G ( x )(Note that G X = Z G ( X ) is connected, c.f. [BP15, p.16]); • For the notion of G -good (resp. G -excellent) open subsets, we refer to [BP15,3.2] (resp. [BP15, 3.3]); • For the notion and basic properties of log-norms σ = σ X on an algebraicvariety X , we refer to [BP15, 1.2]; • Let QC ( G ) (resp. QC ( g )) be the space of quasi-characters on G ( F ) (resp. g ( F )). Let SQC ( g ) be the space of Schwartz quasi-characters on g ( F ). Let QC c ( G ) (resp. QC c ( g )) be the space of quasi-characters that are compactlysupported modulo conjugation. We refer definitions and basic properties ofquasi-characters to [BP15, 4.1,4.2]; • For an irreducible admissible representation π of G , π ∞ is the space of smoothvectors, and θ π is the distribution character of π ; • For the definition of sets R temp ( G ) , X ( G ) , X temp ( G ) , X ell ( G ), see [BP15, 2.7]; • For f ∈ C scusp ( G ), set ([BP15, 5.2]) θ f ( x ) = ( − a G − a M ( x ) ν ( G x ) − D G ( x ) − / J M ( x ) ( x, f ) . Acknowledgement . I would like to thank my advisor D. Jiang for suggesting theproblem to me. I would also like to thank R. Beuzart-Plessis discussing the relationbetween the regular germ formula and the Kostant sections. I am particularly grate-ful to C. Wan for a lot of helpful discussions and various critical corrections when Iwas writing down the paper. The work is supported in part through the NSF Grant:DMS1901802 of D. Jiang.2.
The Gan-Gross-Prasad triples
Following [BP15, Section 6], the concept of Gan-Gross-Prasad triples in the specialorthogonal groups setting is introduced. Various algebraic, geometric and analyticalproperties for the triple are listed.
ZHILIN LUO
Definitions.
Quadratic spaces and orthogonal groups.
Let (
V, q ) be a finite dimensional quadraticspace. Let O( V ) (SO( V )) be the corresponding (special) orthogonal group and so ( V )the Lie algebra defined over F . Example 2.1.1. (1)
Let ( D ν , q ) be an anisotropic line, i.e. D ν ≃ F × with qua-dratic form defined by q ( x ) = νx , where x ∈ F × and ν ∈ F × . (2) Let H be the -dimensional split quadratic space, i.e. H ≃ F v ⊕ F v ∗ withquadratic form defined by q ( v, v ∗ ) = 1 , q ( v, v ) = q ( v ∗ , v ∗ ) = 0 . (3) For any b, c ∈ F × , let ( E = F ( √ b ) , c · N E/F ) be the -dimensional quadraticspace sending m ⊕ n √ b to c ( m − bn ) , where m, n ∈ F × . Definition 2.1.2. ( V, q ) is quasi-split if there exists some positive integer n ∈ N such that ( V, q ) ≃ H n − ⊕ (cid:26) ( E = F ( √ b ) , c · N E/F ) , if dim V is even D ν , if dim V is odd Assume that (
V, q ) is quasi-split. When dim V is odd, ( V, q ) is always split; whendim V is even, ( V, q ) is split if and only if b ∈ F × . Regular nilpotent orbits.
The set of SO( V )( F )-regular nilpotent orbits in so ( V )( F ),which is denoted as Nil reg ( so ( V )) = Nil reg ( so ( V )( F )), admits the following descrip-tion. Following [Wal01, Chaptire I], Nil reg ( so ( V )) is nonempty only when ( V, q ) isquasi-split. When (
V, q ) is quasi-split, the regular nilpotent orbits form a singlestable conjugacy class. Up to conjugation by SO( V )( F ), they are contained in the F -points of a fixed F -rational Borel subgroup B with Levi decomposition B = T U .Since conjugation by U ( F ) stabilizes the regular nilpotent orbits, one only need tocompute the conjugation action of T ( F ). Through straightforward computation thefollowing descriptions for Nil reg ( so ( V )) can be deduced. • When dim V is odd or ≤
2, Nil reg ( so ( V )) contains only one element. • When dim V = 2 m is even and ≥ V, q ) ≃ (cid:26) H m , when ( V, q ) is split , H m − ⊕ ( E = F ( √ b ) , c · N E/F ) , when ( V, q ) is not split . In particular, when (
V, q ) is not split b / ∈ F × .Let N V = (cid:26) F × /F × , when ( V, q ) is split ,c · N E/F ( E × ) /F × , when ( V, q ) is not split . Then N V is in bijection with Nil reg ( so ( V )). LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 9
More precisely, for any ν ∈ N V with fixed representative in F × which is stilldenoted by ν , an explicit representative in Nil reg ( so ( V )) can be constructedas follows. Write V = D ⊕ W such that D is an anisotropic line, and therestriction of q to W has the quadratic form x → νx when restricted tothe anisotropic line in W . Let H = SO( W ) where ( W, q | W ) is split of odddimension. It has a unique regular nilpotent orbit in so ( W )( F ). Let theassociated nilpotent orbit in so ( V )( F ) be O ν . Then one can show that theorbit does not depend on the choice of the representative of ν , and the set {O ν | ν ∈ N V } is in bijection with Nil reg ( so ( V )). Definition of GGP triples.
Let (
W, V ) be a pair of quadratic spaces. (
W, V ) iscalled admissible if there exists an anisotropic line D and a hyperbolic space of Z ofdimension 2 r such that V ≃ W ⊕ D ⊕ Z. In particular, there exists a basis { z i } ± ri = ± of Z such that q ( z i , z j ) = δ i, − j , ∀ i, j ∈ {± , ..., ± r } . There exist ν ∈ F × and z ∈ D such that q ( z , z ) = ν .Let P V be the parabolic subgroup of SO( V ) with Levi decomposition P V = M V N stabilizing the following totally isotropic flag of V h z r i ⊂ h z r , z r − i ⊂ ... ⊂ h z r , ..., z i . Set G = SO( W ) × SO( V ) and P = SO( W ) × P V . Then P is a parabolic subgroup of G with Levi decomposition P = M N where M = SO( W ) × M V . Using the diagonalembedding SO( W ) ֒ → G , SO( W ) can be identified as an algebraic subgroup of G contained in M . In particular SO( W ) acts via conjugation on N . Set H = SO( W ) ⋉ N. Define a morphism λ : N → G a by λ ( n ) = r − X i =0 q ( z − i − , nz i ) , n ∈ N. Then λ is SO( W ) conjugation invariant and hence λ has a unique extension to H that is trivial on SO( W ), which is still denoted as λ . Let λ F : H ( F ) → F be theinduced morphism on F -rational points. Set ξ ( h ) = ψ ( λ F ( h )) , h ∈ H ( F ) . Definition 2.1.3.
The triple ( G, H, ξ ) defined above is called the Gan-Gross-Prasadtriple associated to the admissible pair ( W, V ) . Besides the above notation and definitions, the following notions are introduced. • d = dim( V ), m = dim( W ); • Z + = h z r , ..., z i , Z − = h z − , ..., z − r i ; • D = F · z , V = W ⊕ D ; • H = SO( W ), G = SO( W ) × SO( V ). The triple ( G , H ,
1) is associated to(
W, V ). It is called the codimension one case; • T is the subtorus of SO( V ) stabilizing lines h z i i for i = ± , ..., ± r and actingtrivially on V . In particular M = T × G ; • Let h = Lie( H ). Using the same notation as before, ξ is the character on h ( F ) that is trivial on so ( W )( F ) and equal to ξ ◦ exp on n ( F ); • B ( · , · ) is the following non-degenerate G ( F )-invariant bilinear form on g ( F ), B (( X W , X V ) , ( X ′ W , X ′ V )) = 12 (tr( X W X ′ W ) + tr( X V X ′ V ));The following lemma, whose proof is similar to [BP15, Lemma 6.2.1], will be usefulfor later discussion. Lemma 2.1.4. (1)
The map G → H \ G has the norm descent property. (2) The M -conjugacy orbit of λ in ( n / [ n , n ]) ∗ is Zariski open.The multiplicity m ( π ) . For any π ∈ Temp( G ), let Hom H ( π, ξ ) be the space of con-tinuous linear forms ℓ : π ∞ → C satisfying ℓ ( π ( h ) e ) = ξ ( h ) ℓ ( e ) , ∀ e ∈ π ∞ , h ∈ H ( F ) . Define m ( π ) = dim Hom H ( π, ξ ) . The following theorem is proved in [JSZ11] for archimedean case (for r = 0 case itis also proved in [SZ12]) and through a combination of [AGRS10] ( r = 0 case) and[GGP12] (extending to general r case) for p -adic case. Theorem 2.1.5.
The following inequality holds for any π ∈ Temp( G ) m ( π ) ≤ . Following [BP15, 6.3.1] the identity m ( π ) = m ( π )holds where π is the complex conjugation of π .2.2. Spherical variety structure.
A parabolic subgroup Q of G is called good if HQ is Zariski-open in G . Through taking the generic fiber the condition is equivalentto H ( F ) Q ( F ) being open in G ( F ). LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 11
Proposition 2.2.1. (1)
There exists minimal parabolic subgroups of G that aregood, and they are conjugate under H ( F ) . Moreover, for a good minimalparabolic subgroup P min = M min U min , H ∩ U min = { } , and the complementof H ( F ) P min ( F ) in G ( F ) has zero measure; (2) A parabolic subgroup Q of G is good if and only if it contains a good minimalparabolic subgroup.Proof. The proof follows from the same argument as [BP15, Propostion 6.4.1], exceptthe fact that H ( F, SO( W )) classifies the isomorphism classes of quadratic spaces ofthe same dimension and discriminant as W ([KMRT98, 29.29]) (c.f. [BP15, p.150]). (cid:3) Relative weak Cartan decompositions.
In the codimension one case ( G , H , G admits good minimal parabolic subgroups. Let P = M U bea good minimal parabolic subgroup of G . Let A = A M be the maximal centralsplit torus of M . Set A +0 = { a ∈ A ( F ) | | α ( a ) | ≥ , ∀ α ∈ R ( A , P ) } . Proposition 2.2.2.
There exists a compact subset K ⊂ G ( F ) such that G ( F ) = H ( F ) A +0 K . Proof.
Together with [Wal10, 7.2], the discussion of [BP15, Section 6.6] works inparallel with the current situation. (cid:3)
For a general GGP triple (
G, H, ξ ), let ( P , M , A , A +0 ) be as above. Let P = M N be the parabolic subgroup opposite to P w.r.t. M . Set A min = A T ⊂ M min = M T ⊂ P min = P T N .
Then P min is a parabolic subgroup, M min is the Levi component of it and A min isthe maximal central split subtorus of M min . Moreover, from the proof of [BP15,Proposition 6.4.1], P min is a good parabolic subgroup of G . Set A +min = { a ∈ A min ( F ) | | α ( a ) | ≥ , ∀ α ∈ R ( A min , P min ) } . Let P min be the parabolic subgroup opposite to P min w.r.t. M min . Then P min ⊂ P .Let ∆ be the set of simple roots of A min in P min and set ∆ P = ∆ ∩ R ( A min , N ) be theset of simple roots appearing in n = Lie( N ). For α ∈ ∆ P let n α be the correspondingroot subspace. Lemma 2.2.3.
The following properties hold. (1) A +min = { a ∈ A +0 T ( F ) | | α ( a ) | ≤ , ∀ α ∈ ∆ P } ; (2) There exists a compact subset K G of G ( F ) such that G ( F ) = H ( F ) A +0 T ( F ) K G ;(3) For any α ∈ ∆ P , the restriction of ξ to n α ( F ) is nontrivial.Proof. The proof of [BP15, Lemma 6.6.2] works verbatim. (cid:3)
Analytical estimations.
Various analytical estimations are stated associatedto the GGP triples (
G, H, ξ ). The results will be used to establish the results forexplicit tempered intertwining, the convergence of the distribution J and J Lie , andthe spectral expansion for the distribution J , which are introduced in the next sec-tions. However, as we omit most of the details (those will appear in [Luo21]) thatare similar to those in [BP15], [Wal10] and [Wal12b], they do not show up explicitlyin the context. For the sake of completeness we prefer to keep them here. Lemma 2.3.1.
Let P min = M min U min be a good minimal parabolic subgroup and let A min be the maximal split central torus of M min . Recall that A +min = { a ∈ A min ( F ) | | α ( a ) | ≥ , ∀ α ∈ R ( A min , P min ) } . Then the following inequalities hold. (1) σ ( h ) + σ ( a ) ≪ σ ( ha ) for any a ∈ A +min and h ∈ H ( F ) ; (2) σ ( h ) ≪ σ ( a − ha ) for any a ∈ A +min and h ∈ H ( F ) .Proof. The proof follows from the same argument as [BP15, Proposition 6.4.1 (iii)]. (cid:3)
Estimations for Harish-Chandra Ξ functions. Various estimations are stated for theintegrations of the Harish-Chandra Ξ function of G on H . Lemma 2.3.2. (1)
There exists ǫ > such that the integral Z H ( F ) Ξ G ( h ) e ǫσ ( h ) d h is absolutely convergent. (2) There exists d > such that the integral Z H ( F ) Ξ G ( h ) σ ( h ) − d d h is absolutely convergent. (3) For any δ > there exists ǫ > such that the integral Z H ( F ) Ξ G ( h ) e ǫσ ( h ) (1 + | λ ( h ) | ) − δ d h is absolutely convergent. LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 13 (4)
Let P min = M min U min be a good minimal parabolic subgroup of G . Then thefollowing result holds.For any δ > there exists ǫ > such that the integral I ǫ,δ ( m min ) = Z H ( F ) Ξ G ( hm min ) e ǫσ ( h ) (1 + | λ ( h ) | − δ ) d h is absolutely convergent for any m min ∈ M min ( F ) and there exists d > suchthat I ǫ,δ ( m min ) ≪ δ P min ( m min ) − / σ ( m min ) d for any m min ∈ M min ( F ) . (5) Assume moreover that T is contained in A M min . Then for any δ > thereexists ǫ > such that the integral I ǫ,δ ( m min ) = Z H ( F ) × H ( F ) Ξ G ( hm min )Ξ G ( h ′ hm min ) e ǫσ ( h ) e ǫσ ( h ′ ) (1 + | λ ( h ′ ) | ) − δ d h ′ d h is absolutely convergent for any m min ∈ M min ( F ) , and there exists d > suchthat I ǫ,δ ( m min ) ≪ δ P min ( m min ) − σ ( m min ) d for any m min ∈ M min ( F ) .Proof. The proof of (1) follows from [Wal12b, Lemme 4.9], whose proof works ver-batim for archimedean case.The rest of the proof follows from [BP15, Lemma 6.5.1] verbatim. (cid:3)
The function Ξ H \ G . Let C ⊂ G ( F ) be a compact subset with nonempty interior.Define a function Ξ H \ GC on H ( F ) \ G ( F ) byΞ H \ GC ( x ) = vol H \ G ( xC ) − / for any x ∈ H ( F ) \ G ( F ). By finite cover theorem, for any other C ′ ⊂ G ( F ) compactsubset with nonempty interior, Ξ H \ GC ( x ) ∼ Ξ H \ GC ′ ( x )for any x ∈ H ( F ) \ G ( F ). In the following we will implicitly fix a compact subsetwith nonempty interior C ⊂ G ( F ) and setΞ H \ G ( x ) = Ξ H \ GC ( x )for any x ∈ H ( F ) \ G ( F ).The following estimations for Ξ H \ G hold. Proposition 2.3.3. (1)
For any compact subset
K ⊂ G ( F ) , the following equiv-alence hold (a) Ξ H \ G ( xk ) ∼ Ξ H \ G ( x ) , (b) σ H \ G ( xk ) ∼ σ H \ G ( x ) for any x ∈ H ( F ) \ G ( F ) and k ∈ K . (2) Let P = M U ⊂ G be a good minimal parabolic subgroup of G and A = A M be the split part of the center of M . Set A +0 = { a ∈ A ( F ) | | α ( a ) | ≥ , ∀ α ∈ R ( A , P ) } . Then there exists a positive constant d > such that (a) Ξ G ( a ) δ / P ( a ) σ ( a ) − d ≪ Ξ H \ G ( aa ) ≪ Ξ G ( a ) δ / P ( a ) , (b) σ H \ G ( aa ) ∼ σ G ( aa ) for any a ∈ A +0 and a ∈ T ( F ) . (3) There exists d > such that Z H ( F ) \ G ( F ) Ξ H \ G ( x ) σ H \ G ( x ) − d d x is absolutely convergent. (4) For any d > , there exists d ′ > such that Z H ( F ) \ G ( F ) σ H \ G ≤ c ( x )Ξ H \ G ( x ) σ H \ G ( x ) d d x ≪ c d ′ for any c ≥ . (5) There exists d > and d ′ > such that Z H ( F ) Ξ G ( x − hx ) σ G ( x − hx ) − d d h ≪ Ξ H \ G ( x ) σ H \ G ( x ) d ′ for any x ∈ H ( F ) \ G ( F ) . (6) For any d > , there exists d ′ > such that Z H ( F ) Ξ G ( hx ) σ ( hx ) − d ′ d h ≪ Ξ H \ G ( x ) σ H \ G ( x ) − d for any x ∈ H ( F ) \ G ( F ) . (7) Let δ > and d > . Then the integral I δ,d ( c, x ) = Z H ( F ) Z H ( F ) σ ≥ c ( h ′ )Ξ G ( hx )Ξ G ( h ′ hx ) σ ( hx ) d σ ( h ′ hx ) d (1 + | λ ( h ′ ) | ) − δ d h ′ d h is absolutely convergent for any x ∈ H ( F ) \ G ( F ) and c ≥ . Moreover, thereexist ǫ > and d ′ > such that I δ,d ( c, x ) ≪ Ξ H \ G ( x ) σ H \ G ( x ) d ′ e ǫc for any x ∈ H ( F ) \ G ( F ) and c ≥ .Proof. The proof of [BP15, Proposition 6.7.1] works verbatim. (cid:3)
LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 15
Parabolic degenerations.
Let Q = LU Q be a good parabolic subgroup of G . Let P min = M min U min ⊂ Q be a good minimal parabolic subgroup of G with the Levicomponent chosen so that M min ⊂ L . Let A min = A M min be the maximal central splittorus of M min and set A +min = { a ∈ A min ( F ) | | α ( a ) | ≥ , ∀ α ∈ R ( A min , P min ) } . Let H Q = H ∩ Q and H L be the image of H Q by the natural projection Q ։ L .Define H Q = H L ⋉ U Q . Proposition 2.3.4. (1) H Q ∩ U Q = { } so that the natural projection H Q → H L is an isomorphism. (2) δ Q ( h Q ) = δ H Q ( h Q ) and δ Q ( h L ) = δ H L ( h L ) for any h Q ∈ H Q ( F ) and h L ∈ H L ( F ) . In particular, the group H Q is unimodular.Fix a left Haar measure d L h L on H L ( F ) and a Haar measure dh Q on H Q ( F ) . (3) There exists d > such that the integral Z H L ( F ) Ξ L ( h L ) σ ( h L ) − d δ H L ( h L ) / d L h L converges. Moreover, in the codimension one case (that is G = G and H = H ), the integral Z H L ( F ) Ξ L ( h L ) σ ( h L ) d δ H L ( h L ) / d L h L is convergent for any d > . (4) There exists d > such that the integral Z H Q ( F ) Ξ G ( h Q ) σ ( h Q ) − d d h Q converges. (5) σ ( h Q ) ≪ σ ( a − h Q a ) for any a ∈ A +min and h Q ∈ H Q ( F ) . (6) There exist d > and d ′ > such that Z H Q ( F ) Ξ G ( a − h Q a ) σ ( a − h Q a ) − d d h Q ≪ Ξ H \ G ( a ) σ H \ G ( a ) d ′ for any a ∈ A +min .Proof. The proof of [BP15, Proposition 6.8.1] works verbatim. (cid:3) Explicit tempered intertwining
In this section an explicit relation between the non-vanishing of m ( π ) and anassociated linear form L π for any tempered representation π ∈ Temp( G ) is stated.The relation between m ( π ) and parabolic induction is also stated. As a corollary,we are able to establish Conjecture 1.0.1 for tempered L -parameters when F = C .Moreover, the results will be indispensable for the proof of the spectral expansion ofthe distribution J . Most of the proof appearing in [BP15, Section 7] works verbatim.The details are referred to [Luo21]. The ξ -integral. For any f ∈ C ( G ), the integral Z H ( F ) f ( h ) ξ ( h ) d h is absolutely convergent by Lemma 2.3.1 (2). Moreover, by Lemma 2.3.1 (2) it definesa continuous linear form on C ( G ). Recall that C ( G ) is a dense subspace of the weakHarish-Chandra Schwartz space C w ( G ) (c.f. [BP15, 1.5.1]). Proposition 3.0.1.
The linear form f → C ( G ) → Z H ( F ) f ( h ) ξ ( h ) d h extends continuously to C w ( G ) .Proof. The proof of [BP15, Proposition 7.1.1] works verbatim. (cid:3)
The continuous linear form on C w ( G ) proved above is called the ξ -integral on H ( F )and will be denoted by f ∈ C w ( G ) → Z ∗ H ( F ) f ( h ) ξ ( h ) d h or f ∈ C w ( G ) → P H,ξ ( f ) . The following properties of the ξ -integral hold. Lemma 3.0.2. (1)
For any f ∈ C w ( G ) and h , h ∈ H ( F ) , P H,ξ ( L ( h ) R ( h ) f ) = ξ ( h h − ) P H,ξ ( f ) . (2) Let a : G m → T be a one-parameter subgroup such that λ ( a ( t ) ha ( t ) − ) = tλ ( h ) for any t ∈ G m and h ∈ H . Denote by Ad a the representation of F × on C w ( G ) given by Ad a ( t ) = L ( a ( t )) R ( a ( t )) for any t ∈ F × . Let ϕ ∈ C ∞ c ( F × ) . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 17
Set ϕ ′ ( t ) = | t | − δ P ( a ( t )) ϕ ( t ) for any t ∈ F × and denote by b ϕ ′ its Fouriertransform, that is b ϕ ′ ( x ) = Z F ϕ ′ ( t ) ψ ( tx ) d t, x ∈ F. Then P H,ξ (Ad a ( ϕ ) f ) = Z H ( F ) f ( h ) b ϕ ′ ( λ ( h )) d h for any f ∈ C w ( G ) , where the second integral is absolutely convergent.Proof. Both statements are continuous in f ∈ C w ( G ) (for (2) it follows from Lemma2.3.1 (3)). Hence we only need to verify it for f ∈ C ( G ) where it follows from directcomputation. (cid:3) Definition of L π . Let π be a tempered representation of G ( F ). For any T ∈ End( π ) ∞ ,the function g ∈ G ( F ) → Trace( π ( g − ) T )belongs to C w ( G ) by [BP15, 2.2.4]. Hence one may define a linear form L π :End( π ) ∞ → C by L π ( T ) = Z ∗ H ( F ) Trace( π ( h − ) T ) ξ ( h ) d h, T ∈ End( π ) ∞ . By Lemma 3.0.2 (1), L π ( π ( h ) T π ( h ′ )) = ξ ( h ) ξ ( h ′ ) L π ( T ) L π ( T )for any h, h ′ ∈ H ( F ) and T ∈ End( π ) ∞ . By [BP15, 2.2.5], the map which associatesto T ∈ End( π ) ∞ the function g → Trace( π ( g − ) T )in C w ( G ) is continuous. Since the ξ -integral is a continuous linear form on C w ( G ), itfollows that L π is continuous.Recall that there exists a canonical continuous G ( F ) × G ( F )-equivariant embed-ding with dense image π ∞ ⊗ π ∞ ֒ → End( π ) ∞ , e ⊗ e ′ → T e,e ′ (which is an isomorphismin the p -adic case). In any case, End ∞ ( π ) ∞ is naturally isomorphic to the completedprojective tensor product π ∞ b ⊗ p π ∞ . Thus L π can be identified with the continuoussesquilinear form on π ∞ given by L π ( e, e ′ ) := L π ( T e,e ′ )for any e, e ′ ∈ π ∞ . Expanding the definitions, L π ( e, e ′ ) = Z ∗ H ( F ) ( e, π ( h ) e ′ ) ξ ( h ) d h for any e, e ′ ∈ π ∞ . Fixing e ′ ∈ π ∞ , the map e ∈ π ∞ → L ( e, e ′ ) then lies inHom H ( π ∞ , ξ ). By the density, it follows that L π = 0 ⇒ m ( π ) = 0 . The goal is to show the converse.
Theorem 3.0.3. L π = 0 ⇔ m ( π ) = 0 for any π ∈ Temp( G ) . When F is non-archimedean, the result has been established in [Wal12b, Proposi-tion 5.7]. To establish the theorem, a key result is Proposition 3.0.6, which is goingto be explained in the following sections.Finally some basic properties of L π are listed. Since L π is a continuous sesqulinearform on π ∞ , it defines a continuous linear map L π : π ∞ → π −∞ , e → L π ( e, · )where π −∞ is the topological conjugate-dual of π ∞ endowed with the strong topology.The operator L π has its image contained in π −∞ H,ξ = Hom H ( π ∞ , ξ ). By Theorem2.1.5, this subspace has finite dimension, and is less than or equal to 1 if π is irre-ducible. Let T ∈ End( π ) ∞ . Recall that it extends uniquely to a continuous operator T : π −∞ → π ∞ . Thus, the following two compositions can be formed T L π : π ∞ → π ∞ , L π T : π −∞ → π −∞ which are both finite-rank operators. In particular, their traces are well-defined and(3.0.1) Trace( T L π ) = Trace( L π T ) = L π ( T ) . Lemma 3.0.4.
The following properties hold. (1)
The maps π ∈ X temp ( G ) → L π ∈ Hom( π ∞ , π −∞ ) π ∈ X temp ( G ) → L π ∈ End( π ) −∞ are smooth in the following sense: For every parabolic subgroup Q = LU Q of G , for any σ ∈ Π ( L ) and for every maximal compact subgroup K of G ( F ) that is special in the p -adic case, the maps λ ∈ i A ∗ L → L π λ ∈ End( π λ ) −∞ ≃ End( π K ) −∞ λ ∈ i A ∗ L → L π λ ∈ Hom( π ∞ λ , π −∞ λ ) ≃ Hom( π ∞ K , π −∞ K ) are smooth, where π λ = i GQ ( σ λ ) and π K = i KQ ∩ K ( σ ) . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 19 (2)
Let π ∈ Temp( G ) or X temp ( G ) , Then for any S, T ∈ End( π ) ∞ , SL π ∈ End( π ) ∞ and L π ( S ) L π ( T ) = L π ( SL π T ) . (3) Let
S, T ∈ C ( X temp ( G ) , E ( G )) . Then, the section π ∈ Temp( G ) → S π L π T π ∈ End( π ) ∞ belongs to C ∞ ( X temp ( G ) , E ( G )) . (4) Let f ∈ C ( G ) and assume that its Plancherel transform π ∈ X temp ( G ) → π ( f ) is compactly supported (which is automatic when F is p -adic). Then thefollowing identity holds, Z H ( F ) f ( h ) ξ ( h ) d h = Z X temp ( G ) L π ( π ( f )) µ ( π ) d π where both integrals are absolutely convergent. (5) Let f, f ′ ∈ C ( G ) and assume that the Plancherel transform of f is compactlysupported, then the following identity holds Z X temp ( G ) L π ( π ( f )) L π ( π ( f ′ )) µ ( π ) d π = Z H ( F ) Z H ( F ) Z G ( F ) f ( hgh ′ ) f ′ ( g ) d gξ ( h ′ ) dh ′ ξ ( h ) d h where the first integral is absolutely convergent and the second integral isconvergent in that order but not necessarily as a triple integral.Proof. The proof of [BP15, Lemma 7.2.2] works verbatim. (cid:3)
Asymptotic of tempered intertwinings.
