Intertwining periods and distinction for p-adic Galois symmetric pairs
aa r X i v : . [ m a t h . R T ] J a n INTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOISSYMMETRIC PAIRS
NADIR MATRINGE AND OMER OFFEN
Abstract.
We consider distinction of representations in the context of p -adic Galoissymmetric spaces. We provide new sufficient conditions for distinction of parabolicallyinduced representations in terms of similar conditions on the inducing data and deducea characterization for distinction of representations parabolically induced from cuspidal.We explicate the results further for classical groups and give several applications, in par-ticular, concerning the preservation of distinction via Langlands functoriality. We relateour results with a conjecture of Dipendra Prasad. Contents
1. Introduction 2Acknowledgement 52. Notation and preliminaries 52.1. Parabolic subgroups and Iwasawa decomposition 52.2. Roots and Weyl groups 52.3. Symmetric spaces 72.4. Parabolic induction and standard intertwining operators 72.5. Meromorphic families of linear forms 92.6. The result of Blanc and Delorme 93. Intertwining periods 103.1. An informal introduction 103.2. The strategy 103.3. The set up 113.4. A graph of involutions 113.5. Minimal vertices 113.6. The cone of convergence 123.7. Admissible orbits 123.8. The intertwining periods modulo convergence 133.9. Intertwining periods for minimal vertices 133.10. Convergence in the unramified case 133.11. The main result 153.12. Applications to distinction 164. Classical Galois pairs 18
Date : February 2, 2021. G ◦ W G ( M ) and elementary symmetries. 306.6. A choice of representatives in G for W G ( M ) 316.7. Orbits and stabilizers in X ∩ N G ◦ ( M ◦ ) 326.8. The contribution of an admissible orbit 376.9. Completion of the proof of Theorem 2 377. Distinction and conjugate-selfduality in the split odd orthogonal case 388. On distinction and local Langlands correspondence 418.1. On Galois distinction for GL N Introduction
Let
E/F be a quadratic extension of p -adic fields with Galois involution θ , H a connectedreductive group defined over F , and G = Res E/F ( H ). Set G = G ( F ) and H = H ( F ). In thispaper we provide a step in reducing the study of H -distinction of irreducible representationsof G to that of cuspidal representations.Recall that a smooth representation π of G is called H -distinguished if it admits a nonzero H -invariant linear form. Also, any irreducible representation π of G is the quotient of aninduced representation of the form I GP ( σ ) for σ a cuspidal representation of a Levi part ofthe parabolic subgroup P of G . In particular if π is H -distinguished then so is I GP ( σ ). One NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 3 of our main results is a necessary and sufficient condition on σ for I GP ( σ ) to be distinguished,it is the content of Corollary 2.Of course one is far from understanding when π is distinguished from this result, andto go further one needs to understand when a nonzero H -invariant linear form L on I GP ( σ )descends to π .One of the motiviations of the present work is the paper [Pra] that makes precise pre-dictions on the Langlands parameter of an irreducible representation of G that is H -distinguished, and on the multiplicity of H -invariant linear forms on π in terms of such aparameter. In fact if we want to state results in terms of Langlands parameters, one hasto go via more steps. More precisely, π is the unique irreducible quotient of a standardmodule I GP ( τ ) thanks to the Langlands quotient theorem. Here τ is an essentially temperedrepresentation and therefore a summand of a representation induced from an essentiallydiscrete series representation. Finally, one studies distinction of discrete series representa-tions via distinction of their cuspidal support. At each step one would like a necessary andsufficient condition on the inducing datum of the induced representation considered to be H -distinguished, and an understanding of when an H -invariant linear form on the inducedrepresentation descends to a given irreducible quotient.Necessary conditions for distinction of an induced representation of G in terms of dis-tinction of the inducing datum are usually obtained by means of the Berstein-Zelevinskygeometric lemma, and this technique is nowadays very standard. In fact in the case ofrepresentations induced from a cuspidal representation the necessary condition obtainedfrom the geometric lemma becomes sufficient, as we show in Corollary 2.To show that the necessary condition given by the geometric lemma is sufficient is notas straightforward and we make use of local intertwining periods which in fact allow us toobtain sufficient conditions for induced representations to be distinguished in a more generalcontext, see Proposition 3. This can be seen as the most basic application of intertwiningperiods to the theory. More generally to understand when a nonzero H -invariant linearform on an induced representation descends to an irreducible quotient is a notoriouslydifficult problem in the field. One way to tackle it is to study local intertwining periodson induced representations and their functional equation (see Remark 3). We expectour result to have further applications concerning distinction of generic discrete series ofclassical groups.Intertwining periods were introduced in the global setting in [Jac95] and further studiedin [JLR99] and [LR03]. Informally, suppose for a moment that F is a number field with ringof adeles A . The global intertwining periods are meromorphic families of H ( A )-invariantlinear forms on spaces obtained by parabolic induction from automorphic representationsof Levi subgroups of G ( A ). The linear forms are defined by (the meromorphic continuationof) certain integrals. They arise naturally in the study of period integrals of Eisensteinseries and consequently appear in the spectral expansion of the relative trace formula.They have also been considered for non-Galois symmetric pairs (see e.g. [Off06] wherethey were used to study the Sp n ( A )-period integral on the residual spectrum of GL n ( A )).Here, building on [Off17], we develop in Section 3 their local theory for Galois pairs, inanalogy with the global work [LR03]. Returning to the local setting where F is p -adic, our NADIR MATRINGE AND OMER OFFEN main result on local intertwining periods is their convergence in a cone and meromorphiccontinuation. More precisely, let P be a standard parabolic subgroup of G (with respectto a fixed minimal parabolic subgroup and a fixed θ -stable maximal F -split torus in it)and M the standard Levi subgroup of P . Let x ∈ G be such that θ ( x ) = x − and assumefurther that the involution θ x = Ad( x ) ◦ θ of G stabilizes M . Denote by S x the θ x -fixedpoints in a subset S of G .For a finite length (smooth complex valued) representation σ of M with bounded matrixcoefficients, ℓ ∈ Hom M x ( σ, M x ) and χ an unramified character of M anti-invariant under θ x , we show that for χ in a certain sufficiently positive cone, the integral Z P x \ G x ℓ ( ϕ χ ( h )) dh converges absolutely for all holomorphic sections ϕ χ ∈ I GP ( χ ⊗ σ ) and admits a meromorphiccontinuation with respect to χ . When ℓ is nonzero, by taking a leading term of such anintertwining period we prove in Proposition 3 the existence of a nonzero G x -invariant linearform on I GP ( σ ). The existence of such ℓ is our new sufficient condition for distinction. Itimmediately leads to our characterization of distinction when σ is cuspidal (Corollary 2).The Levi subgroup M acts naturally by twisted conjugation on the set of all x as aboveand the condition on a cuspidal σ in Corollary 2 is expressed in terns of the M -orbits. Theformulation is not always convenient to use in view of the motivation, obtaining results interms of Langlands parametrization. Hence in Section 5, we make our criterion completelyexplicit when H is a classical group. This is the content of Theorem 2. Its proof requiresa careful analysis of orbits and stabilizers that is carried out in Section 6. In particular,for this task we obtain in § G -orbits in the symmetric space X = { x ∈ G : θ ( x ) = x − } with respect to twisted conjugation g · x = gxθ ( g ) − .Let K/k be a quadratic extension of number fields with Galois involution still denotedby θ . As an application of Theorem 2 we prove in Theorem 3 that an irreducible genericcuspidal automorphic representation of the split odd special orthogonal group SO n +1 ( A K )which is SO n +1 ( A k )-distinguished (or more generally, distinguished by a certain K/k -inner form) is automatically θ -conjugate self-dual. The rigidity theorem of [JS03] is akey ingredient in our proof. From this and a globalization result of Beuzart-Plessis [BP]we deduce a local analogue of this result in Theorem 4 that slightly generalizes, for p-adic SO n +1 , a Theorem in [BP] for general p-adic quasi-split groups. We also deduce inProposition 7 similar results for some non-generic representations.In Section 8 we study a relation between Galois distinction for representations of quasi-split classical groups and their transfer to a general linear group. As explained by Proposi-tion 8 this relation is a consequence of Dipendra Prasad’s Conjecture 1. As an applicationof the main results in this work, we prove this relation in Theorem 7 for a certain classof generic irreducible representations of the classical group, that contains all irreduciblegeneric principal series.Finally in Appendix A we provide explicit formulas for Prasad’s quadratic character forclassical groups. Most of these formulas are well-known but we give proofs for convenienceof the reader. We also establish further properties of this character. NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 5
Acknowledgement
We warmly thank Rapha¨el Beuzart-Plessis for his help in the writing of Section A.1 ofAppendix A, and Ahmed Moussaoui for useful conversations and explanations. We alsothank Colette Moeglin for useful correspondence. The first named author thanks the CNRSfor giving him a “d´el´egation” in 2020.2.
