The Local Gan-Gross-Prasad Conjecture for Special Orthogonal Groups over Archimedean Local Fields
aa r X i v : . [ m a t h . N T ] F e b THE LOCAL GAN-GROSS-PRASAD CONJECTURE FOR SPECIALORTHOGONAL GROUPS OVER ARCHIMEDEAN LOCAL FIELDS
CHENG CHEN
Abstract.
In the paper, we prove the local Gan-Gross-Prasad conjecture for special orthogonalgroups over archimedean local fields for generic local L -parameters. The non-archimedean casewas proved by C. Moeglin and J.-L. Waldspurger in [MW12]. When the local L -parameters aretempered, the conjecture was proved by Waldspurger in [Wal09] for non-archimedean local fieldsand by Zhilin Luo in [Luo20] for archimedean local fields. Contents
1. Introduction 22. The Local Gan-Gross-Prasad Conjecture 42.1. The Gan-Gross-Prasad triples 42.2. L -parameters and Vogan packets 52.3. Component groups 62.4. Statements of our results 73. Genericity and irreducibilty 74. Conjectures 1.1 and 2.1 over the complex field 105. Conjectures 1.1 and 2.1 over the real field 115.1. A technical lemma 135.2. Construction and bijectiveness of F ,s F ,s . Introduction
The local Gan-Gross-Prasad conjecture for classical groups over any local field F ( [GGP12]) is anextension to general classical groups of the original Gross-Prasad conjecture for special orthogonalgroups in [GP92] and [GP94].Let ( V, V ′ ) be a pair of non-degenerate quadratic spaces defined over a local field F such that V ′ is a subspace of V and its orthogonal complement V ′⊥ in V is odd dimensional and split. Let e G = SO( V ) × SO( V ′ ) be the product of two special orthogonal groups, and e H the subgroup of e G defined by the semi-direct product of the image ∆ SO ( V ′ ) of the diagonal embedding of SO( V ′ )into b G with the unipotent subgroup e N of SO( V ) defined by the totally isotropic flag determined by V ′⊥ . Fix a generic character e ν of e N which extends to e H . For any irreducible Casselman-Wallachrepresentation e π of e G , define the multiplicity m ( e π ) = dim Hom e H ( e π, e ν ) . The Multiplicity One Theorem that m ( e π ) ≤ F isnonarchimedean and in [SZ12] and [JSZ10] when F is archimedean. The relevant pure inner forms e G α of e G are classified by α ∈ H ( F, SO ( V ′ )) = H ( F, e H ). We write that e G α = SO ( V α ) × SO ( V ′ α ),where ( V α , V ′ α ) is a pair of quadratic spaces such that V α = V ′ α ⊕ V ′⊥ . Then one can define e H α , e ν α and the multiplicity accordingly. Moreover, e G α has the same L -group as e G . For every generic L -parameter e ϕ : W F → L e G where W F is the Weil-Deligne group of F , let Π e ϕ ( e G α ) be the associated L -packet of e G α . Conjecture 1.1 ( [GGP12]) . For every generic L -parameter e ϕ : W F → L G , we have X α ∈ H ( F, e H ) X e π ∈ Π e ϕ ( e G α ) m ( e π ) = 1 . A refinement of Conjecture 1.1 that determines when the multiplicity m ( e π ) is not equal to 0 willbe discussed in Conjecture 2.1.The goal of this paper is to prove Conjecture 1.1, the local Gan-Gross-Prasad conjecture for spe-cial orthogonal groups over archimedean local fields for general generic local L -parameters (Theorem2.1 in Section 2.4), based on the case of tempered local L -parameters. When the local L -parametersare tempered, Conjecture 1.1 was proved by Zhilin Luo in his paper [Luo20], following the idea ofthe pioneering work of J.-L. Waldspurger in [Wal09] for nonarchimedean local fields. We also studyConjecture 2.1. We prove that Conjecture 2.1 for generic L -parameters holds over the complexfield C (Theorem 2.2 in Section 2.4 or Theorem 4.1). However, for the real field case, we deduceConjecture 2.1 for general generic L -parameters from that for tempered L -parameters (Theorem2.1 in Section 2.4). While Conjecture 2.1 for tempered L -parameters is our on-going work jointwith Zhilin Luo.Our proof of Conjecture 1.1 follows philosophically the ideas of C. Moeglin and Waldspurgerin [MW12], which proves the conjecture for general generic local L -parameters over nonarchimedeanlocal fields, based on the work of Waldspurger ( [Wal09]) for tempered local L -parameters. Thereare three key steps in their proof: (1) the relation between the local Bessel functionals and theparabolic induction, (2) the mathematical induction to prove the co-dimension one case with theresults of the tempered case, and (3) the reduction to the co-dimension one case. More specifically,Moeglin and Waldspurger proved a multiplicity formula that shows the relation between the localBessel functionals and the parabolic inductions, and with this multiplicity formula, they can reducethe generic co-dimension one case to the tempered case by mathematical induction. With the ame multiplicity formula, they reduce the general case to the co-dimension one case by using asupercuspidal representation in the relevant parabolic induction.Over archimedean local fields, there are no supercuspidal representations. However, we are ableto establish parallel arguments along the lines of Moeglin and Waldspurger by using a principalseries representation. Such an idea goes back to the work of Jiang-Sun-Zhu in [JSZ10] on theirproof of the uniqueness of Bessel models over archimedean local fields, and has been recently usedby H. Xue in his work on the local Gan-Gross-Prasad conjecture for the unitary group case overarhimedean local fields ( [Xue20]). Another difficulty is the proof of the multiplicity formula thatmanifests the relation between local Bessel functionals and the parabolic inductions, due to thatthe distributional analysis involved in the proof over archimedean local fields is much more subtlethan that over nonarchimedean local fields. The complicated nature of the parameters makes theorthogonal case more difficult than the unitary case, especially in the analysis of the ”descent” onthe open orbit in Section 5.3, and our method can be applied to other cases. In order to demonstrateour arguments, a proof of the complex case is given in Section 4 in the language of distributiontheory, and a nonzero Bessel functional with required properties in this case is constructed explicitly.From our work in the complex case, we observe that the multiplicity formula can be deduced fromthe relation between the equivariant tempered distributions on a given manifold and the equivarianttempered distributions on an open submanifold of the manifold.Although the distributional analysis gets much more involved in the archimedean case, withthe results on Schwartz homologies developed by Chen-Sun in [CS20], we are still able to obtaina sufficient condition for the bijectiveness of the restriction map from equivariant distributions ona Nash manifold to that of its open Nash submanifold in Appendix A and Appendix B. This isthe main technical ingredient in our proof of the multiplicity formula in the archimedean case, andwe use it to prove a techical lemma in Section 5.1. In Section 5.2, we do analysis on the closedorbits and prove a vanishing result which shows the restriction map from the invariant functionalsof the space of Schwartz sections on a Nash manifold to that of its open Nash submanifold isbijective. In Section 5.3, we do analysis on the open Nash submanifold, and prove some vanishingresults that show how the invariant functional of Schwartz sections on the open orbit descendsto Bessel functionals associated to a smaller Gan-Gross-Prasad triple. Our proof of the vanishingresults is based on infinitesimal characters of representations, so it is independent on the choiceof parabolic subgroups in the even orthogonal case (Remark 3.1). With the results in Section 5.2and Section 5.3, we can construct a descent map between the spaces of Bessel functionals for twopairs of representations, whose bijectiveness gives a multiplicity formula at the end of Section 5.3.With this multiplicity formula, in Section 5.5, we can use mathematical induction to prove theco-dimension one case by using Luo’s results of the tempered case in [Luo20], the proof of which issimilar to that in [MW12] for the nonarchimedean case. Finally, we take parabolic induction of thecompleted tensor product of one representation and a principal series representation to reduce thegeneral case to the co-dimension one case in Section 5.6.It is worthwhile to mention that the local Gan-Gross-Prasad conjecture at all local places impliesthe uniqueness in the statement of the global Gan-Gross-Prasad conjecture over number fields( [GGP12, Conjectures 24.1 and 26.1]). As shown in the work of Jiang and Zhang in [JZ20], thelocal Gan-Gross-Prasad conjecture at all local places is one of the key ingredients in their proof ofirreducibility of the twisted automorphic descents.The paper is organized as follows. In Section 2, we introduce concepts pertaining to the localGan-Gross-Prasad conjecture to state our main results, Theorem 2.1 and Theorem 2.2. And inSection 3, we express representations in generic Vogan packets as normalized parabolic inductions f tensor products of tempered representations and essentially discrete series representations. InSection 4, we will give a constructive proof of the local Gan-Gross-Prasad conjecture over thecomplex field in terms of equivariant distributions. In Section 5, we prove a technical lemma thatimplies the vanishing of the Schwartz homologies of certain representations. Then we make use ofSchwartz analysis in our proofs of some multiplicity formulas in the co-dimension one case, whichare essentially some distributional analysis. Finally we reduce the generic case to the tempered caseby using the proved multiplicity formulas. In Appendix A we review the properties of Schwartzinductions and Schwartz homologies discussed in [CS20], and in Appendix B we recall the Harish-Chandra correspondence and use it to parameterize the infinitetisimal characters. Acknowledgement.
I would like express my sincere thanks to my advisor Prof. Dihua Jiangfor encouraging me to study this subject and giving me generous advices while I was working onthis topic and writing this article. I thank Fangyang Tian for very helpful discussions on a fewtechnical issues related to Casselman-Wallach representations. I thank Zhilin Luo for his patientexplanation and clarification of the results in his thesis. I thank Chen Wan for important commentsthat leads my re-organization of Section 4.2.
The Local Gan-Gross-Prasad Conjecture
In this section, we will define every ingredients of the local Gan-Gross-Prasad conjecture ofspecial orthogonal groups over archimedean local fields and give a complete statement of that.2.1.
The Gan-Gross-Prasad triples.