Lemma 3.0.5. (1)
Let π be a tempered representation of G ( F ) and ℓ ∈ Hom H ( π ∞ , ξ ) be a continuous ( H, ξ ) -equivariant linear form. Then, there exist d > anda continuous semi-norm v d on π ∞ such that | ℓ ( π ( x ) e ) | ≤ v d ( e )Ξ H \ G ( x ) σ H \ G ( x ) d for any e ∈ π ∞ and x ∈ H ( F ) \ G ( F ) . (2) For any d > , there exists d ′ > and a continuous semi-norm v d,d ′ on C wd ( G ( F )) such that |P H,ξ ( R ( x ) L ( y ) ϕ ) | ≤ v d,d ′ ( ϕ )Ξ H \ G ( x )Ξ H \ G ( y ) σ H \ G ( x ) d ′ σ H \ G ( y ) d ′ for any ϕ ∈ C wd ( G ( F )) and x, y ∈ H ( F ) \ G ( F ) .Proof. The proof follows from the same argument as [BP15, Lemma 7.3.1]. (cid:3)
Explicit intertwining and parabolic induction.
Let Q = LU Q be a parabolic subgroupof G = SO( W ) × SO( V ). Then there are decompositions(3.0.2) Q = Q W × Q V , L = L W × L V , where Q W and Q V are parabolic subgroups of SO( W ) and SO( V ) respectively and L W and L V are the associated Levi components. By the explicit descriptions ofparabolic subgroups of special orthogonal groups,(3.0.3) L W = GL( Z ,W ) × ... × GL( Z a,w ) × SO( f W ) , (3.0.4) L V = GL( Z ,V ) × ... × GL( Z b,V ) × SO( e V ) , where Z i,W , 1 ≤ i ≤ a (respectively Z i,V , 1 ≤ i ≤ b ) are totally isotropic subspacesof W (respectively of V ) and f W (respectively e V ) is a non-degenerate subspace of W (respectively of V ). Let e G = SO( f W ) × SO( e V ). Up to permutation the pair( e V , f W ) is admissible, hence in particular it yields a GGP triple ( e G, e H, e ξ ) which iswell-defined up to e G ( F )-conjugation. For any tempered representations e σ of e G ( F ),one can consider the continuous linear form L e σ : End( e σ ) ∞ → C .Let σ be a tempered representation of L ( F ) which decomposes according to (3.0.2),(3.0.3), (3.0.4) as a tensor product(3.0.5) σ = σ W ⊠ σ V , (3.0.6) σ W = σ ,W ⊠ ... ⊠ σ a,W ⊠ e σ W , (3.0.7) σ V = σ ,V ⊠ ... ⊠ σ b,V ⊠ e σ V , where σ i,W ∈ Temp(GL( Z i,W )) for 1 ≤ i ≤ a , σ i,V ∈ Temp(GL( Z i,V )) for 1 ≤ i ≤ b , e σ W is a tempered representation of SO( f W )( F ) and e σ V is a tempered representationof SO( e V )( F ). Set e σ = e σ W ⊠ e σ V . It is a tempered representation of e G ( F ). Finallyset π = i GQ ( σ ), π W = i SO( W ) Q W ( σ W ), and π V = i SO( V ) Q V ( σ V ) for the parabolic inductionsof σ, σ W and σ V respectively. Then π = π W ⊠ π V . Proposition 3.0.6.
With the notations as above, L π = 0 ⇔ L e σ = 0 . Proof.
The proof in [BP15, Proposition 7.4.1] works verbatim. (cid:3)
Now the proof of Theorem 3.0.3 follows from the same argument as in [BP15,Section 7]. The details are referred to [Luo21].
LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 21
A corollary.
We adopt the notation and hypothesis of Section 3. In particular, Q = LU Q is a parabolic subgroup of G with L decomposes as in (3.0.2), (3.0.3)and (3.0.4), σ is a tempered representation of L ( F ) admitting decomposition as in(3.0.5), (3.0.6) and (3.0.7) and set e σ = e σ W ⊠ e σ V . This is a tempered representationof e G ( F ) where e G ( F ) = SO( f W ) × SO( e V ). Recall that the admissible pair ( f W , e V ) (upto permutation) defines a GGP triple ( e G, e H, e ξ ). Hence, the multiplicity m ( e σ ) of e σ relative to this GGP triple is defined. Set π = i GQ ( σ ). Corollary 3.0.7. (1)
Assume that σ is irreducible, m ( π ) = m ( e σ ) . (2) Let
K ⊂ X temp ( G ) be a compact subset. There exists a section T ∈ C ( X temp ( G ) , E ( G )) such that L π ( T π ) = m ( π ) for any π ∈ K . Moreover, the same equality is satisfied for every sub-representation π of some π ′ ∈ K .Proof. The proof of [BP15, Corollary 7.6.1] works verbatim. (cid:3)
When F = C , f W and e V can be taken to be zero dimension since tempered repre-sentations are always principal series. It follows that the tempered part of Conjecture1.0.1 holds automatically when F = C .4. The distributions
In this section, the trace distribution J and its Lie algebra variant J Lie is intro-duced.
The distribution J . For any f ∈ C ( G ), define a function K ( f, · ) on H ( F ) \ G ( F ) by K ( f, x ) = Z H ( F ) f ( x − hx ) ξ ( h ) d h, x ∈ H ( F ) \ G ( F ) . From Lemma 2.3.2 (2) the integral is absolutely convergent. The theorem below andProposition 2.3.3 (3) shows that the following integral J ( f ) = Z H ( F ) \ G ( F ) K ( f, x ) d x is absolutely convergent for any f ∈ C scusp ( G ), and J ( · ) defines a continuous linearform on C scusp ( G ). Theorem 4.0.1. (1)
There exists d > and a continuous semi-norm ν on C ( G ) such that | K ( f, x ) | ≤ ν ( f )Ξ H \ G ( x ) σ H \ G ( x ) d for any f ∈ C ( G ) and x ∈ H ( F ) \ G ( F ) . (2) For any d > there exists a continuous semi-norm ν d on C ( G ) such that | K ( f, x ) | ≤ ν d ( f )Ξ H \ G ( x ) σ H \ G ( x ) − d for any x ∈ H ( F ) \ G ( F ) and f ∈ C scusp ( G ) .Proof. The proof in [BP15, Theorem 8.1.1] works verbatim. (cid:3)
The distribution J Lie . Parallel to J the following Lie algebra variant distribution J Lie is introduced.For any f ∈ S ( g ), define a function K Lie ( f, · ) on H ( F ) \ G ( F ) by K Lie ( f, x ) = Z h ( F ) f ( x − Xx ) ξ ( X ) d X, x ∈ H ( F ) \ G ( F ) . The integral is absolutely convergent. The theorem below, whose proof is similar toTheorem 4.0.1, shows that the following integral J Lie ( f ) = Z H ( F ) \ G ( F ) K Lie ( f, x ) d x is absolutely convergent for any f ∈ S scusp ( g ) and defines a continuous linear formon S scusp ( g ). Theorem 4.0.2. (1)
There exists c > and a continuous semi-norm ν on S ( g ) such that | K Lie ( f, x ) | ≤ ν ( f ) e cσ H \ G ( x ) for any x ∈ H ( F ) \ G ( F ) and f ∈ S ( g ) . (2) For any c > , there exists a continuous semi-norm ν c on S ( g ) such that | K Lie ( f, x ) | ≤ ν c ( f ) e − cσ H \ G ( x ) for any x ∈ H ( F ) \ G ( F ) and f ∈ S scusp ( g ) . Spectral expansion for the distribution J In this section the spectral expansion of the distribution J is established. Theproof of the spectral side of the distribution J for unitary groups case ([BP15])carries without hard effort to the special orthogonal groups setting. To save thelength of the paper, we outline the main analytical results and refer the details to[Luo21].We first give the statement of the theorem. LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 23
Theorem 5.0.1.
For any f ∈ C scusp ( G ) , set J spec ( f ) = Z X ( G ) D ( π ) b θ f ( π ) m ( π ) d π. Then the integral is absolutely convergent and J ( f ) = J spec ( f ) for any f ∈ C scusp ( G ) . From [BP15, Lemma 5.4.2] and Theorem 4.0.1, both sides of the equality arecontinuous on C scusp ( G ). Hence by [BP15, Lemma 5.3.1 (ii)] it is sufficient to establishthe equality for functions f ∈ C scusp ( G ) which have compactly supported Fouriertransforms (where the Fourier transform is understood as the spectral transformappearing in matrix Paley-Wiener theorem, see [BP15, p.121]). Throughout thesection a function f ∈ C scusp ( G ) with compactly supported Fourier transform isfixed. Study of an auxiliary distribution.
For any f ′ ∈ C ( G ) , the following integrals areintroduced, K Af,f ′ ( g , g ) = Z G ( F ) f ( g − gg ) f ′ ( g ) d g, g , g ∈ G ( F ) ,K f,f ′ ( g, x ) = Z H ( F ) K Af,f ′ ( g, hx ) ξ ( h ) d h, g, x ∈ G ( F ) ,K f,f ′ ( x, y ) = Z H ( F ) K f,f ′ ( h − x, y ) ξ ( h ) d h, x, y ∈ G ( F ) ,J aux ( f, f ′ ) = Z H ( F ) \ G ( F ) K f,f ′ ( x, x ) d x. The following proposition gives estimations for the auxiliary distributions.
Proposition 5.0.2. (1)
The integral defining K Af,f ′ ( g , g ) is absolutely conver-gent. For any g ∈ G ( F ) the map g ∈ G ( F ) → K Af,f ′ ( g , g ) belongs to C ( G ) . Moreover, for any d > there exists d ′ > such thatfor every continuous semi-norm ν on C wd ′ ( G ( F )) , there exists a continuoussemi-norm µ on C ( G ) satisfying ν ( K Af,f ′ ( g, · )) ≤ ν ( f ′ )Ξ G ( g ) σ ( g ) − d for any f ′ ∈ C ( G ) and g ∈ G ( F ) . (2) The integral defining K f,f ′ ( g, x ) is absolutely convergent. Moreover, for any d > , there exists d ′ > and a continuous semi-norm ν d,d ′ on C ( G ) suchthat | K f,f ′ ( g, x ) | ≤ ν d,d ′ ( f ′ )Ξ G ( g ) σ ( g ) − d Ξ H \ G ( x ) σ H \ G ( x ) d ′ for any f ′ ∈ C ( G ) and g, x ∈ G ( F ) . (3) The integral defining K f,f ′ ( x, y ) is absolutely convergent convergent. More-over K f,f ′ ( x, y ) = Z X temp ( G ) L π ( π ( x ) π ( f ) π ( y − )) L π ( π ( f ′ )) µ ( π ) d π for any f ′ ∈ C ( G ) and x, y ∈ G ( F ) . The integral is absolutely convergent. (4) The integral defining J aux ( f, f ′ ) is absolutely convergent. More precisely, forevery d > there exists a continuous semi-norm ν d on C ( G ) such that | K f,f ′ ( x, x ) | ≤ ν d ( f ′ )Ξ H \ G ( x ) σ H \ G ( x ) − d for any f ′ ∈ C ( G ) and x ∈ H ( F ) \ G ( F ) . In particular, the linear form f ′ ∈ C ( G ) → J aux ( f, f ′ ) is continuous.Proof. The proof of [BP15, Proposition 9.2.1] works verbatim. (cid:3)
From Proposition 5.0.2, the following proposition can be established. Note thatthe upshot of the two propositions is various analytical estimations, which buildsupon the analytical estimations proved in Subsection 2.3.
Proposition 5.0.3.
The following equality holds J aux ( f, f ′ ) = Z X ( G ) D ( π ) b θ f ( π ) L π ( π ( f ′ )) d π for any f ′ ∈ C ( G ) .Proof. The proof of [BP15, Proposition 9.2.2] works verbatim. (cid:3)
Now we end of proof of the theorem 5.0.1.Recall that a function f ∈ C scusp ( G ) with compactly supported Fourier transformhas been fixed. By Lemma 3.0.4 (4),(5.0.1) K ( f, x ) = Z X temp ( G ) L π ( π ( x ) π ( f ) π ( x − )) µ ( π ) d π for any x ∈ H ( F ) \ G ( F ). By Corollary 3.0.7 (2) there exists a function f ′ ∈ C ( G )such that(5.0.2) L π ( π ( f ′ )) = m ( π ) LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 25 for any π ∈ X temp ( G ) such that π ( f ) = 0. Also, by Theorem 3.0.3 and Corollary3.0.7 (1), for any π ∈ X temp ( G ), L π = 0 ⇔ m ( π ) = 1 . Hence by (5.0.1) the equality holds K ( f, x ) = Z X temp ( G ) L π ( π ( x ) π ( f ) π ( x − )) L π ( π ( f ′ )) µ ( π ) d π and by Proposition 5.0.2 (3), it follows that K ( f, x ) = K f,f ′ ( x, x )for any x ∈ H ( F ) \ G ( F ). Consequently, J ( f ) = J aux ( f, f ′ ) . After applying Proposition 5.0.3, J ( f ) = Z X ( G ) D ( π ) b θ f ( π ) L π ( π ( f ′ )) d π. Let π ∈ X ( G ) be such that b θ f ( π ) = 0 and let π ′ be the unique representation in X temp ( G ) such that π is a linear combination of sub-representations of π ′ . Then π ′ ( f ) = 0. Hence by 5.0.2 and Corollary 3.0.7 (2) L π ( π ( f ′ )) = m ( π ) = m ( π ). Itfollows that J ( f ) = Z X ( G ) D ( π ) b θ f ( π ) m ( π ) d π and this ends the proof of Theorem 5.0.1.6. Spectral expansion for the distribution J Lie
In this section a spectral expansion of J Lie is established, which is a key steptowards the geometric expansion of J .6.1. The affine subspace Σ . In Section 2.1 a parabolic subgroup P = M N of G = SO( W ) × SO( V ) has been defined, which can be written as SO( W ) × P V , with P V a parabolic subgroup of SO( V ) . Let P = M N be the parabolic subgroup oppositeto P w.r.t. M . The unipotent radicals N and N can be identified as subgroups ofSO( V ).In Section 2.1 a character ξ on N ( F ) has been defined which extends to a characterof H ( F ) = SO( W )( F ) ⋉ N ( F ) trivial on SO( W )( F ). Using the G ( F )-invariantbilinear pairing B on g ( F ) defined in Section 2.1, there exists a unique elementΞ ∈ n ( F ) such that ξ ( X ) = ψ ( B (Ξ , X )) for any X ∈ n ( F ).There is the following explicit description of Ξ ∈ so ( V ),(6.1.1)Ξ z i = z i − , ≤ i ≤ r, Ξ z − i = − z − i − , ≤ i ≤ r − , Ξ z = − ν − z − , Ξ( W ) = 0 . Set Σ = Ξ + h ⊥ where h ⊥ is the orthogonal complement of h in g w.r.t. B ( · , · ).Fix a Haar measure d µ h ( X ) on h ( F ). From [BP15, Section 1.6], there is a naturalway to associate to d µ h ( X ), using B ( · , · ), a Haar measure d µ ⊥ h on h ⊥ ( F ). Denoteby d µ Σ the translation of the measure to Σ( F ). By Fourier inversion, the followingequality holds(6.1.2) Z h ( F ) f ( X ) ξ ( X ) d µ h ( X ) = Z Σ( F ) b f ( Y ) d µ Σ ( Y )for any f ∈ S ( g ). Conjugation by N . By explicit calculation, there is the following description of h ⊥ :an element X = ( X W , X V ) ∈ g = so ( W ) ⊕ so ( V ) is in h ⊥ if and only if X V = − X W + c ( z , w ) + T + N for some w ∈ W F , T ∈ t and N ∈ n . Thus for every element X = ( X V , X W ) of Σ,(6.1.3) X V = Ξ − X W + c ( z , w ) + T + N where w, T, N are as above. Define the following affine subspaces of g : • so ( W ) − = { ( X W , − X W ) | X W ∈ so ( W ) } ; • Λ is the subspace of so ( V ) ⊂ g generated by c ( z i , z i +1 ) for i = 0 , ..., r − c ( z r , w ) for w ∈ W ; • Λ = Ξ + ( so ( W ) − ⊕ Λ ). Proposition 6.1.1.
Conjugation by N preserves Σ , and induces an isomorphism ofalgebraic varieties: N × Λ → Σ( n, X ) → nXn − . Proof.
First we show that the map N × Λ → Σ(6.1.4) ( n, x ) → nXn − is injective. This amounts to proving that for any n ∈ N and X ∈ Λ, if nXn − ∈ Λ,then n = 1. We let n ∈ N and X = ( X W , X V ) ∈ Λ be such that nXn − ∈ Λ. By
LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 27 the definition of Λ, we may write X V and nX V n − as(6.1.5) X V = Ξ − X W + c ( z r , w ) + r − X i =0 µ i c ( z i , z i +1 )(6.1.6) nX V n − = Ξ − X W + c ( z r , w ′ ) + r − X i =0 µ ′ i c ( z i , z i +1 )where w, w ′ ∈ W F and µ i , µ ′ i ∈ F , 0 ≤ i ≤ r −
1. We first prove the followingstatement,(6.1.7) nz i = z i , ≤ i ≤ r. The proof is by descending induction. The result is trivial for i = r since n ∈ N .Assume now that the equality (6.1.7) is true for some 1 ≤ i ≤ r . Then from (6.1.5)we have ( nX V n − ) z i = nX V z i = n Ξ z i = nz i − and from (6.1.6), we also have( nX V n − ) z i = Ξ z i = z i − , By induction hypothesis we get (6.1.7).We now prove the following(6.1.8) nz − i = z − i , ≤ i ≤ r. We prove by strong induction on i . First we treat the case i = 1. By (6.1.5) and(6.1.7) we have( nX V n − ) z = nX V z = n ( − ν − z − + µ ν z ) = − ν − nz − + µ ν z . On the other hand, from (6.1.6) we get( nX V n − ) z = − ν − z − + µ ′ ν z . It follows that nz − − z − = ( µ − µ ′ ) ν z . After pairing both sides with z − , we have q ( nz − , z − ) = q ( z − , z − ) = 0. The firstidentity q ( nz − , z − ) = 0 follows from the fact that n ∈ N . Hence we can deducethat µ = µ ′ so that we indeed have nz − = z − . Now let 1 ≤ j ≤ r − ≤ i ≤ j . By (6.1.5) and (6.1.7) we have( nX V n − ) z − j = nX V z − j = n ( − z − j − − µ j − z j − + µ j z j +1 )= − nz − j − − µ j − z j − + µ j z j +1 . On the other hand, we have( nX V n − ) z − j = − z − j − − µ ′ j − z j − + µ ′ j z j +1 . It follows that nz − j − − z − j − = ( µ ′ j − − µ j − ) z j − + ( µ j − µ ′ j ) z j +1 . By induction hypothesis, we have nz − j = z − j , nz − j +1 = z − j +1 . Moreover since n ∈ SO( V ), we have q ( nz − j − , z − j ) = q ( z − j − , z − j ) = 0, q ( nz − j − , z − j +1 ) = q ( − z − j − , z − j +1 ) =0. Hence we deduce that µ ′ j − = µ j − and µ ′ j = µ j , and nz − j − = z − j − . This endsthe proof of (6.1.8).From (6.1.7) and (6.1.8) and since n ∈ N , we immediately deduce that n = 1.This ends the proof that the map (6.1.4) is injective. By explicit computation, wehave dim( N × Λ) = dim( N ) + dim(Λ)= ( r + rm ) + m ( m − / m + r = m ( m + 1) / r + mr + r dim(Σ) = dim( h ⊥ ) = dim( G ) − dim( H )= ( m + 2 r + 1)( m + 2 r ) / − ( r + rm )= m ( m + 1) / mr + r + r, where m = dim( W ). Hence dim(Σ) = dim( N × Λ). Since we are in characteristic0 situation, we get that the regular map (6.1.4) induces an isomorphism between N × Λ and a Zariski open subset of Σ. But N × Λ and Σ are both affine spaces hencethe only Zariski open subset of Σ that can be isomorphic to N × Λ is Σ itself. Itfollows that the regular map (6.1.4) is an isomorphism. (cid:3)
Characteristic polynomial.
Let X = ( X W , X V ) ∈ Λ. By the definition of Λ,one may write(6.2.1) X V = Ξ − X W + c ( z r , w ) + r − X i =0 µ i c ( z i , z i +1 )where w ∈ W F and µ i ∈ F . Denote by P X V and P − X W the characteristic polynomialsof X V and − X W acting on V F and W F respectively, both of which are elements of F [ T ]. Let D be the F -linear endomorphism of F [ T ] given by D ( T i +1 ) = T i for i ≥ D (1) = 0.Over the algebraic closure F , we fix a hyperbolic basis for W F which are eigen-vectors for − X W , i.e. when dim W = m is even, a basis { w i } i = ± ,..., ± m satisfying q ( w i , w j ) = δ i, − j ; when dim W = m is odd, a basis { w i } i =0 , ± ,..., ± m − satisfying q ( w i , w j ) = δ i, − j . Moreover we have − X W ( w i ) = s i w i , s i ∈ F with i >
0. Then in
LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 29 particular P − X W ( T ) = (cid:26) Q m i =1 ( T − s i ) , dim W = m is even; T Q m − i =1 ( T − s i ) , dim W = m is odd.Write w ∈ W as w = P ± m i = ± z i w i with z i ∈ F when dim W = m is even, and w = P ± m − i =0 , ± z i w i with z i ∈ F when dim W = m is odd. In the following proposition wewill write w to indicate the associated column vector of w under the above basis, and q ( w, W ) to indicate the row vector P ± m i = ± q ( w, w i ) w i = P ± m i = ± z − i w i when dim W = m is even, and P ± m − i =0 , ± q ( w, w i ) w i = P ± m − i =0 , ± z − i w i when dim W = m is odd. Proposition 6.2.1.
The following equality relating P X V ( T ) and P − X W ( T ) holds.If r = 0 , then P X V ( T ) = det (cid:18) T − ( − X W ) − ν wq ( w, W ) T (cid:19) = T P − X W ( T ) + m − X i =0 ( − j ν q ( w, X jW w ) D j +1 ( P − X W ( T )) . The formula can also be written as P X V ( T ) = T P − X W ( T )+ (cid:26) ν P m i =1 z i z − i T P − XW ( T ) T − s i , when dim W = m is even, ν z P − XW ( T ) T + 2 ν P m − i =1 z i z − i T P − XW ( T ) T − s i , when dim W = m is odd.If r > , then P X V ( T ) = ( − r det (cid:18) T − ( − X W ) wq ( w, W ) 0 (cid:19) + P − X W ( T )( T r +1 − ( ν + ν − ) T r − µ + r − X j =1 ( − j +1 ν − µ j T r − − j ) . Here det (cid:18) T − ( − X W ) wq ( w, W ) 0 (cid:19) = m − X i =0 ( − j +1 q ( w, X jW w ) D j +1 ( P − X W ( T ))= (cid:26) − P m i =1 z i z − i T P − XW ( T ) T − s i , when dim W = m is even, − z P − XW ( T ) T − P m − i =1 z i z − i T P − XW ( T ) T − s i , when dim W = m is odd.Proof. The statement can be proved via induction on r . We left the proof to thereader. (cid:3) Corollary 6.2.2.
The following
SO( W ) -invariant polynomial functions on Λ X = ( X W , X V ) → q ( w, X jW w ) ∈ F , j = 0 , ..., m − (where we have written X V as in (6.2.1)) extend to G -invariant polynomial functionson g defined over F . In particular, the polynomial function X = ( X W , X V ) → det( q ( X iW w, X jW w )) ≤ i,j ≤ m − ∈ F extends to a G -invariant polynomial function on g defined over F . Denote by Q such an extension. Set d G ( X ) := det(1 − Ad( X )) | g / g X for any X ∈ g reg . The d G extends uniquely to a polynomial d G ∈ F [ g ] G . Set Q = Q d G ∈ F [ g ] G and let Λ ′ and Σ ′ be the non-vanishing loci of Q in Λ and Σ respectively. Notice that we haveΛ ′ ⊂ Λ reg and Σ ′ ⊂ Σ reg since d G divides Q .There is the following characterization of Σ ′ . Proposition 6.2.3. Σ ′ is precisely the set of X = ( X W , X V ) ∈ Σ reg such that thefamily z r , X V z r , ..., X d − V z r generates V F as a F -module (Recall that d = dim( V ) ).Proof. The proof in [BP15, Proposition 10.4.1] works verbatim. (cid:3)
Conjugation classes in Σ ′ .Proposition 6.3.1. The conjugation action of H = SO( W ) ⋉ N on Σ ′ is free. Twoelements of Σ ′ are G -conjugate if and only if they are H -conjugate. Moreover, aftertaking the F -rational points, two elements of Σ ′ ( F ) are G -conjugate (in the sense ofstable conjugacy) if and only if they are H ( F ) -conjugate.Proof. For the case when d = dim V ≤
2, the proposition holds automatically. There-fore in the following we only consider the case when dim V ≥
3. In this case wenotice that the adjoint orbit of so ( V ) is determined by its characteristic polynomial,in other words two elements of so ( V ) are conjugate by SO( V ) action if and only ifthey have the same characteristic polynomial. We also notice that when dim V = 2,two elements fo so ( V ) have the same characteristic polynomial if and only if theyare conjugate by O( V ).(1) We look at the case when dim W = 2. We follow closely with [Wal10,Lemme 9.5].Recall that by definition, H acts freely on Σ ′ if the map H × Σ ′ → Σ ′ × Σ ′ ( h, X ) → ( X, hXh − ) LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 31 is a closed immersion. By Proposition 6.1.1, this is equivalent to proving thatSO( W ) × Λ ′ → Λ ′ × Λ ′ (6.3.1) ( h, X ) → ( X, hXh − )is a closed immersion. For X = ( X W , X V ) ∈ g , we define the characteristic polyno-mial of X to be the pair P X = ( P X W , P X V ). Let Y ⊂ Λ ′ × Λ ′ be the closed subset ofpairs ( X, X ′ ) such that P X = P X ′ . We claim the following(6.3.2) The map is a closed immersion whose image is Y .This will prove the two points of the proposition, since if two elements of g are G -conjugate, they share the same characteristic polynomial, therefore they are H -conjugate by (6.3.2). First by definition, the image of (6.3.1) is contained in Y . Let( X, X ′ ) ∈ Y . We may write X V = Ξ − X W + c ( z r , w ) + r − X i =0 µ i c ( z i , z i +1 ) X ′ V = Ξ − X ′ W + c ( z r , w ′ ) + r − X i =0 µ ′ i c ( z i , z i +1 )where X = ( X V , X W ) , X ′ = ( X ′ V , X ′ W ) , w, w ′ ∈ W F , and µ i , µ ′ i ∈ F . Since dim W =2, and P X W = P X ′ W , we know that X W and X ′ W are conjugate by SO( W ) action.Hence up to SO( W )-conjugation we may assume that X W = X ′ W . By Proposition6.2.1, we have µ i = µ ′ i for i = 0 , ..., r − q ( w, X iW w ) = q ( w ′ , X ′ iW w ′ ) i = 0 , ..., m − . Moreover, by definition of Λ ′ , ( w, X W w, ..., X m − W w ) and ( w ′ , X ′ W w ′ , ..., X ′ m − W w ′ ) arebases of W F .We first treat the case when m = dim W is even. We fix a basis { w i | i = ± , ..., ± m } of W F satisfying q ( w i , w j ) = δ i, − j for any i, j = ± , ..., ± m . Since X W lies in Σ reg , up to SO( W )-conjugation we may assume that X W lies in the diagonalmaximal split Cartan subalgebra of so ( W ). In particular, the centralizer of X W in SO( W ) is the diagonal split torus of SO( W ), and the action of any element inthe centralizer of the form diag( z m , ..., z , z − , ..., z − m ) on w i is via scaling z i , where i = ± , ..., ± m . If we write w = P ± m i = ± a i w i , and w ′ = P ± m i = ± a ′ i w i , then from (6.3.3)we have a i a − i = a ′ i a ′− i for any i = ± , ..., ± m . Moreover, since ( w, X W w, ..., X m − W w )and ( w ′ , X ′ W w ′ , ..., X ′ m − W w ′ ) are bases of W F , and X W = X ′ W is diagonal, we noticethat a i and a ′ i are nonzero for any i . Now the element diag( z m , ..., z , z − , ..., z − m )with z i = a ′ i a i conjugates X V to X ′ V . Then we consider the case when m = dim W is odd. Similarly we fix a ba-sis { w , w i | i = ± , ..., ± m − } of W F satisfying q ( w i , w j ) = δ i, − j for any i, j =0 , ± , ..., ± m − . Using the same notation as the even case, we find that a i a − i = a ′ i a ′− i with a i and a ′ i being nonzero for any i = 0 , ± , ..., ± m − . Then the elementdiag( z m − , ..., z , , z − , ..., z − m − with z i = a ′ i a i conjugates X V to X ′′ V , with X ′′ V = X ′ V ,or X ′′ V and X ′ V differ by the conjugation action of the element δ ∈ O( V ) sending w to − w and stabilizing the complement of w in V . We only need to show that thesecond case does not occur. Otherwise, since X V is SO( V )-conjugate to both X ′ V and X ′′ V , and X ′ V is conjugate to X ′′ V by δ ∈ O( V ) \ SO( V ) action, we notice thatthe centralizer of X V in O( V ) is not connected. However, from Proposition 6.2.3,we know that X V is regular and does not contain 0 as its eigenvalues, therefore thecentralizer of X V should be connected, which is a contradiction. Hence in conclusionwe have that the element diag( z m − , ..., z , , z − , ..., z − m − ) ∈ SO( W ) conjugates X V to X ′ V .The uniqueness of g ∈ SO( W ) is easy to derive, which, again using the conjugationof c ( z r , w ) and c ( z r , w ′ ), we know that g must send w to w ′ , and therefore the element g should exactly send the basis ( w, X W w, ..., X m − W w ) to ( w ′ , X ′ W w ′ , ..., X ′ m − W w ′ ),hence unique.Hence, we have proved that the map induces a bijection from SO( W ) × Λ ′ to Y and we have constructed the inverse, which is a morphism of algebraic varieties.Finally, if we take the F -rational points, then the uniqueness of the element g ∈ SO( W ) immediately implies that g ∈ SO( W )( F ) as it sends the F -rational basis( w, X W w, ..., X m − W w ) to ( w ′ , X ′ W w ′ , ..., X ′ m − W w ′ ), from which we directly deduce thatthe two elements in Λ ′ ( F ) are SO( W )( F )-conjugate, and so the parallel statementfor Σ ′ ( F ) holds.(2) We look at the case when dim W = 2. We use same notation as the case whendim W = 2.Similar to the case when dim W ≥
3, since we assume that X = ( X W , X V ) and( X ′ W , X ′ V ) are G -conjugate, we know that X W = X ′ W . The remaining discussion isthe same as the dim W ≥ (cid:3) Corollary 6.3.2.