Notation and preliminaries
Let F be a non-archimedean local field of characteristic zero and |·| the normalizedabsolute value on F . For an affine algebraic variety X defined over F set X = X ( F ).Let G be a connected reductive group defined over F . We denote by X ∗ ( G ) the group of F -rational characters from G to the multiplicative group G m and let a ∗ G = X ∗ ( G ) ⊗ Z R .Let a G = Hom( a ∗ G , R ) be the dual vector space and let H G : G → a G be the map definedby e h χ,H G ( g ) i = | χ ( g ) | , χ ∈ X ∗ ( G ) , g ∈ G. For a a real vector space we denote by a C = a ⊗ R C its compexification.Throughout the paper, we mostly avoid mention of the underlying algebraic groupsdefined over F and consider only their groups of F -points. When we say that P is aparabolic subgroup of G we mean that P = P ( F ) where P is a parabolic subgroup of G defined over F . We use a similar convention for other algebraic properties such as, (split)tori, unipotent subgroups etc.Let A G be the split component of G , that is, the maximal split torus in the center of G .We recall that restriction to A G identifies a ∗ G with a ∗ A G .For groups A ⊆ B denote by Z B ( A ) the centralizer of A in B and by N B ( A ) thenormalizer of A in B . For a locally compact group Q denote by δ Q its modulus function.2.1. Parabolic subgroups and Iwasawa decomposition.
Let P be a minimal para-bolic subgroup of G and T a maximal F -split torus of G contained in P . The group P admits a Levi decomposition P = M ⋉ U where M = Z G ( T ) and U is the unipotentradical of P . Let a = a T = a M and a ∗ = a ∗ T = a ∗ M .A parabolic subgroup P of G is called semi-standard if it contains T , and standard if itcontains P . A semi-standard parabolic subgroup P of G with unipotent radical U admitsa unique Levi decomposition P = M ⋉ U with the Levi subgroup M containing T . Such aLevi subgroup is called semi-standard. If in addition P is standard, M is called a standardLevi subgroup. Whenever we write P = M ⋉ U is a (semi-)standard parabolic subgroupwe implicitly assume the above (semi-)standard Levi decomposition.Fix a maximal compact open subgroup K of G in good position with respect to P (see[MW95, I.1.4]). For a standard parabolic subgroup P = M ⋉ U of G this allows one toextend H M to a function on G = U M K by H M ( umk ) = H M ( m ) , u ∈ U, m ∈ M, k ∈ K. Roots and Weyl groups.
NADIR MATRINGE AND OMER OFFEN
Roots.
Denote by R ( T, G ) the root system of G with respect to T , by R ( T, P ) theset of positive roots of R ( T, G ) with respect to P , and by ∆ the corresponding set ofsimple roots. Note that R ( T, G ) lies in a ∗ . For every α ∈ R ( T, G ) we denote by α ∨ ∈ a the corresponding coroot.There is a unique element ρ ∈ a ∗ (half the sum of positive roots with multiplicities)such that δ P ( p ) = e h ρ ,H M ( p ) i , p ∈ P . For a standard parabolic subgroup P = M ⋉ U of G let R ( A M , G ) be the set of non-trivialrestrictions to A M of elements of R ( T, G ), R ( A M , P ) the subset of non-zero restrictions to A M of elements of R ( T, P ) and ∆ P the subset of non-zero restrictions to A M of elementsin ∆ . For α ∈ R ( A M , G ) we write α > α ∈ R ( A M , P ) and α < Q = L ⋉ V be a parabolic subgroup of G containing P . The restriction map from A M to A L defines a projection a ∗ M → a ∗ L that gives rise to a direct sum decomposition a ∗ M = a ∗ L ⊕ ( a LM ) ∗ (the second component is the kernel of this projection) and a compatibledecomposition a M = a L ⊕ a LM for the dual spaces. For λ ∈ a ∗ M (respectively ν ∈ a M ) wewrite λ = λ L + λ LM (respectively, ν = ν L + ν LM ) for the corresponding decomposition. Set∆ QP = { α ∈ ∆ P , α L = 0 } . (Note that ∆ QP identifies with ∆ P ∩ L defined with respect to ( L, L ∩ P ) replacing ( G, P ).)The coroot α ∨ ∈ a M associated to α ∈ R ( A M , G ) is defined as follows: le α ∈ R ( T, G )be such that α = ( α ) M then one sets α ∨ = ( α ∨ ) M (it is independent of the choice of α ).Let ρ P = ( ρ ) M . We have the relation δ P ( p ) = e h ρ P ,H M ( p ) i , p ∈ P. Weyl groups and Bruhat decomposition.
We denote by W G := N G ( T ) /M the Weylgroup of G . Note that for a semi-standard parabolic subgroup P = M ⋉ U of G theLevi part M contains M . Therefore expressions such as wP and P w are well defined for w ∈ W G . Furthermore, W M is a subgroup of W G . Let P = M ⋉ U and Q = L ⋉ V be semi-standard parabolic subgroups of G . The Bruhat decomposition is the bijection W M wW L P wQ from W M \ W G /W L to P \ G/Q .2.2.3.
Elementary symmetries.
For standard Levi subgroups M ⊆ L of G let W L ( M ) bethe set of elements w ∈ W L such that w is of minimal length in wW M and wM w − is astandard Levi subgroup of G . Note that for w ∈ W L ( M ) and w ′ ∈ W L ( wM w − ) we have w ′ w ∈ W L ( M ). By definition W L ( M ) is the disjoint union over standard Levi subgroups M ′ of G of the sets W L ( M, M ′ ) = { w ∈ W L ( M ) : wM w − = M ′ } . Set W ( M ) = W G ( M ). In [MW95, I.1.7, I.1.8] the elementary symmetries s α ∈ W ( M )attached to each α ∈ ∆ P are introduced and used in order to define a length function ℓ M on W ( M ). There is a unique element of W L ( M ) of maximal length for ℓ M and we denoteit by w LM . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 7
Symmetric spaces.
By a symmetric pair (
G, θ ) we mean that θ is an involution of G defined over F . We say that the symmetric pair ( G, θ ) is a Galois pair if G = Res E/F H is the Weil restriction of scalars of a connected reductive group H defined over F , E/F isa quadratic field extension and θ is the associated Galois action. Thus, for a Galois pair( G, θ ) we have G = H ( E ) and G θ = H ( F ).To a symmetric pair ( G, θ ) we associate the symmetric space X = { g ∈ G, θ ( g ) = g − } equipped with the twisted conjugation G -action g · x = gxθ ( g ) − , g ∈ G, x ∈ X. By [HW93, Proposition 6.15], X consists of a finite number of G -orbits. For x ∈ X wedenote by θ x the involution of G given by θ x ( g ) = xθ ( g ) x − . For x ∈ X and Q a subgroup of G , we denote by Q x the stabilizer of x in Q , so that Q x = ( Q ∩ θ x ( Q )) θ x . Such an involution θ will induce linear involutions on different vector spaces. Whenever θ is an involution acting linearly on a vector space V over R or C , we write V = V + θ ⊕ V − θ for the corresponding decomposition into the θ -eigenspaces V ± θ with corresponding eigen-values ± Parabolic induction and standard intertwining operators.
By a representationof a subgroup of G we always mean a smooth complex valued representation.2.4.1. Induced representations.
Let P = M ⋉ U be a standard parabolic subgroup of G and σ a representation of M . We denote by I GP ( σ ) the representation of G by right translationson the space of functions ϕ on G with values in the space of σ that are right invariant bysome open subgroup of G and satisfy ϕ ( umg ) = δ P ( m ) / σ ( m ) ϕ ( g ) , u ∈ U, m ∈ M, g ∈ G. Holomorphic sections.
For λ ∈ a ∗ M, C we denote by σ [ λ ] the representation of M onthe space of σ given by σ [ λ ]( m ) = e h λ,H M ( m ) i σ ( m ). In order to make sense of meromor-phic families of linear forms on induced representations we realize all the representations I GP ( σ [ λ ]) for λ ∈ a ∗ M, C on the space of I GP ( σ ).For λ ∈ a ∗ M, C and ϕ ∈ I GP ( σ ) write ϕ λ ( g ) = e h λ,H M ( g ) i ϕ ( g ) and let I GP ( σ, λ ) be therepresentation of G on the space I GP ( σ ) defined by( I GP ( g, σ, λ ) ϕ ) λ ( x ) = ϕ λ ( xg ) . The map ϕ ϕ λ is an isomorphism of representations I GP ( σ, λ ) → I GP ( σ [ λ ]). NADIR MATRINGE AND OMER OFFEN
Intertwining operators.
Let w ∈ W ( M ) and select a representative n of w in N G ( T ).Set M = wM w − = nM n − and let P = M ⋉ U be the standard parabolic subgroup of G with Levi subgroup M . For c ∈ R let D M,w ( c ) = { λ ∈ a ∗ M, C : h Re( λ ) , α ∨ i} > c, ∀ α ∈ R ( A M , G ) , α > , wα < } . We denote by nσ the representation of M on the space of σ defined by ( nσ )( nmn − ) = σ ( m ), m ∈ M . Note that the isomorphism class of nσ is independent of the choice ofrepresentative n for w and we denote it by wσ . For λ ∈ a ∗ M, C we have wλ ∈ a ∗ M , C and thestandard intertwining operator M ( n, σ, λ ) : I GP ( σ, λ ) → I GP ′ ( wσ, wλ )is the meromorphic continuation of the operator given by the following convergent integralfor λ ∈ D M,w ( c σ ) for some constant c σ ( M ( n, σ, λ ) ϕ ) wλ ( g ) = Z U ∩ wUw − \ U ϕ λ ( n − ug ) du. By definition, the convergence of the integral in the appropriate domain means that forevery element v ∨ in the smooth dual of σ the scalar valued integral Z U ∩ wUw − \ U v ∨ [ ϕ λ ( n − ug )] du converges. We record the following simple consequence. Lemma 1.