In this subsection, we will define the Gan-Gross-Prasadtriples in the setting of special orthogonal groups over an archimedean local field F .Let F = R or C . For a non-degenerate quadratic space ( V, q ) over F and its non-degeneratequadratic subspace ( V ′ , q ′ ), the pair ( V, V ′ ) is called admissible if the orthogonal complement V ′⊥ of V ′ is odd dimensional and split, that is, V ′⊥ = X r ⊕ Y r ⊕ C v , where X r , Y r are maximal totallyisotropic subspaces in V ′⊥ and v is an anisotropic vector orthogonal to X r ⊕ Y r . There is a basis v , · · · , v r of X r and a basis v − , · · · , v − r of Y r such that q ( v i , v − j ) = δ i, − j , for i, j ∈ {± , · · · , ± r } . Let G = SO ( V, q ) be the special orthogonal group on (
V, q ), and regard G ′ = SO ( V ′ , q | V ′ ) asthe subgroup of G that stabilizes every vector in V ′⊥ . We let P be the parabolic subgroup of G with Levi decomposition P = M N stabilizing the following totally isotropic flag Fh v i ⊂ h v , v i ⊂ · · · ⊂ h v , · · · , v r i . For a nontrivial unitary additive character ψ of F , define ν of N to be ν ( n ) = r − X i =0 q ( v − i − , nv i ) . If we denote X i = h v , · · · , v i i and X r +1 = X r ⊕ C v , we can define a subgroup S r = { g ∈ G : ( g − X i ⊂ X i − , ≤ i ≤ r + 1 } , which is often called a Bessel subgroup of G associated to the pair ( F , v ). And we have S r = G ′ ⋊ N so ν induces a character ν r on S r , which is called a generic Bessel character . efine e G = G × G ′ , e H = ∆ G ′ ⋊ N ⊂ S r × G ′ and e ν is the character of e H induced from thecharacter ν of N . Then ( e G, e H, e ν ) is called the Gan-Gross-Prasad triple associated to the admissiblepair ( V, V ′ ) and the co-dimension of this Gan-Gross-Prasad triple is defined to be 2 r + 1.2.2. L -parameters and Vogan packets. In this subsection, we review local L -parameters andlocal L -packets for special orthogonal groups over archimedean local fields, define pure inner forms ofspecial orthogonal groups over archimedean local fields and the local Vogan packet of a L -parameter.Let G be a real reductive group, K is its maximal compact subgroup, and g C is the complexifi-cation of the real Lie algebra of G . A smooth representation ( π, V ) of G is said to be admissibleif the ( g C , K )-module V K consisting of the K -finite vectors of V is admissible, that is, it is a di-rect sum of irreducible representations of K with finite multiplicities. We call a ( g C , K )-module V K Harish-Chandra if it is admissible and Z ( g C )-finite, where Z ( g C ) is the center of the universalenveloping algebra U ( g C ). A Casselman-Wallach representation ( π, V ) of G is a smooth Fr´echetrepresentation of moderate growth whose ( g C , K )-module is Harish-Chandra (see [Cas89] [Wal94]).The Casselman-Wallach globalization theorem (see [BK14]) gives categorical equivalence betweenthe category of Harish-Chandra ( g C , K )-modules and Casselman-Wallach representations of G .The local Langlands group L F of an archimedean local field F is equal to the Weil-Deligne group W F = (cid:26) C × when F = C , C × ∪ C × j when F = R , where j satisfies j = − jzj = − z . A local L -parameter ϕ of G = SO ( V, q ) is a b G -conjugacyclass of smooth morphism ϕ : L F → L G = b G ⋊ Gal ( E/F ) , such that the elements in Imϕ are semisimple in L G , where E is the splitting field of the non-degenerate quadratic space ( V, q ). A parameter ϕ is called tempered if Imϕ is bounded.The local Langlands correspondence of real reductive groups was established by Langlands in[Lan88], which asserts that every irreducible finitely generated ( g C , K )-module of G can be assignedto a local L -parameter ϕ , and for every L -parameter ϕ , only finitely many equivalence classes of( g C , K )-modules are associated to ϕ , and their completions by Casselman-Wallach globalizationtheorem form the local L -packet Π ϕ ( G ) of the local L -parameter ϕ .For G = SO ( V, q ), the pure inner forms of G are the groups G α ( α ∈ H ( F, G )) obtained by innertwisting by elements α , and they are characterized as special orthogonal groups G α = SO ( V α , q α )of quadratic spaces ( V α , q α ) over F satisfyingdim( V ) = dim( V α ) , disc( V ) = disc( V α ) . When F = C , the special orthogonal group G = SO ( V, q ) is the unique pure inner form of itself.When F = R , the pure inner forms of SO ( p, q ) are G α = SO ( p α , q α ) where p + q = p α + q α and p − p α is even. It is straightforward that every G has a unique quasisplit pure inner form G α qs upto isomorphism.As the pure inner forms of G shares the same L -group, a parameter ϕ of G can be regarded asa parameter ϕ of any pure inner form G α . A parameter ϕ is called generic if for the quasi-splitpure inner form G α qs of G , Π ϕ ( G α qs ) contains a generic representation, that is, a member with anonzero Whittaker model with respect to a certain Whittaker datum for G α qs . The Vogan packetΠ Vogan ϕ for a generic parameter ϕ is defined byΠ Vogan ϕ = a α ∈ H ( F,G ) Π ϕ ( G α ) . hen we consider the Gan-Gross-Prasad triple ( e G, e H, e ν ) for an admissible pair ( V, V ′ ). We calla pure inner form e G α of e G is relevant if e G α = SO ( V α ) × SO ( V ′ α ) where SO ( V ′ α ) is a pure innerform of SO ( V ′ ) and V α = V α ⊕ V ′⊥ . The relevant pure inner forms of e G are parameterized by H ( F, SO ( V ′ )) = H ( F, e H ). With the flag F and the anisotropic vector v , we are able to define e H α and e ν α , then we have a Gan-Gross-Prasad triple ( e G α , e H α , e ν α ). For a L -parameter e ϕ = ϕ × ϕ ′ , e ϕ is called generic if both ϕ and ϕ ′ are generic, and the Vogan packet Π Vogan e ϕ of e ϕ is defined byΠ Vogan e ϕ = a α ∈ H ( F, e H ) Π ϕ ( G α ) × Π ϕ ( G ′ α ) = a α ∈ H ( F, e H ) { π α b ⊗ π ′ α | π α ∈ Π ϕ ( G α ) , π ′ α ∈ Π ϕ ′ ( G ′ α ) } . where π α b ⊗ π ′ α is the e G -representation defined by the completed tensor product of π α ⊗ ⊗ π ′ α .2.3. Component groups.
In this section, we define the component groups for L -parameters anddefine some local root numbers. Define a pairing B of a representation M of a group L with sign b ( b ∈ {± } ) to be a bilinear map: B : M × M → C satisfying B ( lm, ln ) = B ( m, n ) , B ( m, n ) = bB ( n, m ) , f or m, n ∈ M, l ∈ L. In [GGP12, Section 7], the L -group of a special orthogonal group G = SO ( V, q ) can be embeddedinto the invariant group of a pairing B on M with sign b . If dim V = 2 n + 1, then L G = Sp ( M )where M = C n and b = −
1. If dim( V ) = 2 n and disc( V ) ∈ F × then L G = SO ( M )where M = C n and b = 1. If dim( V ) = 2 n and disc( V ) / ∈ F × then L G = O ( M )where M = C n and b = 1.Hence, we can identify a parameter ϕ of G = SO ( V, q ) as a semisimple representation of W F on the space M with a pairing B of sign b . Let S ϕ be the centralizer of the Imϕ in GL ( M ),and the component group S ϕ be the finite group π ( S ϕ ). For an element a ∈ S ϕ , let M a be the( − a on M which is only dependent on the image of a in the component group S ϕ .For the Gan-Gross-Prasad triple ( e G, e H, e ν ) for an admissible pair ( V, V ′ ), we define characters χ V and χ ′ V ′ of the component groups S ϕ and S ϕ ′ as η V ( a ) = det( M a )( − dim( M ′ ) / · det( M ′ )( − dim( M a ) / · ǫ ( 12 , M a ⊗ M ′ , ψ ) , for a ∈ S ϕ , and η V ′ ( a ′ ) = det( M )( − dim( M ′ a ′ ) / · det( M ′ a ′ )( − dim( M ) / · ǫ ( 12 , M ⊗ M ′ a ′ , ψ ) , for a ′ ∈ S ϕ ′ . For the L -parameter e ϕ = ϕ × ϕ ′ of e G = SO ( V ) × SO ( V ′ ), we have S e ϕ = S ϕ × S ϕ ′ and let η e V = η V × η V ′ , be a character of S e ϕ .Vogan conjectured in [Vog93] that, for a given Whittaker datum in the quasisplit pure innerform G α qs of G , there is a non-degenerate pairingΠ Vogan ϕ ( G ) × S ϕ → {± } . herefore, we can parameterize representations in Π Vogan ϕ ( G ) with characters η π : S ϕ → {± } ,which is called Langlands-Vogan parameterization.2.4. Statements of our results.
We are able to state our main results of this article.
Theorem 2.1.
The Conjecture 1.1 is true over archimedean local fields.
Theorem 2.1 will be proved in Section 4 for complex field case, and in Section 5 for the real fieldcase. Moreover, there is a refinement of Conjecture 1.1 in [GGP12, Conjecture 17.3].
Conjecture 2.1.
For a Gan-Gross-Prasad triple ( e G, e H, e ν ) over an archimedean local field F andgeneric parameters e ϕ , the unique representations e π ∈ Π Vogan e ϕ such that m ( e π ) = 1 satisfy η e π = η e V . From [GGP12], the statement of Conjecture 2.1 is independent of the choice of the Whittakerdatum in the Langlands-Vogan parameterization. We are going to prove Conjecture 2.1 over thecomplex field in Section 4. Over the real field, we can deduce Conjecture 2.1 for generic L -parametersfrom that for tempered L -parameters in Section 5. Hence we have Theorem 2.2.
Over the complex field C , Conjecture 2.1 holds for general generic L -parameters.Over the real field R , if Conjecture 2.1 is true for tempered L -parameters, then it is true for generalgeneric L -parameters. Genericity and irreducibilty
In this section, we prove that every representation in generic Vogan packets is isomorphic to anormalized parabolic induction that induces a representation of the Levi subgroup with essentiallydiscrete series representations and tempered representations on the blocks. The nonarchimedeancounterpart of this section was proved in [MW12, Section 3].First of all, we summarize some classical results about the reducibility of parabolically inducedrepresentations. Let G be a quasisplit connected real reductive group and let B = T U be a Borelsubgroup of G , where T is the maximal torus of B and U is the unipotent radical of B . Let A be the maximal split torus of T . Fix a standard parabolic subgroup P of G , that is, a parabolicsubgroup P with Levi decomposition P = M N , with T ⊂ M and N ⊂ U .Let σ be an irreducible tempered representation of M and choose ν ∈ a ∗ C , the complex dual of thereal Lie algebra of the split component A of M . Let I ( ν, σ ) be the normalized induction I GP ( ν ⊗ σ ).Assume that ν is in the positive Weyl chamber. Then I ( ν, σ ) is called a standard module. Let J ( ν, σ ) be the unique Langlands quotient of I ( ν, σ ) from Langlands classification. The followingconjecture is proved by Vogan in [Vog78, Theorem 6.2], which is called the archimedean case of thestandard model conjecture. Proposition 3.1. If J ( ν, σ ) is generic, then I ( ν, σ ) = J ( ν, σ ) . Hence I ( ν, σ ) is irreducible. For an archimedean local field F , it was proved by Knapp and Stein in [KS80] with the irre-ducibility of principal series representation when F = C , by Vogan in his classification of unitarydual of GL n ( R ) in [Vog86] when F = R that Proposition 3.2.
Let G = GL n ( F ) and σ be an irreducible unitary representation of M , then I GP ( σ ) is irreducible. peh and Vogan proved an irreducibility criterion for limits of generalized principal series rep-resentations of Lie groups in [SV80, Theorem 6.19]. Since the proof of Speh and Vogan only usethe structure of the relevant local L -parameters of the induced data, we can deduce the followingirreducibility of the representations in a given local L -packet. Proposition 3.3.
Let G = SO ( V , q ) be the special othogonal group for a non-degenerate qua-dratic space ( V , q ) . Suppose that for essentially discrete series representations σ i of GL r i anda tempered representation π of G with L -parameter ϕ , the normalized induced representation σ × · · · × σ l × π is an irreducible representation. Then σ × · · · × σ l × π ′ is an irreducible repre-sentation for every π ′ ∈ Π Vogan ϕ . Remark 3.1.