The following inequality holds σ G ( t ) ≪ σ H \ G ( t ) σ Σ ′ ( X ) for any X ∈ Σ ′ and t ∈ G X .Proof. The proof of [BP15, Corollary 10.5.2] works verbatim. (cid:3)
Borel sub-algebras and Σ ′ . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 33
Proposition 6.4.1.
Let X ∈ Σ ′ and b be a Borel subalgebra of g defined over F containing X , then b ⊕ h = g . Proof.
Let X ∈ Σ ′ and b ⊂ g be a Borel subalgebra containing X . By Proposition6.1.1, up to N ( F )-conjugation we may assume that X ∈ Λ ′ and we will assumethis so henceforth. Write X = ( X W , X V ) with X W ∈ so ( W ) and X V ∈ so ( V ). Bydefinition of Λ, we have a decomposition(6.4.1) X V = Ξ − X W + c ( z r , w ) + r − X i =0 µ i c ( z i , z i +1 )where w ∈ W F , 0 ≤ i ≤ r and µ i ∈ F , 0 ≤ i ≤ r − b ) + dim( h ) = dim( g ), hence it suffices toprove(6.4.2) b ∩ h = 0 . There exist Borel subalgebras b W and b V of so ( W ) and so ( V ) respectively suchthat X W ∈ b W , X V ∈ b V and b = b W × b V . Then b ∩ h = 0 is equivalent to(6.4.3) ( b W + n ) ∩ b V = 0 . Fix an F -embedding F ֒ → F and set V = V ⊗ F F , W = W ⊗ F F . Then we haveisomorphism of F -vector spaces V F ≃ V , W F ≃ W .
Also, if U is a subspace of V we will set U = U ⊗ F F and view it as a subspace of V . We will adopt similar notation w.r.t. W . We have isomorphisms so ( V ) F ≃ so ( V ) , so ( W ) F ≃ so ( W )and we will identify two sides via the isomorphisms. Then, b W is the stabilizer in so ( W ) of a complete flag 0 = W ⊂ W ⊂ ... ⊂ W m = W and similarly b V is the stabilizer in so ( V ) of a complete flag F : 0 = V ⊂ V ⊂ ... ⊂ V d = V .
We define another complete flag F ′ : 0 = V ′ ⊂ V ′ ⊂ ... ⊂ V ′ d = V of V by setting • V ′ i = h z r , ..., z r − i +1 i for i = 1 , ..., r + 1; • V ′ r +1+ i = Z + ⊕ D ⊕ W i for i = 1 , ..., m ; • V ′ r + m +1+ i = Z + ⊕ V ⊕ h z − , ..., z − i i for i = 1 , ..., r .for any v ∈ V , let us denote by V ( X V , v ) the subspace of V generated by v, X V v, X V v, ... We have the following lemma.
Lemma 6.4.2.
Let ≤ i ≤ d , then we have (1) for any nonzero v ∈ V ′ i , V ′ i − + V ( X V , v ) = V ; (2) V ′ i ∩ V d − i = 0 .Proof. First we prove that (1) implies (2). Indeed, if v ∈ V ′ i ∩ V d − i is nonzero, then by(1), we would have dim( V ( X V , v )) ≥ d + 1 − i . But V ( X V , v ) ⊂ V d − i (since v ∈ V d − i and X V ∈ b V preserves V d − i ), and therefore dim( V ( X V , v )) ≤ dim( V d − i ) = d − i .This is a contradiction.We now prove the first statement. Let v ∈ V ′ i be nonzero. Without loss ofgenerality, we may assume that v ∈ V ′ i \ V ′ i − since otherwise the result with i − i is stronger. We assume this is so henceforth and it follows that(6.4.4) V ′ i − + V ( X V , v ) = V ′ i + V ( X V , v ) . By definition of V ′ i , we have z r ∈ V ′ i + V ( X V , v ) and so by Proposition 6.2.3, itsuffices to show that V ′ i − + V ( X V , v ) is X V -stable. The subspace V ( X V , v ) is X V -stable almost by definition. Hence, we are left with proving that(6.4.5) X V V ′ i − ⊂ V ′ i + V ( X V , v ) . This is clear if 1 ≤ i ≤ r + 1 or r + m + 2 ≤ i ≤ d = 2 r + m + 1 since in thiscase using the decomposition of X V we easily check that X V V ′ i − ⊂ V ′ i . It remainsto show that it also holds for r + 2 ≤ i ≤ r + m + 1. In this case, again using thedecomposition of X V , we can find that(6.4.6) X V v ′ ∈ V ′ i + h z ∗ , v ′ i X V z for any v ′ ∈ V ′ i (where X V z = − ν − z − + µ ν z if r ≥ X V z = 0 if r = 0).Here, we have used the fact that X W ∈ b W so that W i − r − and W i − r − are X W -stable.As v ∈ V ′ i , it suffices to show that the existence of k ≥ h z ∗ , X kV v i 6 = 0.By Proposition 6.2.3, the family z ∗ r , t X V z ∗ r , t X V z ∗ r , ... generates V ∗ . Hence, since v = 0, there exists k ≥ h t X k V z ∗ r , v i = h z ∗ r , X k V v i 6 = 0. This already settles the case where r = 0. In the case r ≥
1, since V ′ i is included in the kernel of z ∗ r this shows that the sequence v, X V v, X V v, ... eventuallydoes not lie in V ′ i and by (6.4.6) this implies also the existence of k ≥ h z ∗ , X kV v i 6 = 0. This ends the proof of (6.4.5) and of the lemma. (cid:3) LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 35
Let us now set D i = V ′ i ∩ V d +1 − i for i = 1 , .., d . By the previous lemma anddimension consideration, these are one dimensional subspaces of V and we have V = d M i =1 D i . Let Y ∈ ( b W + n ) ∩ b V . We want to prove that Y = 0 so as to obtain (6.4.2). Bydefinition Y must stabilize the flags F and F ′ so that Y stabilizes the lines D , ..., D d .We claim that(6.4.7) Y ( D i ) = 0 , i = 1 , .., r + 1 , i = r + m + 2 , ..., d. Indeed, since Y ∈ b W + n , we have Y V ′ i ⊂ V ′ i − for any i = 1 , ..., r + 1 and i = r + m + 2 , ..., d and so Y D i ⊂ V ′ i − ∩ V d +1 − i = 0.To deduce that Y = 0, it only remains to show that(6.4.8) Y ( D i ) = 0 , i = r + 2 , ..., r + m + 1 . Assume, by way of contradiction, that there exists 1 ≤ j ≤ m such that Y D r +1+ j = 0.Since Y ∈ b W + n , we have Y V ′ r +1+ j ⊂ W j ⊕ Z + so that D r +1+ j = Y D r +1+ j ⊂ W j ⊕ Z + . Let v ∈ D r +1+ j be nonzero. We claim that(6.4.9) ( Z + ⊕ W j ) + V ( X V , v ) = V Indeed by the previous lemma, it suffices to prove that z ∈ ( Z + ⊕ W j ) + V ( X V , v ).By the decomposition (6.4.1), we can find that X V ( Z + ⊕ W j ) ⊂ Z + ⊕ W j ⊕ F z so that we only need to check that the sequence v, X V v, ... eventually does not liein Z + ⊕ W j . From Proposition 6.2.3, we know that there exists k ≥ h z ∗ r , X kV v i 6 = 0. Since Z + ⊕ W j is included in the kernel of z ∗ r , this proves (6.4.9).From (6.4.9), we deduce that dim V ( X V , v ) ≥ d − j − r . On the other hand,we have v ∈ V d − r − j since v ∈ D r +1+ j , and X V leaves V d − r − j stable (since X V ∈ b V ).As dim V d − r − j = d − r − j it is a contradiction. This ends the proof of (6.4.8) andof the Proposition. (cid:3) Corollary 6.4.3.
There exists a constant c > such that for any ǫ > sufficientlysmall, any X ∈ Σ ′ ( F ) and parabolic subalgebras p of g defined over F and containing X , exp[ B ( · , ǫe − cσ Σ ′ ( X ) )] ⊂ H ( F ) exp( B (0 , ǫ ) ∩ p ( F )) . Proof.
The proof of [BP15, Corollary 10.6.3] works verbatim. (cid:3)
The quotient Σ ′ ( F ) /H ( F ) . By Proposition 6.1.1, Σ ′ has a geometric quotientby N and Σ ′ /N ≃ Λ ′ . Because H = N ⋊ SO( W ) and SO( W ) is reductive, thegeometric quotient of Σ ′ by H exists and Σ ′ /H ≃ Λ ′ / SO( W ). Denote by g ′ thenon-vanishing locus of Q in g and by g ′ /G the geometric quotient of g ′ by G for theadjoint action. The natural map Σ ′ → g ′ /G factors through the quotient Σ ′ /H andthere is the induced morphism π : Σ ′ /H → g ′ /G. We will also consider the F -analytic counterpart of this map: π F : Σ ′ ( F ) /H ( F ) → g ′ ( F ) /G ( F ) . Put on H ( F ) the Haar measure µ H which lift the Haar measure µ h on h ( F ). Because H ( F ) acts freely on Σ ′ ( F ), we can define a measure µ Σ ′ /H on Σ ′ ( F ) /H ( F ) to be thequotient of (the restriction to Σ ′ ( F ) of) µ Σ by µ H . It is the unique measure onΣ ′ ( F ) /H ( F ) such that Z Σ( F ) ϕ ( X ) d µ Σ ( X ) = Z Σ ′ ( F ) /H ( F ) Z H ( F ) ϕ ( h − Xh ) d h d µ Σ ′ /H ( X )for any ϕ ∈ C c (Σ( F )). One can define a measure d X on g rss ( F ) /G ( F ) = Γ rss ( g ) fol-lowing [BP15, Section 1.7]. Moreover, g ′ ( F ) /G ( F ) is an open subset of g rss ( F ) /G ( F )and we will still denote by d X the restriction of this measure to g ′ ( F ) /G ( F ). Proposition 6.5.1. (1) π is an isomorphism of algebraic varieties and π F is anopen embedding of F -analytic spaces; (2) π F sends the measure d µ Σ ′ /H ( X ) to D G ( X ) / d X ; (3) The natural projection p : Σ ′ → Σ ′ /H has the norm descent property.Proof. The proof of (1) and (2) follows from the same argument as in [BP15, Propo-sition 10.7.1]. We only establish (3).For (3), by Proposition 6.1.1, it is sufficient to show thatΛ ′ → Λ ′ / SO( W )has the norm descent property. Denote by Λ Q the non-vanishing locus of Q in Λ(where Q ∈ F [ g ] G is defined in Section 6.2. Then we have the following Cartesiandiagram where horizontal maps are open immersionsΛ ′ / / (cid:15) (cid:15) Λ Q (cid:15) (cid:15) Λ ′ / SO( W ) / / Λ Q / SO( W ) LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 37
Thus, if we prove that Λ Q → Λ Q / SO( W ) has the norm descent property, we willbe done. By definition of Λ (cf. Section 6.1), we have an SO( W )-equivariant isomor-phism Λ ≃ so ( W ) × W × A r , where the action of SO( W ) on the RHS is the product of the adjoint action on so ( W ),the natural action on W and the trivial action on A r . Denote by ( so ( W ) × W ) the open-Zariski subset of so ( W ) × W consisting of all pairs ( X, w ) ∈ so ( W ) × W such that ( w, Xw, X , ... ) generates W . Then Λ Q corresponds via the previousisomorphism to ( so ( W ) × W ) × A r . Since SO( W ) acts trivially on A r , we arereduced to show that( so ( W ) × W ) → ( so ( W ) × W ) / SO( W )has the norm descent property. Let B be the flag variety of basis of W and let P ol m be the variety of monic polynomial P ∈ F [ T ] of degree m . Consider the action ofSO( W ) on P ol m ×B which is trivial on P ol m and given by g ( e , ..., e m ) = ( ge , ..., ge m )on B . By Proposition 6.3.1 the map( so ( W ) × W ) → P ol m × B ( X, w ) → ( P X , w, Xw, ..., X m − W )is a SO( W )-equivariant closed immersion. Passing to the quotient, we get a commu-tative diagram ( so ( W ) × W ) / / (cid:15) (cid:15) P ol m × B (cid:15) (cid:15) ( so ( W ) × W ) / SO( W ) / / P ol m × B / SO( W )where horizontal maps are closed immersion (since SO( W ) is reductive). More-over the diagram is Cartesian (because all SO( W )-orbits in B are closed and sothe quotient separates all orbits). Thus, we are finally reduced to showing that B → B / SO( W ) has the norm descent property. Choosing a particular basis of W ,this amounts to proving thatGL( W ) → GL( W ) / SO( W )has the norm descent property. Since the map is GL( W )-equivariant for the actionby left translation, by [BP15, Lemma 1.2.2(i)], it suffices to show the existence ofa nonempty Zariski open subset of GL( W ) / SO( W ) over which the previous maphas the norm descent property. Choose an orthogonal basis ( e , ..., e m ) of W anddenote by B the standard Borel subgroup of GL( W ) relative to this basis. Then B ∩ SO( W ) = Z is the subtorus acting by ± e i (so that Z ≃ ( Z / Z ) m ).Let U be the unipotent radical of B , then U ∩ SO( W ) = { } , moreover dim U + dim SO( W ) = dim GL( W ). It follows that we have the canonical open immersion U ֒ → GL( W ) / SO( W ). ThereforeGL( W ) → GL( W ) / SO( W )admits a section on an open dense subset, from which we ends the proof of (3). (cid:3) Spectral expansion of J Lie . Define Γ(Σ) to be the subset of Γ( g ) consistingof the conjugacy classes of the semi-simple parts of elements in Σ( F ). Equip thesubset with the restriction of the measure defined on Γ( g ). Thus, if T ( G ) is a set ofrepresentatives for the G ( F )-conjugacy classes of maximal tori of G and if for any T ∈ T ( G ) we denote by t ( F ) Σ the subset of elements X ∈ t ( F ) whose conjugacyclass belongs to Γ(Σ), then Z Γ(Σ) ϕ ( X ) d X = X T ∈T ( G ) | W ( G, T ) | − Z t ( F ) Σ ϕ ( X ) d X for any ϕ ∈ C c (Γ(Σ)). Recall that in Section 4, a continuous linear form J Lie on S scusp ( g ) has been defined.The theorem below is one of the most technical result of the paper. Fortunatelythe proof of [BP15, § § Theorem 6.6.1.
The following identity holds J Lie ( f ) = Z Γ(Σ) D G ( X ) / b θ f ( X ) d X for any f ∈ S scusp ( g ) . Geometric expansion and multiplicity formula
In this section, we are going to establish the geometric expansion of the tracedistribution J , from which we are able to deduce a geometric multiplicity formula m ( π ) for any tempered representation π of G ( F ).7.1. The linear forms m geom and m Liegeom . Geometric support.
For any x ∈ H ss ( F ), we are going to introduce spaces of semi-simple conjugacy classes in G ( F ), G x ( F ) and g ( F ), which will be denoted by Γ( G, H ),Γ( G x , H x ) and Γ Lie ( G, H ), respectively. The definition is in parallel with [BP15, 11.1].Fix x ∈ H ss ( F ), up to conjugation we may assume that x ∈ SO( W ) ss ( F ). Let W ′ x (resp. V ′ x ) be the kernel of 1 − x in W (resp. V ) and W ′′ x (resp. V ′′ x ) be the image of1 − x . There are orthogonal decompositions W = W ′ x ⊕ ⊥ W ′′ x and V = V ′ x ⊕ ⊥ W ′′ x . Set H ′ x = SO( W ′ x ) ⋉ N x (where N x is the centralizer of x in N ), G ′ x = SO( W ′ x ) × SO( V ′ x ), LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 39 H ′′ x = SO( W ′′ x ) x and G ′′ x = SO( W ′′ x ) x × SO( W ′′ x ) x . Set ξ x = ξ | H x ( F ) . Then there arefollowing decompositionsSO( V ) x = SO( V ′ x ) × SO( W ′′ x ) x , SO( W ) x = SO( W ′ x ) × SO( W ′′ x ) x ,H x = SO( W ) x ⋉ N x , and Z SO( V ) ( x ) = SO( V ′ x ) × Z SO( W ′′ x ) ( x ) , (7.1.1) Z SO( W ) ( x ) = SO( W ′ x ) × Z SO( W ′′ x ) ( x ) . Moreover, it is not hard to verify that SO( W ′′ x ) x commutes with N x . Hence there arealso decompositions(7.1.2) G x = G ′ x × G ′′ x , H x = H ′ x × H ′′ x , where the inclusion H x ⊂ G x is the product of the two inclusions H ′ x ⊂ G ′ x and H ′′ x ⊂ G ′′ x . Since ξ x is trivial on H ′′ x , there is a further decomposition( G x , H x , ξ x ) = ( G ′ x , H ′ x , ξ ′ x ) × ( G ′′ x , H ′′ x , , where ξ ′ x = ξ | H ′ x . Note that the triple ( G ′ x , H ′ x , ξ ′ x ) coincides with the GGP tripleassociated to the admissible pair ( V ′ x , W ′ x ). The second triple ( G ′′ x , H ′′ x ,
1) is also of aparticular shape: the group G ′′ x is the product of two copies of H ′′ x and the inclusion H ′′ x ⊂ G ′′ x is the diagonal one. Following [BP15, 11.1] such a triple is called an Arthurtriple . Finally note that although x ∈ SO( W ) ss ( F ), there is a decomposition similarto (7.1.2) for any x ∈ H ss ( F ) (simply conjugated x inside H ( F ) to an element inSO( W ) ss ( F )) and that if x, y ∈ H ss ( F ) are H ( F )-conjugate, then there are naturalisomorphisms of triples( G ′ x , H ′ x , ξ ′ x ) ≃ ( G ′ y , H ′ y , ξ ′ y ) , ( G ′′ x , H ′′ x , ≃ ( G ′′ y , H ′′ y , H ′ x ( F ) and H ′′ x ( F ) respectively.Let x ∈ H ss ( F ). Denote by Γ( H ), Γ( H x ), Γ( G ) and Γ( G x ) the sets of semi-simple conjugacy classes in H ( F ) , H x ( F ) , G ( F ) and G x ( F ) respectively and they areequipped with topologies.By the definition of GGP triples, the two canonical maps Γ( H x ) → Γ( G x ) andΓ( H ) → Γ( G ) are injective. Since these maps are continuous and proper ([BP15,Section 1.7]) and Γ( G x ), Γ( H ), Γ( G ) are all Hausdorff and locally compact, it turnsout that Γ( H x ) → Γ( G x ) and Γ( H ) → Γ( G ) are closed embeddings.We are going to define a subset Γ( G, H ) of Γ( H ) as follows: x ∈ Γ( G, H ) if andonly if H ′′ x is an anisotropic torus (and hence G ′′ x also). Since Γ( H ) → Γ( G ) is a closedembedding, Γ( G, H ) can also be viewed as a subset of Γ( G ). Notice that Γ( G, H ) is asubset of Γ ell ( G ) containing 1. We now equip Γ( G, H ) with a topology, which is finerthan the one induced from Γ( G ), and a measure. For this, we need to give a more concrete description of Γ( G, H ). Consider the following set T of subtori of SO( W ): T ∈ T if and only if there exists a non-degenerate subspace W ′′ ⊂ W (possibly W ′′ = 0) such that T is a maximal elliptic subtorus of SO( W ′′ ). For such a torus T ,denote by T ♮ the open Zariski subset of elements t ∈ T which are regular in SO( W ′′ )acting with distinct eigenvalues on W ′′ . Then Γ( G, H ) is the set of conjugacy classesthat meet S T ∈T T ♮ ( F ) (c.f. [BP15, p.259]). Let W ( H, T ) = Norm H ( T ) /Z H ( T ).Then, by definition there is a natural bijection(7.1.3) Γ( G, H ) ≃ G T ∈T T ♮ ( F ) /W ( H, T ) . Now the RHS of (7.1.3) has a natural topology and we transfer it to Γ(
G, H ). More-over, we equip Γ(
G, H ) with the unique regular Borel measure such that Z Γ( G,H ) ϕ ( x ) dx = X T ∈T | W ( H, T ) | − ν ( T ) Z T ( F ) ϕ ( t ) dt for any ϕ ∈ C c (Γ( G, H )). Recall that ν ( T ) is the only positive factor such that thetotal mass of T ( F ) for the measure ν ( T ) d t is one. Note that 1 is an atom for themeasure whose mass is equal to 1 (this corresponds to the contribution of the trivialtorus in the formula above).More generally, for any x ∈ H ss ( F ) we may construct a subset Γ( G x , H x ) of Γ( G x )which is equipped with its own topology and measure as follows. By (7.1.2) wehave a decomposition Γ( G x ) = Γ( G ′ x ) × Γ( G ′′ x ). Since the triple ( G ′ x , H ′ x , ξ ′ x ) is aGGP triple, the previous construction provides us with a space Γ( G ′ x , H ′ x ) of semi-simple conjugacy classes in G ′ x ( F ). On the other hand, we define Γ( G ′′ x , H ′′ x ) tobe the image of Γ ani ( H ′′ x ) (the set of anisotropic conjugacy classes in H ′′ x ( F ), c.f.[BP15, Section 1.7]) by the diagonal embedding Γ( H ′′ x ) ⊂ Γ( G ′′ x ). Following [BP15,Section 1.7], the set Γ( G ′′ x , H ′′ x ) = Γ ani ( H ′′ x ) can be equipped with a topology and ameasure. Set Γ( G x , H x ) = Γ( G ′ x , H ′ x ) × Γ( G ′′ x , H ′′ x )and we equip this set with the product of the topologies and the measures definedon Γ( G ′ x , H ′ x ) and Γ( G ′′ x , H ′′ x ). Note that Γ( G x , H x ) = ∅ unless x ∈ G ( F ) ell (becauseotherwise Γ ani ( H ′′ x ) = ∅ ).There is a parallel Lie algebra variant, Γ Lie ( G, H ) is a subset of Γ( g ), againequipped with a topology and a measure, as follows. for any X ∈ so ( W ) ss ( F ),we have decompositions(7.1.4) G X = G ′ X × G ′′ X , H X = H ′ X × H ′′ X LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 41 where G ′ X = SO( W ′ X ) × SO( V ′ X ) , G ′′ X = SO( W ′′ X ) X × SO( W ′′ X ) X ,H ′ X = SO( W ′ X ) ⋉ N X , H ′′ X = SO( W ′′ X ) X for W ′ X , V ′ X the kernels of X acting on W and V respectively, W ′′ X the image of X in W and N X the centralizer of X in N . Again the decomposition (7.1.4) still hold forevery X ∈ h ss ( F ) and they depend on the choice of representatives in the conjugacyclass of X only up to an inner automorphism. We now define Γ Lie ( G, H ) to be theset of semi-simple conjugacy classes X ∈ Γ( h ) such that H ′′ X is an anisotropic torus.Similarly we can identify Γ Lie ( G, H ) with a subset of Γ( g ). Notice that Γ Lie ( G, H ) isa subset of Γ ell ( g ) containing 0. Moreover, fixing a set of tori T as before we have aidentification(7.1.5) Γ Lie ( G, H ) = G T ∈T t ♮ ( F ) /W ( H, T )where for T ∈ T , t ♮ denotes the Zariski open subset consisting of elements X ∈ t that are regular in so ( W ′′ T ) and acting with distinct eigenvalues in W ′′ T . By theidentification (7.1.5), Γ Lie ( G, H ) inherits a natural topology. Moreover, we equipΓ
Lie ( G, H ) with the unique regular Borel measure such that Z Γ Lie ( G,H ) ϕ ( X ) d X = X T ∈T | W ( H, T ) | − ν ( T ) Z t ( F ) ϕ ( X ) d X for any ϕ ∈ C c (Γ Lie ( G, H )). Note that 0 is an atom for this measure whose associatedmass is 1. The following lemma establishes a link between Γ
Lie ( G, H ) and Γ(
G, H ). Lemma 7.1.1.
Let ω ⊂ g ( F ) be a G -excellent open neighborhood of and set Ω = exp( ω ) . Then, the exponential map induces a topologicalisomorphism ω ∩ Γ Lie ( G, H ) ≃ Ω ∩ Γ( G, H ) preserving measures.Proof. The proof of [BP15, Lemma 11.1.2] works verbatim. (cid:3)
We are ready to introduce the geometric multiplicity formula and its Lie algebravariant.
The germs c θ . We first define the germ c θ associated to a quasi-character θ of G ( F ).Here we would like to mention that the germ c θ we are considering are of differentnature from the unitary group case considered in [BP15].Let θ be a quasi-character of G ( F ) = SO( W ) × SO( V ) . For convenience we onlyconsider the case when G is quasi-split. Recall from Section 2.1 that the regularnilpotent orbits of G ( F ) has the following description. • If m = dim W ≥ G ( F ) areparametrized by N W ; • If d = dim V ≥ G ( F ) areparametrized by N V ; • For other cases, G ( F ) has a unique regular nilpotent orbit O reg .In each situation, for any ν ∈ N W (resp. ν ∈ N V ), let O ν be the associatedregular nilpotent orbit of G , which admits explicit description as in Section 2.1.We first consider the case when G has only one regular nilpotent orbit O reg . Inthis case, we simply set c θ = c θ, O reg .Then we consider the case when d = dim V ≥ G ( F ) are parametrized by N V . Since dim W is odd and SO( W )is quasi-split, we have that q W, an is of dimension one. Let q D be the restriction of thequadratic form q V to the associated line D . Then ( V, q V ) has the same anisotropickernel as q W, an ⊕ q D . In particular, ν = q ( v ) ∈ N V . Then we set c θ = c θ, O ν .Finally when m = dim W ≥ G ( F ) areparametrized by the set N W . Since G ( F ) is quasi-split, the quadratic form q W, an ⊕ q D is actually split. In particular, − ν = − q ( v ) ∈ N W . Then set c θ = c θ, O − ν . Discriminants.