With the above notation, let λ ∈ D M,w ( c σ ) and ℓ a linear form on the space of σ such that for every ϕ ∈ I GP ( σ ) and g ∈ G the integral Z U ∩ wUw − \ U (cid:12)(cid:12) ℓ ( ϕ λ ( n − ug )) (cid:12)(cid:12) du converges. Then ℓ (( M ( n, σ, λ ) ϕ ) wλ ( g )) = Z U ∩ wUw − \ U ℓ ( ϕ λ ( n − ug )) du. Proof.
Replacing ϕ with I GP ( g, σ, λ ) ϕ we assume without loss of generality that g is theidentity e of G . Let M be a compact open subgroup of M such that ϕ is right M -invariant and M = n − M n a compact open subgroup of M . Let ℓ ( v ) = Z M ℓ ( σ ( m ) v ) dm for every v in the space of σ where integration is over the probability measure of M . Then ℓ lies in the smooth dual of σ . For m ∈ M we have σ ( m )[( M ( n, σ, λ ) ϕ ) wλ ( e )] = ( M ( n, σ, λ ) ϕ ) λ ( nmn − ) = ( M ( n, σ, λ ) ϕ ) wλ ( e )and therefore ℓ (( M ( n, σ, λ ) ϕ ) wλ ( g )) = ℓ (( M ( n, σ, λ ) ϕ ) wλ ( g )) = Z U ∩ wUw − \ U ℓ ( ϕ λ ( n − ug )) du. NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 9
Let ( U k ) k ∈ N be an increasing sequence of compact open subgroups of U with union U andsuch that M normalizes each U k . (It is easy to see that such a sequence exists.) By thedominated convergence theorem we have Z U ∩ wUw − \ U ℓ ( ϕ λ ( n − u )) du = lim k →∞ Z U k ∩ wUw − \U k ℓ ( ϕ λ ( n − u )) du. Note that for every k we have Z U k ∩ wUw − \U k ℓ ( ϕ λ ( n − u )) du = Z U k ∩ wUw − \U k Z M ℓ ( σ ( m )[ ϕ λ ( n − u ))]) dm du = Z U k ∩ wUw − \U k Z M ℓ ( ϕ λ ( mn − u )) dm du. After the change of variables m n − mn this becomes Z U k ∩ wUw − \U k Z M ℓ ( ϕ λ ( n − mu )) dm du. Changing order of integration, making the change of variables u m − um and applyingthe right M -invariance of ϕ this becomes Z U k ∩ wUw − \U k ℓ ( ϕ λ ( n − u )) du. Applying the dominated convergence theorem again we have thatlim k →∞ Z U k ∩ wUw − \U k ℓ ( ϕ λ ( n − u )) du = Z U ∩ wUw − \ U ℓ ( ϕ λ ( n − u )) du and the lemma follows. (cid:3) Meromorphic families of linear forms.
For a finite dimensional complex vectorspace V and a complex vector space V we say that L ( λ ) λ ∈ V is a meromorphic family oflinear forms on V if there is a non-zero polynomial function f on V such that f ( λ ) L ( λ ) isa linear form on V for every λ ∈ V and for all ν ∈ V the map λ f ( λ ) L ( λ ) ν from V to C is holomorphic.2.6. The result of Blanc and Delorme.
For (
G, θ ) a symmetric pair, let P = M ⋉ U be a parabolic subgroup of G such that P ∩ θ ( P ) = M . Note that in this case P θ = M θ isunimodular. Furthermore, since M is θ -stable, θ acts as a linear involution on a ∗ M so that a ∗ M = ( a ∗ M ) + θ ⊕ ( a ∗ M ) − θ . There is a similar decomposition to the dual space a M so that ( a M ) ± θ is dual to ( a ∗ M ) ± θ . Inparticular, we have h λ, H M ( m ) i = 0 , λ ∈ ( a ∗ M ) − θ , m ∈ M θ . For c > D M,θ ( c ) = { λ ∈ ( a ∗ M ) − θ : h λ, α ∨ i > c, ∀ α ∈ R ( A M , P ) } . The following is a consequence of [BD08]. Let σ be a representation of M of finite length.There exists c > ϕ ∈ I GP ( σ ), ℓ ∈ Hom M θ ( σ, M θ ) and λ ∈ D M,θ ( c ) theintegral J GP ( ϕ ; θ, ℓ, σ, λ ) = Z M θ \ G θ ℓ ( ϕ λ ( g )) dg is absolutely convergent. Furthermore, it admits a meromorphic continuation to a mero-morphic family of linear forms J GP ( θ, ℓ, σ, λ ) ∈ Hom G θ ( I GP ( σ, λ ) , G θ ), λ ∈ ( a ∗ M, C ) − θ .3. Intertwining periods
Our goal in this section is to construct certain meromorphic families of invariant linearforms, local intertwining periods, on induced representations associated to a p-adic Galoispair. In the global setting, intertwining periods were introduced in [Jac95] and studiedfurther in a more general setting in [JLR99] and [LR03].3.1.
An informal introduction.
The local intertwining periods emerge as follows. Con-sider a symmetric pair (
G, θ ). Let χ be a character of G θ . For a representation π of G wedenote by Hom G θ ( π, χ ) the space of ( G θ , χ )-equivariant linear forms on the space of π .Consider a parabolic subgroup P of G with a θ -stable Levi subgroup M and let U bethe unipotent radical of P so that P θ = M θ ⋉ U θ . Let σ be a representation of M and ℓ ∈ Hom M θ ( σ, δ P θ δ − / P χ ). For ϕ ∈ I GP ( σ ), the map g χ ( g ) − ℓ ( ϕ ( g )) is left ( P θ , δ P θ )-equivariant and therefore at least formally the integral L ( ϕ ) = Z P θ \ G θ χ ( g ) − ℓ ( ϕ ( g )) dg makes sense. If it converges it defines a linear form L ∈ Hom G θ ( I GP ( σ ) , χ ). In general,however, the integral fails to converge. By considering unramified twists of σ one hopes todefine the linear form by a convergent integral on some cone of unramified twists and bymeromorphic continuation in general.In this section we achieve this goal when ( G, θ ) is a Galois pair and χ is the trivialcharacter.3.2. The strategy.
Our proof of convergence in a cone and meromorphic continuation isbased on the study of the geometry and combinatorics of the P -orbits on G/G θ carried outin [LR03, § P -orbits. Roughly speaking, vertices in the graph areassociated to certain P -orbits and edges to a relation between them defined via twistedconjugation. For a directed edge one expresses the intertwining period associated withthe origin as a double integral that formally corresponds with the composition of theintertwining period associated with the destination and an intertwining operator. Theseare the local analogues of the simple functional equations obtained in [LR03, Proposition10.1.1]. They allow reduction of the meromorphic continuation from origin to destination.In order to reduce convergence of the origin intertwining period to the destination one it isrequired to prove that the above composition expressed as a double integral is absolutely NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 11 convergent. It is done in two steps. First, we prove it for unramified representationsinduced from real exponents. In this case the integrands at hand are positive and it isenough to prove convergence as an iterated integral. This reduction is carried out applyingthe Gindikin-Karpelevich formula and convergence of the standard intertwining operators.In the second step we use the bounds obtained by Lagier in [Lag08, Theorem 4] in orderto reduce to the first case. This inductive process along the graph allows us to reduceconvergence in a cone and meromorphic continuation to certain minimal cases. By alocal analogue of [LR03, Lemma 5.4.1], the minimal case is reduced to the main resultsestablished by Blanc and Delorme in [BD08]. The entire proof is a local analogue of the onecarried by Lapid and Rogawski in [LR03]. We remark that Lapid and Rogawski assumethat G admits a θ -stable minimal parabolic subgroup. Thanks to the analysis of P -orbitson G/G θ for a general involution θ of G in [Off17] we do not need to make this assumption.3.3. The set up.
For the rest of this section assume that (
G, θ ) is a Galois symmetricpair. Fix the minimal parabolic subgroup P of G . According to [HW93, Lemma 2.4]there exists a θ -stable maximal split torus T of G contained in P . Thus, P admits a Levidecomposition P = M ⋉ U where M = Z G ( T ) is a θ -stable Levi subgroup of P .3.4. A graph of involutions.
For a standard Levi subgroup M of G let X [ M ] = { x ∈ X : θ x ( M ) = M } . Note for example that X [ M ] = X ∩ N G ( T ).We define a directed labeled graph G = G ( G, θ, P , T ) as follows. The vertices of G arethe pairs ( M, x ) where M is a standard Levi subgroup of G and x ∈ X [ M ]. The edges of G are given by ( M, x ) n ց ( M , x )if there is α ∈ ∆ P with − α = θ x ( α ) < n ∈ s α M where s α ∈ W ( M ) isthe elementary symmetry associated to α , M = nM n − and x = n · x . Note that( M ) x = nM x n − .3.5. Minimal vertices.
Recall that the Weyl group W G acts (by conjugation) simplytransitively on the set of minimal semi-standard parabolic subgroups of G . There is there-fore a unique τ ∈ W G such that θ ( P ) = τ P τ − .Let ( M, x ) be a vertex in G and P the standard parabolic subgroup of G with Levisubgroup M . Set P ′ = τ − θ ( P ) τ . Then P ′ = M ′ ⋉ U ′ is a standard parabolic subgroupof G with standard Levi subgroup M ′ = τ − θ ( M ) τ = τ − x − M xτ and unipotent radical U ′ = τ − θ ( U ) τ . We say that ( M, x ) is minimal if there exists a standard parabolic subgroup Q = L ⋉ V of G containing P such that • θ ( L ) = τ Lτ − ; • M xτ M ′ = M w LM ′ M ′ ; • θ x ( α ) = − α , α ∈ ∆ QP . This definition coincides with [Off17, Definition 6.6]. Note that the conditions M ⊆ L together with θ ( L ) = τ Lτ − imply that M ′ ⊆ L and therefore w LM ′ makes sense. Thefollowing is a straightforward consequence of the definition. Lemma 2.