In the literature, σ ×· · ·× σ l × π represents the normalized induction of σ b ⊗ · · · b ⊗ σ l b ⊗ π from a parabolic subgroup P of G = SO ( V, q ) whose Levi component M is isomorphic to GL r ( F ) ×· · · × GL r l ( F ) × SO ( V , q ) , where V = V ⊕ D r for a hyperbolic space D r orthogonal to V . When F = R , dim V is even and the quasi-split inner form of G is SO ( n, n − , the parabolic subgroupswith Levi component M are conjugate to each other only by O ( V, q ) but not by SO ( V, q ) , and theinduced representations from different parabolic subgroups may not be isomorphic as representationsof SO ( V, q ) . Therefore, we need to clarify the meaning of our notation σ × · · · × σ l × π in thiscase.Let V = X r ⊕ Y r ⊕ V an be a generalized polar decomposition, where X r , Y r are maximal totallyisotropic in V and q | X r × Y r is non-degenerate, and V an is anisotropic. Then there is a basis { v i } ≤ i ≤ r of X r and A basis { v − i } ≤ i ≤ r of Y r , such that q ( v i , v − j ) = δ ij . Let I j , ˇ I j (1 ≤ j ≤ c ) be disjointsubsets of {± , ± , · · · , ± r } and ˇ I j = {− i | i ∈ I j } . The parabolic subgroup associated to I , I , · · · , I c is the parabolic subgroup stabilizing X ⊂ X ⊂ · · · ⊂ X l , where X j = Span { v i ∈ I k | ≤ k ≤ j } . The Levi component of P is isomorphic to GL r ( F ) × · · · × GL r l ( F ) × GL ( V , q ) , where r i = | I i | and V = V an ⊕ L i/ ∈∪ lj =1 I j ( C v i ⊕ C v − i ) . For admissiblerepresentations τ i of GL r i and admissible representation π of GL ( V , q ) , denote ( τ , I ) × ( τ , I ) × · · · × ( τ l , I l ) × π = I GP ( τ b ⊗ · · · b ⊗ τ c b ⊗ π ) , where I GP is the normalized induction from P to G . We abbreviate it as τ × τ × · · · × τ l × π whenthe I j ’s associated to τ j ’s are clear. In the analysis in Section 5, the choice of the parabolic subgroupwill not affect the analysis as the induced representations from different parabolic subgroups havethe same infinitesimal characters. With these results we are ready to prove that representations in generic Vogan packets are fullyinduced. For a generic parameter ϕ of the Weil-Deligne group W F , it can be regarded a semisimplerepresentation with a pairing (see Section 2.3). Then ϕ can be decomposed as a direct sum ofirreducible representations(3.1) ϕ = M i ∈ I ϕ ,i + l M j =1 ( ϕ j + ϕ ∨ j )where ϕ ,i ’s are distinct self-dual irreducible representations, and ϕ i ’s are not self-dual. One canfind the classification of the irreducible pieces in [Kna94]. et ϕ be the direct sum of self-dual representations in the decomposition (3.1), that is, ϕ = L i ∈ I ϕ ,i . Since each self-dual representation of W F has bounded image, ϕ ,i are tempered parame-ters, and thus the parameter ϕ itself is a tempered parameter. It is well known (see [GGP12, Section4]) that the component group S ϕ of the parameter ϕ depends only on ϕ .Let r j = dim ϕ j and r = P lj =1 r j . Under the local Langlands correspondence of the generallinear groups (see [Kna94]), ϕ j corresponds to | det( · ) | s j τ j , where τ j is equal to the character ρ m j ( z ) = ( z | z | ) m j of GL ( F ) or a discrete series representation of GL ( R ). We may assume that Re ( s ) ≥ · · · ≥ Re ( s l ) >
0, then under the local Langlands correspondence of the general lineargroup GL r ( F ), the parameter ϕ + = l M j =1 ϕ j corresponds to the unique quotient τ of | det( · ) | s τ × · · · × | det( · ) | s l τ l . The decomposition (3.1) can be simplified as(3.2) ϕ = ϕ ⊕ ϕ + ⊕ ( ϕ + ) ∨ . It is well known (see [GGP12, Section 4]) that the component group S ϕ of the parameter ϕ depends only on ϕ . And there is commutative diagramΠ Vogan ϕ / / (cid:15) (cid:15) Hom( S ϕ , {± } ) (cid:15) (cid:15) Π Vogan ϕ / / Hom( S ϕ , {± } )where the first vertical arrow maps every tempered representation π ∈ Π Vogan ϕ to the uniquequotient π of τ × π , and the horizontal arrows are defined by Langlands-Vogan parameterization.Then each π ∈ Π Vogan ϕ is the unique quotient of a normalized induction | det( · ) | s τ × · · · × | det( · ) | s l τ l × π , Let λ > · · · > λ t > Re ( s i )(1 ≤ i ≤ l ). And for 1 ≤ c ≤ t , let σ c = | det( · ) | iIm ( s i ) τ i +1 × · · · × | det( · ) | iIm ( s ic + kc ) τ i c + k c where s j ( j = i c + 1 , · · · , i c + k c ) are all s j such that Re ( s j ) = λ c . From Proposition 3.2, σ c is anirreducible tempered unitary representation.When G is quasisplit and π is generic, from Proposition 3.1, this induced representation I π = | det( · ) | s τ × · · · × | det( · ) | s l τ l × π = | det( · ) | λ σ × · · · × | det( · ) | λ t σ t × π is irreducible. In general, as there exists a generic representation in Π Vogan ϕ , from Proposition 3.3,we have the irreducibility of the induced representation I π , and thus π = I π . Proposition 3.4.
For a generic L -parameter ϕ = ϕ ⊕ ϕ + ⊕ ( ϕ + ) ∨ , every representation π in Π V oganϕ can be expressed as π = τ × π where π is a tempered representation with L -parameter ϕ and τ is the representation of GL r ( F ) with L -parameter ϕ + . Moreover, τ = | det( · ) | s τ × · · · × | det( · ) | s l τ l here Re ( s ) ≥ Re ( s ) ≥ · · · ≥ Re ( s l ) > and τ i is a discrete series representation of GL r i ( G ) . In particular,
Corollary 3.1.
When F = C , representations in generic Vogan packets are principal series repre-sentations. Conjectures 1.1 and 2.1 over the complex field
The co-dimension one case of the local Gan-Gross-Prasad conjecture for special orthogonal groupsover the complex field was proved by Jan M¨ollers in [M¨ol17]. And in this section, we generalize hisresults and give the proof for the general case by constructing a nonzero Bessel functional using themethod in [GSS19].For the Gan-Gross-Prasad triple ( e G, e H, e ν ) for an admissible pair ( V, V ′ ). We first show that wecan choose the Borel subgroup e B of e G appropriately such that the double coset e B e H is open densein e G and the intersection e B ∩ e H is the trivial group.Recall that in the notations in Section 2.1, P is the parabolic subgroup of G = SO ( V, q ) stabilizingthe flag F with the Levi decomposition P = M N . The Levi part M can be decomposed as A × G + ,where A ∼ = ( C × ) r and G + = SO ( V ′ ⊕ C v ). Let e G + = SO ( V ′ ⊕ C v ) × SO ( V ′ ) and e H + is theimage of the diagonal embedding from G ′ = SO ( V ′ ) into e G + . Then the group e P = P × G ′ is aparabolic subgroup of e G , with Levi part f M = ( A × e G + and unipotent radical e N = N ×
1. Let e P be the opposite of the parabolic subgroup e P and e N be the unipotent radical of e P . Then e N f M e N isopen dense in e G and the multiplication map from e N × f M × e N to e N f M e N is an isomorphism.From [M¨ol17, Section 6.2.4], e G + has a Borel subgroup e B + such that the double coset e B + e H + isopen dense and e B + ∩ e H + = 1. Notice that e H = e H + e N , so if we choose e B = e N ( A × e B + , we have e B ∩ e H = e N ( A × e B + ∩ e H + e N = ( A × e B + ∩ e H + = 1and e B e H = e N ( A × e B + e H + ) e N is open dense in e N f M e N , which implies e B e H is open dense in e G .For a character e σ of e B that trivialize the unipotent radical of e B , from [DC91, Remark 2.1.4], as P \ G is compact, Hom e H ( I e G e B ( e σ ) , e ν ) is equal to the space of ( e B × e H, δ e B e σ × e ν )-equivariant distributions.Then we construct such a distribution by ”extending” an equivariant tempered measure on the opendouble coset e B e H to e G with results of Gourevitch, Sahi and Sayag in [GSS19].Since e B ∩ e H = 1, we can construct a ( e B × e H, δ e B e σ × e ν )-equivariant measure µ on e B e H as(4.3) µ = δ − e B e σ − ( e b ) e ν ( e h ) d l e bd r e h. Recall that e ν is a unitary character and δ − e B e σ is a multiplicative character of e B which is trivial inthe unipotent radical of e B so it is of moderate growth. This implies that µ is a tempered measure,that is, a measure that yields finite value on Schwartz functions of e B e H .Because e B is solvable, from [GSS19, Corollary 2.10], there exists a character χ of e B × e H and a( e B × e H, χ )-equivariant regular function f of e G such that e B e H = { g ∈ e G | f ( g ) = 0 } . For Re ( λ ) ≫ µ ( λ ) = µ | f | λ can be extended by 0 to a ( e B × e H, | χ | λ ⊗ ( δ e B e σ × e ν ))-equivariant distribution µ ( λ ) on e G . From [GSS19, Lemma 4.1], µ ( λ ) can be meromorphically continued to λ ∈ C . In onsequence, the leading term of the Laurent series of µ ( λ ) is a nonzero ( e B × e H, δ e B e σ × e ν )-equivariantdistribution on G , which corresponds to a nonzero element in Hom e H ( I e G e B ( e σ ) , e ν ).For every principal series representation e π = I e G e B ( e σ ) of e G , by combining the above constructionwith [JSZ10, Theorem A], we can conclude that m ( e π ) = dim Hom e H ( I e G e B ( e σ ) , e ν ) = 1 . For a generic parameter e ϕ of e G , since there is no discrete series representation of special or-thogonal groups over the complex field, each L -packet Π ϕ ( e G α ) contains only one representation.Besides, there is only one pure inner form of SO ( V, q ), so the Vogan packet Π
Vogan e ϕ contains onlyone representation. From Corollary 3.1, this representation is equal to π b ⊗ π ′ where π and π ′ areprincipal series representations of G and G ′ respectively, and from [War12, Appendix 2.3], π b ⊗ π ′ isa principal series representation. Therefore, Conjecture 1.1 and Conjecture 2.1 are proved over thecomplex field. Theorem 4.1.