Set ∆( x ) = D G ( x ) D H ( x ) − for any x ∈ H ss ( F ) where D G ( x ) = | det(1 − Ad( x )) | g / g x | and D H ( x ) = | det(1 − Ad( x )) | h / h x | . Then ([BP15, Lemma 3.1.1])(7.1.6) ∆( x ) = | det(1 − x ) | W ′′ x | for any x ∈ H ss ( F ). Similarly define∆( X ) = D G ( X ) D H ( X ) − for any X ∈ h ss ( F ) and(7.1.7) ∆( X ) = | det( X | W ′′ X ) | for any X ∈ h ss ( F ). Let ω ⊂ g ( F ) be a G -excellent open neighborhood of 0. Then ω ∩ h ( F ) is an H -excellent open neighborhood of 0 (cf. the remark at the end of[BP15, Section 3.3]). We may thus set j HG ( X ) = j H ( X ) j G ( X ) − for any X ∈ ω ∩ h ( F ). By [BP15, 3.3.1],(7.1.8) j HG ( X ) = ∆( X )∆( e X ) − for any X ∈ ω ∩ h ss ( F ). Note that j HG is a smooth positive and H ( F )-invariantfunction on ω ∩ h ( F ). It actually extends (not uniquely) to a smooth, positive and G ( F )-invariant function on ω . This can be seen as follows. We can embed the groups LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 43 H = SO( W ) and H = SO( V ) into a GGP triple ( G , H , ξ ) and ( G , H , ξ ). Thenthe function X = ( X W , X V ) ∈ ω → j H G ( X W ) / j H G ( X V ) / can be seen as such anextension through using the equality (7.1.8). We will always assume that such anextension has been chosen and we will still denote it by j HG .Let x ∈ H ss ( F ). Define ∆ x ( y ) = D G x ( y ) D H x ( y ) − for any y ∈ H x, ss ( F ). On the other hand, since the triple ( G ′ x , H ′ x , ξ ′ x ) is a GGPtriple, the previous construction yields a function ∆ G ′ x on H ′ x, ss ( F ). Then(7.1.9) ∆ x ( y ) = ∆ G ′ x ( y ′ )for any y = ( y ′ , y ′′ ) ∈ H x, ss ( F ) = H ′ x, ss ( F ) × H ′′ x, ss ( F ).Let Ω x ⊂ G x ( F ) be a G -good open neighborhood of x . Then, Ω x ∩ H ( F ) ⊂ H x ( F )is a H -good open neighborhood of x . This allows us to set η HG,x ( y ) = η Hx ( y ) η Gx ( y ) − for any y ∈ Ω x ∩ H ( F ). By [BP15, 3.2.4](7.1.10) η HG,x ( y ) = ∆ x ( y )∆( y ) − for any y ∈ Ω x ∩ H ss ( F ). Note that η HG,x is a smooth, positive and H x ( F )-invariantfunction on Ω x ∩ H ( F ). It also extends (not uniquely) to a smooth, positive and G x ( F )-invariant function on Ω x . We will always still denote by η HG,x such an extension.The definitions of the distributions m geom and m Liegeom are contained in the followingproposition.
Proposition 7.1.2. (1)
Let θ ∈ QC ( G ) . Then the following integral is abso-lutely convergent for any Re( s ) > Z Γ( G,H ) D G ( x ) / c θ ( x )∆( x ) s − / d x and the following limit m geom ( θ ) := lim s → + Z Γ( G,H ) D G ( x ) / c θ ( x )∆( x ) s − / d x. exists. Similarly, for any x ∈ H ss ( F ) and θ x ∈ QC ( G x ) , the following integralis absolutely convergent for any Re( s ) > Z Γ( G x ,H x ) D G x ( y ) / c θ x ( y )∆ x ( y ) s − / d y and the following limit m x, geom ( θ x ) := lim s → + Z Γ( G x ,H x ) D G x ( y ) s − / c θ x ( y )∆ x ( y ) s − / d y exists. In particular, m geom is a continuous linear form on QC ( G ) and forany x ∈ H ss ( F ) , m x, geom is a continuous linear form on QC ( G x ) . (2) Let x ∈ H ss ( F ) and let Ω x ⊂ G x ( F ) be a G -good open neighborhood of x andset Ω = Ω Gx . Then, if Ω x is sufficiently small, m geom ( θ ) = 2[ Z H + ( x )( F ) : H x ( F )] m x, geom (( η HG,x ) / θ x, Ω x ) for any θ ∈ QC c (Ω) . (3) Let θ ∈ QC c ( g ) . Then the following integral is absolutely convergent for any Re( s ) > Z Γ Lie ( G,H ) D G ( X ) / c θ ( X )∆( X ) s − / d X and the following limit m Liegeom ( θ ) := lim s → + Z Γ Lie ( G,H ) D G ( X ) / c θ ( X )∆( X ) s − / d X exists. In particular, m Liegeom is a continuous linear form on QC c ( g ) that ex-tends continuously to SQC ( g ) and m Liegeom ( θ λ ) = | λ | δ ( G ) / m Liegeom ( θ ) for any θ ∈ SQC ( g ) and λ ∈ F × . Here we recall that θ λ ( X ) = θ ( λ − X ) forany X ∈ g reg ( F ) . (4) Let ω ⊂ g ( F ) be a G -excellent open neighborhood of and set Ω = exp( ω ) .Then m geom ( θ ) = m Liegeom (( j HG ) / θ ω ) for any θ ∈ QC c (Ω) .Proof. (1) When F is p -adic, the proof follows from [Wal10, § § F = R , the only difference between our situation and that of [BP15, Propo-sition 11.2.1 (i)] is the last part ([BP15, (11.2.22)]) of the proof in [BP15], where inthe unitary group situation the regular nilpotent orbits can be permuted by scaling([BP15, Section 6.1]), which is not the case for special orthogonal groups. However,following the proof of [BP15, Proposition 11.2.1 (i)], indeed, the problem can bereduced to show the limit(7.1.11) lim s → + m Liegeom ,s ( θ ) LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 45 exist for any θ = ϕ b j ( O , · ), where ϕ ∈ C ∞ ( g ) G is an invariant smooth functioncompactly supported modulo conjugation and equals 1 in some neighborhood of 0,and O ∈
Nil reg ( g ).The proof of (7.1.11) follows from the same argument as in [Wal10, § § • In the proof of [Wal10, Lemme 7.4], change Ω s to be the subset of ω withcoordinates ( λ j ) j =1 ,...,m in the basis ( e j ) j =1 ,...,m satisfying ≤ | λ j | ≤
1, andchange ̟ kF to ( ) k ; • The proof of [Wal10, Lemme 7.6] does not work in general in the archimedeancase, as the germ expansions for quasi-characters is not an exact identityparametrized by nilpotent orbits. However, the quasi-character θ appearingin (7.1.11) is equal to ϕ b j ( O , · ). In particular the proof of [Wal10, Lemme 7.6]indeed applies to our situation.(2) We notice that θ x, Ω x ∈ QC c (Ω x ) Z G ( x ) (cf. [BP15, Proposition 4.4.1 (iii)]).Hence the proof follows from [Wal10, Lemme 8.3] directly.(3) The absolute convergence can be proved in parallel with (1) using [BP15,Lemma B.1.2(ii)]. The existence of the limit follows from the same argument asin [BP15, Proposition 11.2.1 (iii)]. The homogeneity of m Liegeom ( θ λ ) follows from thedirect computation as in [BP15, Proposition 11.2.1]. Finally by [BP15, Proposi-tion 4.6.1(ii)], m Liegeom can be extended to
SQC ( g ).(4) The proof is parallel to (2), where we apply Lemma 7.1.1 to our situation. (cid:3) Geometric multiplicity and parabolic induction.
Let L be a Levi sub-group of G . Then as in Section 3, L can be decomposed as a product L = L GL × e G where L GL is a product of general linear groups over F and e G belongs to a GGPtriple ( e G, e H, e ξ ) which is well-defined up to e G ( F ) conjugation. In particular, there iscontinuous linear form m e G geom on QC ( e G ). Define a continuous linear form m L geom on QC ( L ) = QC ( L GL ) b ⊗ p QC ( e G ) by setting m L geom ( θ GL ⊗ e θ ) = m e G geom ( e θ ) c θ GL (1)for any θ GL ∈ QC ( L GL ) and e θ ∈ QC ( e G ).Before stating the relation between geometric multiplicity and parabolic induction,we note the following lemma, which is the analogue result for [Wal12b, Lemme 2.3]associated with regular nilpotent germs in the archimedean case. For the notationappearing in the lemma below we refer to [Wal12b, Lemme 2.3]. Lemma 7.2.1.
Suppose θ L ∈ QC ( L ) and θ = i GL θ L (c.f. [BP15, (3.4.2)] ). Then (1) θ ∈ QC ( G ) ; (2) Suppose x ∈ G ss ( F ) and O ∈
Nil reg ( g x ) , then D G ( x ) / c θ, O ( x ) = X y ∈X L ( x ) X g ∈ Γ y /G x ( F ) X O ′ ∈ Nil reg ( l y ) [ g O : O ′ ][ Z L ( y ) : L y ( F )] D L ( y ) / c θ L , O ′ ( y ) . Proof. (1) follows from [BP15, Proposition 4.7.1 (i)].For (2), through the limit formula in [BP15, p.98, Section 4.5] together with the ex-plicit induction formula ([BP15, (3.4.2)]), we can reduce θ L ∈ QC ( L ) to θ L ∈ QC ( l )which is finite linear combinations of distributions { b j ( O ′ , · ) | O ′ ∈ Nil reg ( l ) } . Sincethe parabolic induction of finite linear combinations in { b j ( O ′ , · ) | O ′ ∈ Nil reg ( l ) } is finite linear combinations of { b j ( O , · ) | O ∈ Nil reg ( g ) } , the proof of [Wal12b,Lemme 2.3] applies verbatim. (cid:3) Now combing with the lemma above, the following lemma follows from the sameargument as [Wal12b, Lemme 7.2].
Lemma 7.2.2.
Let θ L ∈ QC ( L ) and set θ = i GL ( θ L ) . Then m geom ( θ ) = m L geom ( θ L ) . The theorems.
Set J geom ( f ) = m geom ( θ f ) , f ∈ C scusp ( G ) ,m geom ( π ) = m geom ( θ π ) , π ∈ Temp( G ) ,J Liegeom ( f ) = m Liegeom ( θ f ) , f ∈ S scusp ( g ) . Theorem 7.3.1. J ( f ) = J geom ( f ) for any f ∈ C scusp ( G ) . Theorem 7.3.2. m ( π ) = m geom ( π ) for any π ∈ Temp( G ) . Theorem 7.3.3. J Lie ( f ) = J Liegeom ( f ) for any f ∈ S scusp ( g ) . The proof is by induction on dim( G ) (the case dim( G ) = 1 is trivial). Hence, wemake the following induction hypothesis.(HYP) Theorem 7.3.1, Theorem 7.3.2 and Theorem 7.3.3 hold for any GGP triples( G ′ , H ′ , ξ ′ ) such that dim( G ′ ) < dim( G ). LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 47
Equivalence of theorem 7.3.1 and theorem 7.3.2.
Proposition 7.3.4.
Assume the induction hypothesis (HYP) . (1) Let π ∈ R ind ( G ) . Then m ( π ) = m geom ( π ) . (2) For any f ∈ C scusp ( G ) , there is the following equality J ( f ) = J geom ( f ) + X π ∈X ell ( G ) D ( π ) b θ f ( π )( m ( π ) − m geom ( π )) , and the sum on the RHS is absolutely convergent. (3) Theorem 7.3.1 and Theorem 7.3.2 are equivalent. (4)
There exists a unique continuous linear form J qc on QC ( G ) such that • J ( f ) = J qc ( θ f ) for any f ∈ C scusp ( G ) ; • Supp( J qc ) = G ( F ) ell .Proof. The proof of [BP15, Proposition 11.5.1] works verbatim. (cid:3)
Semisimple descent and the support of J qc − m geom . Proposition 7.3.5.
Assume the induction hypothesis (HYP) . Let θ ∈ QC ( G ) andassume that / ∈ Supp( θ ) . Then J qc ( θ ) = m geom ( θ ) . Proof.
Since both J qc and m geom are both supported on Γ ell ( G ), by partition of unity,we only need to prove the equality for θ ∈ QC c (Ω) where Ω is a completely G ( F )-invariant open subset of G ( F ) of the form Ω Gx for some x ∈ G ( F ) ell , x = 1, and some G -good open neighborhood Ω x ⊂ G x ( F ) of x . Moreover, we may take Ω x as smallas we want. In particular, we will assume that Ω x is relatively compact moduloconjugation.Assume first that x is not conjugate to any element of H ss ( F ). Then since Γ( H )is closed in Γ( G ), if Ω x is chosen sufficiently small, we would have Ω ∩ Γ( H ) = ∅ . Inthis case, both sides of the equality are zero for any θ ∈ QC c (Ω).Assume now that x is conjugate to some element of H ss ( F ). We may simplyassume x ∈ H ss ( F ). Then, G x = G ′ x × G ′′ x , G ′′ x = H ′′ x × H ′′ x . By (7.1.1), Z G ( x ) = G ′ x × Z SO( W ′′ x ) ( x ) × Z SO( W ′′ x ) ( x ) . Shrinking Ω x if necessary, we may assume that Ω x decomposes as a productΩ x = Ω ′ x × (Ω ′′ x × Ω ′′ x ) where Ω ′ x ⊂ G ′ x ( F ) (resp. Ω ′′ x ⊂ H ′′ x ( F )) is open and completely G ′ x -invariant (resp.completely Z SO( W ′′ x ) ( x )-invariant). Note that x is elliptic in both G ′ x and H ′′ x . Hence,by [BP15, Corollary 5.7.2(i)], shrinking Ω x if necessary, we may assume that thelinear maps f ′ x ∈ S scusp (Ω ′ x ) → θ f ′ x ∈ QC c (Ω ′ x ) f ′′ x ∈ S scusp (Ω ′′ x ) → θ f ′′ x ∈ QC c (Ω ′′ x )have dense image. Since ([BP15, Proposition 4.4.1(v)]) QC c (Ω x ) = QC c (Ω ′ x ) b ⊗ p QC c (Ω ′′ x ) b ⊗ p QC c (Ω ′′ x ) ,QC c (Ω x ) Z G ( x ) = QC c (Ω ′ x ) b ⊗ p QC c (Ω ′′ x ) Z SO( W ′′ x ) ( x ) b ⊗ p QC c (Ω ′′ x ) Z SO( W ′′ x ) ( x ) , and J qc and m geom are continuous linear forms on QC c (Ω), we only need to provethe equality of the proposition for quasi-characters θ ∈ QC c (Ω) such that θ x, Ω x = θ f x ∈ QC c (Ω x ) Z G ( x ) for some f x ∈ S scusp (Ω x ) which further decomposes as a tensorproduct f x = f ′ x ⊗ ( f ′′ x, ⊗ f ′′ x, ) where f ′ x ∈ S scusp (Ω ′ x ) and f ′′ x, , f ′′ x, ∈ S scusp (Ω ′′ x ). Upto translation we may assume that the functions f ′′ x, and f ′′ x, are invariant underconjugation by the Weyl elements showing up in Z SO( W ′′ x ) ( x ) /H ′′ x ( F ).Consider a map as in [BP15, Proposition 5.7.1], and set f = e f x ∈ S scusp (Ω). Then( θ f ) x, Ω x = X z ∈ Z G ( x ) /G x ( F ) z θ f x = [ Z G ( x ) : G x ] θ f x . In particular up to translation we can assume that( f ) x, Ω x = 1[ Z G ( x ) : G x ] f x . We also have(7.3.1) J ( f ) = J qc ( θ f ) = J qc ( θ ) . We denote by J G ′ x the continuous linear form on C scusp ( G ′ x ) associated to the GGPtriple ( G ′ x , H ′ x , ξ ′ x ). Also we denote by J A,H ′′ x the continuous bilinear form on C scusp ( H ′′ x )introduced in [BP15, Section 5.5] (where we replace G by H ′′ x ). We show the following(7.3.2) If Ω x is sufficiently small, we have J ( f ) = 2[ Z H + ( x )( F ) : H x ( F )] J G ′ x ( f ′ x ) J A,H ′′ x (( η Hx,G ) / f ′′ x, , f ′′ x, ) . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 49
The intersection Ω x ∩ H x ( F ) ⊂ H x ( F ) is a H -good open neighborhood of x (cf. theremark at the end of [BP15, Section 3.2]). By [BP15, 3.2.5] and [Wal10, Lemme 8.2], J ( f ) = Z H ( F ) \ G ( F ) Z H ( F ) g f ( h ) ξ ( h ) d h d g (7.3.3) = 2[ Z H + ( x )( F ) : H x ( F )] Z H x ( F ) \ G ( F ) Z H x ( F ) η Hx ( h x ) g f ( h x ) ξ x ( h x ) d h x d g = 2[ Z H + ( x )( F ) : H x ( F )] Z H ( F ) \ G ( F ) Z H x ( F ) \ H ( F ) Z H x ( F ) η Hx,G ( h x ) / ( hg f ) x, Ω x ( h x ) ξ x ( h x ) d h x d h d g. Assume one moment that the exterior double integral above is absolutely convergent.Then J ( f ) = 2[ Z H + ( x )( F ) : H x ( F )] Z H x ( F ) \ G ( F ) Z H x ( F ) η Hx,G ( h x ) / ( g f ) x, Ω x ( h x ) ξ x ( h x ) d h x d g = 2[ Z H + ( x )( F ) : H x ( F )] Z Z G ( x )( F ) \ G ( F ) Z H x ( F ) \ Z G ( x )( F ) Z H x ( F ) η Hx,G ( h x ) / ( g f ) x, Ω x ( g − x h x g x ) ξ x ( h x ) d h x d g x d g. Introduce a function on Z G ( x )( F ) \ G ( F ) as in [BP15, Proposition 5.7.1]. Let g ∈ G ( F ). Up to translating g by an element of Z G ( x )( F ), we may assume that ( g f ) x, Ω x = Z G ( x ): G x ] α ( g ) f x . Then the interior integral above decomposes as α ( g )[ Z G ( x ) : G x ] Z H x ( F ) \ Z G ( x )( F ) Z H x ( F ) η Hx,G ( h x ) / f x ( g − x h x g x ) ξ x ( h x ) d h x d g x = α ( g )[ Z G ( x ) : G x ] Z H ′ x ( F ) \ G ′ x ( F ) Z H ′ x ( F ) f ′ x ( g ′− x h ′ x g ′ x ) ξ ′ x ( h ′ x ) d h ′ x d g ′ x × Z Z SO( W ′′ x )( x )( F ) Z Z SO( W ′′ x )( x )( F ) η Hx,G ( h ′′ x ) / f ′′ x, ( g ′′− x h ′′ x g ′′ x ) f ′′ x, ( h ′′ x ) d h ′′ x d g ′′ x = α ( g )[ Z SO( W ′′ x )( x ) ( F ) : H ′′ x ( F )] [ Z G ( x ) : G x ] Z H ′ x ( F ) \ G ′ x ( F ) Z H ′ x ( F ) f ′ x ( g ′− x h ′ x g ′ x ) ξ ′ x ( h ′ x ) d h ′ x d g ′ x × Z H ′′ x ( F ) Z H ′′ x ( F ) η Hx,G ( h ′′ x ) / f ′′ x, ( g ′′− x h ′′ x g ′′ x ) f ′′ x, ( h ′′ x ) d h ′′ x d g ′′ x , since by (7.1.1), [ Z SO( W ′′ x )( x ) ( F ) : H ′′ x ( F )] = [ Z G ( x ) : G x ] , we finally can write the interior integral as α ( g ) Z H ′ x ( F ) \ G ′ x ( F ) Z H ′ x ( F ) f ′ x ( g ′− x h ′ x g ′ x ) ξ ′ x ( h ′ x ) d h ′ x d g ′ x × Z H ′′ x ( F ) Z H ′′ x ( F ) η Hx,G ( h ′′ x ) / f ′′ x, ( g ′′− x h ′′ x g ′′ x ) f ′′ x, ( h ′′ x ) d h ′′ x d g ′′ x . We recognize the two integrals above: the first one is J G ′ x ( f ′ x ) and the second oneis J A,H ′′ x (( δ Hx,G ) / f ′′ x, , f ′′ x, ) (Notice that the center of H ′′ x ( F ) is compact since x iselliptic). By Theorem 4.0.1 (2) and [BP15, Theorem 5.5.1(ii)] and since the function α is compactly supported, this shows that the exterior double integral of the last line(7.3.3) is absolutely convergent. Moreover, since we have Z Z G ( x )( F ) \ G ( F ) α ( g ) dg = 1 , this also proves (7.3.2).We assume from now on that Ω x is sufficiently small so that (7.3.2) holds. By theinduction hypothesis (HYP), we have J G ′ x ( f ′ x ) = m G ′ x geom ( θ f ′ x ) . On the other hand, by [BP15, Theorem 5.5.1(iv)], we have J H ′′ x A (( η Hx,G ) / , f ′′ x, , f ′′ x, ) = Z Γ( H ′′ x ) η HG,x ( y ) / D G ′′ x ( y ) / θ f ′′ x, ( y ) θ f ′′ x, ( y ) d y. Notice that here both f ′′ x, and f ′′ x, are strongly cuspidal, hence the terms correspond-ing to Γ( H ′′ x ) − Γ ani ( H ′′ x ) vanishes. Moreover, we can verify that m geom ,x (( η HG,x ) / θ f x ) = m G ′ x geom ( θ f ′ x ) × Z Γ ani ( H ′′ x ) η Hx,G ( y ) / D G ′′ x ( y ) / θ f ′′ x, ( y ) θ f ′′ x, ( y ) d y. Hence, by (7.3.1), (7.3.2) and Proposition 7.1.2 (2), we have J qc ( θ ) = J ( f ) = m geom ,x (( η HG,x ) / θ f x ) = m geom ( θ ) . This ends the proof of the proposition. (cid:3)
Descent to the Lie algebra and equivalence of theorem 7.3.1 and theorem 7.3.3.
Let ω ⊂ g ( F ) be a G ( F )-excellent open neighborhood of 0 and set Ω = exp( ω ). Recallthat for any quasi-character θ ∈ QC ( g ) and all λ ∈ F × , θ λ denotes the quasi-character given by θ λ ( X ) = θ ( λ − X ) for any X ∈ g rss ( F ). Proposition 7.3.6.
Assume the induction hypothesis (HYP) . (1) For any f ∈ S scusp (Ω) , J ( f ) = J Lie (( j HG ) / f ω ) . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 51 (2)
There exists a unique continuous linear form J Lieqc on SQC ( g ) such that J Lie ( f ) = J Lieqc ( θ f ) for any f ∈ S scusp ( g ) . Moreover, J Lieqc ( θ λ ) = | λ | δ ( G ) / J Lieqc ( θ ) for any θ ∈ SQC ( g ) . Moreover, J Lieqc ( θ λ ) = | λ | δ ( G ) / J Lieqc ( θ ) for any θ ∈ SQC ( g ) and λ ∈ F × . (3) Theorem 7.3.1 and Theorem 7.3.3 are equivalent. (4)
Let θ ∈ SQC ( g ( F )) and assume that / ∈ Supp( θ ) . Then, J Lieqc ( θ ) = m Liegeom ( θ ) . Proof.
The proof follows from the same argument as [BP15, Proposition 11.7.1]. (cid:3)
End of the proof.
Proposition 7.3.7.