Let ( M, x ) be a minimal vertex in G . In the above notation Q , L and V are θ x stable.Proof. Let n ∈ N G ( T ) be a representative of τ . It follows from the second point of thedefinition that xn ∈ L and from the first that L = n − θ ( L ) n . Conjugating with xn weconclude that L = θ x ( L ). Since n − θ ( Q ) n is a standard parabolic subgroup of G withLevi subgroup n − θ ( L ) n = L we conclude that n − θ ( Q ) n = Q and conjugation with xn similarly shows that θ x ( Q ) = Q . Since V is the unipotent radical of Q it immediatelyfollows that θ x ( V ) = V . (cid:3) As a consequence of [LR03, Lemma 3.2.1 and Proposition 3.3.1] (see [Off17, Corollary6.7] ) we have the following.
Proposition 1.
Let ( M, x ) be a vertex in the graph G . There exists a path ( M, x ) n ց ( M , θ ) n ց · · · n k ց ( M k , x k ) in G such that ( M k , x k ) is a minimal vertex. (cid:3) The cone of convergence.
Let (
M, x ) be a vertex in G . Recall that θ x acts as aninvolution on a ∗ M and let ( a ∗ M ) − x = ( a ∗ M ) − θ x . For c > D M,x ( c ) = { λ ∈ ( a ∗ M ) − x : h λ, α ∨ i > c, ∀ α ∈ R ( A M , G ) , α > , θ x ( α ) < } . For an edge (
M, x ) n ց ( M , x ) in G let α ∈ ∆ P be such that n ∈ s α M . By [LR03, Lemma5.2.1] we have D M,x ( c ) = s − α D M ,x ( c ) ∩ { λ ∈ ( a ∗ M ) − x : h λ, α ∨ i > c } . Admissible orbits.
Let P = M ⋉ U be a standard parabolic subgroup of G . For x ∈ X let w ∈ W G be such that P xθ ( P ) = P wθ ( P ). (By the Bruhat decomposition, thedouble coset W M wW θ ( M ) is uniquely determined by x .)We say that x (or its P -orbit in X ) is P -admissible if M = wθ ( M ) w − . If a P -orbit Oin X is M admissible then it contains an element of X [ M ]. In fact O O ∩ X [ M ] is abijection from the admissible P -orbits in X to the M -orbits in X [ M ]. (See [Off17, Lemma3.2]). It further follows from [Off17, Corollary 6.9] that(1) δ / P | P x = δ P x , x ∈ X [ M ] . As a consequence, for a representation σ of M , x ∈ X [ M ], ℓ ∈ Hom M x ( σ, M x ), ϕ ∈ I GP ( σ )and λ ∈ ( a ∗ M, C ) − x we have(2) ℓ ( ϕ λ ( pg )) = δ P x ( p ) ℓ ( ϕ λ ( g )) , p ∈ P x , g ∈ G. NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 13
The intertwining periods modulo convergence.
It follows from (2) in its nota-tion that the following integral formally makes sense J GP ( ϕ ; x, ℓ, σ, λ ) = Z P x \ G x ℓ ( ϕ λ ( g )) dg. If λ is such that the integral is convergent for all ϕ ∈ I GP ( σ ) then it defines a linear form J GP ( x, ℓ, σ, λ ) ∈ Hom G x ( I GP ( σ, λ ) , G x ).Assuming that σ is of finite length and with bounded matrix coefficients we prove thatthere exists c > λ ) ∈D M,x ( c ) and admit meromorphic continuation to λ ∈ ( a ∗ M, C ) − x . We will prove this byinduction along a path as in Proposition 1 in the graph G .3.9. Intertwining periods for minimal vertices.
Our first step is the observation thatthe results of Blanc and Delorme ( § Corollary 1.
Let ( M, x ) be a minimal vertex in G and σ a representation of M of finitelength. Then for any ℓ ∈ Hom M x ( σ, M x ) the linear form J GP ( x, ℓ, σ, λ ) is defined by a con-vergent integral for λ ∈ D M,x ( c ) and admits a meromorphic continuation to λ ∈ ( a ∗ M, C ) − x .Proof. We use the notation of § J GP ( ϕ ; x, ℓ, σ, λ ) instages. We have J GP ( ϕ ; x, ℓ, σ, λ ) = Z P x \ G x ℓ ( ϕ λ ( g )) dg = Z Q x \ G x Z P x \ Q x δ Q x ( m ) − ℓ ( ϕ λ ( mg )) dm dg. Recall that Q , L and V are θ x -stable (Lemma 2). It follows that Q x \ G x is compact (see[GO16, Lemma 3.1]). By smoothness, the outer integral may therefore be replaced by afinite sum and it is enough to prove convergence and meromorphic continuation of theinner integral.It further follows from [Off17, Lemma 6.3] that Q x = L x ⋉ V x and P x = M x ⋉ U x . Since U = ( U ∩ L ) ⋉ V we conclude that U x = ( U x ∩ L ) ⋉ V x . By the definition of minimality U x ∩ L = { e } and therefore P x = M x ⋉ V x . The inner integral is therefore Z M x \ L x δ Q x ( m ) − ℓ ( ϕ λ ( mg )) dm. Recall that by (1) we have δ / Q | Q x = δ Q x and that the function m δ − / Q ( m ) ϕ ( mg ), m ∈ L lies in I LP ∩ L ( σ ). Thus θ x is an involution on L and by minimality of the vertex, theparabolic subgroup P ∩ L of L satisfies θ x ( P ∩ L ) ∩ ( P ∩ L ) = M . Note further that since V is θ x -stable we have D M,x ( c ) = D M,θ x | L ( c ) where the right hand side is defined in § § (cid:3) Convergence in the unramified case.
Let P = M ⋉ U be a standard parabolicsubgroup of G and M the trivial representation of M . For any λ ∈ a ∗ M we consider the induced representation I GP ( M , λ ) realized on the space I GP ( M ). Let x ∈ X [ M ] and λ ∈ ( a ∗ M ) − x . For ϕ ∈ I GP ( M ) consider the integral J GP ( ϕ ; x, λ ) = Z P x \ G x ϕ λ ( g ) dg. Note that ϕ λ ( pg ) = δ / P ( p ) ϕ λ ( g ), p ∈ P x and g ∈ G and by (1) the integral formally makessense. Proposition 2.
There exists c > such that J GP ( ϕ ; x, λ ) is defined by a convergent integralfor every standard parabolic subgroup P = M ⋉ U of G , ϕ ∈ I GP ( M ) , x ∈ X [ M ] and λ ∈ D M,x ( c ) .Proof. Let M be the finite set of standard Levi subgroups of G . For a parabolic subgroup P of G there are finitely many P -orbits in X , [HW93, Proposition 6.15]. Consequently,there are finitely many M -orbits in X [ M ] for every M ∈ M (see § P = M ⋉ U , x ∈ X [ M ] and m ∈ M we have J GP ( ϕ ; m · x, λ ) = e h λ + ρ P ,H M ( m ) i J GP ( I GP ( m − , M , λ ) ϕ ; x, λ )where the left hand side is defined by a convergent integral if and only if the right side is.It therefore follows from Corollary 1 that there exists c > J GP ( ϕ, x, λ ) converges for all standard parabolic subgroups P = M ⋉ U of G , ϕ ∈ I GP ( M ), x ∈ X [ M ] such that ( M, x ) is a minimal vertex in G and λ ∈ D M,x ( c ). We may furtherassume that c is large enough so that the intertwining operators M ( n, M , λ ) are definedby a convergent integral for every standard parabolic subgroup P = M ⋉ U of G , α ∈ ∆ P , n ∈ s α M and λ ∈ D M,s α ( c ). Note further that if x ∈ X [ M ] is such that − α = θ x ( α ) < D M,x ( c ) ⊆ D M,s α ( c ) (see § § M, x ) n ց ( M , x ) is an edge on the graph G and α ∈ ∆ P is such that n ∈ s α M . If J GP ( ϕ ; x , s α λ ) is defined by a convergent integral for every ϕ ∈ I GP ( M ) and λ ∈ D M ,x ( c )then J GP ( ϕ ; x, λ ) is defined by a convergent integral for every ϕ ∈ I GP ( M ) and λ ∈ D M,x ( c ).Let ξ be the spherical vector in I GP ( M ) normalized by taking the value one at theidentity. Thanks to the Iwasawa decomposition, for every ϕ ∈ I GP ( M ) there is a positiveconstant A such that | ϕ | ≤ Aξ hence | ϕ λ | ≤ Aξ λ for all λ ∈ a ∗ M . It therefore suffices toconsider ϕ = ξ .Let P = M ⋉ U be the standard parabolic subgroup with Levi subgroup M and Q = L ⋉ V the standard parabolic subgroup containing P such that ∆ QP = { α } . For λ ∈ D M,x ( c ) we have J GP ( ξ ; x, λ ) = Z P x \ G x ξ λ ( g ) dg = Z nP x n − \ G x ξ λ ( n − gn ) dg. It follows from [Off17, Lemma 6.4] that nU x n − ⊆ ( U ) x and from [Off17, Lemma 6.3]that P x = M x ⋉ U x and ( P ) x = ( M ) x ⋉ ( U ) x .Since ( M ) x = nM x n − we deduce that nP x n − ⊆ ( P ) x and the inclusion ( U ) x ⊆ ( P ) x induces an isomorphism nU x n − \ ( U ) x ≃ nP x n − \ ( P ) x . Applying [Off17, Lemma NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 15 Q to L defines an isomorphism nU x n − \ ( U ) x ≃ L ∩ U ≃ ( U ∩ nU n − ) \ U . The local analogues of [LR03, Lemma 4.3.1 (3) and (4)] easily followfrom [Off17, Lemma 6.4]. Proceeding as in the proof of [LR03, Proposition 10.1.1] we getthat(3) J GP ( ξ ; x, λ ) = Z ( P ) x \ G x Z ( U ∩ nUn − ) \ U ξ λ ( n − ugn )) du dg. By our assumption on c , the inner integral is absolutely convergent and we have Z ( U ∩ nUn − ) \ U ξ λ ( n − ugn )) du = ( M ( n, M , λ ) ξ ) s α λ ( gn ) . By the Gindikin-Karpelevich formula we have ( M ( n, M , λ ) ξ ) s α λ ( g ) = c ( α, λ ) ξ s α λ ( g ), g ∈ G where c ( α, λ ) = L ( h λ, α ∨ i ) L (1 + h λ, α ∨ i ) ,L ( s ) = (1 − q − s ) − and q is the size of the residual field of F . By assumption the integral Z ( P ) x \ G x ξ s α λ ( gn ) dg converges (see § (cid:3) Remark . It follows from the above proposition and its proof that there exists c = c ( G, P ) > • the linear forms J GP ( x, λ ) and • the intertwining operators M ( n, M , λ )are absolutely convergent for every standard parabolic subgroup P = M ⋉ U of G , x ∈ X [ M ], λ ∈ D M,x ( c ), α ∈ ∆ P such that − α = θ x ( α ) < n ∈ s α M . For the next sectionwe fix such a c and write D M,x = D M,x ( c ).3.11. The main result.