Conjectures 1.1 and 2.1 hold over the complex field C . Conjectures 1.1 and 2.1 over the real field
In this section, we prove the generic part of the conjectures based on the tempered part overthe real field. We construct a descent map between Bessel functionals for representations of twoGan-Gross-Prasad triples to manifest the relation between local Bessel functionals and parabolicinductions. We will first prove a technical lemma in Section 5.1. Then we use our technical lemmato prove some vanishing results in Section 5.2 and 5.3, which implies the descent map is bijective.With the bijectiveness of the descent map, we have some multiplicity formula. Finally, in Section5.5 and 5.6, we apply the multiplicity formula to prove the Theorem 2.1 and 2.2 following Moeglinand Waldspurger’s idea in [MW12].We work in the category R ( G ) of smooth Fr´echet representations of moderate growth for analmost linear Nash group G . A representation π in R ( G ) is called Casselman-Wallach if the ( g C , K )-module π K is Harish-Chandra (see Section 2.2). The reason for not working in the category ofCasselman-Wallach representations is that the representation ” η π | SO ( V ′ ,q ′ ) ” in Section 5.2 is notadmissible. When no confusion is possible, we do not distinguish a representation with its underlyingspace.For the Gan-Gross-Prasad triple ( e G, e H, e ν ) of an admissible pair ( V, V ′ ), a Casselman-Wallach rep-resentation π of SO ( V ) and a Casselman-Wallach representation π ′ of SO ( V ′ ), we denote m ( π, π ′ )to be the dimension of the space of Bessel functionals Hom S r ( π b ⊗ π ′ , ν r ) of the pair ( π, π ′ ), where S r is the Bessel subgroup and ν r is the generic Bessel character defined in Section 2.1. Then wehave(5.4) m ( π, π ′ ) = dim Hom S r ( π b ⊗ π ′ , ν r ) = dim Hom e H ( π b ⊗ π ′ , e ν ) = m ( π b ⊗ π ′ ) . For a closed subgroup P of G , we denote by S-Ind GP σ the Schwartz induction of σ from H to G , which is defined in Appendix A.1. One can check that S-Ind GP σ is canonically isomorphic tothe unnormalized smooth induction when P \ G is compact. We denote by I GP ( σ ) the normalizedinduction of σ from P to G , which is isomorphic to S-Ind GP ( δ − G δ P σ ). Here δ P , δ G are the modularcharacters of P and G respectively.Then we review our basic settings and introduce the notations we would use in our proof. Take theGan-Gross-Prasad triple ( e G ′ , e H ′ , e ν ′ ) for an admissible pair ( V ′ , V ). Let the orthogonal complement ⊥ = ( X ′ r ⊕ Y ′ r ) ⊕ C v ′ for totally isotropic subspaces X ′ r , Y ′ r and anisotropic vector v ′ , and wedenote X ′ r +1 = X ′ r ⊕ C v ′ . Now we construct an anisotropic vector v such that ( v, v ) = − ( v ′ , v ′ )and v is orthogonal to V ′ , and define V = V ′ ⊕ C v . Then we have V = V ′ ⊕ C v = ( X r +1 ⊕ Y r +1 ) ⊕ V . where X r +1 = X r ⊕ C ( v + v ′ ) and Y r +1 = Y r ⊕ C ( v − v ′ ). By taking G = SO ( V, q ), we obtain aco-dimension one Gan-Gross-Prasad triple ( e G, e H, e ν ), where e G = G × G ′ , e H is the image ∆ G ′ ofthe diagonal embedding of G ′ into e G , and e ν is the trivial character of e H . Let P be the parabolicsubgroup of G stabilizing X r +1 , and P = M N be the Levi decomposition of P where M = GL ( X r +1 ) × G . From the calculation in [M¨ol17, Section 6.3.5], P G ′ is open dense in G . Take P ′ = P ∩ G ′ . Then P ′ is the subgroup of G ′ consisting of elements p ′ that satisfy p ′ X ′ r ⊂ X ′ r , p ′ ( v + v ′ ) ⊂ X ′ r ⊕ C v ′ , that is, p ′ X ′ r ⊂ X ′ r , ( p ′ − v ⊂ X ′ r . Therefore, P ′ can be decomposed as(5.5) ( R r, × G ) ⋊ N ′ where G = SO ( V , q ), R r, is the mirabolic subgroup of GL ( X r +1 ) that consists of m ′ satisfying( m ′ − v + v ′ ) ⊂ X ′ r , m ′ X ′ r ⊂ X ′ r , and N ′ is the unipotent subgroup of G ′ satisfying( g − X ′ r +1 = 0 , ( g − V ⊂ X ′ r +1 . Then there is a decomposition S ′ r = ( N r +1 × G ) ⋊ N ′ ⊂ P ′ , where N r +1 is the subgroup of GL r +1 = GL ( X ′ r +1 ) consisting of g ∈ GL ( X ′ r +1 ) such that( g − X ′ i +1 ⊂ X ′ i , for i = 0 , , · · · , r. Let ψ r +1 be the (unitary) character of N r +1 obtained by the restriction of ν ′ r to N r +1 , then we have(5.6) S-Ind P ′ S ′ r ν ′ = S-Ind ( R r, × G ) ⋊ N ′ ( N r +1 × G ) ⋊ N ′ ν ′ = S-Ind R r, N r +1 ψ r +1 . Let π ′ (resp. π , σ ) be a Casselman-Wallach representation of G ′ (resp. G , GL r +1 ). For s ∈ C ,let π s be the representation | det( · ) | s σ × π = I GP ( | det( · ) | s σ b ⊗ π ).When σ is a principal series, our construction of the descent map F s ( s ∈ C ) is the composite F ,s ◦ F ,s of F ,s : Hom G ′ ( π s b ⊗ π ′ , C ) → Hom G ′ (S-Ind G ′ P ′ ( | det( · ) | s δ P σ (cid:12)(cid:12) R r, b ⊗ π ) b ⊗ π ′ , C )constructed in Section 5.2, and F ,s : Hom G ′ (S-Ind G ′ P ′ ( | det( · ) | s δ P σ (cid:12)(cid:12) R r, b ⊗ π ) b ⊗ π ′ , C ) → Hom S ′ r ( π ′ b ⊗ π , ν ′ r )constructed in Section 5.3. And we prove that for general s , that is, for s ∈ C − E where E is acountable subset of C , F ,s and F ,s are isomorphisms in Section 5.2 and 5.3, which implies that m ( π ′ , π ) = m ( π s , π ′ ) for general s . In order to use the mathematical induction as in [MW12],the multiplicity formulas should be proved under some stronger conditions (which will be called as”Condition (P)” in the rest of this section) when π , π ′ are in generic Vogan packets, that is,
1) for general s , if σ is a principal series representation;(2) for Re ( s ) ≥ s ′ , if r = 0 and σ = sgn m ( m = 0 ,
1) of GL ( R ), where sgn ( x ) = 1 when x > sgn ( x ) = − x < Re ( s ) + n ≥ s ′ , if r = 1 and σ is the discrete series representation D n of GL ( R ), whichis the unique subrepresentation of | · | n sgn n +1 × | · | − n .Here s ′ is a real number associated to π ′ defined by (5.7) in Section 5.1.When r = 0 and σ = sgn m ( m = 0 , Re ( s ) ≥ s ′ .When r = 1 and σ is the discrete series representation D n of GL ( R ), we have to construct F ,s differently to ensure it is bijective when Re ( s ) + n ≥ s ′ , and in this case the decent map ”descends”Bessel functionals of the pair ( σ × π , π ′ ) to Bessel functionals of the pair ( π ′ , χ × π ), where χ isthe character of GL ( R ) defined by χ ( x ) = | x | n sgn n +1 ( x ).5.1. A technical lemma.
In this subsection, we apply the results in Appendix A and Appendix Bto prove Lemma 5.1, our main technical lemma. We would freely use the notations about Schwartzhomologies in Appendix A and Harish-Chandra parameters in Appendix B.
Lemma 5.1.
Let p, q, l be non-negative integers and n = [ p + q ] . Suppose we have a representation π ∈ R ( SO ( p, q )) , an irreducible Casselman-Wallach representation τ of GL l ( R ) with a Harish-Chandra parameter ( s , · · · , s l ) and an irreducible Casselman-Wallach representation π ′ of SO ( p + l, q + l ) with a Harish-Chandra parameter ( λ ′ , · · · , λ ′ n ) , suppose we have s = ± λ ′ i for ≤ i ≤ n ,then the Schwartz homologies of ( τ × π ) b ⊗ π ′ vanish.Proof. From Corollary A.2, it suffices to find an element z ∈ Z ( g C ) such that χ π ′∨ ( z ) = 0 and z.v = 0 , for every v ∈ τ × π . Let h the Cartan subalgebra of GL l ( R ) × SO ( p, q ). With the parameterization of the complexdual h ∗ C of h in Appendix B.1, we define a polynomial f in the polynomial algebra P ( h ∗ C ) by f ( λ , · · · , λ n ) = n Y i =1 ( λ i − s )and f is invariant under W G , so f ∈ P ( h ∗ C ) W G . Then we take the z ∈ Z ( g C ) corresponding to f under Harish-Chandra isomorphism.On the one hand, for every irreducible component e π of π , suppose the Harish-Chandra conju-gacy class of e π is ( e λ , e λ , · · · . e λ n − l ), then the Harish-Chandra conjugacy class for every irreduciblecomponent e π of τ × e π is [( s , s , · · · , s l , e λ , e λ , · · · . e λ n − l )] . Then z.v = 0 for every v ∈ e π , so z.v = 0 for every v ∈ τ × π . On the other hand, since χ π ′∨ ( z ) = n Y i =1 ( − λ ′ i + s )( − λ ′ i − s )and s = ± λ ′ i for i = 1 , · · · , n , we have χ π ′∨ ( z ) = 0 . Therefore, the Schwartz homologies of ( τ × π ) b ⊗ π ′ vanish. (cid:3) or a Casselman-Wallach representation π ′ in the general packets of G ′ = SO ( V ′ , q ′ ), fromProposition 3.4, π ′ can be expressed as the parabolic induction | det( · ) | s ′ τ ′ × · · · × | det( · ) | s ′ l τ ′ l ′ × π ′ , where Re ( s ′ ) ≥ · · · ≥ Re ( s ′ l ′ ) > τ ′ j are isomorphic to sgn m ′ j ( m ′ j = 0 ,
1) of GL ( R ) or discreteseries representations D n ′ j of GL ( R ) and π ′ is tempered. Then the Harish-Chandra conjugacyclass [ v π ′ ] can be expressed as follows[ v π ′ ] = [( v , · · · , v l , v π ′ )] , where v π ′ is a Harish-Chandra parameter of π ′ and v j = ( ( s ′ j ) if τ ′ j = sgn m ′ j , ( s ′ j + n ′ j , s ′ j − n ′ j ) if τ ′ j = D n ′ j . Now we define the complex number s ′ by(5.7) s ′ = max { sup { Re ( s ′ i ) + n ′ i } , } . In particular, s ′ is equal to zero when π ′ is tempered. It is worthwhile to mention that the s ′ definedabove depends only on the parameter ϕ + in (3.2) from Section 2.4. And from Lemma 5.1 we have Corollary 5.1.
With the notations in Proposition 5.1, if s > s ′ , then the Schwartz homologies of ( τ × π ) b ⊗ π ′ vanish. By combining Lemma 5.1 with Proposition B.2 and Proposition B.3,
Corollary 5.2.
With the notations in Proposition 5.1, and let ρ be a finite-dimensional represen-tation of GL l ( R ) . Then the Schwartz homologies of (( | det( · ) | s τ ⊗ ρ ) × π ) b ⊗ π ′ vanish for general s . Construction and bijectiveness of F ,s . In this subsection, we first construct F ,s and thendo the Schwartz analysis on the closed orbits to obtain a vanishing result in Proposition 5.2, whichimplies F ,s is isomorphic under Condition(P).On the one hand, if we denote π s to be the representation | det( · ) | s σ × π , we can express π s asthe space of Schwartz sections on P \ G , which gives π s b ⊗ π ′ = I GP ( | det( · ) | s σ b ⊗ π ) b ⊗ π ′ = S-Ind GP ( | det( · ) | s δ P σ b ⊗ π ) b ⊗ π ′ = Γ S ( P \ G, | det( · ) | s δ P σ b ⊗ π ) b ⊗ π ′ . Here δ P is the modular character of P , and we have δ P ( p ) = | det( m ) | d − − r where d = dim V and p = ( m × g ) ⋊ n ∈ P . For simplicity, we regard δ P as a character of GL r +1 .On the other hand, we haveS-Ind G ′ P ′ ( | det( · ) | s δ P σ (cid:12)(cid:12) R r, b ⊗ π ) b ⊗ π ′ = Γ S ( P ′ \ G ′ , | det( · ) | s δ P σ (cid:12)(cid:12) R r, b ⊗ π ) b ⊗ π ′ = Γ S ( P \ P G ′ , | det( · ) | s δ P σ b ⊗ π ) b ⊗ π ′ Since P \ P G ′ is an open orbit of P \ G , there is a natural embedding(5.8) i U , X : Γ S ( P \ P G ′ , | det( · ) | s δ P σ b ⊗ π ) → Γ S ( P \ G, | det( · ) | s δ P σ b ⊗ π ) efined by the extension by zero. With this embedding i U , X , we define F ,s : Hom G ′ ( π s b ⊗ π ′ , C ) → Hom G ′ (S-Ind G ′ P ′ ( | det( · ) | s δ P σ (cid:12)(cid:12) R r, b ⊗ π ) b ⊗ π ′ , C )induced by i U , X . We will prove that F ,s is an isomorphism under Condition(P) using Schwartzhomologies. For this purpose, we first analyze the structure of G ′ -orbits of P \ G .Let X = P \ G , then X is the topological subspace of the Grassmannian manifold Gr(r + 1 , V)consisting of totally isotropic subspaces V r +1 of dimension r + 1 in V and it can be regarded as a G ′ -space under the right action of G ′ . Note that U = P \ P G ′ is the open orbit of X consisting of V r +1 that is not contained in V ′ and Z = X \U is the closed subspace of X consisting of V r +1 thatis contained in V ′ .If V ′ = V ⊕ C v ′ is anisotropic, then X ′ r is maximal isotropic in V ′ . In this case, Z is empty. If V ′ is isotropic, then there is v ∈ V such that ( v , v ) = − ( v ′ , v ′ ). In this case, Z is the disjointunion of two connected G ′ -orbits { [ C ( v + v ′ )] } and { [ C ( v − v ′ )] } when dim V ′ = 2. If V ′ isisotropic and dim V ′ >
2, then Z is a single G ′ -orbit P \ P ηG ′ , where η is an element in G satisfying P \ P η = [ X ′ r ⊕ C ( v + v ′ )]. Then η − P η is the parabolic subgroup of G stabilizing X ′ r ⊕ C ( v + v ′ ) and Q ′ = η − P η ∩ G ′ is the parabolic subgroup of G ′ stabilizing X ′ r ⊕ C ( v + v ′ ). The Levi decomposition Q ′ = M Q ′ N Q ′ indicates that M Q ′ is isomorphic to GL ( X ′ r ⊕ C ( v + v ′ )) × SO ( V ′ , q ′ ).The complexified conormal bundle N ∨Z / X has dimension r + 1, and gives rise to a representationof Q ′ on its fiber over [ X ′ r ⊕ C ( v + v ′ )] such that N Q ′ and SO ( V ′ , q ′ ) acts trivially and GL ( X ′ r ⊕ C ( v + v ′ )) acts as the standard representation ρ on the fiber.Let Γ S Z ( X , | det( · ) | s δ P σ b ⊗ π )) = Γ S ( X , E ) / Γ S ( U , E ), where E is the tempered bundle P ′ \ ( G ′ × ( | det( · ) | s δ P σ b ⊗ π )) . Proposition 5.1.