Assume the induction hypothesis (HYP) , then J Lieqc ( θ ) = m Liegeom ( θ ) for any θ ∈ SQC ( g ) .Proof. We first establish the following fact.(7.3.4) There exists constants c O , O ∈
Nil reg ( g ), such that J Lieqc ( θ ) − m Liegeom ( θ ) = X O∈ Nil reg ( g ) c O c θ, O (0)for any θ ∈ SQC ( g ).Let θ ∈ SQC ( g ) be such that c θ, O (0) = 0 for any O ∈
Nil reg ( g ). We want to show that J Lieqc ( θ ) = m Liegeom ( θ ). Let λ ∈ F × be such that | λ | 6 = 1. Denote by M λ the operator on SQC ( g ) given by M λ θ = | λ | − δ ( G ) / θ λ . Then, from [BP15, Proposition 4.6.1 (i)], wemay find θ , θ ∈ SQC ( g ) such that θ = ( M λ − θ + θ and θ is compactly supportedaway from 0. By Proposition 7.3.6 (4), we have J Lieqc ( θ ) = m Liegeom ( θ ). On the otherhand, by the homogeneity property of J Lieqc and m Liegeom (cf. Proposition 7.3.6 (2) andProposition 7.1.2 (2)), we also have J Lieqc (( M λ − θ ) = m Liegeom (( M λ − θ ) = 0. Thisproves (7.3.4).To end the proof of the proposition, it remains to show that the coefficients c O for O ∈
Nil reg ( g ), are all zero. We separate the discussion to the following cases. • If G is not quasi-split then there is nothing to prove. • When G has a unique regular nilpotent orbit, we follow the strategy of [BP15,Section 11.9] as follows. Fix a Borel subgroup B ⊂ G and a maximal torus T qd ⊂ B , both of which are defined over F . Let Γ qd ( g ) be the subset of Γ( g )consisting of conjugacy classes that meet t qd ( F ). Recall that in Section 6.6,we have defined a subset Γ(Σ) ⊂ Γ( g ). It consists of conjugacy classes of thesemi-simple parts of elements in the affine subspace Σ( F ) ⊂ g ( F ) defined inSection 6.1. We claim that(7.3.5) Γ qd ( g ) ⊂ Γ(Σ) . Up to G ( F ) conjugation, we may assume that B is a good Borel subgroup(c.f. Section 2.2). Then we have h ⊕ b = g and therefore(7.3.6) h ⊥ ⊕ u = g where u denotes the unipotent radical of b . Recall that Σ = Ξ + h ⊥ . From(7.3.6) we find that the restriction of the natural projection b → t qd to Σ ∩ b induces an affine isomorphism Σ ∩ b = t qd and it implies (7.3.5).Let θ ∈ C ∞ c ( t qd , reg ) be W ( G, T qd )-invariant and such that(7.3.7) Z t qd ( F ) D G ( X ) / θ ( X ) dX = 0 . We may extend θ to a smooth invariant function on g rss ( F ), still denotedby θ , which is zero outside t qd , reg ( F ) G . Then θ is a compactly supportedquasi-character. We consider its Fourier transform θ = b θ . By [BP15, 3.4.5,Lemma 4.2.3 (iii), Proposition 4.1.1 (iii)], θ is supported on Γ qd ( g ). SinceΓ qd ( g ) ∩ Γ( G, H ) = { } , by definition of m Liegeom , we have m Liegeom ( θ ) = c θ (0) . On the other hand, by [BP15, Proposition 4.1.1 (iii), Lemma 4.2.3(iii), Propo-sition 4.5.1.2 (v)], we have(7.3.8) c θ (0) = Z Γ( g ) D G ( X ) / θ ( X ) c b j ( X, · ) (0) dX = Z Γ qd ( g ) D G ( X ) / θ ( X ) dX. By definition of J Lieqc and (7.3.5), the last term is also equal to J Lieqc ( θ ). Hencewe have J Lieqc ( θ ) = m Liegeom ( θ ). Combining (7.3.7) and (7.3.8), we find that c θ (0) = 0 and it follows that we have proved the proposition. • Assume now that G is quasi-split with more than one regular nilpotent orbitparametrized by the set N W (i.e. d = dim W is even. The case for dim V even can be treated similarly). We use the notation from Section B.3 thenrecall the following results from Lemma B.3.1. LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 53 (7.3.9) We haveΓ O ν ( X + F ) − Γ O ν ( X − F ) = sgn F /F ( ην ) , ν ∈ N V . The elements X + F and X − F are stably conjugate but not F -conjugate,and from Proposition 6.3.1 only one of them lies in Γ(Σ)( F ).In the following we will determine exactly which one of X ± F lies in Γ(Σ)( F ).We will follow the strategy of [Wal10, 11.5].From Proposition 6.2.1, we know that when dim W is even, an element X = ( X W , X V ) ∈ g ( F ) lies in Σ ′ ( F ) if and only if there exists elements( z ± i ) m/ i =1 of F , where m = dim W , such that(7.3.10) (cid:26) z i z − i = P XV ( s i )2 ν s i P − XW ,i ( s i ) , for any i = 1 , ..., m , P ± m i = ± z i w i ∈ W where P − X W ,i ( T ) = P − XW ( T ) T − s i . Moreover we notice that P X V ( s i ) = 0 for any i .Now following the notation from Section B.3, there exists – a decomposition of W as a direct sum F ⊕ F ⊕ e Z, where F i is a quadratic extension of F for i = 1 , – element c ∈ F × such that q W is the direct sum of the quadratic forms c N F /F on F , − c N F /F and e Z is a hyperbolic space; – elements a j ∈ F × j such that τ F j ( a j ) = − a j for j = 1 , e S belonging to the Lie algebra of a maximal split subtorus of the specialorthogonal group of e Z , such that X F acts by multiplication by a j on F j and by e S on e Z .Following Lemma B.1.1, for any j = 1 ,
2, there exists a basis { e j , e − j } of F j ⊗ F F such that any element x ∈ F j can be written as xe j ⊗ τ F j ( x ) e − j .Moreover we have τ F j ( e j ) = e − j . We have the equality q ( e j , e − j ) = c j with c = c and c = − c . We can set w j = e j and w − j = c − j e − j for j = 1 ,
2. Wealso fix a hyperbolic basis ( w j ) j = ± ,..., ± m for e Z . It turns out that we have s j = a j for j = 1 ,
2. Now the property (7.3.10) can be decomposed into m separate relations (7 . . j for the pairs ( z j , z − j ) by taking z − j = 1 and z j isequal to the RHS of the first relation appearing in (7.3.10). For j ≤
2, thesecond relation is equivalent to z j ∈ F × j and τ F j ( z j ) = c − j z − j . Therefore thecondition (7 . . j is equivalent tosgn F j /F ( P X V ( a j ) c j ν a j P − X W ( a j ) ) = 1 . Following the same explicit computation as done in the proof of Lemma B.3.1,for X ζF with ζ = ± , the above formula is equal to sgn F /F ( − ην ) = ζ if j = 1and sgn F /F ( − ην ) = ζ if j = 2. But since sgn E/F ( − ην ) = 1, therefore thetwo equalities for j = 1 , X ζF lies in Σ ′ ( F ) if and only if sgn F ( − ην ) = ζ . Then, parallel to the previous case, for any T ∈ T ( G ) with Lie algebra t ,we can find θ ,T ∈ C ∞ c ( t reg ) that is W ( G, T )-invariant, such that Z t ( F ) D G ( X ) / θ ,T ( X ) dX = 1 , and we can extend θ ,T to a smooth invariant function on g rss ( F ) still de-noted by θ that is zero outside t reg ( F ) G . It is a compactly supported quasi-character. We let the associated Fourier transform be θ T = d θ ,T . In particular,the torus associated to X + F (resp. X − F ) is denoted by T +1 (resp. T − ). More-over, from (7.3.8) up to scaling we can further assume that c θ T +1 , O ( X + F ) =Γ O ( X + F ) (resp. c θ T − , O ( X − F ) = Γ O ( X − F ) ) for any O ∈ N V . Then we set e θ = |N V | − ( θ + X F ∈F V (sgn F /F ( νη ))( θ T +1 − θ T − )) , where the quasi-character θ is the one constructed from the case when G ( F )has a unique regular nilpotent orbit. Here F V is the set of collections of alldegree two field extensions of F if SO( W ) is split. When SO( W ) is quasi-splitbut not split, following the notation in Section B.3, we consider the pairs ofquadratic extensions F , F of F such that sgn F /F sgn F /F = sgn E/F and nonof F nor F is equal to E . We pick up either element in the pair and let theassociated set be F V . We note that |N V | = |F V | + 1. Then by definition weimmediately find that m Liegeom ( e θ ) = δ ν, − ν . On the other hand, using the same argument as previous case, we havethat J Lie ( θ ) = 1, and J Lie ( θ T +1 ) = 1 (resp. J Lie ( θ − T )=1) if and only ifsgn F /F ( − ην ) = 1 (resp. sgn F /F ( − ην ) = −
1) and 0 otherwise. There-fore we have J Lie ( θ T +1 ) − J Lie ( θ T − ) = sgn F /F ( − ην ) LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 55 and hence J Lie ( e θ ) = |N V | − ( J Lie ( θ ) + X F ∈F V (sgn F /F ( νη )( J Lie ( θ T +1 ) − J Lie ( θ T − ))= |N V | − (1 + X F ∈F V sgn F /F ( νη )sgn F /F ( − ην )) , which again is equal to δ ν, − ν . It follows that c O = 0 for any O ∈
Nil reg ( g ( F ))in (7.3.4) and we have completed the proof.The situation for dim V being even can be treated similarly following[Wal10, 11.6], and we omit the details. (cid:3) An application to the Gan-Gross-Prasad conjecture
We are going to prove that there is exactly one distinguished representation π (i.e.one such that m ( π ) = 1) in every (extended) tempered L -packet of G ( F ). In the p -adic case, this result was already proved by Waldspurger ([Wal10], [Wal12b]).8.1. Strongly stable conjugacy classes, transfer between pure-inner formsand the Kottwitz sign.
Let G be any connected connected group defined over F .In the discussion below we assume that F is not algebraically closed, i.e. F is either p -adic or real.Recall that a pure inner form for G is a triple ( G ′ , ψ, c ) where • G ′ is a connected reductive group defined over F ; • ψ : G F ≃ G ′ F is an isomorphism defined over F ; • c : σ ∈ Γ F → c σ ∈ G ( F ) is a 1-cocycle such that ψ − σ ψ = Ad( c σ ) for any σ ∈ Γ F .The set of pure inner forms of G can be parametrized by H ( F, G ). Moreover, insidean equivalence class of pure inner forms ( G ′ , ψ, c ), the group G ′ is well-defined up to G ′ ( F )-conjugacy. We will always assume for any α ∈ H ( F, G ) a pure inner form inthe class of α that we will denote by ( G α , ψ α , c α ) is fixed.Let ( G ′ , ψ, c ) be a pure inner form of G . Following [BP15, 12.1], say that twosemi-simple elements x ∈ G ss ( F ) and y ∈ G ′ ss ( F ) are strongly stably conjugate andwrite x ∼ stab y if there exists g ∈ G ( F ) such that y = ψ ( gxg − ) and the isomorphism ψ ◦ Ad( g ) : G x ≃ G y is defined over F . The last condition has an interpretation in terms ofcohomological classes: it means that the 1-cocycle σ ∈ Γ F → g − c σ σ ( g ) takes its values in Z ( G x ). For x ∈ G ss ( F ) the set of semi-simple conjugacy classes in G ′ ( F )that are strongly stably conjugate to x is naturally in bijection withIm( H ( F, Z ( G x )) → H ( F, Z G ( x ))) ∩ p − x ( α )where α ∈ H ( F, G ) parametrizes the equivalence class of pure inner form ( G ′ , ψ, c )and p x denotes the natural map H ( F, Z G ( x )) → H ( F, G ). The following fact willbe needed (c.f. [BP15, 12.1.1])(8.1.1) Let y ′ ∈ G ′ ss ( F ) and assume that G and G ′ y are both quasi-split. Then, theset { x ∈ G ss ( F ) | x ∼ stab y } is non-empty.Continue to fix a pure inner form ( G ′ , ψ, c ) of G . Say a quasi-character θ on G ( F ) is stable if for any regular elements x, y ∈ G rss ( F ) that are stably conjugate, θ ( x ) = θ ( y ). Let θ and θ ′ be stable quasi-characters on G ( F ) and G ′ ( F ) respectivelyand assume moreover that G is quasi-split. Then θ ′ is called a transfer of θ if for anyregular points x ∈ G rss ( F ) and y ∈ G ′ rss ( F ) that are stably conjugate, θ ′ ( y ) = θ ( x ).Note that if θ ′ is a transfer of θ then θ ′ is entirely determined by θ . Theorem 8.1.1.
Let ( V, q V ) be a quasi-split quadratic space of even dimension, θ astable quasi-character on G ( F ) = SO( V )( F ) . Then D G ( x ) / c θ ( x ) = | W ( G x , T x ) | − lim x ′ ∈ T x ( F ) → x D G ( x ′ ) / θ ( x ′ ) , In particular, for a stable quasi-character θ on G ( F ) , the definition c θ introduced in7.1 coincides with the definition introduced in [BP15, Section 4.5] .Proof. When G ( F ) contains only one regular nilpotent orbit, there is nothing toprove. Therefore we focus on the case when G ( F ) has more than one regular nilpotentorbits.When F is p -adic, it follows from the same argument as [Wal10, Section 13.4]and [Wal10, Section 13.6] (one may replace the character distribution θ π by anyquasi-character θ to arrive at the conclusion).When F = R . By the description of regular nilpotent orbits in Section 2.1, when d ≥ | Nil reg ( g ) | = | F × /F × | = 2. From [BP15, Proposition 4.4.1 (vi)] and [BP15,1.8.1], we have the following identity for any x ∈ G ss ( F ) and Y ∈ g rss ( F ),lim t ∈ F × ,t → D G ( xe tY ) / θ ( xe tY ) = lim t ∈ F × ,t → D G ( xe tY ) / X O∈ Nil reg ( g x ) c θ, O ( x ) b j ( O , tY )= lim t ∈ F × ,t → D G ( xe tY ) / | t | − δ Gx / X O∈ Nil reg ( g x ) c θ, O ( x ) b j ( O , Y )where δ G x = dim G x − dim T x and dim T x is the dimension of maximal torus in G x . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 57
In particular, since θ is a stable quasi-character, X O∈ Nil reg ( g x ) c θ, O ( x ) b j ( O , Y )is a stable distribution supported on the regular nilpotent elements inside g x ( F )(Recall that a distribution T on g ( k ) is called stable if for f ∈ C ∞ c ( g ), T ( f ) = 0whenever the stable orbital integral of f at any regular semi-simple elements is equalto 0).Note that whenever x is not regular semi-simple (otherwise there is nothing toprove), the set of regular nilpotent orbits in g x ( F ) can also be parametrized by N V . Write N V = {± } , and choose regular semi-simple elements { X + F , X − F } as inappendix B.3, which is determined by the fact that Γ O + ( X + F ) = 1 (resp. Γ O − ( X − F ) =1) and zero otherwise.Since b j ( X + F , · ) + b j ( X − F , · ) is a stable distribution on g x ( F ), applying the limitformula [BP15, p.98, Section 4.5] to b j ( X + F , · ) + b j ( X − F , · ), we get that b j ( O + , · ) + b j ( O − , · ) is a stable distribution on g x ( F ). But for a single distribution b j ( O + , · )(resp. b j ( O − , · )), it is not a stable distribution, it follows that c θ, O + = c θ, O − .It follows that we have established the fact. (cid:3) We need the following fact ([BP15, 12.1.2]).(8.1.2) Let θ and θ ′ be stable quasi-characters on G ( F ) and G ′ ( F ) respectively andassume that θ ′ is a transfer of θ . Then, for any x ∈ G ss ( F ) and y ∈ G ′ ss ( F )that are strongly stably conjugate, c θ ′ ( y ) = c θ ( x )Now assume that G is quasi-split. Following Kottwitz ([Kot83]), we may associateto any class of pure inner forms α ∈ H ( F, G ) a sign e ( G α ). When F is either p -adicor real, let Br ( F ) = H ( F, {± } ) = {± } be the 2-torsion subgroup of the Brauergroup of F . The sign e ( G α ) will more naturally be an element of Br ( F ). To defineit, we need to introduce a canonical algebraic central extension(8.1.3) 1 → {± } → e G → G → F is classified up to conjugation by its(canonical) based root datum Ψ ( G ) = ( X G , ∆ G , X ∨ G , ∆ ∨ G ) together with the naturalaction of Γ F on Ψ ( G ). For any Borel pair ( B, T ) of G that is defined over F , we havea canonical Γ F -equivariant isomorphism Ψ ( G ) ≃ ( X ∗ ( T ) , ∆( T, B ) , X ∗ ( T ) , ∆( T, B ) ∨ )where ∆( T, B ) ⊂ X ∗ ( T ) denotes the set of simple roots of T in B and ∆( T, B ) ∨ ⊂ X ∗ ( T ) denotes the corresponding sets of simple coroots. Fix such a Borel pair and set ρ = 12 X β ∈ R ( G,T ) β ∈ X ∗ ( T ) ⊗ Q for the half sum of the roots of T in B . The image of ρ in X G ⊗ Q does not dependon the choice of ( B, T ) chosen and we still denote by ρ this image. Consider now thefollowing based root datum(8.1.4) ( e X G , ∆ G , e X ∨ G , ∆ ∨ G )where e X G = X G + Z ρ ⊂ X G ⊗ Q and e X G = { λ ∨ ∈ X ∨ G | h λ ∨ , ρ i ∈ Z } . Note thatwe have ∆ ∨ G ⊂ e X ∨ G since h α ∨ , ρ i = 1 for any α ∨ ∈ ∆ ∨ G . The based root datum (8.1.4)with its natural Γ F -action, it the base root datum of a unique quasi-split group e G over F well-defined up to conjugacy. Moreover, we have a natural central isogeny e G → G , well-defined up to G ( F )-conjugacy, whose kernel is either trivial or {± } .If the kernel is {± } , we set e G = e G otherwise we simply set e G = G × {± } . In anycase, we obtain a short exact sequence like (8.1.3) well-defined up to G ( F )-conjugacy.The last term of the long exact sequence associated to (8.1.3) yields a canonical map(8.1.5) H ( F, G ) → H ( F, {± } ) = Br ( F ) ≃ {± } We now define the sign e ( G α ) for α ∈ H ( F, G ), simply to be the image of α by thismap. We will need the following fact (c.f. [BP15, 12.1.6])(8.1.6) Let T be a (not necessarily maximal) subtorus of G . Then, the composition of(8.1.5) with the natural map H ( F, T ) → H ( F, G ) is a group homomorphism H ( F, T ) → Br ( F ). Moreover, if T is anisotropic this morphism is onto ifand only if the inverse image e T of T in e G is a torus (i.e. is connected). Pure inner forms of a GGP triple.
Let V be an quadratic space. There is thefollowing explicit description of the pure inner forms of SO( V ). The cohomology set H ( F, SO( V )) naturally classifies the isomorphism classes of quadratic spaces of thesame dimension and same discriminant as V . Let α ∈ H ( F, SO( V )) and choose anquadratic space V α in the isomorphism class corresponding to α . Set V F = V ⊗ F F and V α,F = V α ⊗ F F . Fix an isomorphism φ α : V F ≃ V α,F of F -quadratic spaces.Then, the triple (SO( V α ) , ψ α , c α ), where ψ α is the isomorphism SO( V ) F ≃ SO( V α ) F given by ψ α ( g ) = φ α ◦ g ◦ φ − α and c α is the 1-cocycle given by σ ∈ Γ F → φ − σα φ α , isa pure inner form of SO( V ) in the class of α . Moreover, the 2-cover f SO( V ) has thefollowing description • f SO( V ) = Spin( V ), when dim V is odd; • f SO( V ) = SO( V ) × {± } , when dim V is even. LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 59
We now return to the GGP triple (
G, H, ξ ) that we have fixed. Recall that theGGP triple comes from an admissible pair (
V, W ) of quadratic spaces and that weare assuming in this chapter that G and H are quasi-split. Let α ∈ H ( F, H ).We are going to associate to α a new GGP triple ( G α , H α , ξ α ) well-defined up toconjugacy. Since H ( F, H ) = H ( F, SO( W )), the cohomology class α correspondsan isomorphism class quadratic space of the same dimension and same discriminantas W . Let W α be an quadratic space in this isomorphism class and set V α = W α ⊕ ⊥ Z . Then the pair ( V α , W α ) is an admissible pair and hence there is a GGP triple( G α , H α , ξ α ) associated to it. This GGP triple is well-defined up to conjugacy. Wecall such a GGP triple a pure inner form of ( G, H, ξ ). By definition, these pure innerforms are parametrized by H ( F, H ). Note that for any α ∈ H ( F, H ), G α is a pureinner form of G in the class corresponding to the image of α in H ( F, G ) and thatthe natural map H ( F, H ) → H ( F, G ) is injective.8.2.
The local Langlands correspondence.
In this section, we recall the localLanglands correspondence in a form that will be used in the following. Let G bea quasi-split connected reductive group over F and denote by L G = b G × W F itsLanglands dual, where W F denotes the Weil group of F . Recall that a Langlandsparameter for G is a homomorphism from the group L F = (cid:26) W F × SL ( C ) if F is p -adic W F if F is archimedean to L G satisfying the usual conditions of continuity, semi-simplicity, algebraicity andcompatibility with the projection L G → W F . A Langlands parameter is said to be tempered if ϕ ( W F ) is bounded. By the hypothetical local Langlands correspondence,a tempered Langlands parameter ϕ for G should give rise to a finite set Π G ( ϕ ), calleda L -packet , of (isomorphism classes of) tempered representations of G ( F ). Actually,such a parameter ϕ should also give rise to tempered L -packets Π G α ( ϕ ) ⊂ Temp( G α )for any α ∈ H ( F, G ). Among them, we expect the following properties to hold forevery tempered Langlands parameter ϕ of G :(STAB) for any α ∈ H ( F, G ), the character θ α,ϕ = X π ∈ Π Gα ( ϕ ) θ π is stable.See Section 8.1 for the definition of stable; also notice that our different notion of”strongly stable conjugate” does not affect this property since it only involves thevalues of θ α,ϕ at regular semi-simple elements (for which the two notions of stableconjugacy coincides). For α = 1 ∈ H ( F, G ), in which case G α = G , we shall simplyset θ ϕ = θ ,ϕ . (TRANS) for any α ∈ H ( F, G ), the stable character θ α,ϕ is the transfer of e ( G α ) θ ϕ where e ( G α ) is the Kottwitz sign whose definition has been recalled in Section 8.1.(WHITT) For every O ∈
Nil reg ( g ), there exists exactly one representation in the L -packet Π G ( ϕ ) admitting a Whittaker model of type O .Notice that these conditions are far from characterizing the composition of the L -packets uniquely. However, by the linear independence of characters, conditions(STAB) and (TRANS) uniquely characterize the L -packets Π G α ( ϕ ), α ∈ H ( F, G ),in terms of Π G ( ϕ ).When F is archimedean, the local Langlands correspondence has been constructedby Langlands himself ([Lan89]) building on previous results of Harish-Chandra. Thiscorrespondence indeed satisfies the three conditions stated above. That (STAB)and (TRANS) hold is a consequence of early work of Shelstad ([She79, Lemma 6.2,Theorem 6.3]. The property (WHITT) for its part, follows from the result of Kostant([Kos78, Theorem 6.7.2]) and Vogan ([Vog78, Theorem 6.2]). When F is p -adic, thelocal Langlands correspondence is known in a variety of cases. In particular, forspecial orthogonal groups, the existence of the Langlands correspondence is nowfully established thanks to Arthur ([Art13]) for quasi-split case, with supplement by[AG17] and a conjectural description for the inner forms, which in principle shouldfollow from the last chapter of [Art13]. That the tempered L -packets constructedin these references verify the conditions (STAB) and (TRANS) follows from [Art13].Moreover, the L -packets on the quasi-split form G satisfy condition (WHITT) by[Art13].8.3. The theorem.
Recall that we have fixed a GGP triple (
G, H, ξ ) with the re-quirement that G and H being quasi-split. Also, we have fixed in Section 8.1 thepure inner forms ( G α , H α , ξ α ) of ( G, H, ξ ). These are also GGP triples, they areparametrized by H ( F, H ) and G α is a pure inner form of G corresponding to theimage of α in H ( F, G ) via the map H ( F, H ) → H ( F, G ). Stable conjugacy classes inside Γ( G, H ) . Recall that in Section 7.1, we have defineda set Γ(
G, H ) of semi-simple conjugacy classes in G ( F ). It consists in the G ( F )-conjugacy classes of elements x ∈ SO( W ) ss ( F ) such that T x := SO( W ′′ x ) x is an anisotropic torus (where we recall that W ′′ x denotes the image of x − W ). Two elements x, x ′ ∈ SO( W ) ss ( F ) are G ( F )-conjugate if and only if they areSO( W )( F )-conjugate and moreover if it is so, any element g ∈ SO( W )( F ) conjugat-ing x to x ′ induces an isomorphismSO( W ′′ x ) x ≃ SO( W ′′ x ′ ) x ′ LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 61
Moreover, this isomorphism depends on the choice of g only up to inner automor-phism. From this it follows that any conjugacy class x ∈ Γ( G, H ) determines theanisotropic torus T x up to a unique isomorphism so that we can speak of ”the torus” T x associated to x .These considerations apply verbatim to the pure inner forms ( G α , H α , ξ α ) , α ∈ H ( F, H ), of the GGP triple (
G, H, ξ ) that were introduced in Section 8.1. In partic-ular, for any α ∈ H ( F, H ), we have a set Γ( G α , H α ) of semisimple conjugacy classesin G α ( F ) and to any y ∈ Γ( G α , H α ) is associated an anisotropic torus T y . Proposition 8.3.1. (1)
Let α ∈ H ( F, H ) and y ∈ Γ( G α , H α ) be such that G α,y is quasi-split. Then, the set { x ∈ Γ( G, H ) | x ∼ stab y } is non-empty. (2) Let α ∈ H ( F, H ) , x ∈ Γ( G, H ) and y ∈ Γ( G α , H α ) be such that x ∼ stab y .Choose g ∈ G α ( F ) such that gψ α g − = y and Ad( g ) ◦ ψ α : G x ≃ G y is definedover F . Then, Ad( g ) ◦ ψ α restricts to an isomorphism T x ≃ T y that is independent of the choice of g . (3) Let x ∈ Γ( G, H ) . Then, for any α ∈ H ( F, H ) there exists a natural bijectionbetween the set { y ∈ Γ( G α , H α ) | x ∼ stab y } and the set q − x ( α ) where q x denotes the natural map H ( F, T x ) → H ( F, G ) . (4) Let x ∈ Γ( G, H ) , x = 1 . Then the composition of the map α ∈ H ( F, G ) → e ( G α ) ∈ Br ( F ) with the natural map H ( F, T x ) → H ( F, G ) gives a surjec-tive morphism of groups H ( F, T x ) → Br ( F ) .Proof. (1) and (2): The proof of [BP15, Proposition 12.5.1] works verbatim.(3) : Besides the fact that the set H ( F, SO( V ′ x )) classifies the (isomorphism classesof) quadratic spaces of the same dimension and discriminant as V ′ x , the proof of[BP15, Proposition 12.5.1] works verbatim.(4) Let us denote by e G the 2-cover of G at the end of Section 8.1 and let e T x be theinverse image of T x in this 2-cover. Then, by (8.1.6), it suffices to check that e T x isconnected. By the precise description of f SO( V ) and f SO( W ) given at the beginning foSection 8.1 and since exactly one of the quadratic spaces V and W is odd dimensional,we have e T x is equal to the inverse image of T x in the associated product of the spingroup with another special orthogonal group, which is connected. (cid:3) Now we are ready to state our main theorem. The proof in [BP15, 12.6] worksverbatim. We refer the details to [Luo21].
Theorem 8.3.2.
Let ϕ be a tempered Langlands parameter for G . Then, there existsa unique representation π in the disjoint union of L -packets G α ∈ H ( F,H ) Π G α ( ϕ ) such that m ( π ) = 1 . Appendix A. A formula for the regular nilpotent germs
In this section, we are going to give a formula for the germs of Lie algebra orbitalintegrals associated to regular nilpotent orbits in a quasi-split connected reductivealgebraic group G over any local field F of characteristic zero in terms of endoscopicinvariants. The formula was first proved by Shelstad over p -adic fields ([She89]). Wealso discuss the relation between the formula and Kostant’s sections.The organization of the section is as follows. We first review the definition of Lie al-gebra endoscopic transfer factors, following the work of Langlands-Shelstad ([LS87]),Waldspurger ([Wal97]) and Kottwitz ([Kot99]), and recall a theorem in [LS87, The-orem 5.5.A.] which relates the transfer factors for regular semi-simple elements andregular nilpotent orbits. Then we establish the formula. The definition of germ ex-pansion for Lie algebra orbital integrals is known over p -adic fields ([Sha72]). Overarchimedean local fields, there are also asymptotic expansions ([Bou94], [Art16])relating the orbital integrals for regular semi-simple elements and distributions sup-ported on the nilpotent cone. Since we only care about the germs associated to theregular nilpotent orbits, we will simply use the results from [BP15, p.98, Section 4.5].Finally we establish the relation between the formula and Kostant’s sections basedon [Kot99, Theorem 5.1]. Notation and conventions . Let F be a local field of characteristic zero with fixedvaluation | · | and Galois group Γ F = Gal( F /F ), where F is a fixed algebraic closureof F . Fix an additive character ψ of F and a Haar measure on F that is self-dualw.r.t. ψ .Fix a quasi-split connected reductive algebraic group G defined over F . Over F ,fix a maximal torus T of G and a Borel subgroup B of G containing T with Levidecomposition B = T N , where N is the unipotent radical of B . Let B ∞ be theunique Borel subgroup of G containing T that is opposite to B . Gothic letters areused to denote the Lie algebras of the associated algebraic groups.Let W = W ( G, T ) be the associated Weyl group. Let R G = R ( T, G ) be the set ofroots of T in G . For any α ∈ R G , let g α be the root space of t in g corresponding to LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 63 α . In particular n = L α> g α and n ∞ = L α< g α . For any α ∈ R G , let ˇ α be thecoroot dual to α . Let ∆ = ∆( T, B ) be the set of positive simple roots determinedby B .Let g rss ( F ) be the subset of regular semi-simple elements in g ( F ). The definitionof Weyl discriminant D G ( X ) is given by D G ( X ) = | det ad( X ) g ( F ) / g X ( F ) | for any X ∈ g rss ( F ), where g X ( F ) is the centralizer of X in g ( F ).Let S ( g ) = S ( g ( F )) be the space of Schwartz-Bruhat functions on g ( F ), where g ( F ) is viewed as a vector space over F .Fix a G ( F )-invariant non-degenerate bilinear form h· , ·i on g ( F ) . Endow g ( F ) withthe self-dual measure w.r.t. h· , ·i , which is the unique Haar measure d X on g ( F )such that the Fourier transform F ( f )( Y ) = b f ( Y ) = Z g ( F ) f ( X ) ψ ( h X, Y i ) d X, f ∈ S ( g )satisfies F ( F ( f ))( X ) = f ( − X ) . Equip G ( F ) with the unique Haar measure d g such that the exponential map hasJacobian equal to 1 at identity.Let Nil reg ( g ) = Nil reg ( g ( F )) be the set of regular nilpotent orbits in g ( F ) under theadjoint action of G ( F ). For any O ∈
Nil reg ( g ) and X ∈ O , the bilinear map ( Y, Z ) → B ( Y, [ X, Z ]) yields a non-degenerate symplectic form on g ( F ) / g X ( F ), which can beviewed as the tangent space of O at X . This gives O a structure of symplectic F -analytic manifold. By the Haar measure on F , O can be equipped with a naturalmeasure that is G ( F )-invariant.For X ∈ g rss ( F ), there is the normalized orbital integral at X defined by J G ( X, f ) = D G ( X ) / Z G X \ G f ( g − Xg ) d g, f ∈ S ( g ) . Similarly, for
O ∈
Nil reg ( g ), J O ( f ) = Z O f ( X ) d X, f ∈ S ( g ) . The distribution J G ( X, · ) (resp. J O ( · )) is tempered. Denote the Fourier transformby b j ( X, · ) (resp. b j ( O , · )). It is locally integrable on g ( F ) × g ( F ) (resp. g ( F )) andsmooth on g rss ( F ) × g rss ( F ) (resp. g rss ( F )) (c.f. [BP15, 1.8]). A.1.
Transfer factors.
In this subsection, we review the definition of Lie algebraendoscopic transfer factors.
Definition A.1.1.