Next, we show that the intertwining periods can be defined in theGalois setting. The method of proof also provides us with the simple functional equationsthat they satisfy.
Theorem 1.
Let P = M ⋉ U be a standard parabolic subgroup of G , σ a representation of M of finite length with bounded matrix coefficients, x ∈ X [ M ] and ℓ ∈ Hom M x ( σ, M x ) . Forevery x ∈ X such that θ x ( M ) = M , λ ∈ ( a ∗ M, C ) − x such that Re( λ ) ∈ D M,x and ϕ ∈ I GP ( σ ) the integral J GP ( ϕ ; x, ℓ, σ, λ ) = Z P x \ G x ℓ ( ϕ λ ( g )) dg converges. Furthermore, the linear form J GP ( x, ℓ, λ ) admits a meromorphic continuationto λ ∈ ( a ∗ M, C ) − x and satisfies the following functional equations. Whenever α ∈ ∆ P and n ∈ s α M are such that ( M, x ) n ց ( M , x ) in G we have J GP ( ϕ ; x, ℓ, σ, λ ) = J GP ( M ( n, σ, λ ) I GP ( n, σ, λ ) ϕ ; x , ℓ, s α σ, s α λ ) . Proof.
We first prove the convergence for λ ∈ ( a ∗ M, C ) − x such that Re( λ ) ∈ D M,x . Taking theabsolute integral we may assume that λ = Re( λ ). For every g ∈ G write g = u g m g k g with u g ∈ U , m g ∈ M and k g ∈ K . Then ℓ ( ϕ λ ( g )) = e h λ + ρ P ,H M ( m g ) i ℓ ( σ ( m g ) ϕ ( k g )) = e h λ + ρ P ,H M ( g ) i ℓ ( σ ( m g ) ϕ ( k g )) . It follows from [Lag08, Th´eor`eme 4] that the function m ℓ ( σ ( m ) v ) is bounded for every v in the space of σ . Since ϕ takes finitely many values on K it follows that there exists C > Z P x \ G x | ℓ ( ϕ λ ( g )) | dg ≤ C Z P x \ G x e h λ + ρ P ,H M ( g ) i dg. The convergence therefore follows from Proposition 2.If (
M, x ) is a minimal vertex in G the meromorphic continuation follows from Corollary1. Now that convergence is granted, we show that if α ∈ ∆ P and n ∈ s α M are such that( M, x ) n ց ( M , x ) in G and P = M ⋉ U is the standard parabolic subgroup of G withstandard Levi subgroup M then the meromorphic continuation of J GP ( x , ℓ, λ ) implies thatof J GP ( x, ℓ, λ ). In the process we establish the desired functional equation (which alreadyappeared in the proof of the unramified case). In fact the computation hereunder is acompletely parallel generalization of that done in Proposition 2. We will use the notationof its proof and will not repeat all the arguments. In particular, Q = L ⋉ V is the standardparabolic subgroup containing P such that ∆ QP = { α } .Since σ has bounded matrix coefficients, it is straightforward that the integral defining M ( n, σ, λ ) is absolutely convergent for λ ∈ D M,x (see Remark 1). If Re( λ ) ∈ D M,x thearguments in the proof of Proposition 2 imply that J GP ( ϕ ; x, ℓ, σ, λ ) = Z P x \ G x ℓ ( ϕ λ ( g )) dg = Z nP x n − \ G x ℓ ( ϕ λ ( g )) dg = Z ( P ) x \ G x Z ( U ∩ nUn − ) \ U ℓ ( ϕ λ ( n − ugn ))) du dg and by Lemma 1 this equals Z ( P ) x \ G x ℓ [( M ( n, σ, λ ) ϕ ) s α λ ( gn )] dg. The theorem now follows by induction based on Proposition 1. (cid:3)
Applications to distinction.
In this section we continue to denote by (
G, θ ) aGalois symmetric pair and by X = { x ∈ G : θ ( x ) = x − } the associated symmetric space.Fix a standard parabolic subgroup P = M ⋉ U of G and a representation σ of M and let π = I GP ( σ ).For every P orbit O in X there exists a standard parabolic subgroup Q = L ⋉ V of G contained in P and x ∈ O such that M ∩ θ x ( M ) = L . We say that O contributes to σ ifthe normalized Jacquet module r L,M ( σ ) is L x -distinguished. This property depends onlyon O. NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 17
Let x ∈ X . As a consequence of [Off17, Theorem 4.2 and Corollary 6.9], applied tothe Galois symmetric pair ( G, θ x ), if π is G x -distinguished then there exists a P -orbit in G · x ⊆ X that contributes to σ . The following is a partial converse for this necessarycondition for distinction. Proposition 3.
With the above notation, if σ is of finite length with bounded matrixcoefficients and there exists x ′ ∈ G · x such that θ x ′ ( M ) = M and σ is M x ′ -distinguishedthen I GP ( σ, λ ) is G x -distinguished for every λ ∈ ( a ∗ M, C ) − x ′ .Proof. We construct a meromorphic family of linear forms L λ ∈ Hom G x ( I GP ( σ, λ ) , G x ), λ ∈ ( a ∗ M, C ) − x ′ on I GP ( σ ) as follows. Let η ∈ G be such that x ′ = η · x (so that G x ′ = ηG x η − ),0 = ℓ ∈ Hom M x ′ ( σ, M x ′ ) and L λ = J GP ( x ′ , ℓ, σ, λ ) ◦ I GP ( η, σ, λ ). By Theorem 1, indeed, L λ is a meromorphic family of linear forms as required and it is clearly non zero. Theproposition now follows by taking a leading term of L λ along a generic direction. That is,for λ ∈ ( a ∗ M, C ) − x ′ there exist λ ∈ ( a ∗ M, C ) − x ′ (a generic direction) and a non-negative integer m such that s s m L λ + sλ ∈ Hom G x ( I GP ( σ, λ + sλ ) , G x ) is holomorphic from C to thelinear dual of I GP ( σ ) and non-zero at s = 0. Its value at s = 0 gives a non-zero linear formin Hom G x ( I GP ( σ, λ ) , G x ). (cid:3) As a consequence, for a representation of G induced from cuspidal, contribution of a P -orbit characterizes distinction. Corollary 2.
Let x ∈ X and σ a cuspidal representation of M of finite length. Then I GP ( σ ) is G x -distinguished if and only if there exists x ′ ∈ G · x such that θ x ′ ( M ) = M and σ is M x ′ -distinguished.Proof. If L is a proper Levi subgroup of M then r L,M ( σ ) = 0. The only if direction istherefore immediate from the necessary condition obtained by [Off17, Theorem 4.2 andCorollary 6.9].Assume now that x ′ ∈ G · x ∩ X [ M ] and σ is M x ′ -distinguished. By cuspidality σ is adirect sum of irreducible representations and it is enough to assume that σ is irreducible.Let ω σ be the central character of σ and note that ω σ is trivial on ( Z M ) x ′ . Therefore, thereexists µ ∈ ( a ∗ M ) − x ′ such that | ω σ | = e h− µ,H M ( · ) i and therefore the unramified twist σ [ µ ] of σ isunitary and in particular has bounded matrix coefficients. Furthermore, Hom M x ′ ( σ, M x ′ ) =Hom M x ′ ( σ [ µ ] , M x ′ ). Since I GP ( σ ) ≃ I GP ( σ [ µ ] , − µ ) the corollary follows from Proposition 3applied to σ [ µ ]. (cid:3) We end this section with a somewhat weaker variant of Proposition 3 that, nevertheless,removes the bounded hypothesis on the inducing representation and avoids the use ofintertwining periods. For irreducibly induced representations it provides the same sufficientcondition for distinction.
Proposition 4.