The Schwartz homologies of Γ S Z ( X , | det( · ) | s δ P σ b ⊗ π )) b ⊗ π ′ vanish under Condition (P).Proof. From Proposition A.6, Γ S Z ( X , | det( · ) | s δ P σ b ⊗ π ) has a complete decreasing filtrationΓ S Z ,k ( X , | det( · ) | s δ P σ b ⊗ π )whose graded pieces are isomorphic toΓ S ( Z , Sym k N ∨Z / X ⊗ E| Z ) ∼ = V ′ is anisotropic, I G ′ Q ′ ( | det( · ) | s + ( σ ⊗ Sym k ρ ) b ⊗ η π | SO ( V ′ ,q ′ ) ) if dim V = 2 , and V ′ is isotropic,( I G ′ Q ′ ( | det( · ) | s + ( σ ⊗ Sym k ρ ) b ⊗ η π | SO ( V ′ ,q ′ ) )) ⊕ otherwise . for k = 0 , , · · · , where η π is the representation of η − G η defined by η π ( g ′ ) = π ( ηg ′ η − ). Thefactor | det( · ) | comes from the fact that δ η − P η ( p ′ ) /δ Q ′ ( p ′ ) = | det( m ′ ) | , for every p ′ = ( m ′ × g ′ ) ⋊ n ′ ∈ Q ′ .Then we use our technical lemma to prove that, under Condition(P), the Schwartz homologiesof I G ′ Q ′ (( | det( · ) | s + σ ⊗ Sym k ρ ) b ⊗ η π | SO ( V ′ ,q ′ ) ) b ⊗ π ′ (5.9)vanish. f σ is the principal series representation with a Harish-Chandra parameter ( s , s , · · · , s l ), thenthe parameters of the irreducible components of σ ⊗ Sym k ρ are [( s + a , · · · , s l + a l )] from Propo-sition B.3, where a i are integers such that P li =1 a i = k . Hence, for general s , s + s + k + = ± λ ′ i for i = 1 , · · · , n and non-negative integer k . From Lemma 5.1 and Proposition B.2, the Schwartzhomologies of I G ′ Q ′ (( | det( · ) | s + σ ⊗ Sym k ρ ) b ⊗ η π | SO ( V ′ ,q ′ ) ) b ⊗ π ′ vanish for general s .If r = 0 and σ = sgn m ( m = 0 , | det( · ) | s + σ ⊗ Sym k ρ is [( s + k + )]. If Re ( s ) ≥ s ′ , we have Re ( s ) + > s ′ , thenfrom Corollary 5.1, the Schwartz homologies of I G ′ Q ′ (( | det( · ) | s + σ ⊗ Sym k ρ ) b ⊗ η π | SO ( V ′ ,q ′ ) ) b ⊗ π ′ vanish.If r = 1 and σ is the discrete series representation D n of GL ( R ), the Harish-Chandra parametersfor irreducible components of | det( · ) | s + σ ⊗ Sym k ρ are[( s + n a + 12 , s − n a + 12 )]such that a , a are integers satisfying a + a = k . If Re ( s ) + n ≥ s ′ , we have Re ( s ) + n + > s ′ ,from Corollary 5.1 and Proposition B.2, the Schwartz homologies of I G ′ Q ′ (( | det( · ) | s + σ ⊗ Sym k ρ ) b ⊗ η π | SO ( V ′ ,q ′ ) ) b ⊗ π ′ vanish.Finally, from Corollary A.1, we can conclude thatΓ S Z ( X , | det( · ) | s δ P σ b ⊗ π ) b ⊗ π ′ vanish under Condition(P). (cid:3) From the long exact sequence of the Schwartz homologies for the short exact sequence0 → Γ S ( U , E ) b ⊗ π ′ → Γ S ( X , E ) b ⊗ π ′ → Γ S Z ( X , | det( · ) | s δ P σ b ⊗ π ) b ⊗ π ′ → , the above proposition implies F ,s is isomorphic under Condition (P).5.3. Construction and bijectiveness of F ,s . In this subsection, we construct F ,s and doSchwartz analysis on the open orbit to obtain vanishing results in Proposition 5.2 and Proposi-tion 5.3, which shows that F ,s is isomorphic under Condition(P). Proposition 5.2. If σ is a principal series representation of GL r +1 , there is a R r, -homomorphism T p : S-Ind R r, N r +1 ψ r +1 → | det( · ) | s δ P ( det ( · )) σ (cid:12)(cid:12) R r, that induces an isomorphism H S ( G ′ , S-Ind G ′ P ′ (S-Ind R r, N r +1 ψ r +1 b ⊗ π ) b ⊗ π ′ ) ∼ = H S ( G ′ , S-Ind G ′ P ′ ( | det( · ) | s δ P ( · ) σ (cid:12)(cid:12) R r, b ⊗ π ) b ⊗ π ′ ) for general s .Proof. The proof for [Xue20, Proposition 5.1] over the complex field can be generalized to principalseries representations over the real field word by word. Hence, for a principal series representation σ , | det( · ) | s σ (cid:12)(cid:12) R r, has a R r, -subrepresentation isomorphic to S-Ind R r, N r +1 ψ r +1 , and their quotient hasa R r, -stable complete filtration whose graded pieces σ α satisfy H S i ( G ′ , Ind G ′ P ′ ( σ α b ⊗ π ) b ⊗ π ′ ) = 0 or general s (see [Xue20, (6.2)]) and k = 0 , , · · · . Therefore the proposition follows from CorollaryA.1 and the long exact sequence. (cid:3) When σ is a principal series representation, the above proposition shows that T p induces a map(5.10)Hom G ′ (S-Ind G ′ P ′ ( | det( · ) | s δ P σ (cid:12)(cid:12) R r, b ⊗ π ) b ⊗ π ′ , C ) → Hom G ′ (S-Ind G ′ P ′ (S-Ind R r, N r +1 ψ r +1 b ⊗ π ) b ⊗ π ′ , C ) . which is isomorphic for general s . In particular, if r = 0, R r, is the trivial group, and thus themap in (5.10) is the identity map. Recall that S-Ind R r, N r +1 ψ r +1 = S-Ind P ′ S ′ r ν ′ r . Since δ P ′ | S ′ r = 1, fromProposition A.4 and Corollary A.3, we haveHom G ′ (S-Ind G ′ P ′ (S-Ind P ′ S ′ r ν ′ r b ⊗ π ) b ⊗ π ′ , C ) = Hom G ′ (S-Ind G ′ P ′ (S-Ind P ′ S ′ r ( ν ′ r b ⊗ π b ⊗ π ′ | S ′ r )) , C )= Hom G ′ (S-Ind G ′ S ′ r ( ν ′ r ⊗ π b ⊗ π ′ | S ′ r ) , C )= Hom S ′ r ( π b ⊗ π ′ ⊗ ν ′ r , C )= Hom S ′ r ( π b ⊗ π ′ , ν ′ r )(5.11)Now we can define F ,s : Hom G ′ (S-Ind G ′ P ′ ( | det( · ) | s δ P σ b ⊗ π ) b ⊗ π ′ , C ) → Hom S ′ r ( π ′ b ⊗ π , ν ′ r )to be the composition of (5.10) and (5.11) when σ is a principal series (including the case when r = 0 and σ = ρ m ), and F ,s is an isomorphism under Condition(P).When r = 1 and σ is the discrete series representation D n of GL ( R ), let χ be the characterof GL ( R ) defined by χ ( x ) = | x | s + d + n − sgn n +1 ( x ), where d = dim V . We would define a R , -homomorphism T d : S-Ind R , R × × χ → | det( · ) | s δ P ( det ( · )) σ (cid:12)(cid:12) R , . Denote GL = GL ( R ), w = (cid:18) (cid:19) ∈ GL , and B is the (upper-triangular) Borel subgroupof GL with Levi decomposition B = T N . For a character χ of GL ( R ), χ ⊗ T and also as a character of B that trivialize N . Let X = B \ GL , U = B \ B w B ⊂ X , and Z = B \ B . Recall that R , is the mirabolic subgroup of GL consisting of matrices with (0 ,
1) on the last row. And we have the following lemmas.
Lemma 5.2.
Extension by zero gives a natural embedding Γ S ( U , χ ⊗ → Γ S ( X , χ ⊗ , and Γ S ( X , χ ⊗ / Γ S ( U , χ ⊗ has a complete decreasing filtration Γ SZ ( X , χ ⊗ k with gradedpieces isomorphic to χ ( det ( · )) sgn k ( det ( · )) | det( · ) | k (cid:12)(cid:12) R , , for k = 0 , , · · · .Proof. This lemma follows from the Borel’s Lemma. (cid:3)
Lemma 5.3.
We have an R , -isomorphism Γ S ( U , χ ⊗ ∼ = S-Ind R , R × × . roof. Γ S ( U , χ ⊗
1) = Γ S ( B \ B w B , χ ⊗ S ( T \ B , ⊗ χ )= Γ S ( R × × \ R , ,
1) = S-Ind R , R × × . (cid:3) Recall the modular character δ P (( m × g ) ⋊ n ) = | det( m ) | d − − r = | det( m ) | d − . Define σ ′ = δ P ( · ) | det( · ) | s σ , and it is the unique subrepresentation of δ P ( · ) | det( · ) | s ⊗ ( | · | n sgn n +1 × | · | − n ) = S-Ind GL B ( | · | s + d + n − sgn n +1 ⊗ | · | s + d − n − ) , so it is isomorphic to the unique quotient π I /F n of π I = S-Ind GL B ( | · | s + d − n − ⊗ | · | s + d + n − sgn n +1 ) = χ ( det ( · )) ⊗ S-Ind GL B ( χ ⊗ , where χ = | · | − n +1 sgn − n +1 and F n is the unique subrepresentation of π I which has dimension n . Now we define a finite-dimensional representation F ′ = χ − (det( · )) ⊗ F n . It is the uniquesubrepresentation of π I ′ = S-Ind GL B ( χ ⊗
1) = χ − ( det ( · )) π I . It is clear that their quotient D ′ = π I ′ /F ′ ∼ = χ − (det( · )) σ ′ .Now we define a R , -homomorphism T ′ : S-Ind R , R × × → D ′ by the composition ofS-Ind R , R × × S ( U , χ ⊗ → Γ S ( X , χ ⊗
1) = π I ′ → π I ′ /F ′ = D ′ . Lemma 5.4.