The triple spl := ( B , T, { X α } ) is called an F -splitting for G . More precisely, by the Jacobson-Morozov theorem ([Jac51, Theorem 3]), for any α ∈ ∆, it can be associated with a standard sl (2) triple { X α , H α , X − α } . Fix such atriple for any α ∈ ∆, and let X + = P α ∈ ∆ X α , X − = P α ∈ ∆ X − α . Both X + and X − are regular nilpotent elements in g ( F ).Let ( H, H , s, ξ ) be an endoscopic data for G ([LS87, p.9]). For a quasi-split con-nected reductive algebraic group G defined over F , Langlands and Shelstad definedthe notion of transfer factor ∆ ( γ H , γ G ) for any γ H ∈ H ( F ) that is G ( F )-regularsemi-simple ([LS87, 1.3]) and γ G ∈ G ( F ) that is regular-simple. The transfer factor∆ ( γ H , γ G ) depends on the choice of the F -splitting spl . The corresponding Lie al-gebra variant of ∆ ( γ H , γ G ) introduced below, which is denoted by ∆ ′ ( X H , X G ), isanalogous to the one defined by Langlands and Shelstad, with the factor ∆ IV (Weyldiscriminant) removed.To introduce ∆ ′ ( X H , X G ), a set of a -data and χ -data need to be fixed. We recallthe notions from [Wal97, 2.3]. a -data. Let T H be a maximal torus of H defined over F . From the definition ofendoscopic data, there exists a canonical G -conjugacy class of embeddings T H ֒ → G .Moreover we can fix an embedding T H ֒ → G that is defined over F . Let T G bethe image of T H in G , where T G is a maximal torus of G defined over F . Let R G = R ( T G , G ) be the set of roots of T G in G . Similarly the set R H can be defined.Identify T H with T G so that R H becomes a subset of R G . Definition A.1.2. An a -data for T G is a collection of elements { a α } α ∈ R G with a α ∈ F × such that • a σα = σ ( a α ) for any α ∈ R G and σ ∈ Γ F ; • a − α = − a α for any α ∈ R G . Fix an a -data for T G . χ -data. For any α ∈ R G , let F α (resp. F ± α ) be the field of definition of α (resp. theset {± α } ). Then F ⊂ F ± α ⊂ F α ⊂ F , and [ F α : F ± α ] = 1 or 2. Following [LS87,2.5], α (and its Γ F -orbit in R G ) is called symmetric if [ F α : F ± α ] = 2 and let χ α be the quadratic character on F ×± α associated to the quadratic extension F α /F ± α vialocal class field theory ([Ser67, Theorem 2]). Simpler than the definition in [LS87,2.5], it is not needed to extend χ α to a character of F × α .In the following the Lie algebra transfer factors is introduced. Fix X H ∈ t H ( F )and assume that its image X G in t G ( F ) is regular semi-simple. LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 65 ∆ I ( X H , X G ) . First consider the case when G is semi-simple and simply connected.With the fixed Borel pair ( B , T ), we can fix a canonical section n : W = W ( G, T ) → Norm G ( F )( T ) ([LS87, 2.1]). Fix an element x ∈ G ( F ) such that xT x − = T G . Thenthe conjugation action of x yields an ordering on R G = R ( T G , G ). For any σ ∈ Γ F ,the element x − σ ( x ) normalizes T , hence it provides an element w σ ∈ W . Let n σ = n ( w σ ). For any α ∈ R G , let ˇ α be the associated coroot. Set a σ = Y α ∈ R G ,α> ,σ − ( α ) < ˇ α ⊗ a α ∈ X ∗ ( T G ) ⊗ Z F × ≃ T G ( F ) , σ ∈ Γ F , and define λ ( T G )( σ ) = a σ xn σ σ ( x − ) ∈ T G ( F ) , σ ∈ Γ F . It can be shown that σ → λ ( T G )( σ ) is a 1-cocycle of Γ F valued in T G ( F ) ([LS87,2.3]). We use the same notation λ ( T G ) to denote the associated cohomology class in H ( F, T G ( F )).In general, let G sc be the simply connected cover of the derived group of G , and T sc G be the inverse image of T G under the canonical morphism G sc → G . Denote λ ( T G )the image of λ ( T sc G ) under the map induced by the canonical morphism T sc G → T G . Remark A.1.3.
To make the notation in consistent with [She89] , we will also denotethe invariant λ ( T G ) by inv( T G ) . Let b T G be the complex torus dual to T G . The element s appearing in the endoscopicdata is a Γ F -fixed element in the center of the Langlands dual group b H of H , andthus can be viewed as a Γ F -fixed element s T G in b T H = b T G .Recall the Tate-Nakayama pairing ([Kot86]) h· , ·i : H ( F, T G ( F )) × b T Γ F G → C × . Definition A.1.4.
Define the transfer factor ∆ I ( X H , X G ) to be h inv( T G ) , s T G i . ∆ II ( X H , X G ) . Definition A.1.5.
Define ∆ II ( X H , X G ) := Y α ( α ( X G ) a α ) , where the product is taken over a set of representatives for the symmetric orbits of Γ F in the set R H \ R G . Following [LS87, Lemma 2.2.B, 2.2.C, 3.2.D], [She89], the factor ∆ II also admitsthe following interpretation. The morphisminv( X G ) : σ ∈ Γ F → Y α ∈ R G ,α> ,σ − α< ˇ α ◦ ( exp( α ( X G )2 ) − exp( − α ( X G )2 ) a α )is a 1-cocycle of Γ F in T G ( F ). From [LS87, Lemma 3.2.D], there is the equality∆ II ( X H , X G ) = h inv( X G ) , s T G i whenever X G is sufficiently close to 0.Now the Lie algebra transfer factor is defined as the product of ∆ I and ∆ II . Definition A.1.6.
Define ∆ ′ ( X H , X G ) = ∆ I ( X H , X G )∆ II ( X H , X G ) . Remark A.1.7.
Following [LS87, Lemma 3.2.C] , we can show that ∆ ′ ( X H , X G ) isindependent of the choice of a -data. In particular ∆ ′ ( X H , X G ) depends only on thechoice of the F -splitting.Set ∆ ′ ( γ H , γ G ) := ∆ ( γ H , γ G ) · ∆ IV ( γ H , γ G ) − where the terms are group version ofthe transfer factors defined in [LS87] . Then by definition, whenever X G is sufficientlyclose to , ∆ ′ ( X H , X G ) = ∆ ′ (exp( X H ) , exp( X G )) . Moreover, from the above definitions, the following equality holds for any a ∈ F × , ∆ ′ ( a X H , a X G ) = ∆ ′ ( X H , X G ) . Remark A.1.8.
Suppose that X ′ G ∈ g ( F ) is stably conjugate to X G , i.e. thereexists h ∈ G ( F ) such that Ad( h )( X ′ G ) = X G . Then σ → hσ ( h ) − is a -cocycle of Γ F in T G ( F ) whose cohomology class will be denoted by inv( X G , X ′ G ) . From [LS87,Lemma 3.2.B, 3.4.A] , the following equality holds ∆ ′ ( X H , X ′ G ) · h inv( X G , X ′ G ) , s T G i − = ∆ ′ ( X H , X G ) . ∆( O ) . Finally we recall the transfer factor associated to the regular nilpotent (unipo-tent) elements in G ( F ). Following [LS87, Section 5.1], for any regular nilpotentconjugacy class O ∈
Nil reg ( g ), we can attach to it an F -splitting spl ( O ). More-over, the correspondence O → spl ( O ) induces a bijection between Nil reg ( g ) andthe G -conjugacy classes of F -splittings of G . Following Langlands and Shelstad, if spl g = spl ∞ , where spl ∞ is the F -splitting opposite to the fixed one in the beginning,and g ∈ G sc ( F ), then inv( O ) : σ → gσ ( g ) − is a 1-cocycle of Γ F in Z sc ( F ), where Z sc is the center of G sc . After composing the canonical morphisms Z sc → T sc G → T G ,we obtain a cohomology class inv T G ( O ) in H ( F, T G ). Set∆( O ) = h inv T G ( O ) , s T G i . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 67
There is the following theorem connecting the transfer factors introduced above,which can be obtained from [LS87, Theorem 5.5.A] through descending to Lie algebradirectly.
Theorem A.1.9.
The following identity holds lim X H → X X G ∆ ′ ( X H , X G ) J G ( X G , f ) = X O∈ Nil reg ( g ( F )) ∆( O ) J O ( f ) , f ∈ S ( g ) . The sum X G runs over elements in g rss ( F ) such that ∆ ′ ( X H , X G ) = 0 . In particularit is a finite sum. A.2.
The formula.
We are going to establish the following theorem. Over p -adicfields, it was first proved by Shelstad ([She89]). Theorem A.2.1.
For any X ∈ g rss ( F ) and O ∈
Nil reg ( g ) , let T G = G X , then Γ O ( X ) = (cid:26) , if inv( X )inv( T G ) = inv T G ( O ) , , otherwise . From [BP15, p.98, Section 4.5], the following asymptotic expansion for any Y ∈ g rss ( F ) holds(A.2.1) lim t ∈ F × ,t → D G ( tY ) / b j ( X, tY ) = D G ( Y ) / X O∈ Nil reg ( g ) Γ O ( X ) b j ( O , Y ) . The constants Γ O ( X ) appearing in the statement of Theorem A.2.1 are exactly theterms showing up in (A.2.1).We first establish the following lemma. Lemma A.2.2.
For any f ∈ S ( g ) , lim t ∈ F × ,t → J G ( X, b f t ) = X O∈ Nil reg ( g ) Γ O ( X ) J O ( f ) , where f t ( Y ) = | t | δ G / − dim G f ( t − Y ) , and δ G = dim G − dim T G .Proof. From (A.2.1),lim t ∈ F × ,t → D G ( tY ) / D G ( Y ) / b j ( X, tY ) = X O∈ Nil reg ( g ) Γ O ( X ) b j ( O , Y ) . Integrate both sides against a function f ( Y ) ∈ S ( g ) on g rss ( F ). ThenRHS = Z g rss ( F ) X O∈ Nil reg ( g ) Γ O ( X ) b j ( O , Y ) f ( Y ) d Y. Since b j ( O , Y ) is locally integrable on g ( F ),RHS = X O∈ Nil reg ( g ) Γ O ( X ) J O ( b f ) . For the LHS, using the fact that D G ( Y ) / b j ( X, Y ) is globally bounded on g rss ( F ) × g rss ( F ) (c.f. [BP15, 1.8]), and the function D G ( Y ) − / defines a tempered distributionon S ( g ) (c.f. [BP15, 1.7.1]), by the dominated convergence theorem ([SS05, p.67]),the LHS can be written aslim t ∈ F × ,t → Z g rss ( F ) D G ( tY ) / D G ( Y ) / b j ( X, tY ) f ( Y ) d Y. Since Y ∈ g rss ( F ), D G ( tY ) / D G ( Y ) / = | t | δ G . After a change of variable Y → t − Y ,LHS = lim t ∈ F × ,t → Z g rss ( F ) | t | δ G / − dim G b j ( X, Y ) f ( t − Y ) d Y, which is equal to lim t ∈ F × ,t → J G ( X, b f t )by the local integrability of b j ( X, Y ) in variable Y .It follows that we have established the lemma. (cid:3) As a corollary, we can show that Γ O ( X ) is invariant under the scaling action of F × . Over p -adic fields it is already known ([Wal10, 2.6]). Corollary A.2.3. Γ O ( aX ) = Γ O ( X ) for any a ∈ F × , X ∈ g rss ( F ) and O ∈
Nil reg ( g ) .Proof. For any f ∈ S ( g ), b f t ( Y ) = Z g ( F ) | t | δ G / − dim G f ( t − X ) ψ ( h X, Y i ) d X. After a change of variable X → tX , b f t ( Y ) = Z g ( F ) | t | δ G / f ( X ) ψ ( h X, tY i ) d X = | t | δ G / b f ( tY ) . Therefore J G ( X, b f t ) = | t | δ G / D G ( X ) / Z G X \ G b f ( gtXg − ) d g (A.2.2) = D G ( tX ) / Z G X \ G b f ( gtXg − ) d g = J G ( tX, b f ) . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 69
In particular Lemma A.2.2 can be reformulated as(A.2.3) lim t ∈ F × ,t → J G ( tX, b f ) = X O∈ Nil reg ( g ) Γ O ( X ) J O ( b f ) . From (A.2.3) we get the desired identity. (cid:3)
We also have the following lemma.
Lemma A.2.4.
For any f ∈ S ( g ) , t ∈ F × , and O ∈
Nil reg ( g ) , J O ( b f t ) = J O ( b f ) . Proof.
By definition, J O ( b f t ) = Z g ( F ) | t | δ G / − dim G b j ( O , Y ) f ( t − Y ) d Y. After a change of variable Y → tY , J O ( b f t ) = Z g ( F ) | t | δ G / b j ( O , tY ) f ( Y ) d Y. Since t ∈ F × , from [BP15, 1.8.1], b j ( O , tY ) = | t | − dim( O ) / b j ( O , Y ) = | t | − δ G / b j ( O , Y ) . It follows that we have established the desired equality. (cid:3)
For any f ∈ S ( g ) and t ∈ F × , plug the function b f t into Theorem A.1.9. Then thefollowing identity holdslim X H → X X G ∆ ′ ( X H , X G ) J G ( X G , b f t ) = X O∈ Nil reg ( g ) ∆( O ) J O ( b f t ) . Applying (A.2.2) and Lemma A.2.4 to the above identity, the following identity holds,(A.2.4) lim X H → X X G ∆ ′ ( X H , X G ) J G ( tX G , b f ) = X O∈ Nil reg ( g ) ∆( O ) J O ( b f ) . On the other hand, from Lemma A.2.2 and (A.2.2), the following identity holds(A.2.5)lim t ∈ F × ,t → X X G ∆ ′ ( X H , X G ) J G ( tX G , b f ) = X X G ∆ ′ ( X H , X G ) X O∈ Nil reg ( g ) Γ O ( X G ) J O ( b f ) . It is natural to expect that the RHS of the two equations are equal to each other,which is going to be established in the next lemma.
Lemma A.2.5.
For any f ∈ S ( g ) , X O∈ Nil reg ( g ) ∆( O ) J O ( b f ) = X X G ∆ ′ ( X H , X G ) X O∈ Nil reg ( g ) Γ O ( X G ) J O ( b f ) . Proof.
For any a ∈ F × , by Remark A.1.7, ∆ ′ ( aX H , aX G ) = ∆ ′ ( X H , X G ). FromLemma A.2.4, Γ O ( X G ) = Γ O ( aX G ).Consider the limit,lim a ∈ F × ,a → lim t ∈ F × ,t → X aX G ∆ ′ ( aX H , aX G ) J G ( atX G , b f )= lim a ∈ F × ,a → lim t ∈ F × ,t → X X G ∆ ′ ( X H , X G ) J G ( atX G , b f )= X X G ∆ ′ ( X H , X G ) X O∈ Nil reg ( g ( F ) Γ O ( aX G ) J O ( b f )= X X G ∆ ′ ( X H , X G ) X O∈ Nil reg ( g ( F )) Γ O ( X G ) J O ( b f ) . In particular, the limit is uniform in a ∈ F × . Hence by Moore-Osgood theorem([Tay85, p139]), the order of the limit can be switched, from which we get X O∈ Nil reg ( g ( F )) ∆( O ) J O ( b f ) . It follows that we have establish the desired identity. (cid:3)
Using the linear independence of the distributions { J O | O ∈ Nil reg ( g ) } (c.f.[BP15, 1.8.2]), the following corollary holds. Corollary A.2.6.
For any
O ∈
Nil reg ( g ) , ∆( O ) = X X G ∆ ′ ( X H , X G )Γ O ( X G ) . Now we are ready to establish Theorem A.2.1. We follow the argument of [She89].For any character κ of E ( T ) = Im( H ( F, T sc G ( F )) → H ( F, T G ( F ))), we can attachto it an endoscopic group H = H ( T G , κ ), together with an admissible embedding T H → T G = G X whose underlying Lie algebra homomorphism sends X H to X . Forconvenience we may assume that L H embeds admissibly into L G .By Remark A.1.8, for the identity in Corollary A.2.6, we may write each X G appearing in the summation as X G = g − Xg = X (inv( X G , X )), where g ∈ G ( F )and σ → σ ( g ) g − represents the element inv( X G , X ) in E ( T ). Then ∆ ′ ( X H , X G ) = LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 71 h inv( X G , X ) , κ i · ∆ ′ ( X H , X ). It follows that we arrive at the following identity X X G h inv( X, X G ) , κ i Γ O ( X G ) = ∆( O )∆ ′ ( X H , X ) . On the other hand, we may write∆( O ) = h inv T G ( O ) , κ i , ∆ I ( X H , X ) = h inv( T G ) , κ i , ∆ II ( X H , X ) = h inv T G ( X ) , κ i . Hence we get X X G h inv( X, X G ) , κ i Γ O ( X G ) = h inv T G ( O )inv( T G )inv( X ) , κ i . After summing over κ , we obtain Theorem A.2.1.A.3. Relation with the Kostant’s sections.
We are going to point out the rela-tion between Theorem A.2.1 and Kostant’s sections.For the presentation of Kostant’s sections, we follow [Kot99], [Dri] and [Kos63].In particular we assume that G is split over F and T is a fixed split maximal torusin G defined over F .We first recall the Chevalley’s isomorphism ([CG97, 6.7]). Under the adjoint actionof G on g and W on t , the restriction morphism F [ g ] → F [ t ] yields an isomorphism F [ g ] G ≃ F [ t ] W . The associated morphism u : g → t /W sends Z ∈ g ( F ) to the W -orbit in t ( F ) consisting of elements that are G ( F )-conjugate to the semi-simplepart Z s of the Jordan decomposition Z = Z s + Z n . Here Z s is semi-simple, Z n isnilpotent, and [ Z s , Z n ] = 0. Definition A.3.1.
An element Z ∈ g ( F ) is called regular if the dimension of itscentralizer in g ( F ) is equal to the dimension of t . It is known that the set of regular elements is open and dense in g ([Ste65, I.3]).In [Kos63], over the algebraic closure F , B. Kostant showed that Z is regular ifand only if the nilpotent part Z n of Z is a regular element in the centralizer of Z s in g . The map Z → Z s induces a bijection between the set of regular Ad( G )-orbitsin g and the set of semi-simple Ad( G )-orbits in g . Moreover, using the morphism u ,both sets of orbits can be identified with t /W . Kostant’s sections.
Let B ∞ = T N ∞ be the Borel subgroup of G defined over F thatis opposite to B .Kostant proved that every element in the F -points of the affine space b ∞ ( F ) + X + ⊂ g ( F ) is regular. For any H ∈ t ( F ) ⊂ b ∞ ( F ), the semi-simple part of H + X + is conjugate to H . In particular, over the algebraic closure the set t + X + meetsevery regular Ad( G )-orbit in g . Let a = Cent g ( X − ) (the choice of a has more freedom, see [Kot99, 2.4]). Thenover the algebraic closure F , a + X + meets every regular Ad( G )-orbit exactly once.Over the rational field F , the composition of the closed embedding a + X + ֒ → g andthe morphism u : g → t /W is an isomorphism of algebraic varieties. For any Z ∈ t ( F ) + X + , there exists a unique element n ( Z ) ∈ N ∞ ( F ) such that Ad( n ( Z ))( Z ) ∈ a ( F ) + X + , and the map Z → n ( Z ) gives a morphism of algebraic varieties t + X + → N ∞ . It can be deduced from [Kos63] that for any Z ∈ b ∞ ( F ) + X + there existsa unique element n ( Z ) ∈ N ∞ ( F ) such that Ad( n ( Z )) ∈ a ( F ) + X + , and the map Z → n ( Z ) gives a morphism of algebraic varieties b ∞ + X + → N ∞ . It followsthat the map ( n, Y ) → Ad( n ) Y gives an isomorphism of algebraic varieties from N ∞ × ( a + X + ) to b ∞ + X + .In summary, we obtain the following commutative diagram over F . a + X + ≃ & & ▲▲▲▲▲▲▲▲▲▲ / / X + + b ∞ Ad- N ∞ trivial bundle (cid:15) (cid:15) / / g (cid:15) (cid:15) t /W ≃ / / g // G . (A.3.1) Submersion.
Following the diagram (A.3.1), let Σ be the image of the followingmorphism G × ( a + X + ) → g ( g, Y ) → g − Y g.
Then there is the following commutative diagram G × ( a + X + ) / / p ' ' PPPPPPPPPPPP Σ / / p (cid:15) (cid:15) g (cid:15) (cid:15) ( a + X + ) ≃ / / t /W . Here p is the canonical projection onto the second variable, and p is the canonicalmorphism induced from p . After composition, there is a morphism c : Σ → t /W .After passing to the F -rational points, we get an F -analytical smooth submersionbetween two smooth F -analytic manifolds c F : Σ( F ) → ( t /W )( F ) . The F -analytical structure of Σ( F ) is inherited from g ( F ), together with the factthat the set of regular elements in g are open and dense. Denote the measure onΣ( F ) induced from g ( F ) by µ Σ( F ) . Similarly there is the canonical measure on( t /W )( F ) which is denoted by µ ( t /W )( F ) ([FLN10, Proposition 3.29]). The fibersof the morphism c F are given by the G ( F )-orbits associated to the elements in( a + X + )( F ) via the isomorphism ( a + X + ) ≃ t /W . In particular, the fibers of c F are LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 73 all of the same dimension. By the theory of integration along fibers ([BT82, p.61] forarchimedean, [Igu00, 7.6] for p -adic), for any Y ∈ ( t /W )( F ), there exists a canonicalchosen measure µ Y on the G ( F )-orbit associated to Y , such that the measure µ Y varies F -analytically in Y with the the following identity Z Σ( F ) f ( X ) µ Σ( F ) ( X ) = Z ( t /W )( F ) F f ( Y ) µ ( t /W )( F ) ( Y ) . From [FLN10, 3.30], when the associated element in ( a + X + )( F ) is regular semi-simple, which is still denoted by Y by abuse of notation, F f ( Y ) = J G ( Y, f ) . It follows that if we choose a sequence { Y i } i ≥ , Y i ∈ ( a + X + ) rss ( Y ), such thatlim i →∞ Y i = N ∈ ( a + X + )( F ) where N is the unique regular nilpotent element in( a + X + )( F ), then lim i →∞ µ Y i = µ N , i.e. lim i →∞ J G ( Y i , f ) = J O N ( f ) , f ∈ C ∞ c (Σ( F )) , where O N ∈ Nil reg ( g ) is the regular nilpotent element associated to N . Remark A.3.2.
There is an alternative proof of the fact that lim i →∞ µ Y i = µ N given in [Wal10, Lemme 11.4] . It is worth pointing out that in Waldspurger’s proof,he chose an arbitrary sequence { Y i } i ≥ where Y i ⊂ g ( F ) rss and lim i →∞ Y i = N . Butwe can observe that the sequence satisfies the property that Y i ⊂ Σ( F ) whenever i issufficiently large. This follows from the fact that Σ( F ) is open in g ( F ) .The relation. Now we discuss the relation between Theorem A.2.1 and the Kostantsections. Recall that we have fixed an endoscopic data ( H, H , s, ξ ) for G and an F -splitting spl = ( B , T, { X α } ).We recall the following theorem and corollary proved by Kottwitz ([Kot99, Theo-rem 5.1, Corollary 5.2]). Theorem A.3.3.
The transfer factor ∆ ′ ( X H , X G ) is equal to whenever X G liesin the set of F -rational points of b + X − . Corollary A.3.4. ∆ ′ ( X H , X G ) = h inv( X G , X ′ G ) , s T G i where X ′ G is any F -rational element in b + X − that is stably conjugate to X G . Following the construction of Kostant’s sections, we can define a linear subspace a of b such that a + X − yields a section for the adjoint quotient map g → t /W . Now as before let Σ be the image of G × ( a + X − ) → g where G acts on a + X − via adjoint action. Then Corollary A.3.4 says that ∆ ′ ( X H , X G ) = 1 if and only if X G lies in Σ( F ).By Theorem A.2.1, for any O ∈
Nil reg ( g ) and X ∈ g rss ( F ),Γ O ( X ) = (cid:26) , inv( X )inv( T G ) = inv T G ( O ) , , otherwise . Note that the invariant inv T G ( O ) measures the difference between spl ∞ , the opposite F -splitting to spl , and the F -splitting spl ( O ) determined by O . In particular, upto G ( F )-conjugation, for convenience we may set spl ( O ) = spl ∞ , i.e. X − ∈ O andinv T G ( O ) = 1. Then the orbit O lies in Σ( F ), and the formula can be simplified tosay that Γ O ( X ) = 1 if and only if inv( X )inv( T G ) = 1, which, is equivalent to saythat ∆ ′ ( X H , X ) = 1 identically. On the other hand, by Kottwitz’s theorem A.3.3,it is also equivalent to say that X lies in Σ( F ). In particular, both the G ( F )-orbitof X and O are contained in Σ( F ), i.e. they lie in the image of the G ( F )-orbit of acommon Kostant’s section.Conversely, assume both the G ( F )-orbit of X and O lie in the image of the G ( F )-orbit of a common Kostant’s section, which again is denoted by Σ( F ). Upto G ( F )-conjugation for spl ∞ , again we may assume that inv T G ( O ) = 1. Then The-orem A.3.3 says that ∆ ′ ( X H , X G ) is identically 1, which is equivalent to say thatinv( X )inv( T G ) = 1. Hence Γ O ( X ) = 1.In conclusion, the following theorem has been proved. Theorem A.3.5.
With the above notations, Γ O ( X ) = 1 if and only if the G ( F ) -orbitof X and O lie in the G ( F ) -orbit of a common Kostant’s section. Appendix B. Calculation of germs
In this section, we are going to compute the endoscopic invariants defined in Sec-tion A.1 for even dimensional quasi-split special orthogonal Lie algebras. We followthe strategy of [Wal01, Chapitre I,X]. Using the regular germ formula proved in Sec-tion A.2, we give explicit formulas for the regular nilpotent germs for some regularsemi-simple conjugacy classes in an even dimensional quasi-split special orthogonalLie algebra. Finally using the explicit formulas we can determine the intersection ofNil reg ( g ) and Γ(Σ) following the strategy of [Wal10, 11.5,11.6]. The results estab-lished in the section will be important determining the nilpotent orbit support of thegeometric expansion of the trace formula.Throughout the section, fix an even dimensional quasi-split quadratic space ( V, q V )of dimension d defined over F , with associated (special) orthogonal group O( V )(SO( V )) and corresponding Lie algebra so ( V ). LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 75
B.1.
Some regular semi-simple conjugacy classes.
We are going to review theparametrization of some regular semi-simple conjugacy classes (without eigenvalue0) in so ( V )( F ), and prove some basic properties for the parametrization.The following data is fixed. • A finite set I ; • For any i ∈ I , fix a finite field extension F i of F together with a 2-dimensional F i commutative algebra F i . Denote τ i the unique nontrivial automorphismof F i over F i . • For any i ∈ I , fix constants a i , c i ∈ F × i .For fix data as above, we make further assumptions as follows. • For any i ∈ I , a i generates F i over F ; • For any i, j ∈ I with i = j , there do not exist any F -linear isomorphismsbetween F i and F j that send a i to a j ; • For any i ∈ I , τ i ( a i ) = − a i , and τ i ( c i ) = c i ;For any i ∈ I , note sgn F i /F i the quadratic character of F × i associated to F i vialocal class field theory ([Ser67, Theorem 2]). Let I ∗ be the subset of I consisting of i ∈ I such that F i is a field, i.e. sgn F i /F i is nontrivial.With the above assumptions set W = L i ∈ I F i and define a quadratic space ( W, q W )via q W ( X i ∈ I w i , X i ∈ I w ′ i ) = X i ∈ I [ F i : F ] − tr F i /F ( τ i ( w i ) w ′ i c i ) , w i , w ′ i ∈ F i . Using the fact that τ i ( c i ) = c i we immediately find that the bilinear form q W isindeed symmetric.Note X W the element in End F ( W ) defined by X W ( X i ∈ I w i ) = X i ∈ I a i w i , w i ∈ F i . Then X W ∈ so ( W )( F ) which follows directly from the fact that τ i ( a i ) = − a i .We establish some basic properties for the quadratic space ( W, q W ). Lemma B.1.1.