Suppose that x ′ ∈ G · x is such that θ x ′ ( M ) = M and σ is M x ′ -distinguished.Then there exists w ∈ W ( M ) such that I GP ( wσ ) is G x -distinguished. (Here P is thestandard parabolic subgroup of G with Levi subgroup wM w − .) In particular, if in addition I GP ( σ ) is irreducible then it is G x -distinguished. Proof.
Note that for w ∈ W ( M ) and a representative n of w in G we haveHom ( wMw − ) n · x ′ ( nσ, ( wMw − ) n · x ′ ) = Hom M x ′ ( σ, M x ′ ) . Consequently, applying Proposition 1 it is enough to assume that (
M, x ′ ) is in fact minimaland prove that I GP ( σ ) is G x -distinguished. But then let L and Q be as in Section 3.5. Therepresentation σ ′ = I LP ∩ L ( σ ) is L x ′ -distinguished thanks to the proof of [Off17, Proposition7.2]. Now I GP ( σ ) ≃ I GQ ( σ ′ ) is G x ′ -distinguished, hence G x -distinguished, by the proof of[Off17, Proposition 7.1]. (cid:3) Remark . The results of § m ( D ) , C E (GL m ( D ))) where D is an F -central divisionalgebra of index d with md even, and C E (GL m ( D )) is the centralizer of an embedding of E in the space M m ( D ) of m × m matrices with entries in D . This follows from the factthat Equation (1) holds in this setting (see [BM19, Equation (5.3) and Remark 5.4]). Remark . In particular, Proposition 3 and Corollary 2 become more powerful when I GP ( σ )is reducible. Note that any irreducible representation π of G is the quotient of a representa-tion of the form I GP ( σ ) for σ as in Proposition 3 or Corollary 2, and if π is G x -distinguished,certainly I GP ( σ ) is as well. Moreover, Proposition 3 provides a particular G x -invariant lin-ear form L on I GP ( σ ). In the situation where Hom G x ( I GP ( σ ) , G x ) is one dimensional, thismeans that π is distinguished if and only if L descends to π . Such an observation is inpractice very useful and has been used to study distinction of Langlands quotients in avariety of situations, see [FLO12], [Gur15], [Mat17a] or [Mat17b].If I GP ( σ, λ ) satisfies that Hom G x ( I GP ( σ, λ ) , G x ) is one dimensional for a generic choiceof λ then the corresponding intertwining periods should satisfy functional equations moregeneral then those obtained in this work. Namely, for any w ∈ W ( M ) (represented by n ∈ G ) and P the standard parabolic subgroup of G with Levi subgroup wM w − , theintertwining period J GP ( x ′ , ℓ, σ, λ ) satisfies an identity of the form J GP ( n · x ′ , ℓ, wσ, wλ ) ◦ M ( n, σ, λ ) ◦ I GP ( n, σ, λ ) = γ w ( σ, λ ) J GP ( x ′ , ℓ, σ, λ )for some proportionality constant γ w that is meromorphic in λ . In some cases γ w canbe explicitely expressed in terms of local invariants associated to σ , and again this hasturned out very helpful to understand when the linear form L descends to π , see [FLO12],[Mat17b] or [SX]. 4. Classical Galois pairs
In the next section we explicate Corollary 2 for a Galois symmetric pair (
G, θ ) where G is a classical group. In this section we introduce the setting and classify the G -orbits in X . This classification is new and of interest for its own sake.4.1. Further notation and preliminaries.
Let k be a non Archimedean local field ofcharacteristic zero, K/k a field extension of degree one or two and ρ a generator of theGalois group Gal( K/k ). Let(
K/k ) = { a ∈ K ∗ : aa ρ = 1 } and N ( K/k ) = { aa ρ : a ∈ K ∗ } . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 19
That is, (
K/k ) is the subgroup ker( N K/k ) if
K/k is a quadratic extension and N K/k : K ∗ → k ∗ is the norm map associated with K/k and the group {± } otherwise while N ( K/k ) isthe subgroup N K/k ( K ∗ ) of k ∗ if K/k is a quadratic extension and the group k ∗ of squaresin k ∗ otherwise.For ε ∈ {± } we denote by Y n ( K/k, ε ) = { y ∈ GL n ( K ) : t y ρ = ε y } the space of ǫ -hermitian invertible matrices with respect to the field extension K/k . Itcomes naturally equipped with the following right GL n ( K )-action y ⋆ g = t g ρ yg, y ∈ Y n ( K/k, ε ) , g ∈ GL n ( K ) . We denote by G ( y, K/k ) the stabilizer of y ∈ Y n ( K/k, ε ) in GL n ( K ).This set up is a standard unified notation for considering symplectic, orthogonal andunitary groups. We recall the classification of GL n ( K )-orbits in Y n ( K/k, ε ) and explicatethe stabilizers in each case.(1)
The symplectic case: K = k , ε = − and n is even. The space Y n ( k/k, − n is even and in that case it is the space of alternatingmatrices in GL n ( k ). For m ∈ N we have that Y m ( k/k, −
1) = (cid:0) I m − I m (cid:1) ⋆ GL m ( k )is a unique GL m ( k )-orbit.The stabilizer G ( y, k/k ) is the symplectic group Sp( y, k ) associated to an element y = − t y in GL n ( k ).(2) The orthogonal case: K = k and ε = 1 . The space Y n ( k/k,
1) is the space ofsymmetric matrices in GL n ( k ). The GL n ( k )-orbits in Y n ( k/k,
1) are determined bythe discriminant and the Hasse invariant. The discriminant of y ∈ Y n ( k/k,
1) isthe square class det y k ∗ . The Hasse invariant is defined as follows. There exists g ∈ GL n ( k ) such that y ⋆ g = diag( a , . . . , a n ) is diagonal and the Hasse invariantis defined by Hasse k ( y ) = Y ≤ i 1) are in the same GL n ( k )-orbit if and only if det y k ∗ = det y k ∗ andHasse k ( y ) = Hasse k ( y ).For n = 1 we have that Y ( k/k, 1) = k ∗ and the GL ( k ) = k ∗ -orbits are thesquare classes in k ∗ /k ∗ . For n = 2 there is a unique orbit of discriminant − k ∗ (ithas Hasse invariant one) and two orbits for every other discriminant. For n ≥ G ( y, k/k ) is the orthogonal group O( y, k ) associated to an element y = t y in GL n ( k ).(3) The unitary case: K/k is a quadratic extension. The space Y n ( K/k, ε ) isthe space of ε -hermitian matrices in GL n ( K ) with respect to K/k . We have that y , y ∈ Y n ( K/k, ε ) are in the same GL n ( K )-orbit if and only if y N ( K/k ) = y N ( K/k ). By local class field theory N ( K/k ) is an index two subgroup of k ∗ and therefore Y n ( K/k, ε ) consists of two GL n ( K )-orbits. The stabilizer G ( y, K/k ) is theunitary group U( y, K/k ) associated to the element y = ε t y ρ in GL n ( K ).Let y ∈ Y n ( K/k, ε ). In the symplectic case, every element of G ( y, K/k ) has determinantone. In either the unitary or orthogonal case det maps G ( y, K/k ) to ( K/k ) . We observebellow that this map is in fact surjective. We apply the well known fact that in both casesevery GL n ( K )-orbit in Y n ( K/k, ε ) contains a diagonal element. Observation 1. In the orthogonal and unitary cases for every y ∈ Y n ( K/k, ε ) and a ∈ ( K/k ) there exists h ∈ G ( y, K/k ) such that det h = a . Furthermore, in the orthogonalcase there exists such an h that satisfies h = I n . Indeed, y ⋆ g is diagonal for some g ∈ GL n ( K ) and we can take h = g diag( a, I n − ) g − .4.2. Classical Galois pairs-The set up. Fix a quadratic extension of non Archimedeanlocal fields E/F of characteristic zero and let σ ( x ) = ¯ x be the associated Galois action. Inparticular, E = F [ ı ] for some ı ∈ E \ F such that ı ∈ F .We would like to consider distinction for unitary, special orthogonal and symplecticGalois pairs. In order to unify notation let F ′ = F and τ be the identity on F ′ in thesymplectic or orthogonal case while F ′ is a quadratic extension of F such that E ∩ F ′ = F and τ is the Galois action associated to F ′ /F in the unitary case. Let E ′ = EF ′ and notethat σ (resp. τ ) has a unique extension to an automorphisms of E ′ over F ′ (resp. over E )that, by abuse of notation, we denote