The R , -homomorphism T ′ is injective.Proof. Suppose T ′ is not injective then there exist e f ∈ Γ S ( U , χ ⊗
1) such that the image e f G of e f in π I ′ is contained in F ′ .On the one hand, f ( x ) = e f ( w (cid:18) x (cid:19) ) is a Schwartz function. For θ ∈ (0 , π ), we have (cid:18) cosθ sinθ − sinθ cosθ (cid:19) = (cid:18) /sinθ cosθsinθ (cid:19) w (cid:18) − cotθ (cid:19) . So e f (cid:18) cosθ sinθ − sinθ cosθ (cid:19) = χ (1 /sinθ ) f ( − cotθ ) . Therefore, for every positive integer l , we have( ddθ ) l e f G (cid:18) cosθ sinθ − sinθ cosθ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 = lim θ → χ (1 /sinθ ) f ( − cotθ ) θ l = lim θ → ( cotθ ) − n +12 + l f ( − cotθ ) · ( tanθ ) l θ l = 0 . On the other hand, from [God74, Section 2.3], the space of F ′ is generated by ϕ − n +1 , ϕ − n +3 , · · · , ϕ n − , ϕ n − , here ϕ k ∈ S-Ind GL B ( χ ⊗ ϕ k (cid:18) cosθ sinθ − sinθ cosθ (cid:19) = e ikθ , which determines ϕ k uniquely due to the Iwasawa decomposition GL ( R ) = B · SO (2 , R ).So e f G is a linear combination of ϕ k , that is, there is a nonzero n -tuple ( λ , · · · , λ n ) such that e f G = n X k =1 λ k ϕ k − n − . Then we have ( ddθ ) l e f G (cid:18) cosθ sinθ − sinθ cosθ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 = n − X k =0 λ k ((2 k − n − i ) l so there exists a positive integer l such that ( ddt ) l ( e f G (cid:18) cosθ sinθ − sinθ cosθ (cid:19) ) | θ =0 does not equal to 0,which leads to a contradiction. Therefore, the R , -homomorphism T ′ : Γ S ( U , χ ⊗ → D ′ isinjective. (cid:3) Lemma 5.5.
The finite-dimensional representation F ′ is isomorphic to χ ( det ( · )) Sym n − ( C ) as GL -modules.Proof. By the action of O (2 , R ) and R × I on ϕ k , we have(5.12) (cid:18) cosθ sinθ − sinθ cosθ (cid:19) ϕ k = e ikθ ϕ k , and(5.13) (cid:18) − (cid:19) ϕ k = ϕ − k , and(5.14) (cid:18) a a (cid:19) ϕ k = χ ( a ) ϕ k . Therefore, from (5.12) and the ”unitarian trick”, F ′ is isomorphic to Sym n − ( C ) as representationof SL ( R ), where C = C v ⊕ C v is the standard representation of SL ( R ). We may also write ϕ k = ( v + iv ) k + n − ⊗ ( v − iv ) n − − k ∈ Sym n − ( C ) . From (5.13), F ′ is isomorphic to Sym n − ( C ) as a representation of SL ± ( R ), where C is thestandard representation of SL ± ( R ). From (5.14), F ′ ∼ = χ ( det ( · )) Sym n − ( C ). (cid:3) Lemma 5.6.
The irreducible components of the R , -composition series of F ′ are | det( · ) | k sgn k , for k = 0 , , · · · , k − . Proof.
This lemma follows from the fact that the restriction of F ′ ≃ χ ( det ( · )) Sym n − ( C ) to R × × ⊂ M can be decomposed as n characters, that is, | · | sgn , | · | − sgn , · · · , | · | − ( n − sgn n − . (cid:3) roposition 5.3. When σ is a discrete series representation D n of GL ( R ) , there is a R , -homomorphism T d : S-Ind R , R × × χ → | det( · ) | s δ P ( det ( · )) σ (cid:12)(cid:12) R , such that T d induces an isomorphism H S ( G ′ , S-Ind G ′ P ′ (S-Ind R , R × × χ ) b ⊗ π b ⊗ π ′ ) ∼ = H S ( G ′ , S-Ind G ′ P ′ ( | det( · ) | s δ P ( · ) σ (cid:12)(cid:12) R , b ⊗ π ) b ⊗ π ′ ) . for Re ( s ) + n ≥ s ′ , where χ ( · ) = | · | s + d + n − sgn n +1 ( · ) . Proof.
We first compute the structure of D ′ / Γ S ( U , χ ⊗ π I ′ / Γ S ( U , χ ⊗
1) = Γ S ( X , χ ⊗ / Γ S ( U , χ ⊗
1) has a complete decreasing filtration Γ SZ ( X , χ ⊗ k with graded pieces isomorphic to(5.15) | det( · ) | k sgn ( · ) k χ (det( · )) (cid:12)(cid:12) R , , for k = 0 , , · · · . On the other hand, from Lemma 5.6, the finite dimensional representation F ′ in π ′ I = Γ SZ ( X , χ ⊗ R , -composition series with irreducible pieces σ ′ k = | det( · ) | k sgn k ( det ( · )) χ (det( · )) (cid:12)(cid:12) R , , for k = 0 , , · · · , n − , So the projection to the quotient W = Γ SZ ( X , χ ⊗ / Γ SZ ( X , χ ⊗ n gives an isomorphism between F ′ and W . Therefore, we haveΓ SZ ( X , χ ⊗
1) = F ′ ⊕ Γ SZ ( X , χ ⊗ n . The quotient D ′ / Γ S ( U , χ ⊗
1) = ( π I ′ / Γ S ( U , χ ⊗ /F ′ ∼ = Γ SZ ( X , χ ⊗ n has a complete decreasingfiltration with graded pierces isomorphic to | det( · ) | k sgn k ( det ( · )) χ (det( · )) (cid:12)(cid:12) R , , for k = n, n + 1 , · · · that is,(5.16) | det( · ) | k sgn k ( det ( · )) (cid:12)(cid:12) R , , for k = 1 , , · · · We can define a R , -homomorphism T d fromS-Ind R , R × × χ ∼ = χ (det( · )) S-Ind R , R × × δ P ( · ) | det( · ) | s σ (cid:12)(cid:12) R r, ∼ = χ ( det ( · )) D ′ induced by T ′ and its cokernel has complete decreasing filtration with graded pieces isomorphic to σ k = | det( · ) | k + s + d + n − sgn k (det( · )) (cid:12)(cid:12) R , , for k = 1 , · · · , Recall the totally isotropic flag X ′ ⊂ X ′ in Section 2.1. Let P ′ be the subgroup of G ′ stabilizing X ′ with Levi decomposition P ′ = M ′ N ′ where M ′ ∼ = R × × G ′ . Since for t ∈ C ,S-Ind GL R , | det( · ) | t sgn m ( det ( · )) (cid:12)(cid:12) R , ∼ = S-Ind GL R , | · | t sgn m ( · ) ⊗ , one has S-Ind G ′ P ′ ( | det( · ) | t sgn m (det( · )) (cid:12)(cid:12) R , b ⊗ π ) ∼ = S-Ind G ′ P ′ ( | · | t ⊗ S-Ind G ′ G π ) . And for every irreducible component π ′ of S-Ind G ′ G π , the Harish-Chandra parameter forS-Ind G ′ P ′ ( | · | t b ⊗ π ′ ) = | · | t δ − P ′ × π ′ s ( t − d − , v π ′ ). So when t = k + s + d + n − , the Harish-Chandra parameters of S-Ind G ′ P ′ ( | · | t b ⊗ π ′ )are ( k + s + n − , v π ′ ). For Re ( s ) + n ≥ s ′ , we have Re ( s ) + n + k − > s ′ . From Lemma 5.1,we have H S i ( G ′ , S-Ind G ′ P ′ ( σ k b ⊗ π ) b ⊗ π ′ ) = 0 . for i = 0 , , · · · and k = 1 , , · · · . Recall that σ k are the graded pieces of a complete decreasingfiltration of the cokernel of T d , then it follows from Corollary A.1 that T d induces an isomorphism H S ( G ′ , S-Ind G ′ P ′ ((S-Ind R , R × × χ ) b ⊗ π ) b ⊗ π ′ ) ∼ = H S ( G ′ , S-Ind G ′ P ′ ( | det( · ) | s δ P (det( · )) σ (cid:12)(cid:12) R r, b ⊗ π ) b ⊗ π ′ ) . when Re ( s ) + n ≥ s ′ . (cid:3) The above proposition shows that T d induces an isomorphism(5.17)Hom G ′ (S-Ind G ′ P ′ ( | det( · ) | s δ P (det( · )) σ b ⊗ π ) b ⊗ π ′ , C ) → Hom G ′ (S-Ind G ′ P ′ (S-Ind R , R × × χ b ⊗ π ) b ⊗ π ′ , C )when s + n ≥ s ′ . Let G = SO ( V − , q − ) ⊂ G ′ where V − = X ′ r ⊕ Y ′ r ⊕ V is the orthogonal complementof v ′ in V ′ . Denote P − = (( R × × × G ) ⋊ N ′ ⊂ G − , which is the parabolic subgroup of G − thatstabilizes X ′ r . Then from Proposition A.4 and Corollary A.3, we haveHom G ′ (S-Ind G ′ P ′ (S-Ind P ′ e P χ b ⊗ π ) b ⊗ π ′ , C ) = Hom P − ( δ − P − ⊗ χ b ⊗ π b ⊗ π ′ | P − , C )= Hom G − ((S-Ind G − P − χ b ⊗ π ) b ⊗ π ′ | G − , C ) . (5.18)We define F ,s : Hom G ′ (S-Ind G ′ P ′ ( | det( · ) | s δ P σ b ⊗ π ) b ⊗ π ′ , C ) → Hom G − (S-Ind G − P − ( χ b ⊗ π ) b ⊗ π ′ | G − , C )by compositing (5.17) and (5.18), which is isomorphic when Re ( s ) + n ≥ s ′ .5.4. Descent maps and multiplicity formula.
Now we have our descent map F s , the composi-tion F ,s ◦ F ,s , which is bijective map under Condition(P). When σ is a principal series represen-tation (including the case when r = 0 and σ = ρ m ), the descent map F s : Hom G ′ ( π s b ⊗ π ′ , C ) → Hom S ′ r ( π b ⊗ π ′ , ν ′ r )and when σ is a discrete series representation, the descent map F s : Hom G ′ ( π s b ⊗ π ′ , C ) → Hom G − ((S-Ind G − P − χ b ⊗ π ) b ⊗ π ′ | G − , C ) . From the bijectiveness of the descent map F s under Condition(P), we have the following multi-plicity formulas. Theorem 5.1. (1) If σ is a principal series representation, m ( π s , π ′ ) = m ( π ′ , π ) for general s . (2) If σ is a discrete series representation of GL ( R ) , m ( π s , π ′ ) = m ( π ′ , π ) for Re ( s ) ≥ s ′ . (3) If σ is the discrete series representation D n , then m ( π s , π ′ ) = m ( π ′ , b π ) for Re ( s ) + n ≥ s ′ , where b π = χ ( · ) | · | − d − × π = | · | s + n sgn n +1 × π . .5. Proofs for co-dimension one cases.
In this subsection, we will make use of the multiplicityformula to prove the co-dimension one case of the local Gan-Gross-Prasad conjecture for genericparameters of special orthogonal groups over the real field by reducing it to the tempered case usingmathematical induction.Let ( e G, e H, e ν ) be a co-dimension one Gan-Gross-Prasad triple, where e G = SO ( V, q ) × SO ( V ′ , q ′ ).For a generic L -parameter e ϕ = ϕ × ϕ ′ of e G , and π × π ′ = π b ⊗ π ′ ∈ Π Vogan e ϕ . Let ϕ = ϕ ⊕ ϕ + ⊕ ( ϕ + ) ∨ and ϕ ′ = ϕ ′ ⊕ ϕ ′ + ⊕ ( ϕ ′ + ) ∨ be the decompositions in (3.2), and let τ and τ ′ be representationscorresponding to the L -parameter ϕ + and ϕ ′ + respectively. From Proposition 3.4, π = τ × π , π ′ = τ ′ × π ′ where π , π ′ are tempered representations with L -parameter equal to ϕ and ϕ ′ respectively.We write τ = | det( · ) | s τ × | det( · ) | s τ × · · · × | det( · ) | s l τ l τ ′ = | det( · ) | s ′ τ ′ × · · · × | det( · ) | s ′ l ′ τ ′ l ′ where Re ( s ) ≥ Re ( s ) ≥ · · · ≥ Re ( s l ) > Re ( s ′ ) ≥ · · · ≥ Re ( s ′ l ′ ) >
0. Here τ i arerepresentations of GL r i ( R ), such that when r i = 2, τ i ∼ = D n i for an positive integer n i , when r i = 1, τ i ∼ = sgn m i and we define n i = 0. And we define r ′ i , n ′ i similarly based on τ ′ i . Proposition 5.4.