For any finite-dimensional F -algebra E , let Σ( E ) be the set ofnonzero F -algebra homomorphisms from E to F . Then there exists a basis { f σ } σ ∈ Σ( E ) of F ⊗ F E , such that for any v ∈ E ⊂ F ⊗ F E , v = X σ ∈ Σ( E ) σ ( v ) f σ . Proof.
Let k = [ E : F ], which is also the cardinality of the set Σ( E ). We fix any F -basis { b σ } σ ∈ Σ( E ) of E , and assume that we have f σ = X τ ∈ Σ( E ) c τσ ⊗ b τ for some constant c τσ ∈ F . Then the equality v = X σ ∈ Σ( E ) σ ( v ) f σ is equivalent to the following identities X σ ∈ Σ( E ) σ ( b τ ) c τσ = 1 , ∀ τ ∈ Σ( E ) , X σ ∈ Σ( E ) σ ( b τ ) c ξσ = 0 , ∀ ξ ∈ Σ( E ) , τ = ξ. Let B = ( σ ( b τ )) ∈ M k × k ( F ), where the rows are indexed by σ ∈ Σ( E ), and columnsare indexed by τ ∈ Σ( E ). Similarly let C = (( c τσ )) ∈ M k × k ( F ) where the rowsare indexed by τ and the columns are indexed by τ . Then the above equalities areequivalent to the following matrix identity BC = I k . Using the fact that { b τ } τ ∈ Σ( E ) is an F -basis of E , we realize that the matrix B isinvertible, hence we can simply let C = B − and the lemma follows. (cid:3) As a corollary, we can establish the following fact.
Corollary B.1.2.
For any ζ ∈ Γ F , we have ζ ( f σ ) = f ζσ . Proof.
We use the same notation as in Lemma B.1.1. We write f σ = X τ ∈ Σ( E ) c τσ ⊗ b τ . Then for any ζ ∈ Γ F , ζ ( f σ ) = P τ ∈ Σ( E ) ζ ( c τσ ) ⊗ b τ .Since ζ ( B ) ζ ( C ) = ζ (I k ) = I k , and we know that ζ ( B ) = ( ζ σ ( b τ )), we get ζ ( C ) = (( c τζσ )). It follows that ζ ( f σ ) = f ζσ . (cid:3) LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 77
Now for any i ∈ I , we define Σ( F i ) to be the set of nonzero F -algebra homomor-phisms from F i to F . We let Σ( W ) = { ( i, σ ) | i ∈ I, σ ∈ Σ( F i ) } . From LemmaB.1.1, there exists a basis for F ⊗ F W , which we denote by { f i,σ | ( i, σ ) ∈ Σ( W ) } ,such that for any i ∈ I and v i ∈ F i ⊂ V , v i = X σ ∈ Σ( F i ) σ ( v i ) f i,σ . Lemma B.1.3.
The following properties hold, (1)
For any ( i, σ ) ∈ Σ( W ) , X ( f i,σ ) = σ ( a i ) f i,σ . (2) For any ( i, σ ) and ( i ′ , σ ′ ) ∈ Σ( W ) , q W ( f i ′ ,σ ′ , f i,σ ) = (cid:26) , ( i ′ , σ ′ ) = ( i, στ i ) ,σ ( c i )[ F i : F ] − , ( i ′ , σ ′ ) = ( i, στ i ) . Proof.
We establish (1). By definition, Xv i = a i v i . We write v i = X σ ∈ Σ( F i ) σ ( v i ) f i,σ . Then we get the following equality X σ ∈ Σ( F i ) σ ( v i ) X ( f i,σ ) = X σ ∈ Σ( F i ) σ ( a i v i ) f i,σ . It follows that we have X ( f i,σ ) = σ ( a i ) f i,σ .Now for (2), when i = i ′ , q W ( f i ′ ,σ ′ , f i,σ ) = 0 holds automatically. We only considerthe case when i = i ′ . For any v i , w i ∈ F i , by definition, q W ( v i , w i ) = [ F i : F ] − tr F i /F ( τ i ( v i ) w i c i ) = [ F i : F ] − X σ ∈ Σ( F i ) ( στ i )( v i ) σ ( w i ) σ ( c i ) . On the other hand, by linearity, we have q W ( v i , w i ) = X ξ,σ ∈ Σ( F i ) ξ ( v i ) σ ( w i ) q W ( f i,ξ , f iσ ) . Comparing the two identities, we get the desired result. (cid:3)
We establish the following lemma.
Lemma B.1.4.
The element X W is regular semi-simple, and the centralizer of X W in SO( W ) ( actually O( W )) is isomorphic to Y i ∈ I \ I ∗ F × i Y i ∈ I ∗ ker(N F i /F i : F × i → F × i ) . Proof.
Over the algebraic closure, under the fixed basis { f i,σ | ( i, σ ) ∈ Σ( W ) } , wehave X ( f i,σ ) = σ ( a i ) f i,σ , therefore X is semi-simple.Now by definition, for any i, j ∈ I , there do not exist any F -linear isomorphismsbetween F i and F j sending a i to a j , and a i generates F i over F . Hence the eigenvaluesof X , which we denote by { σ ( a i ) } ( i,σ ) ∈ Σ( W ) , are pairwise different and do not contain0. It follows that X W is regular semi-simple, and the centralizer of X W in O( W )coincides with its centralizer in SO( W ).For any g ∈ GL( W ) that commutes with X W , we have X W ( X i ∈ I g ( w i )) = X i ∈ I a i g ( w i ) , w i ∈ F i . Since there are no F -linear isomorphisms between F i and F j sending a i to a j , we get g ( w i ) ∈ F i . It follows that g ∈ Q i ∈ I GL( F i ). Moreover, since g ∈ SO( W ), we realize that g ∈ Q i ∈ I O( F i ). Here O( F i ) is the orthogonal group associated to the quadratic form( F i , q F i ), where q F i ( w i , w ′ i ) = [ F i : F ] − tr F i /F ( τ i ( w i ) w ′ i c i ) , w i , w ′ i ∈ F i . . For any i ∈ I , F × i is a maximal torus in GL( F i ) that commutes with X W bydefinition. Hence we only need to compute F × i ∩ O( F i ). For any a ∈ F × i , the identity q F i ( aw i , aw ′ i ) = q F i ( w i , w ′ i ) , w i , w ′ i ∈ F i is equivalent to τ i ( a ) a = 1 , which is equivalent to say that a ∈ (cid:26) ker(N F i /F i : F × i → F × i ) , if F i /F is a field extension , { ( z, z − ) | z ∈ F × i } , if F i /F is not a field extension . It follows that we have established the lemma. (cid:3)
Now we assume that d = P i ∈ I [ F i : F ] = dim W = dim V . If there exists anisomorphism between ( W, q W ) and ( V, q V ), we fix such an isomorphism, and then X W can be identified with a regular semi-simple element X in so ( V )( F ). We note thatthe SO( V )-orbit of X depends on the choice of the isomorphism between ( W, q W ) and( V, q V ). We note O( V ) ≃ SO( V ) ⋉ { , O } , where O is the outer automorphism. Thenas we have seen in the proof of Lemma B.1.4, the centralizer of X in O( V ) coincideswith the centralizer of X in SO( V ). Hence conjugation by the outer automorphism LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 79 O yields another SO( V )-orbit, which can be obtained from X W as well throughcomposing the original isomorphism between ( W, q W ) and ( V, q V ) with the outerautomorphism O . In particular we notice that the two orbits are not SO( V ) stableconjugate.For any triple ( I, ( a i ) , ( c i )), assume that there exists an isomorphism ( W, q W ) ≃ ( V, q V ). Then there are two possible F -rational SO( V )-orbits permuted by the outerautomorphism O , which are denoted by O + ( I, ( a i ) , ( c i )) and O − ( I, ( a i ) , ( c i )). Wedenote the associated F -rational SO( V ) stable conjugacy classes by O st , + ( I, ( a i ) , ( c i ))and O st , + ( I, ( a i ) , ( c i )). Finally let the union of the two (stable) conjugacy classes by O ( I, ( a i ) , ( c i )) ( O st ( I, ( a i ) , ( c i ))).We have the following lemma. Lemma B.1.5.
Given two triples ( I, ( a i ) , ( c i )) and ( I ′ , ( a ′ i ) , ( c ′ i )) together with iso-morphisms between ( W, q W ) , ( W ′ , q W ′ ) and ( V, q V ) . After fixing isomorphisms be-tween them, we have the associated orbits in SO( V ) .Then O st ( I, ( a i ) , ( c i )) = O st ( I ′ , ( a ′ i ) , ( c ′ i )) if and only if the following propertieshold. • There exists a bijection φ : I → I ′ ; • For any i ∈ I , an F -linear isomorphism σ i : F ′ φ ( i ) → F i satisfying σ i ( a ′ φ ( i ) ) = a i , ∀ i ∈ I. Furthermore, O ( I, ( a i ) , ( c i )) = O ( I ′ , ( a ′ i ) , ( c ′ i )) if the following additional propertyis satisfied. • For any i ∈ I , we have sgn F i /F i ( c i σ i ( c ′ φ ( i ) ) − ) = 1 . Proof.
We notice that O st ( I, ( a i ) , ( c i )) = O st ( I ′ , ( a ′ i ) , ( c ′ i )) if and only if X W and X W ′ share the same characteristic polynomial. Using the notation from LemmaB.1.4, the roots of the characteristic polynomial of X W (resp. X W ′ ) are given by theset { σ ( a i ) } ( i,σ ) ∈ Σ( W ) (resp. { σ ′ ( a ′ i ′ ) } ( i ′ ,σ ′ ) ∈ Σ( W ′ ) ). It follows that the if part followsimmediately. For the only if part, we notice that by definition a i generates F i over F , and for any i, j ∈ I , there do not exist F -linear isomorphism between F i and F j sending a i to a j . Moreover, for any i ∈ I , the roots related to F i are exactly givenby { σ ( a i ) } i ∈ Σ( F i ) . Then the only if part is also straightforward.Now to determine the conjugacy classes within the stable conjugacy classes, wemay assume that I = I ′ , a i = a ′ i , and σ i = Id F i . Then for the two orbits O ( I, ( a i ) , ( c i ))and O ( I, ( a i ) , ( c ′ i )), again from Lemma B.1.4, over the algebraic closure F , underthe basis { f i,σ } ( i,σ ) ∈ Σ( W ) both X W and X W ′ are diagonal with same eigenvalues.Hence the two orbits are the same if and only if there exists an element in the centralizer of X W giving rise to an isomorphism between the two quadratic spaces( W, q W ) ≃ ( W ′ , q ′ W ). But this follows directly from Lemma B.1.4. (cid:3) B.2.
Computation of the endoscopic invariants.
We are going to compute theendoscopic invariants defined in Section A.1 for the conjugacy classes defined inprevious section. We will freely use notations from Section A.1.
Coordinates and notations.
We first fix the coordinates for the quadratic space(
V, q V ).When ( V, q V ) is split, we fix η ∈ F × and an F -basis { e j | j = 1 , ..., d } of V suchthat for any j, k ∈ { , ..., d } we have q V ( e j , e k ) = (cid:26) , if j + k = d + 1 , ( − j +1 η/ , if j + k = d + 1 , j ≤ d/ . Let B = T N be the F -rational Borel subgroup stabilizing the totally isotropic flag h e i ⊂ ... ⊂ h e , ..., e d/ i . Then T acts on the basis { e i } di =1 via scaling.With the choice of the Borel pair ( B , T ), we have a natural set of simple roots∆ = ∆( B , T ) = { α j | j = 1 , ..., d/ } as follows. For j ∈ { , ..., d/ − } , we let X α j be the element in n that annihilates e k for k = j + 1 , ..., d + 1 − j and X α j ( e j +1 ) = e j , X α j ( e d +1 − j ) = e d − j . The element X α d/ ∈ n annihilates e k for k = d/ , d/ X α d/ ( e d/ ) = e d/ − , X α d/ ( e d/ ) = e d/ . We can identify the Weyl group W = W ( G, T ) as the group of permutations w ofthe set { , ..., d } corresponding to the basis { e j | j = 1 , ..., d } such that • for any j ∈ { , ..., d } , w ( j ) + w ( d + 1 − j ) = d + 1; • the set { j | ≤ j ≤ d/ , d/ ≤ w ( j ) ≤ d } has even cardinality.We also define a function r : { , ..., d } × W → N via r ( j, w ) = |{ k ∈ N | j < k ≤ d, w ( j ) > w ( k ) , j + k = d + 1 }| , j ∈ { , ..., d } , w ∈ W. When (
V, q V ) is quasi-split but not split, following the notation in Section 2.1 thereexists a unique quadratic extension E of F such that V ⊗ F E, q V ) is split. The groupGal( E/F ) = { , τ } acts on V ⊗ F E in the following way. As in the split case, wemay fix η ∈ F × and a basis { e j | j = 1 , ..., d } of V ⊗ F E with prescribed propertiesas above, then • for any j ∈ { , ..., d }\{ d/ , d/ } , e j is fixed by τ ; LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 81 • τ e d/ = e d/ and τ ( e d/ ) = e d/ .We first establish the following lemma. Lemma B.2.1.
For any w ∈ W and j ∈ { , ..., d } , we have n ( w )( e j ) = ( − r ( j,w ) e w ( j ) . Proof.
For any α ∈ ∆ = ∆( B , T ), we let X − α be the unique element in n which isthe radical space associated to − α such that[ X α , X − α ] = d ˇ α (1)and w α the simple reflection associated to α .We note that the Weyl group W has a length function ℓ ([Hum90, 5.2]). EveryWeyl element in W has a reduced expression which can be written as the product ofsimple reflections.The section n is characterized uniquely by the following two properties ([LS87,2.1]). • For any α ∈ ∆, n ( w α ) = exp( X α ) exp( − X − α ) exp( X α ); • For any w, w ′ ∈ W satisfying ℓ ( w ) + ℓ ( w ′ ) = ℓ ( ww ′ ), we have n ( ww ′ ) = n ( w ) n ( w ′ ) . Therefore we only need to establish the lemma for simple reflections and the fol-lowing fact.(B.2.1) For any j ∈ { , ..., d } , and w , w ∈ W satisfying ℓ ( w ) + ℓ ( w ) = ℓ ( w w ) and w being a simple reflection, we have r ( j, w w ) + r ( j, w ) + r ( w ( j ) , w ) ≡ . We first establish the lemma for simple reflections. For any t ∈ { , ..., d/ } , we let w t be the simple reflection associated to α t . By explicit computation, we have n ( w t )( e j ) = ( e t , j = t + 1 , r ( j, w ) = |∅| = 0 , − e t +1 , j = t, r ( j, w ) = |{ t + 1 }| = 1 ,e d − t , j = d + 1 − t, r ( j, w ) = |∅| = 0 , − e d +1 − t , j = d − t, , r ( j, w ) = |{ d + 1 − t }| = 1 ,e j , others , r ( j, w ) = |∅| = 0 . , t ∈ { , ..., d/ − } n ( w d/ )( e j ) = ( − e d/ , j = d/ − , r ( j, w ) = |{ d/ }| = 1 , − e d/ , j = d/ , r ( j, w ) = |{ d/ }| = 1 ,e d/ − , j = d/ , r ( j, w ) = |∅| = 0 ,e d/ , j = d/ , r ( j, w ) = |∅| = 0 ,e j , others , r ( j, w ) = |∅| = 0 . It follows that the lemma holds for any simple reflections.Now we are going to establish (B.2.1).By definition, r ( j, w w ) = |{ α ∈ N | j < α, w w ( j ) > w w ( α ) , j + α = d + 1 }| ,r ( j, w ) = |{ β ∈ N | j < β, w ( j ) > w ( β ) , j + β = d + 1 }| ,r ( w ( j ) , w ) = |{ γ ∈ N | w ( j ) < γ, w w ( j ) > w ( γ ) , w ( j ) + γ = d + 1 }| . For the set appearing in the definition of r ( w ( j ) , w ), we let ξ = w − ( γ ), then r ( w ( j ) , w ) = |{ ξ ∈ N | w ( j ) < w ( ξ ) , w w ( j ) > w w ( ξ ) , j + ξ = d + 1 }| . We can separate the set { ξ ∈ N | w ( j ) < w ( ξ ) , w w ( j ) > w w ( ξ ) , j + ξ = d + 1 } to two disjoint parts, for abbreviation we denote them by { ξ < j } and { ξ > j } .Then { ξ < j } is a subset of the one appearing in the definition of r ( j, w w ). Aftersubtracting { ξ < j } from the set in the definition of r ( j, w w ), we are reduced toprove the following identity |{ α ∈ N | j < α, w ( j ) > w ( α ) , w w ( j ) > w w ( α ) , j + α = d + 1 }|≡ |{ β ∈ N | j < β, w ( j ) > w ( β ) , j + β = d + 1 }| + |{ ξ ∈ N | j > ξ, w ( j ) < w ( ξ ) , w w ( j ) > w w ( ξ ) , j + ξ = d + 1 }| mod 2 . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 83
The above set involving α is a subset of the one involving β . Hence after subtractionwe are reduced to prove the following identity whenever ℓ ( w ) + ℓ ( w ) = ℓ ( w w ), |{ α ∈ N | j < α, w ( j ) > w ( α ) , w w ( j ) < w w ( α ) , j + α = d + 1 }|≡ |{ ξ ∈ N | j > ξ, w ( j ) < w ( ξ ) , w w ( j ) > w w ( ξ ) , j + ξ = d + 1 }| mod 2 . For notational convenience, we let A α = { α ∈ N | j < α, w ( j ) > w ( α ) , w w ( j ) < w w ( α ) , j + α = d + 1 } ,B ξ = { ξ ∈ N | j > ξ, w ( j ) < w ( ξ ) , w w ( j ) > w w ( ξ ) , j + ξ = d + 1 } . We assume that w = w t is a simple reflection, where t ∈ { , ..., d/ } .When t ∈ { , ..., d/ − } , the sets A α and B ξ have the chance to be nonempty onlywhen j = t, t +1 , d − t, d − t +1. When j = t , A α = { α ∈ N | α = t +1 and w ( t +1)
1, since the length ℓ ( w ) of a Weyl element w is equalto the number of positive roots sent to negative roots by w [Hum90, 5.6], whichcontradicts the fact that ℓ ( w w ) = ℓ ( w ) + ℓ ( w ). It follows that A α = ∅ as well.Hence we have the equality | A α | = | B ξ | . Similar arguments adapt to the situationwhen j = t + 1 , d − t, d − t + 1.When t = d/
2, the sets A α and B ξ have the chance to be nonempty only when j = d/ − , d/ , d/ , d/ j = d/ − B ξ = ∅ , and A α = { α ∈ N | α = d/ w ( d/ < w ( d/ − } , then the same argument as aboveimplies that A α = ∅ , and in particular | A α | = | B ξ | as well. Similarly we can obtainthe results for other cases.Hence we have established (B.2.1) and it follows that we have established thelemma. (cid:3) Now we fix a regular semi-simple element X in so ( V )( F ) parametrized by a triple( I, ( a i ) , ( c i )), with a fixed identification ( V, q V ) ≃ ( W, q W ). We let T G be the cen-tralizer of X in SO( V )( F ). We may choose an element x ∈ SO( V )( F ) such that T G = xT x − . In particular, up to scaling the element x should send the basis { e i } di =1 to { f i,σ } ( i,σ ) ∈ Σ( V ) . Using Lemma B.1.3 we may construct an element x as follows.We fix a bijection δ : Σ( V ) → { , ..., d } such that for any ( i, σ ) ∈ Σ( V ) we havethe equality δ ( i, στ i ) + δ ( i, σ ) = d + 1 . We also fix a mapping µ : { , ..., d } → F × such that for any j ∈ { , ..., d/ } if weput ( i, σ ) = δ − ( j ), then we have the equality µ ( j ) µ ( d + 1 − j ) = η/ − j +1 [ F i : F ] σ ( c i ) − , j ≤ d/ . Now we define x ∈ GL( F ⊗ F V ) by x ( e j ) = µ ( j ) f δ − ( j ) , j ∈ { , ..., d } . By Lemma B.1.3 x lies in O( V )( F ). Up to composing δ with the simple reflectioninterchanging 1 and d stabilizing { , ..., d − } , we can assume that x ∈ SO( V )( F ).Then by construction x is a desired element satisfying T G = xT x − .We recall that we have defined a set of roots R = R ( T G , G ) for T G , which hasan ordering inherited from ∆ = ∆( T, B ) via x . We set the a -data { a α } α ∈ R by theequality ˇ α ⊗ a α = 1 ∈ T G ( F ) , ∀ α ∈ R. Then we assume that X is close to 0, and the endoscopic invariants defined in SectionA.1 can be written asinv( X )( σ ) = Y σ ∈ R,α> ,σ − ( α ) < ˇ α ◦ α ( X ) ∈ T G ( F ) , σ ∈ Γ F , inv( T G )( σ ) = xn ( w σ ) σ ( x − ) ∈ T G ( F ) , σ ∈ Γ F . Cohomology classes construction.
Before proving explicit formulas for the endoscopicinvariants, we given an explicit construction of some 1-cocycles.We consider a chain of finite field extensions F ⊂ F ⊂ F , where F is a quadraticextension of F . We fix the nontrivial involution τ ∈ Gal( F /F ) and let T =ker(N F /F : F × → F × ). By local class field theory [Ser67, Theorem 2] we knowthat H ( F, T ( F )) ≃ {± } . We are going to give an explicit construction of theisomorphism.We let Σ F be the set of nonzero embeddings τ : F ֒ → F extending the fixedembedding F ֒ → F . We fix an element 1 = 1 F ∈ Σ( F ), which we view as thecanonical embedding of F into F . Lemma B.2.2.
We have the following identification T ( F ) = { y ∈ Σ( F ) → F × | y ( σ ) y ( στ ) = 1 } . The group Γ F acts on T ( F ) in the following way. For any τ ∈ Γ F and y ∈ T ( F ) , ( τ ( y ))( σ ) = τ ( y ( τ − σ )) , σ ∈ Σ( F ) . Proof.
We first verify that Γ F acts on T ( F ) using the formula defined in the state-ment. For any τ , τ ∈ Γ F , y ∈ T ( F ) and σ ∈ Σ( F ), we have( τ τ ( y ))( σ ) = ( τ τ )( y ( τ − τ − σ )) = τ ( τ ( y )( τ − σ )) = τ ( τ ( y ))( σ ) . Hence Γ F acts on T ( F ). LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 85
The group structure of T ( F ) is straightforward. We only need to verify that T ( F ) Γ F ≃ T = { z ∈ F × | zτ ( z ) = 1 } . For any τ ∈ Γ F and y ∈ T ( F ) Γ F , τ ( y )(1) = τ ( y ( τ − )) = y (1) , hence y ( τ ) = τ ( y (1)). In particular if τ ∈ Γ F , then τ F = 1 F . Hence y (1) ∈ F × , Γ F = F × . It follows that T ( F ) Γ F = { y : Σ( F ) → F × | y (1) τ ( y (1)) = 1 , σ ( y )(1) = y ( σ ) } , which can exactly be identified as the set { z ∈ F × | zτ ( z ) = 1 } via the morphism y → y (1) = z . (cid:3) Suppose that τ → y τ is a 1-cocycle of Γ F valued in T ( F ). Then for any σ, τ ∈ Γ F , we have y στ = y σ σ ( y τ ). Hence y στ (1) = y σ (1) σ ( y τ )(1) = y σ (1) σ ( y τ ( σ − )). Inparticular, when σ, τ ∈ Γ F , we have y στ (1) = y σ (1) σ ( y τ (1)). Hence the morphism τ → y τ (1) is a 1-cocycle of Γ F valued in F × . By Hilbert 90 [Gru67, 2.7], there exists z ∈ F × such that y τ (1) = τ ( z ) z − for any τ ∈ Γ F . We fix such a z . For θ ∈ Γ F such that θ | F is nontrivial, we consider the element y θ (1) zθ ( z ) ∈ F × . Lemma B.2.3.
The element y θ (1) zθ ( z ) ∈ F × is independent of θ .Proof. For θ , θ ∈ Γ F whose restriction to F are nontrivial, we are going to estab-lish the identity y θ (1) θ ( z ) z = y θ (1) θ ( z ) z . Equivalently, we only need to show y θ (1) y θ (1) = θ ( z ) θ ( z ) . By the cocycle condition on y τ , we have y θ (1) = y θ (1) θ ( y θ − θ ( θ − )) . We notice that θ − θ ∈ Γ F , hence y θ − θ ( θ − ) = y θ − θ (1) − = z ( θ − θ )( z ) − . It follows that y θ (1) y θ (1) = θ ( z ( θ − θ )( z − )) = θ ( z ) θ ( z − )and we have proved the lemma. (cid:3) We note the element y θ (1) zθ ( z ) ∈ F × by e y . Lemma B.2.4.
The element e y lies in F × . Proof.
To show e y = y θ (1) zθ ( z ) lies in F × , equivalently we are going to show that σ ( e y ) = σ ( y θ (1))( σθ )( z ) σ ( z ) = e y = y θ (1) θ ( z ) z, ∀ σ ∈ Γ F . By 1-cocycle condition, y σθ (1) = y σ (1) σ ( y θ ( σ − )) . When σ ∈ Γ F , we have σ ( y θ ( σ − )) = σ ( y θ (1)), and y σ (1) = σ ( z ) z − . Hence σ ( e y ) = y σθ (1) σ ( z ) z − ( σθ )( z ) σ ( z ) = y σθ (1)( σθ )( z ) z = e y. When σ ∈ Γ F and the restriction to F is nontrivial, we have σ ( y θ ( σ − )) = σ ( y θ (1) − ), and y σθ (1) = ( σθ )( z ) z − . Hence σ ( e y ) = y σ (1)( σθ )( z ) z − ( σθ )( z ) σ ( z ) = y σ (1) σ ( z ) z = e y. It follows that e y ∈ F × . (cid:3) We consider the element e y N F /F ( F × ) in F × / N F /F ( F × ). Lemma B.2.5.
The element e y N F /F ( F × ) is independent of the choice of z , F ,and the cocycle class τ → y τ in H ( F, T ( F )) .Proof. We first show the element is independent of z . If τ ( z ) z − = τ ( w ) w − for any τ ∈ Γ F , then zw − ∈ F , and y θ (1) θ ( z ) z = y θ (1) θ ( w ) w ( θ ( w − z ) w − z ) . Hence the claim follows.Then we show that the element is independent of the cocycle class of τ → y τ in H ( F, T ( F )). We let τ → z τ be a 1-coboundary. In other words, there ex-ists y ∈ T ( F ) such that z τ = τ ( y ) y − . In particular, z τ (1) = ( τ ( y ))(1) y − (1) = τ ( y ( τ − )) y − (1). When τ ∈ Γ F , we have z τ (1) = τ ( y (1)) y − (1). Hence we noticethat the element associated to y τ z τ with τ ∈ Γ F whose restriction to F is nontrivialis given by y τ (1) z τ (1) zy (1) τ ( zy (1)) . Therefore we have y τ (1) z τ (1) zy (1) τ ( zy (1)) y τ (1) zτ ( z ) = z τ (1) y (1) τ ( y (1)) = τ ( y ( τ − )) τ ( y (1)) = τ ( y (1) − y (1)) = 1 . Hence the claim follows.Finally we show the element is independent of 1 F . Different choices of 1 F yieldan automorphism of T ( F ), which, after applying the automorphism, do not affectthe cocycle class of τ → y τ in H ( F, T ( F )). Hence the claim follows as well. LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 87
It follows that the lemma holds. (cid:3)
From the discussion above we realize that we have a well-defined homomorphismfrom H ( F, T ( F )) to F , × / N F /F ( F × ) ≃ {± } . Lemma B.2.6.