by σ (resp. τ ) and that E ′ /F is a Klein extension(that is, a Galois extension such that Gal( E ′ /F ) is isomorphic to the Klein group ( Z / Z ) ).As a consequence, for x ∈ E ′ we can write ¯ x τ for σ ( τ ( x )) = τ ( σ ( x )) with no ambiguity.4.2.1. The space of ε -hermitian forms. Fix n ∈ N and ε ∈ {± } with the convention that if ε = − F ′ = F and n is even and set Y n ( E ) = Y n ( E ′ /E, ε ) and Y n ( F ) = Y n ( F ′ /F, ε ). Remark . Note that if F ′ /F is a quadratic extension and ∈ F ′ is such that τ ( ) = − then y y is a stabilizer preserving bijection between Y n ( E ′ /E, 1) and Y n ( E ′ /E, − 1) andtherefore in the unitary case it is enough to consider ε = 1.4.2.2. The general Galois symmetric space. We consider another symmetric space Z n ( E ) = { g ∈ GL n ( E ′ ) : g ¯ g = I n } with the left GL n ( E ′ )-action g · z = gz ¯ g − by twisted conjugation. Since the first cohomol-ogy set H (Gal( E ′ /F ′ ) , GL n ( E ′ )) is trivial, we have that Z n ( E ) = GL n ( E ′ ) · I n is a uniqueGL n ( E ′ )-orbit.4.2.3. The symmetric space associated to a classical Galois pair. Fix ∈ Y n ( F ) and let G be the algebraic group, defined over F , of isometries of the ε -hermitian form . That is G ( F ) = { g ∈ GL n ( F ′ ) : t g τ g = } and G ( E ) = { g ∈ GL n ( E ′ ) : t g τ g = } . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 21 Consider the G ( E )-space X = X ( E/F ) = { x ∈ G ( E ) : x ¯ x = I n } with the G ( E ) action g · x = gx ¯ g − , g ∈ G ( E ) , x ∈ X . Note that X is a G ( E )-invariant subspace of Z n ( E ).4.3. A classification of orbits. We parameterize G ( E )-orbits in X in terms of certainGL n ( F ′ )-orbits in Y n ( F ) by the following lemma. Lemma 3. For x ∈ X let z ∈ GL n ( E ′ ) be such that x = z · I n (see § y = ⋆ z ∈ Y n ( E ) . Then, in fact, y ∈ Y n ( F ) and the assignment G ( E ) · x y ⋆ GL n ( F ′ ) iswell defined and defines a bijection between the G ( E ) -orbits in X and the GL n ( F ′ ) -orbitsin ⋆ GL n ( E ′ ) ∩ Y n ( F ) .Proof. Note first that for z ∈ GL n ( E ′ ) we have that z · I n = zz − ∈ G ( E ) if and only if ⋆ z = t z τ z ∈ Y n ( F ). Moreover, for z, z ′ ∈ GL n ( E ′ ) we have that z · I n = z ′ · I n if and onlyif z − z ′ ∈ GL n ( F ′ ) if and only if ⋆ z and ⋆ z ′ belong to the same GL n ( F ′ )-orbit. Hencethe formula p ( z · I n ) = ( ⋆ z ) ⋆ GL n ( F ′ ) well defines a map p from X to the GL n ( F ′ )-orbitsin ⋆ GL n ( E ′ ) ∩ Y n ( F ). For x = z · I n and g ∈ G ( E ) we have g · x = ( gz ) · I n and ⋆ z = ⋆ ( gz ) and therefore p ( x ) = p ( g · x ). Consequently, the map p descends to a map p from the G ( E )-orbits in X to the GL n ( F ′ )-orbits in ⋆ GL n ( E ′ ) ∩ Y n ( F ).Assume now that z, z ′ ∈ GL n ( E ′ ) are such that ⋆z = ⋆z ′ ∈ Y n ( F ). Then z ′ z − ∈ G ( E )and therefore z · I n and z ′ · I n are in the same G ( E )-orbit in X . Consequently, the formula q ( ⋆ z ) = G ( E ) · ( z · I n ) well defines a map q from ⋆ GL n ( E ′ ) ∩ Y n ( F ) to the G ( E )-orbitsin X . It is again straightforward that q is fixed on GL n ( F ′ )-orbits and therefore descendsto a map q from the GL n ( F ′ )-orbits in ⋆ GL n ( E ′ ) ∩ Y n ( F ) to the G ( E )-orbits in X . Itis also straight forward to verify that p and q are inverses of one another and the lemmafollows. (cid:3) We indicate the relation between orbits in X for G ( E ) and for its intersection withSL n ( E ′ ). Of course in the symplectic case G ( E ) is a subgroup of SL n ( E ′ ), the followinglemma is relevant to the case ε = 1.Let SG ( E ) = G ( E ) ∩ SL n ( E ′ ). Next we observe that determinant is the only invariantof an SG ( E )-orbit in a G ( E )-orbit in X . Lemma 4. For every x ∈ X we have SG ( E ) · x = { x ′ ∈ G ( E ) · x : det x ′ = det x } . Inparticular, in the orthogonal case SG ( E ) · x = G ( E ) · x .Proof. Clearly, all elements of SG ( E ) · x have the same determinant det x and the inclusion ⊆ follows. For the other inclusion, note that if g ∈ G ( E ) is such that det( g · x ) = det x thendet g ∈ F ′ ∩ ( E ′ /E ) = ( F ′ /F ) . It is therefore enough to show that for any a ∈ ( F ′ /F ) there exists h ∈ G ( E ) with det h = a such that h · x = x . Indeed, for a = det g − andsuch h we have g · x = ( gh ) · x and gh ∈ SG ( E ). Now let z ∈ GL n ( E ′ ) be such that x = z · I n (see § y = ⋆ z ∈ Y n ( F )(see Lemma 3). Applying Observation 1 let h ′ ∈ G y ( F ) be such that det h ′ = a and set h = zh ′ z − ∈ G ( E ). Then h · x = ( zh ′ ) · I n = z · I n = x. (cid:3) We record separately consequences of Lemmas 3 and 4 in the three cases we consider.4.3.1. The symplectic case. Recall that in the symplectic case Y n ( F ) = ⋆ GL n ( F ) is aunique GL n ( F )-orbit. As an immediate consiquence of Lemma 3 we obtain Corollary 3. In the symplectic case X = G ( E ) · I n is a unique G ( E ) -orbit. (cid:3) The unitary case. We begin with the following simple observation. Lemma 5. Let L/F be a Klein extension of local fields and let E be a quadratic extensionof F in L . Then F ∗ ⊆ N ( L/E ) .Proof. Let E and E be the two different quadratic extensions of F contained in L anddifferent from E . Restriction to E i defines an isomorphism from Gal( L/E ) to Gal( E i /F )and therefore N L/E ( E ∗ i ) = N ( E i /F ) and in particular N ( E i /F ) is a subgroup of N ( L/E ), i = 1 , 2. By local class field theory N ( E i /F ), i = 1 , F ∗ ofindex two and therefore their product equals F ∗ . The lemma follows. (cid:3) Let(4) Γ = { a − σ : a ∈ ( E ′ /E ) } and note that Γ is a subgroup of ( E ′ /E ) ∩ ( E ′ /F ′ ) . Note further that det maps X to( E ′ /E ) ∩ ( E ′ /F ′ ) . Observation 2. In the unitary case, for every a ∈ ( E ′ /E ) ∩ ( E ′ /F ′ ) there exists x ∈ X such that det x = a . Indeed, if is diagonal then diag( a, I n − ) ∈ X for every a ∈ ( E ′ /E ) ∩ ( E ′ /F ′ ) .Otherwise, there exists g ∈ GL n ( F ′ ) such that ⋆ g is diagonal and we then have X = gX ⋆g g − so that det( X ) = det( X ⋆g ). Corollary 4. In the unitary case Y n ( F ) ⊆ ⋆ GL n ( E ′ ) . In particular, X consists ofexactly two G ( E ) -orbits. Namely, Γ is of index two in ( E ′ /E ) ∩ ( E ′ /F ′ ) and the two G ( E ) -orbits in X are { x ∈ X : det x ∈ Γ } = G ( E ) · I n and { x ∈ X j : det x Γ } .Proof. Consider first the case n = 1. Note that Y ( F ) = F ∗ ⊆ Y ( E ) = E ∗ and ⋆ GL ( E ′ ) = N ( E ′ /E ) so that ⋆ GL ( E ′ ) ∩ Y ( F ) = N ( E ′ /E ) ∩ F ∗ = ( N ( E ′ /E ) ∩ F ∗ ) . It follows from Lemma 5 that ⋆ GL ( E ′ ) ∩ Y ( F ) = F ∗ . Note further that GL ( F ′ ) actson ⋆ GL ( E ′ ) ∩ Y ( F ) by y ⋆ g = yN F ′ /F ( g ) and since N ( F ′ /F ) is of index two in F ∗ itfollows that ⋆ GL ( E ′ ) ∩ Y ( F ) consists of two GL ( F ′ )-orbits. NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 23 As a consequence of Lemma 3 we obtain that X consists of two G ( E )-orbits. Note that ∈ F ∗ and G ( E ) = ( E ′ /E ) acts on X = ( E ′ /E ) ∩ ( E ′ /F ′ ) by g · x = g − σ x . It followsthat, indeed, Γ has index two in ( E ′ /E ) ∩ ( E ′ /F ′ ) .We now consider a general n . We first show that ⋆ GL n ( E ′ ) ∩ Y n ( F ) contains twodifferent GL n ( F ′ )-orbits. After replacing with an appropriate element of ⋆ GL n ( F ′ ),we may assume without loss of generality that is diagonal. Now let a ∈ ( E ′ ) ∗ be suchthat N E ′ /E ( a ) = aa τ ∈ F ∗ \ N ( F ′ /F ) be given by Lemma 5. Then y = ⋆ diag( a, I n − ) ∈ ⋆ GL n ( E ′ ) ∩ Y n ( F ) but y ⋆ GL n ( F ′ ). Since Y n ( F ) consists of two GL n ( F ′ )-orbits itfollows that Y n ( F ) = ⋆ GL n ( E ′ ) ∩ Y n ( F ). By Lemma 3 we get that X consists of two G -orbits.It follows from Observations 1 (resp. 2) that det( G · x ) = Γ det x (resp. det( X ) =( E ′ /E ) ∩ ( E ′ /F ′ ) ). The lemma follows. (cid:3) The orthogonal case. Consider the orthogonal case and let SX = X ∩ SG ( E ).Note that SX (and its complement X \ SX ) are G ( E )-invariant spaces since the G ( E )-action preserves determinant. We observe the following refinement of Lemma 3. Lemma 6. Let GL Dn ( E ) = { g ∈ GL n ( E ) : det g ∈ D } for a subset D of E . Consider theorthogonal case.