With the above notations, there are two tempered representations τ temp and τ ′ temp of some general linear groups over R such that (5.19) m ( τ × π , τ ′ × π ′ ) = m ( τ temp × π , τ ′ temp × π ′ ) for every π ∈ Π Vogan ϕ and every π ′ ∈ Π Vogan ϕ ′ .Proof. We prove proposition by mathematical induction on N ( τ, τ ′ ) = X Re ( s i ) =0 r i + X Re ( s ′ i ) =0 r ′ i . If N ( τ, τ ′ ) = 0, we take τ temp = τ and τ ′ temp = τ ′ and then (5.19) holds for every π ∈ Π Vogan ϕ , π ′ ∈ Π Vogan ϕ ′ .If we change the order of the normalized induction, we may assume Re ( s ) + n ≥ Re ( s ) + n ≥ · · · ≥ Re ( s l ) + n l ≥ ,Re ( s ′ ) + n ′ ≥ Re ( s ′ ) + n ′ ≥ · · · ≥ Re ( s ′ l ′ ) + n ′ l ′ ≥ . Suppose the proposition is true when N ( τ, τ ′ ) ≤ k , then when N ( τ, τ ′ ) = k + 1, we consider thefollowing cases.Case 1: If l = 0 and Re ( s ) + n ≥ Re ( s ′ ) + n ′ , then let e τ = | det( · ) | s τ × · · · × | det( · ) | s l τ l .If dim τ = 1, from part (2) of Theorem 5.1 we have m ( τ × π , τ ′ × π ′ ) = m ( τ ′ × π ′ , e τ × π ) , for every π ∈ Π Vogan ϕ and π ′ ∈ Π Vogan ϕ ′ . Now we define τ ( s ) = | · | s × e τ , then from part (1)of Theorem 5.1, for general s , and thus for some s ∈ i R , we have m ( τ ′ × π ′ , e τ × π ) = m ( τ ( s ) × π , τ ′ × π ′ ) . or every π ∈ Π Vogan ϕ and π ′ ∈ Π Vogan ϕ ′ .If dim τ = 2, let b τ = | · | s + n sgn n +1 × e τ , and from part (3) of Theorem 5.1, we have m ( τ × π , τ ′ × π ′ ) = m ( τ ′ × π ′ , b τ × π ) , for every π ∈ Π Vogan ϕ and π ′ ∈ Π Vogan ϕ ′ . Now we define τ ( s ) = | · | s × b τ , then from part (1)of Theorem 5.1, for general s , and thus for some s ∈ i R , we have m ( τ ′ × π ′ , b τ × π ) = m ( τ ( s ) × π , τ ′ × π ′ ) . for every π ∈ Π Vogan ϕ and π ′ ∈ Π Vogan ϕ ′ .As N ( τ ( s ) , τ ′ ) ≤ N ( τ, τ ′ ) − k , we have tempered representation τ temp and τ ′ temp suchthat m ( τ ( s ) × π , τ ′ × π ′ ) = m ( τ temp × π , τ ′ temp × π ′ )for every π ∈ Π Vogan ϕ and π ′ ∈ Π Vogan ϕ ′ , which implies m ( τ × π , τ ′ × π ′ ) = m ( τ temp × π , τ ′ temp × π ′ )Case 2: If l = 0 or Re ( s ) + n < Re ( s ′ ) + n ′ , then take τ ′ ( s ) = | · | s × τ ′ . From [SV80, Theorem1.1] and Langlands classification, for every π ′ ∈ Π Vogan ϕ ′ , τ ′ ( s ′ ) × π ′ is irreducible for general s . so there is a s ′ ∈ i R , such that τ ′ ( s ′ ) × π ′ is irreduciblefor every π ′ ∈ Π Vogan ϕ ′ and m ( τ × π , τ ′ × π ′ ) = m ( τ ′ ( s ′ ) × π ′ , τ × π ) . for every π ∈ Π Vogan ϕ and π ′ ∈ Π Vogan ϕ ′ .Then the pair ( τ ′ ( s ′ ) , τ ) belongs to Case 1 and N ( τ ′ ( s ′ ) , τ ) = N ( τ, τ ′ ) = k + 1, so thereexists τ ′ temp and τ temp such that m ( τ ′ ( s ′ ) × π ′ , τ × π ) = m ( τ ′ temp × π ′ , τ temp × π )for every π ∈ Π Vogan ϕ and π ′ ∈ Π Vogan ϕ ′ , which implies m ( τ × π , τ ′ × π ′ ) = m ( τ ′ temp × π ′ , τ temp × π ) . By combining both cases, we can conclude that the proposition is true when N ( τ, τ ′ ) = k + 1.Finally the proposition follows from mathematical induction. (cid:3) With this proposition, we are able to reduce the co-dimensional one case of local Gan-Gross-Prasad conjecture for generic L -parameters of special orthogonal groups over archimedean localfields to that for tempered L -parameters.Let ϕ + temp and ϕ ′ temp + be the L -parameter of τ temp and τ ′ temp respectively. And we definetempered parameters ϕ temp = ϕ ⊕ ϕ + temp ⊕ ( ϕ + temp ) ∨ ϕ ′ temp = ϕ ′ ⊕ ϕ ′ temp + ⊕ ( ϕ ′ temp + ) ∨ hen from the tempered case proved in [Luo20], we have X π b ⊗ π ′ ∈ Π Vogan ϕ × ϕ ′ m ( π, π ′ ) = X π b ⊗ π ′ ∈ Π Vogan ϕ × ϕ ′ m ( τ × π , τ ′ × π ′ )= X π b ⊗ π ′ ∈ Π Vogan ϕ × ϕ ′ m ( τ temp × π , τ ′ temp × π ′ )= X π temp b ⊗ π ′ temp ∈ Π Vogan ϕtemp × ϕ ′ temp m ( π temp , π ′ temp )= 1 . which proves the co-dimension one case of Theorem 2.1.Now we assume Conjecture 2.1 for tempered parameters. Then for every generic L -parameter e ϕ of e G , from Proposition 5.4, there exists τ temp and τ ′ temp such that m ( τ × π , τ ′ × π ′ ) = m ( τ temp × π , τ ′ temp × π ′ ) . for every π ∈ Π Vogan ϕ and every π ′ ∈ Π Vogan ϕ ′ .On the one hand, the unique pair( π temp , π ′ temp ) = ( τ temp × π , τ ′ temp × π ′ ) ∈ Π Vogan ϕ temp × Π Vogan ϕ ′ temp such that m ( π temp , π ′ temp ) = 1satisfy η π temp = η V temp , η π ′ temp = η V ′ temp . On the other hand, from [GGP12, Proposition 5.1], we have η V = η V = η V temp , η V ′ = η V ′ = η V ′ temp , and for π = τ × π and π ′ = τ ′ × π ′ , we have η π = η π = η π temp , η π ′ = η π ′ = η π ′ temp . Hence m ( π, π ′ ) = 1if and only if η π = η V , η π ′ = η V ′ . Therefore, the co-dimension one case of Theorem 2.2 is proved.5.6.
Reduction to co-dimension one cases.
Finally, we will prove the reduction of the generalcase to the co-dimension one case.Let ( e G ′ , e H ′ , e ν ′ ) be a Gan-Gross-Prasad triple and use the notations in the introductory part ofthis section. Let σ be a principal series of GL r +1 , then for every irreducible Casselman-Wallachrepresentation π ′ in generic L -packets of G ′ and irreducible Casselman-Wallach representation π in the generic L -packets of G , from [SV80, Theorem 1.1] and Langlands classification, for general s , π s = | det( · ) | s σ × π is irreducible.From Theorem 5.1, for general s ∈ C , we have m ( π s , π ′ ) = m ( π ′ , π ) . e choose s such that π s is irreducible and the above equation holds for all π ′ ∈ Π Vogan ϕ ′ and π ∈ Π Vogan ϕ . Then with the same arguments as the last subsection, the local Gan-Gross-Prasadconjecture for ( e G ′ , e H ′ , e ν ′ ) can be reduced to the local Gan-Gross-Prasad conjecture for ( e G, e H, e ν )which is a co-dimension one Gan-Gross-Prasad triple. Therefore, Theorem 2.1 and Theorem 2.2are proved over the real field. Appendix A. Schwartz inductions and Schwartz homologies
In this appendix, we introduce Schwartz inductions and Schwartz homologies, our main technicaltools. We work in the setting of almost linear Nash groups G (see [Sun15]) and in the category R ( G ) of smooth Fr´echet representations of moderate growth. A representation π in R ( G ) is calledCasselman-Wallach if the ( g C , K )-module π K is Harish-Chandra (see Section 2.2).A.1. Schwartz inductions.
For an almost linear Nash group G and a Casselman-Wallach repre-sentation ( π, V ), the Casselman-Wallach globalization theorem implies that V is a nuclear Fr´echetspace (see [BK14]), and the completed tensor product with a nuclear Fr´echet space is an exactfunctor in the category of nuclear Fr´echet spaces ( [CHM00, Lemma A.3]). Then we have Proposition A.1.
The completed tensor product with a Casselman-Wallach representation is anexact functor in R ( G ) . In order to define the Schwartz induction in the sense of [DC91, Section 2], we let H be a Nashsubgroup of G , ( σ, V ) ∈ R ( H ) and I be the continuous map(A.20) I : S ( G, V ) → C ∞ ( G, V ) , f (cid:0) g Z H σ ( h ) f ( h − g ) d l h (cid:1) . The Schwartz induction S-Ind GH V is defined to be the image of this map. Chen and Sun usedin [CS20] another way to give the Schwartz induction with the Schwartz sections of the temperedbundle H \ G × V , where the right action of H is given by h · ( g, v ) = ( h − g, h − v ), and from [CS20,Propositions 6.7, 6.11], we have Proposition A.2.
There is an isomorphism
S-Ind GH V ∼ = Γ S ( H \ G, V ) ∈ R ( G ) , where Γ S ( H \ G, V ) stands for the space of Schwartz sections of the tempered bundle H \ G × V . From [DC91, Proposition 2.2.7], the Schwartz induction S-Ind GH is an exact functor from R ( H ) to R ( G ), and there is a straightforward transitivity of Schwartz inductions proved in [DC91, Lemma2.1.6], that is, Proposition A.3 (Transitivity) . For H ⊂ H ⊂ H , and V ∈ R ( H ) , we have S-Ind H H (cid:16) S-Ind H H V (cid:17) = S-Ind H H V. And Chen and Sun proved in [CS20, Proposition 7.4] that the completed tensor product com-mutes with the Schwartz induction:
Proposition A.4.
Let W ∈ R ( H ) and V ∈ R ( G ) . Assume that one of W and V is nuclear. Then S-Ind GH (cid:0) W b ⊗ V | H (cid:1) = (S-Ind GH W ) b ⊗ V. .2. Schwartz homologies.
The Schwartz homology H S i ( G, V ) is defined to be the left derivedfunctors of the coinvariant functor V V G , where V G = V / P g ∈ G ( g − V . In particular, H S ( G, V ) = V G . Like other kinds of homologies, the Schwartz homologies also produces a longexact sequence from a short exact sequence. Proposition A.5 (Long exact sequence) . Given a short exact sequence of representations in R ( G )0 → V → V → V → Then there is a long exact sequence · · · → H S i +1 ( G, V ) → H S i ( G, V ) → H S i ( G, V ) → H S i ( G, V ) → · · · Because for every f ∈ Hom G ( V, C ), we have f ( P g ∈ G ( g − V ) = P g ∈ G f (( g − V ) = 0. Thenwe have(A.21) Hom G ( H S ( G, V ) , C ) = Hom G ( V, C ) . Suppose that we have
V, W ∈ R ( G ) and a G -homomorphism f : V → W that induces anisomorphism of H S ( G, f ) : H S ( G, V ) → H S ( G, W ) . Then the induced map D f : Hom G ( W, C ) → Hom G ( V, C )is an isomorphism as well.For instance, if W is equal to the space of left equivariant Schwartz functions on a Nash G -manifold X and V is equal to the space of left equivariant Schwartz functions on a open G -submanifold U of X and f is the extension by zero, then the map Hom G ( W, C ) to Hom G ( V, C )induced from f is the restriction map from the space of left equivariant tempered distributions on X to the space of left equivariant tempered distributions on U , and the restriction map is an iso-morphism of G -modules if we assume H S ( G, f ) is an isomorphism. Because f is injective, the longexact sequence gives a sufficient condition for H S ( G, f ) to be bijective, that is, H S i ( G, W/V ) = 0for i = 0 , Borel’s lemma.