The above construction gives an explicit isomorphism between H ( F, T ( F )) and F , × / N F /F ( F × ) .Proof. Following the above notations, we only need to show that if e y ∈ N F /F ( F × ),then the 1-cocycle τ → y τ is actually a 1-cobundary. Now for θ ∈ Γ F whoserestriction to F is trivial, we have e y = y θ (1) θ ( z ) z, and by assumption e y = θ ( w ) w for some w ∈ F × . On the other hand, since y τ (1) = τ ( z ) z − for any τ ∈ Γ F , up to replacing z by zw − , we may assume that w = 1.Hence e y = y θ (1) θ ( z ) z = 1 . In other words, we have y θ (1) = (cid:26) θ ( z − ) z − , θ ∈ Γ F and θ | F is nontrivial .θ ( z ) z − , θ ∈ Γ F . We define a 1-coboundary τ → z τ as follows. We let z τ = τ ( ξ ) ξ − , where for any θ ∈ Σ( F ), ξ ( θ ) = (cid:26) z, θ = 1 ,z − , θ = τ , , otherwise .. Then we can verify that y τ ( z τ ) − (1) = 1. Up to the 1-coboundary, we may assumethat y τ (1) = 1 for any τ ∈ Γ F . Similarly, since for any α ∈ Σ( F ), τ → y τ ( α ) isa 1-coboundary of Γ F in F × , we can run the same procedure as above, and up tomultiplying 1-coboundaries of the above form, we may assume that τ → y τ is a trivial1-cocycle from Γ F in T ( F ), i.e. y τ ( α ) = 1 for any α ∈ Σ( F ) and τ ∈ Γ F . Nowsince we are in characteristic zero situation, we know that the restriction morphism H ( F, T ( F )) → H ( F , T ( F ))is an injection [Ser13, p.15], whose morphism is exactly given by restricting cocyclesin H ( F, T ( F )) to H ( F , T ( F )). Since τ → y τ is trivial in H (Γ F , T ( F )), itfollows that τ → y τ is trivial in H ( F, T ( F )).It follows that the lemma has been proved. (cid:3) The computation.
Now we are ready to compute the transfer factors defined in thebeginning of the section. We let P X be the characteristic polynomial of X acting onEnd F ( V ), and we let P ′ ( X ) be its derivative. For any i ∈ I ∗ , we set C i = η [ F i : F ] − c − i a − i P ′ X ( a i ) . From Lemma B.1.4, we know that inv( X )inv( T G ) ∈ H ( T G ) ≃ {± } I ∗ . Proposition B.2.7.
We have inv( X )inv( T G ) = (sgn F i /F i ( C i )) i ∈ I ∗ . Proof.
We will use the explicit construction of cohomological invariants above tocompute inv( X )inv( T G ).We fix an element 1 F i of Σ( F i ), which we use to identify a canonical F -embeddingof F i in F . For any 1-cocycle λ of Γ F valued in T G ( F ), we define a function λ ( i, · ) : Γ F i → F × by λ ( τ ) f i, Fi = λ ( i, τ ) f i, Fi for any τ ∈ Γ F i . Then following the discussion above Lemma B.2.3, there exists z i ∈ F × such that λ ( i, τ ) = τ ( z i ) z − i for any τ ∈ Γ F i . Fix such a z i and θ i ∈ Γ F i whose restriction to F i is nontrivial.Then from Lemma B.2.6 we have the following equality λ = (sgn F i /F i ( λ ( i, θ i ) z i θ i ( z i ))) i ∈ I ∗ . In the following, we are going to compute the terms for inv( X ) and inv( T G ) explicitly.For any i ∈ I , we let b = δ ( i, T G ) and inv( X )separately.We first treat the computation for inv( T G ). By definition, for any τ ∈ Γ F i ,inv( T G )( τ ) f i, = ( xn ( w τ ) τ ( x − )) f i, = ( xn ( w τ ) τ x − τ − ) f i, . (1) When τ ∈ Γ F i ,( xn ( w τ ) τ x − τ − ) f i, = ( xn ( w τ ) τ x − ) f i, = ( xn ( w τ ) τ )( µ ( b ) − e b )= ( τ ◦ µ ( b ) − )( xn ( w τ )) e b . Here we recall that w τ is the image of x − τ ( x ) under the natural projectionfrom Norm T (SO( V )) → W . By definition, for any α ∈ { , ..., d } , x − τ ( x ) e α = x − τ xτ − e α = x − τ xe α = x − τ ( µ ( α ) f δ − ( α ) ) . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 89
When δ − ( α ) = ( j, σ ) ∈ Σ( V ), we have x − ( τ ( µ ( α ) f j,σ )) = τ ◦ µ ( α ) x − f j,τσ = τ ◦ µ ( α ) µ ( δ ( j, τ σ )) − e δ ( j,τσ ) . Hence, if we let w ′ τ be the bijection of Σ( V ) sending ( j, σ ) to ( j, τ σ ), then theimage of x − τ ( x ) in W is given by δ ◦ w ′ τ ◦ δ − . In particular w τ ( b ) = b . ByLemma B.2.1, we have n ( w τ )( e b ) = ( − r ( τ ) e b , where r ( τ ) = |{ ( j, σ ) ∈ Σ( V ) | δ ( j, τ σ ) < b < δ ( j, σ ) , δ ( j, σ ) + b = d + 1 }| . Therefore finally we get the following identityinv( T G ) f i, = µ ( b ) τ ◦ µ ( b ) − ( − r ( τ ) f i, , τ ∈ Γ F i . (2) Similarly, when τ ∈ Γ F i and the restriction of τ to F i is nontrivial, we haveinv( T G )( τ ) f i, = ( xn ( w τ ) τ x − ) f i,τ − = ( xn ( w τ ) τ )( µ ( d + 1 − b ) − e d +1 − b ) . By parallel argument as above situation, we have n ( w τ )( e d +1 − b ) = ( − r ( τ ) e b , where r ( τ ) = |{ ( j, σ ) ∈ Σ( V ) | d + 1 − b < δ ( j, σ ) , δ ( j, τ σ ) < b, δ ( j, b ) = b }| . Therefore we getinv( T G )( τ ) f i, = µ ( b ) τ ◦ µ ( d + 1 − b ) − ( − r ( τ ) f i, . Then we turn to the computation for inv( X ). By definition, for any τ ∈ Γ F i ,inv( X )( τ ) f i, = ( Y α ∈ R,α> ,τ − ( α ) < ˇ α ◦ α ( X )) f i, . The set of roots R = R ( T G , G ) has the following description. From Lemma B.1.3,any y ∈ T G acts via scalar on the basis { f i,σ } ( i,σ ) ∈ Σ( V ) . In other words, there exists y i,σ ∈ F × such that the following identity holds, y ( f i,σ ) = y i,σ f i,σ , ( i, σ ) ∈ Σ( V ) . Moreover we have y i,σ y i,στ i = 1. For any pair (( j, σ ) , ( j ′ , σ ′ )) ∈ Σ( V ) we let α (( j,σ ) , ( j ′ ,σ ′ )) be the homomorphism from T G to F × sending y ∈ T G to y j,σ y − j ′ ,σ ′ . Inparticular, the pairs (( j, σ ) , ( j ′ , σ ′ )) and (( j ′ , σ ′ τ j ′ ) , ( j, στ j )) give rise to the same ho-momorphism from T G to F × . Under the identification, we notice that R has thefollowing description. We consider the set of pairs (( j, σ ) , ( j ′ , σ ′ )) ∈ Σ( V ) such that ( j, σ ) = ( j ′ , σ ′ ) and δ ( j, σ ) + δ ( j ′ , σ ′ ) = d + 1. We equip the set with the followingequivalent relation (( j, σ ) , ( j ′ , σ ′ )) ∼ (( j ′ , σ ′ τ j ′ ) , ( j, στ j )) . Then the set R can be identified with the set of equivalence classes of such pairs.Suppose that (( j, σ ) , ( j ′ , σ ′ )) is a pair associated to a root α ∈ R . Then we have thefollowing properties. • α ( X ) = σ ( a j ) − σ ′ ( a j ′ ); • α > ⇔ δ ( j, σ ) < δ ( j ′ , σ ′ ); • For any z ∈ F × and ( j ′′ , σ ′′ ) ∈ Σ( V ),ˇ α ( z ) f j ′′ ,σ ′′ = (cid:26) zf j ′′ ,σ ′′ , if ( j ′′ , σ ′′ ) = ( j, σ ) or ( j ′ , σ ′ τ j ′ ) ,z − f j ′′ ,σ ′′ , if ( j ′′ , σ ′′ ) = ( j, στ j ) or ( j ′ , σ ′ ) ,f j ′′ ,σ ′′ , otherwise . By the description we find that for any τ ∈ Γ F i and α ∈ { α ∈ R | α > , τ − ( α ) < } represented by (( j, σ ) , ( j ′ , σ ′ )) ∈ Σ( V ) , the relation α > , τ − ( α ) < δ ( j, σ ) < δ ( j ′ , σ ′ ) , δ ( j, τ − σ ) > δ ( j ′ , τ − σ ′ ) . Moreover, since α ( X ) = σ ( a j ) − σ ′ ( a j ′ ) , we find that the inequality ˇ α ◦ α ( X ) f i, = f i, happens only when ( i, ∈ { ( j, σ ) , ( j, στ j ) , ( j ′ , σ ′ τ j ′ ) , ( j ′ , σ ′ ) } . Modulo the equiva-lence, we may assume that ( i,
1) = ( j, σ ) or ( j ′ , σ ′ τ j ′ ).When ( i,
1) = ( j, σ ), the root associated to the pair (( i, , ( j ′ , σ ′ )) givesˇ α ◦ α ( X ) f i, = ( a i − σ ′ ( a j ′ )) f i, , and the possible pairs of ( j ′ , σ ′ ) are parametrized by the set R + ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( i, < δ ( j, σ ) , δ ( i, τ − ) > δ ( j, τ − σ ) , δ ( j, σ )+ δ ( i, = d +1 } . When ( i,
1) = ( j ′ , σ ′ ) the root associated to the pair (( j, σ ) , ( i, α ◦ α ( X ) f i, = ( σ ( a j ) − a i ) − f i, , and the possible pairs of ( j, σ ) are parametrized by the set R − ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, σ ) < δ ( i, , δ ( j, τ − σ ) > δ ( i, τ − ) , δ ( j, σ )+ δ ( i, = d +1 } . LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 91
It follows that for any τ ∈ Γ F i , we haveinv( X )( τ ) f i, = Y ( j,σ ) ∈ R + ( τ ) ( a i − σ ( a j )) Y ( j,σ ) ∈ R − ( τ ) ( σ ( a j ) − a i ) − f i, . Based on above calculations, we find that whenever τ ∈ Γ F i , we haveinv( T G )( τ )inv( X )( τ ) f i, = µ ( b ) τ ◦ µ ( b ) − ( − r ( τ ) Y ( j,σ ) ∈ R + ( τ ) ( a i − σ ( a j )) Y ( j,σ ) ∈ R − ( τ ) ( σ ( a j ) − a i ) − f i, , where r ( τ ) = |{ ( j, σ ) ∈ Σ( V ) | δ ( j, τ σ ) < b < δ ( j, σ ) , δ ( j, σ ) + δ ( i, = d + 1 }| ,R + ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, τ − σ ) < δ ( i, < δ ( j, σ ) , δ ( j, σ ) + δ ( i, = d + 1 } ,R − ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, σ ) < δ ( i, < δ ( j, τ − σ ) , δ ( j, σ ) + δ ( i, = d + 1 } . In the following we are going to find an element z i ∈ F × such that for any τ ∈ Γ F i ,we have τ ( z i ) z − i = inv( X )( τ )inv( T G )( τ ) f i, . We let S = { ( j, σ ) ∈ Σ( V ) | δ ( i, < δ ( j, σ ) , δ ( j, σ ) + δ ( i, = d + 1 } ,z i = µ ( b ) − Y ( j,σ ) ∈ S ( a i − σ ( a j )) − . Then by definition, we have τ ( z i ) z − i = µ ( b ) τ ◦ µ ( b ) − Y ( j,σ ) ∈ S ( a i − σ ( a j )) Y ( j,σ ) ∈ S ( τ ) ( a i − σ ( a j )) − , where S ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( i, < δ ( j, τ − σ ) , δ ( j, τ − σ ) + δ ( i, = d + 1 } . We notice that by definition, S − S ∩ S ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, τ − σ ) ≤ δ ( i, < δ ( j, σ ) , δ ( j, σ ) + δ ( i, = d + 1 } G { ( j, σ ) ∈ Σ( V ) | δ ( i, < δ ( j, σ ) , δ ( j, σ ) + δ ( i, = d + 1 , δ ( j, τ − σ ) + δ ( i,
1) = d + 1 } . But when δ ( i,
1) = δ ( j, τ − σ ) we have j = i, σ = 1, which contradicts the fact that δ ( i, < δ ( j, σ ), and when δ ( j, τ − σ ) + δ ( i,
1) = d + 1 we have ( j, σ ) = ( i, τ i ), whichalso contradicts the fact that δ ( j, σ ) + δ ( i, = d + 1. It follows that S − S ∩ S ( τ ) = R + ( τ ) . Similarly, we find that S ( τ ) − S ∩ S ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, σ ) ≤ δ ( i, < δ ( j, τ − σ ) , δ ( j, τ − σ ) + δ ( i, = d + 1 } G { ( j, σ ) ∈ Σ( V ) | δ ( i, < δ ( j, τ − σ ) , δ ( j, τ − σ ) + δ ( i, = d + 1 , δ ( j, σ ) + δ ( i,
1) = d + 1 } . When δ ( i,
1) = δ ( j, σ ), we have ( j, σ ) = ( i,
1) = ( j, τ − σ ), hence we also have acontradiction. Similarly, when δ ( j, σ ) + δ ( i,
1) = d + 1, we have ( j, σ ) = ( i, τ i ) whichalso contradicts the fact that δ ( j, τ − σ ) + δ ( i, = d + 1. Hence S ( τ ) − S ∩ S ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, σ ) < δ ( i, < δ ( j, τ − σ ) , δ ( j, τ − σ )+ δ ( i, = d +1 } . But the condition δ ( j, τ − σ ) + δ ( i, = d + 1 is equivalent to ( j, τ − σ ) = ( i, τ i ), whichin turn is equivalent to ( j, σ ) = ( i, τ i ), hence we realize that S ( τ ) − S ∩ S ( τ ) = R − ( τ ) . It follows that τ ( z i ) z − i = µ ( b ) τ ◦ µ ( b ) − ( − | R − ( τ ) | Y ( j,σ ) ∈ R + ( τ ) ( a i − σ ( a j )) Y ( j,σ ) ∈ R − ( τ ) ( σ ( a j ) − a i ) − . Finally by the bijective map ( j, σ ) → ( j, τ σ ) and the fact that δ ( j, σ ) + δ ( i, = d + 1is equivalent to δ ( j, τ σ ) + δ ( i, = d + 1, we arrive at the equality r ( τ ) = | R − ( τ ) | . It follows that τ ( z i ) z − i f i, = inv( T G )( τ )inv( X )( τ ) f i, , ∀ τ ∈ Γ F i . Now by Lemma B.2.6, we are going to compute the following element(inv( T G )( τ )inv( X )( τ ) f i, ) z i τ ( z i )for some τ ∈ Γ F i whose restriction to F i is nontrivial.By the computation that we have done above, whenever τ ∈ Γ F i whose restrictionto F i is nontrivial, we haveinv( T G )( τ ) f i, = µ ( b ) τ ◦ µ ( d + 1 − b ) − ( − r ( τ ) f i, , where r ( τ ) = |{ ( j, σ ) ∈ Σ( V ) | δ ( i, τ i ) < δ ( j, σ ) , δ ( j, τ σ ) < b, δ ( j, σ ) = δ ( i, }| , and inv( X )( τ ) f i, = Y ( j,σ ) ∈ R + ( τ ) ( a i − σ ( a j )) Y ( j,σ ) ∈ R − ( τ ) ( σ ( a j ) − a i ) − f i, , LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 93 where R + ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( i, < δ ( j, σ ) , δ ( j, τ − σ ) < δ ( i, τ i ) , δ ( j, σ ) + δ ( i, = d + 1 } ,R − ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, σ ) < δ ( i, , δ ( j, τ − σ ) > δ ( i, τ i ) , δ ( j, σ ) + δ ( i, = d + 1 } . Moreover, by definition, τ ( z i ) = τ ◦ µ ( b ) − Y ( j,σ ) ∈ S ( τ ) ( − a i − σ ( a j )) − . We set S ′ ( τ ) = { ( j, σ ) ∈ Σ( V ) | b < δ ( j, τ − στ j ) , δ ( j, τ − στ j ) + δ ( i, = d + 1 } . Using the fact that τ j ( a j ) = − a j , we realize that τ ( z i ) = ( − | S ′ ( τ ) | τ ◦ µ ( b ) − Y ( j,σ ) ∈ S ′ ( τ ) ( a i − σ ( a j )) − . It follows that(inv( T G )( τ )inv( X )( τ ) f i, ) z i τ ( z i )= τ ( µ ( b ) µ ( d + 1 − b )) − ( − r ( τ )+ | R − ( τ ) | + | S ′ ( τ ) | Y ( j,σ ) ∈ Σ( V ) ( a i − σ ( a j )) − m ( j,σ ) f i, . Here we set m = 1 R − ( τ ) + 1 S ′ ( τ ) + 1 S − R + ( τ ) , where for a subset U of Σ( V ), 1 U is the characteristic function of U .By definition, R + ( τ ) ⊂ S , and S − R + ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( i, < δ ( j, σ ) , δ ( i, τ i ) ≤ δ ( j, τ − σ ) , δ ( j, σ )+ δ ( i, = d +1 } . Now since R − ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, σ ) < δ ( i, , δ ( j, τ − σ ) > δ ( i, τ i ) , δ ( j, σ )+ δ ( i, = d +1 } , we notice that when δ ( j, τ − σ ) = δ ( i, τ i ), we have ( j, σ ) = ( i, R − ( τ )can also be written as R − ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, σ ) < δ ( i, , δ ( j, τ − σ ) ≥ δ ( i, τ i ) , δ ( j, σ )+ δ ( i, = d +1 } . It follows that the sets S − R + ( τ ) and R − ( τ ) are disjoint, and their union gives { ( j, σ ) ∈ Σ( V ) | δ ( i, = δ ( j, σ ) , δ ( j, τ − σ ) ≥ δ ( i, τ i ) , δ ( j, σ ) + δ ( i, = d + 1 } . On the other hand, S ′ ( τ ) = { ( j, σ ) ∈ Σ( V ) | b < δ ( j, τ − στ j ) , δ ( j, τ − στ j ) + δ ( i, = d + 1 } can also be written as S ′ ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, τ − σ ) < δ ( i, τ i ) , ( j, τ − στ j ) = ( i, τ i ) } . Moreover, the relation ( j, τ − στ j ) = ( i, τ i ) is equivalent to ( j, σ ) = ( i, τ i ). Hence S ′ ( τ ) = { ( j, σ ) ∈ Σ( V ) | δ ( j, τ − σ ) < δ ( i, τ i ) , δ ( j, σ ) + δ ( i, = d + 1 } . It follows that the sets S − R + ( τ ), R − and S ′ ( τ ) are disjoint to each other, andtheir union can be written as Σ( V ) − { ( i, , ( i, τ i ) } .It follows that the product is equal to Y ( j,σ ) ∈ Σ( V ) −{ ( i, , ( i,τ i ) } ( a i − σ ( a j )) − , which is equal to ( P ′ X ( a i ) / a i ) − .Using the bijection ( j, σ ) → ( j, τ − σ ) and the same argument as before we havethat | R − ( τ ) | = r ( τ ). From the construction of the element x ∈ T G , we have µ ( b ) µ ( d + 1 − b ) = η/ − b +1 [ F i : F ] σ ( c i ) − = η/ − b +1 [ F i : F ] c − i , if b ≤ d/ ,µ ( b ) µ ( d + 1 − b ) = η/ − b [ F i : F ] σ ( c i ) − = η/ − b [ F i : F ] c − i , if b > d/ . We also have τ ( c i ) = c i . It follows that the desired invariant is equal to τ ( η − b +1 [ F i : F ] c − i ) − ( − | S ′ ( τ ) | a i P ′ X ( a i ) − = (cid:26) ( η/ − b +1 [ F i : F ] c − i ) − ( − | S ′ ( τ ) | a i P ′ X ( a i ) − , if b ≤ d/ , ( η/ − b [ F i : F ] c − i ) − ( − | S ′ ( τ ) | a i P ′ X ( a i ) − , if b > d/ , Finally by the fact that | S ′ ( τ ) | = d − b − b ≤ d/
2, and d − b whenever b > d/
2, it follows that the desired invariant is equal to4 η − [ F i : F ] − c i a i P ′ X ( a i ) − . Up to square class we may ignore the 4 part. Through taking the inverse and weobtain the eventual formula η [ F i : F ] c − i a − i P ′ X ( a i ) . (cid:3) B.3.
Explicit examples.
In this section, following [Wal10, 11.4], we are going tocalculate the regular germs for explicit regular semi-simple elements using TheoremA.2.1. We compute the invariants in Theorem A.2.1 explicitly via Proposition B.2.7.We focus on the case when d is even and d ≥
4, since only in this case will we have | Nil reg ( so ( V )) | > V, q ) is split and quasi-split butnot split separately. We are going to pick up particular regular semi-simple elementsfollowing the description in Section B.1.
LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 95 • When (
V, q ) is split. We fix two quadratic extensions F , F of F such that theproduct of the two quadratic characters sgn F /F sgn F /F is the trivial characteron F × , i.e. F ≃ F as a field extension of F . For any i = 1 ,
2, we fix a i ∈ F × i such that τ i ( a i ) = − a i and a = ± a , where τ i is the nontrivial elementin Gal( F i /F ). For any c ∈ F × , we consider F (resp. F ) equipped withquadratic form c N F /F (resp. − c N F /F ). Let e Z be a hyperbolic space ofdimension d −
4. Then we can find that F ⊕ F ⊕ e Z ≃ V as split quadraticspace. We fix a maximal split subtorus e T of the special orthogonal groupassociated to e Z . Let e t ( F ) be the F -points of the Lie algebra of e T . We fix aregular semi-simple element e S in e t ( F ).We consider the regular semi-simple element X F ∈ so ( V )( F ) that actsby multiplication by a (resp. a ) on F (resp. F ) and by e S on e Z . FromLemma B.1.5 we know that the conjugacy classes of X F within the stableconjugacy class is determined by c ∈ F × / N F /F ( F × ). In the following we willnote X + F to be the element corresponding to c = c + such that sgn F /F (2 c ) =sgn F /F (N F /F ( a ) − N F /F ( a )) and X − F corresponds to c = c − such thatsgn F /F ( c − ) = − sgn F /F ( c + ).To keep the notation consistent with the following quasi-split but not splitcase, we let η = 1 in our situation. • When (
V, q ) is quasi-split but not split. There exists a quadratic extension
E/F and an element η ∈ F × such that the anisotropic kernel of q is of theform η · N E/F . We fix two quadratic extensions F , F of F such that theproduct of two quadratic characters sgn F /F sgn F /F is equal to sgn E/F . For i = 1 ,
2, suppose that a i ∈ F × i such that τ F i ( a i ) = − a i and a = ± a .We fix c ∈ F × such that sgn E/F (2 ηc N F /F ( a )) = 1. Then we can find that F ⊕ F ⊕ e Z ≃ V as quadratic space where we consider F (resp. F ) equippedwith quadratic form c N F /F (resp. − c N F /F ).Parallel with the split situation, we consider the regular semi-simple ele-ment X F ∈ so ( V )( F ) that acts by multiplication by a (resp. a ) on F (resp. F ) and by e S on e Z . Again from Lemma B.1.5 we note X + F the element cor-responding to c = c + such that sgn F /F (2 c ) = sgn F /F ( η )sgn F /F (N F /F ( a ) − N F /F ( a )) and X − F corresponds to c = c − such that sgn F /F ( c − ) = − sgn F /F ( c + ).Our goal is to compute the regular germ associated to X ± F . We follow the proofof [Wal10, 11.4]. Lemma B.3.1.
Suppose that
O ∈
Nil reg ( so ( V )( F )) . (1) We have Γ O ( X qd ) = 1 for any X qd ∈ t qd . (2) Suppose that d is even and d ≥ . We have the equality Γ O ( X + F ) − Γ O ( X − F ) = sgn F /F ( νη ) if O = O ν with ν ∈ N V .Proof. The first statement follows from [BP15, Section 3.4]. The second statementfollows from Theorem A.2.1 and the explicit computation of the invariants followingProposition B.2.7 . We will go over the proof of [Wal10, 11.4] in the following.For the sign ζ = ± , we note T ζ ( F ) the maximal subtorus of SO( V )( F ) such that X ζF ∈ t ζ ( F ). Suppose that ν ∈ N V and N ∈ O ν . We are going to evaluate the in-variant inv( X ζF )inv( T ζ ) / inv T ζ ( N ). All the elements belong to the cohomology group H ( T ζ ). From Lemma B.1.4 we have H ( T ζ ) = {± }×{± } . The invariants dependon the choice of pining. Following the normalization of the basis for quadratic spacein the beginning of Section B.2 we make choice following above discussion by makingthe element η as above and multiplying by 2( − d/ − . Then from Proposition B.2.7we have the following explicit formula for inv( X ζF )inv( T ζ ). We haveinv( X ζF )inv( T ζ ) =(sgn F /F (( − d/ − η ( c ζ ) − a − P ′ ( a )) , sgn F /F (( − d/ − η ( c ζ ) − a − P ′ ( a )))where P is the characteristic polynomial of X F acting on V and P ′ is the derivative.We let ( ± e s j ) j =3 ,...,d/ be the eigenvalues of e S acting on e Z . They belong to F × since e T is split. Then we have P ( T ) = ( T + N F /F ( a ))( T + N F /F ( a )) Y j =3 ,...,d/ ( T − e s j ) . Then a − P ′ ( a ) = 2( − N F /F ( a ) + N F /F ( a )) Y j =3 ,...,d/ ( − N F /F ( a ) − e s j )= 2( − d/ − (N F /F ( a ) − N F /F ( a )) Y j =3 ,...,d/ (N F /F ( a ) + e s j )Note here we have e s j +N F /F ( a ) = N F /F ( e s j + a ), therefore sgn F /F ( e s j +N F /F ( a )) =1. Similar calculation can be obtained when one switch F to F . It turns out thatwe have inv( X ζF )inv( T ζ ) = (sgn F /F (2 η ( c ζ ) − (N F /F ( a ) − N F /F ( a ))) , sgn F /F (2 η ( c ζ ) − (N F /F ( a ) − N F /F ( a )))) . By the definition of c ζ , we getinv( X ζF )inv( T ζ ) = ( ζ , ζ ) , LOCAL TF FOR THE LOCAL GGP FOR SPECIAL ORTHOGONAL GROUPS 97 where one can identify ζ as an element of {± } .The pining is determined by the standard fixed regular element N ∗ , where fromthe notation in Section B.2, we have N ∗ = P j =1 ,...,d/ X α j . Note ν ∗ the element of N V such that N ∗ ∈ O ν ∗ . We can define a cocycle d N of Gal( F /F ) in T ζ as follows.If ν = ν ∗ , d N = 1. If ν = ν ∗ we set E N = F ( p νν ∗ ). Then, for σ ∈ Gal(
F /F ), we have d N ( σ ) = 1 if σ ∈ Gal(
F /E N ) and d N ( σ ) = − T ζ ( N ) isthe image in H ( T ) of the cocycle d N . We can calculate the imageinv T ζ ( N ) = (sgn F /F ( ν/ν ∗ ) , sgn F /F ( ν/ν ∗ )) . In fact, we can force that sgn E ( ν/ν ∗ ) = 1 following the parametrization of Nil reg ( so ( V ))in Section 2.1. Theninv T ζ ( N ) = (sgn F /F ( ν/ν ∗ ) , sgn F /F ( ν/ν ∗ )) . The element N ∗ stabilize the hyperplane generated by e , ..., e d/ − , e d/ + e d/ , e d/ , ..., e d .The anisotropic kernel of the restriction of the form q to the hyperplane is the re-striction of q to the line e d/ + e d/ , or say q ( e d/ + e d/ ) = η , therefore we have ν ∗ = η . Finallyinv( X ζF )inv( T ζ ) / inv T ζ ( N ) = ( ζ sgn F /F ( νη ) , ζ sgn F /F ( νη )) . It follows that we have Γ O ν ( X ζF ) = (cid:26) , sgn F /F ( νη ) = ζ , , sgn F /F ( νη ) = − ζ . This is exactly the identity that we want. (cid:3)
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School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
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