The Borel’s Lemma depicts the structure of
W/V in a more general context.Let X be a Nash manifold, Z be a closed Nash manifold of X , and U be the open complementof Z in X . For a tempered bundle E over X , the extension by zero gives a natural injectionΓ S ( U , E ) → Γ S ( X , E ). As defined [CS20, Section 6.1], the complexification of the conormal bundle N ∨Z / X is a tempered bundle over Z . From [CS20, Propositions 8.2, 8.3], we have a proposition thatgives a complete decreasing filtration of Γ SZ ( X , E ) = Γ S ( X , E ) / Γ S ( U , E ): Proposition A.6 (Borel’s lemma) . There is a complete decreasing filtration on Γ SZ ( X , E ) , denotedby Γ SZ ( X , E ) k , satisfying Γ SZ ( X , E ) = lim ←− Γ SZ ( X , E ) k and the graded pieces are isomorphic to (A.22) Γ S (cid:16) Z , Sym k N ∨Z / X ⊗ E (cid:12)(cid:12)(cid:12) Z (cid:17) , k = 0 , , , · · · Moreover, if X is a G -Nash manifold, Z is stable under the action of G and E is a tempered G -bundle, then this filtration is stable under G . n order to apply the Borel’s lemma to prove that the Schwartz homology H S i ( G, Γ SZ ( X , E )) = 0,we need the following property about the relation between Schwartz homologies and the inverselimits, which follows from the exactness of the inverse limits in inverse systems with Mittag-Lefflerconditions, and is elaborated in [Xue20, Proposition 2.13]. Proposition A.7.
For representation V ∈ R ( G ) , assume that V admits a complete decreasingfiltration V n , for n = 0 , , · · · , and that for some i , the Schwartz homologies H S i +1 ( G, V /V n ) arefinite-dimensional for all n . Then the canonical map H S i ( G, V ) → lim ←− H S i ( G, V /V n ) is an isomorphism. Suppose that a representation V has a complete decreasing filtration of subspaces V n such that H S i ( G, V n /V n +1 ) = 0 for i = 0 , , · · · . Then from the long exact sequence and mathematicalinduction, we have H S i ( G, V /V n +1 ) = 0 for i = 0 , , · · · , and hence(A.23) H S i ( G, V ) = lim ←− H S i ( G, V /V n ) = 0 . We call the Schwartz homologies of a representation V of G vanish if and only if H i ( G, V ) = 0,for i = 0 , , · · · . The arguments above can be summarized as Corollary A.1.
Suppose that V ∈ R ( G ) has a complete decreasing filtration such that the Schwartzhomologies of the graded pieces vanish. Then the Schwartz homologies of V vanish. A.4.
Vanishing theorem.
For
V, V ∨ ∈ R ( G ), if there is a non-degenerate G -invariant bilinearmap B : V × V ∨ → C , we call V and V ∨ are contragredient to each other and ( V, V ∨ ) is acontragredient pair of G . Suppose V has an infinitesimal character χ V , that is, a character of Z ( g C ) such that z.v = χ ( z ) v for every z ∈ Z ( g C ) and v ∈ V , then Proposition A.8.
The representation V ∨ has an infinitesimal character χ V ∨ which is defined by (A.24) χ V ∨ ( z ) = χ V ( z ) for every z ∈ Z ( g C ) , where z z is the automorphism of the universal enveloping algebra U ( g C ) such that X = − X for X ∈ g C .Proof. For every v ∈ V , v ∨ ∈ V ∨ and z ∈ Z ( g C ), we have B ( v, z.v ∨ ) = B ( zv, v ∨ ) = χ V ( z ) B ( v, v ∨ ) = B ( v, χ V ∨ ( z ) v ∨ ) . And as the bilinear form is non-degenerate, we have z.v ∨ = χ V ∨ ( z ) v ∨ . (cid:3) We are going to give a sufficient condition about when the Schwart homologies of π b ⊗ σ vanish,where π, σ ∈ R ( G ). The following theorem is the counterpart of the vanishing theorem of continuouscohomologies, which can be proved by the bridge between Schwartz homologies and continuouscohomologies built by the ( g C , K )-homologies and cohomologies and Poincar´e duality when G isconnected. And it is also proved directly in [Xue20, Proposition 2.7]. Corollary A.2 (Vanishing theorem) . Let ( π, V ) and ( σ, W ) be representations in R ( G ) , supposethat there are two different complex number c = c and an element z ∈ Z ( g C ) such that z.v = c v and z.w = c w , for every v ∈ V , w ∈ W , then the Schwartz homologies of V b ⊗ W vanish. .5. Shapiro’s lemma.
Chen and Sun proved in [CS20, Proposition 7.5] the following lemmawhich shows a profound relation between Schwartz inductions and Schwartz homologies.
Proposition A.9 (Shapiro’s lemma) . Let H be a closed Nash subgroup of G and V ∈ R ( H ) . Then H S i ( G, (S-Ind GH ( V ⊗ δ H )) ⊗ δ − G ) = H S i ( H, V ) for all i ≥ . In particular, when i = 0, it gives an isomorphism between coinvariants (constructed in [CS20,Propositions 6.9, 6.12])(A.25) T H : ( Ind GH ( V ⊗ δ H ) ⊗ δ − G ) G ∼ = V H , which induces an isomorphism(A.26) Hom G (S-Ind GH ( V ⊗ δ H ⊗ δ − G | H ) , C ) = Hom H ( V, C ) . By combining these arguments with Proposition A.4, we have
Corollary A.3.
Let H be a Nash subgroup of G and V ∈ H . For a Casselman-Wallach represen-tation W of G , the map T H induces an isomorphism Hom G (S-Ind GH ( V ⊗ δ H ⊗ δ − G | H ) b ⊗ W, C ) = Hom H ( V b ⊗ W, C ) . Appendix B. Harish-Chandra parameters
We combine the Harish-Chandra’s parameterization of infinitesimal characters and the vanishingtheorem in Corollary A.2 to obtain a sufficient condition for vanishing of Schwartz homologies of arepresentation. Because the results in this subsection are classical results, we work in the languageof real reductive groups and admissible representations.B.1.
Definitions.
Harish-Chandra gave a full classification of discrete series representations for aconnected semisimple group G with a maximal compact subgroup K . The group G has a discreteseries representation if and only if G and K have the same rank, equivalently, the maximal torus T of K is a compact Cartan subgroup of G . In [HC65] and [HC66], Harish-Chandra classifieddiscrete series representations with Harish-Chandra parameters v π in L + ρ such that v π is notorthogonal to any roots of G , where L is the weight lattice of G contained in the complex dual t ∗ C of the Lie algebra of T and ρ is the Weyl vector, that is, the half sum of of all positive roots inthe root system of G . For v π ∈ L + ρ that is orthogonal to a root of G , it also corresponds to afinite set of irreducible representations, and these representations are called limits of discrete seriesrepresentations of G . For simplicity, in this article, a limit of discrete series representations meansa discrete series representation or a limit of discrete series representations. For limits of discreteseries representations π and π of G , π and π have the same infinitesimal character if and onlyif v π is conjugate to v π by W G , the Weyl group of G . Moreover, the infinitesimal character χ π of π and the character χ v of the polynomial algebra P ( t ∗ C ) W G defined by the evaluation at v π arerelated to each other through the equation χ π ( z ) = χ v ( HC ( z ))for every z ∈ Z ( g C ), where HC is the Harish-Chandra isomorphism HC : U ( g C ) G = Z ( g C ) → S ( t ) W G = P ( t ∗ ) W G . And the correspondence between v π and χ π is called the Harish-Chandra correspondence. or G = GL n ( R ), elements in the dual a ∗ C of Cartan subalgebra of g C can be parametrized by( λ , λ , · · · , λ n ) ∈ C n and the Weyl group W G is isomorphic to the permutation group S n that permutes the n -entries.By the Langlands classification, each irreducible admissible representation π is a subquotient ofsome | · | s sgn m × · · · × | · | s n sgn m n where s i ∈ C , m i = 0 , sgn is the sign representation of GL ( R ) such that sgn ( x ) = 1 when x > sgn ( x ) = − x <
0. Then define the Harish-Chandra conjugacy class [ v π ] of π asthe W G -conjugacy class of ( s , s , · · · , s n ) ∈ a ∗ C . The Harish-Chandra correspondence gives a one to one map from the W G -conjugacy classes of theHarish-Chandra parameters to the set of infinitesimal characters of irreducible admissible represen-tations of G , which shows [ v π ] is well defined. And an element v π ∈ [ v π ] is called a Harish-Chandraparameter.We now come back to the case when G = SO ( V, q ), where (
V, q ) is a non-degenerate quadraticspace over R with a decomposition V = X r ⊕ Y r ⊕ V an such that X r , Y r are maximal totally isotropicsubspaces of V and V an is anisotropic. Let A be a maximal torus of GL ( X r ) and T be a maximaltorus of SO ( V an ). When dim V = 2 n + 1, the weight space h ∗ C = a ∗ , C + t ∗ C are parameterized as λ = ( λ , λ , · · · , λ n ) ∈ C n , where the action of W G is generated by switching two entries and changing signs. When dim V = 2 n the weight space is parameterized as λ = ( λ , λ , · · · , λ n ) ∈ C n , and the action of W G is generated by switching two entries and changing signs for two entries. FromLanglands classification, each irreducible admissible representation π is isomorphic to a subquotientof(B.27) | · | s sgn m × · · · × | · | s l sgn m l × π an where s i ∈ C , m i = 0 , π an is a finite-dimensional representation of SO ( V an ). In this case,the Harish-Chandra conjugacy class [ v π ] is defined as the W G -conjugacy class of ( v τ , v π an ), where v τ = ( s , s , · · · , s l ) is a Harish-Chandra parameter of τ = | · | s sgn m × · · · × | · | s l sgn m l and v π an isthe sum of the highest weight of π an with the Weyl vector of SO ( V an ). Then the Harish-Chandracorrespondence gives that for admissible representations π and π ′ , they have the same infinitesimalcharacters if and only if they have the same Harish-Chandra conjugacy class. And an element v π in the conjugacy class [ v π ] is called a Harish-Chandra parameter.B.2. Relations with the vanishing theorem.
For an irreducible admissible representation π of SO ( V, q ) such that σ is a subquotient of I = | · | s sgn m × · · · × | · | s n sgn m n × π an as in (B.27). We take I ∨ = | · | − s sgn m × · · · × | · | − s n sgn m n × π ∨ an . And there is a non-degenerate P -invariant form of the induced data of I and I ∨ , which gives a non-degenerate G -invariant bilinear form between I and I ∨ by integrating over the maximal compact ubgroup K of G . Therefore there is a unique irreducible subquotient σ ∨ of I ∨ such that B inducesa non-degenerate bilinear form of σ and σ ∨ . Then Proposition A.8 implies that Proposition B.1.
The Harish-Chandra conjugacy class [ v π ∨ ] of the contragredient representation π ∨ of π is the same as [ − v π ] , where v π is a Harish-Chandra parameter of π . From the above discussions, χ π does not equal to χ π ′∨ if and only if [ v π ] = [ − v π ′ ]. When π and π ′ are Casselman-Wallach, from Corollary A.2, we have Corollary B.1. If [ v π ] = [ − v π ′ ] , then the Schwartz homologies of π b ⊗ π ′ vanish. B.3.
Tensor products with finite-dimensional representations.
Finally, we review the Kostant’sresults about the composition series of the tensor product of a finite-dimensional representation withan irreducible Casselman-Wallach representation. It follows from [Kos75, Section 1.3] that
Proposition B.2.
The tensor product of an irreducible finite-dimensional representation with anirreducible Casselman-Wallach representation has a finite composition series.
And [Kos75, Corollary 5.6] shows that
Proposition B.3.
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School of Mathematics, University of Minnesota, USA
Email address : [email protected]@umn.edu