(1) The bijection defined by Lemma 3 restricts to a bijection between SG ( E ) -orbits in SX and GL n ( F ) -orbits in ⋆ GL Fn ( E ) ∩ Y n ( F ) and to a bijection between SG ( E ) -orbits in X \ SX and GL n ( F ) -orbits in ⋆ GL ıFn ( E ) ∩ Y n ( F ) .(2) Each of SX and X \ SX contains at most two SG ( E ) -orbits. For n = 2 if det j ∈ − F ∗ then SX is a unique SG ( E ) -orbit and if det ∈ − ı F ∗ then X \ SX is a unique SG ( E ) -orbit.Proof. Note first that a ∈ E ∗ satisfies a ∈ F ∗ if and only if a ∈ F ∗ ⊔ ıF ∗ . As a consequence E ∗ ∩ F ∗ = F ∗ ⊔ ı F ∗ and therefore comparing determinants we obtain the disjoint union ⋆ GL n ( E ) ∩ Y n ( F ) = ( ⋆ GL Fn ( E ) ∩ Y n ( F )) ⊔ ( ⋆ GL ıFn ( E ) ∩ Y n ( F )) . Let x ∈ X and z ∈ GL n ( E ) be such that x = z · I n . Recall that G ( E ) · x corresponds to( ⋆ z ) ⋆ GL n ( F )) under the bijection of Lemma 3. Note that if x ∈ SX then 1 = det x =det z ¯ z − and therefore z ∈ GL Fn ( E ) and if x ∈ X \ SX then − x = det z ¯ z − andtherefore z ∈ GL ıFn ( E ). By Lemma 4 every SG ( E )-orbit is a G ( E )-orbit. The first partof the lemma follows.Note further that every element in ⋆ GL Fn ( E ) ∩ Y n ( F ) has the fixed discriminant det jF ∗ and every element of ⋆ GL ıFn ( E ) ∩ Y n ( F ) has the fixed discriminant ı det F ∗ . The secondpart of the lemma follows from the first part and the classification of GL n ( F )-orbits in Y n ( F ). (cid:3) Proposition 5. (1) The space SX is a unique SG ( E ) -orbit if and only if there is aunique GL n ( F ) -orbit in Y n ( F ) with discriminant det F ∗ . That is, if and only ifeither n = 1 or n = 2 and det ∈ − F ∗ . (2) The space X \ SX is a unique SG ( E ) -orbit if and only if there is a unique GL n ( F ) -orbit in Y n ( F ) with discriminant ı det F ∗ . That is, if and only if either n = 1 or n = 2 and det ∈ − ı F ∗ .(3) If either n ≥ or n = 2 and det j 6∈ − F ∗ (resp. n ≥ or n = 2 and det j i F ∗ ) then SX (resp. X \ SX ) consists of exactly two orbits.Proof. By Lemma 6(1) the number of SG ( E )-orbits in SX equals the number of GL n ( F )-orbits in ⋆ GL Fn ( E ) ∩ Y n ( F ) and the number of SG ( E )-orbits in X \ SX equals thenumber of GL n ( F )-orbits in ⋆ GL ıFn ( E ) ∩ Y n ( F ). Note thatGL ıFn ( E ) = diag( ı, I n − )GL Fn ( E ) = GL Fn ( E ) diag( ı, I n − ) . Replacing by an appropriate element of ⋆ GL n ( F ) we assume without loss of generalitythat is diagonal. When this is the case ⋆ GL ıFn ( E ) ∩ Y n ( F ) = ( diag( ı , I n − )) ⋆ GL Fn ( E ) ∩ Y n ( F ) . As a consequence, the second part of the proposition follows from the first and Lemma6(1). The third part follows from the first two and Lemma 6(2). It remains to prove thefirst part.When n = 1 we have that SG ( E ) = {± } acts trivially on SX = { } and the statementis obvious. Consider the case n = 2. If det ∈ − F ∗ the statement follows from Lemma6(2). Assume that det 6∈ − F ∗ . Without loss of generality assume that is diagonal andwrite = diag( a, b ). Let K be a quadratic extension of F such that − a − b = u for some u ∈ K ∗ and ρ the Galois action of K/F . Note that for η = (cid:0) u − u (cid:1) ∈ GL ( K ) we have ⋆ η = (cid:0) b b (cid:1) and η − η ρ = (cid:0) (cid:1) . Consider first the case that − ab ∈ E ∗ so that K = E and ρ = σ . Then SG ( E ) = ηSG ⋆η ( E ) η − = η { t a : a ∈ E ∗ } η − where t a = diag( a, a − ), a ∈ E ∗ . For a ∈ E ∗ and g = ηt a η − we have g ¯ g = ηt a ¯ a − η − . Hence SX = η { t a : a ∈ F ∗ } η − . Let a ∈ E ∗ and b ∈ F ∗ and write g = ηt a η − ∈ SG ( E ) and x = ηt b η − ∈ SX . Then g · x = ηt a ¯ ab η − and since N ( E/F ) is of index two in F ∗ it follows that SX consists of two different SG ( E )-orbits.Assume now that − ab E ∗ and let L = KE . Then L/F is a Klein extension. Byabuse of notation we continue to denote by ρ the Galois action of L/E and by σ the Galoisaction of L/K . Note that η σ = η .It follows from the previous case (with L/E replacing E/F ) that SG ( L ) = η { t a : a ∈ L ∗ } η − . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 25 For a ∈ L ∗ we have ( ηt a η − ) ρ = ηt a − ρ η − and therefore SG ( E ) = SG ( L ) ∩ GL ( E ) = η { t a : a ∈ ( L/E ) } η − . For a ∈ ( L/E ) and g = ηt a η − we have g ¯ g = ηt aa σ η − and therefore SX = η { t a : a ∈ ( L/E ) ∩ ( L/K ) } η − . For a ∈ ( L/E ) and b ∈ ( L/E ) ∩ ( L/K ) set g = ηt a η − ∈ SG ( E ) and x = ηt b η − ∈ SX ( E ). It follows that g · x = ηt a − σ b η − . It follows from Corollary 4 that { a − σ : a ∈ ( L/E ) } is of index two in ( L/E ) ∩ ( L/K ) and therefore SX consists of exactly two SG ( E )-orbits.Next we show that for n = 3, the space ⋆ GL F ( E ) ∩ Y ( F ) contains two differentGL ( F )-orbits. Note that for a ∈ F ∗ the map y ay defines a GL ( F )-equivariantbijection between ⋆ GL F ( E ) ∩ Y ( F ) and ( a ) ⋆ GL F ( E ) ∩ Y ( F ). We may thereforeassume without loss of generality that det j ∈ − F ∗ . Let u ∈ F ∗ \ N ( E/F ) and recallthat ( u, ı ) F = − 1. It follows that y = diag(1 , , − 1) and y = diag( ı , u, − ı u ) both havediscriminant − F ∗ , are not in the same GL ( F )-orbit but are in the same GL ( E )-orbit.There is therefore z ∈ GL ( E ) such that y = y ⋆ z and comparing determinants it followsthat z ∈ GL F ( E ). Since ⋆ GL ( F ) = y i ⋆ GL ( F ) for some i ∈ { , } our claim follows.Next we show that for n > ⋆ GL Fn ( E ) ∩ Y n ( F ) contains two differentGL n ( F )-orbits. We may again assume without loss of generality that is diagonal. Write = diag( , ) where ∈ Y ( F ). It follows from the n = 3 case that there exists z ∈ GL F ( E ) such that ⋆ z ⋆ GL ( F ). It follows from Witt’s cancellation theoremthat ⋆ diag( z, I n − ) ⋆ GL n ( F ) while clearly diag( z, I n − ) ∈ GL Fn ( E ). All together thisshows that for n ≥ ⋆ GL Fn ( E ) ∩ Y n ( F ) contains two differentGL n ( F )-orbits. The proposition now follows from Lemma 6(1). (cid:3) Invariants for the SG ( E ) -orbits in X in the orthogonal case. Continue to considerthe orthogonal case. It will be convenient to introduce invariants that classify the SG ( E )-orbits in X based on the above results. Let x ∈ X , z ∈ GL n ( E ) such that x = z · I n and y = ⋆ z ∈ Y n ( F ). Note thatdet yF ∗ = ( det F ∗ x ∈ SX ı det F ∗ x ∈ X \ SX and Hasse F ( y ) are invariants of SG ( E ) · x that uniquely determine it. Let ∂ ( x ) = det yF ∗ and ~ ( x ) = Hasse F ( y ) . For x, x ′ ∈ X we have SG ( E ) · x = SG ( E ) · x ′ if and only if ( ∂ ( x ) , ~ ( x )) = ( ∂ ( x ′ ) , ~ ( x ′ )). Distinction for classical Galois pairs-statement of the main result Let ∈ Y n ( F ) be anisotropic (that is, for 0 = v ∈ ( F ′ ) n we also have t v τ v = 0). Thisimplies that in the symplectic case n = 0, in the unitary case n ∈ { , , } and in theorthogonal case n ∈ { , , , , } . For every n ∈ Z ≥ let w n = ( δ i,n +1 − j ) ∈ GL n ( F ), [ n ] = w n ε w n , G n = G [ n ] ( E ) , X n = X [ n ] and H n = G [ n ] ( F ) . The group G is compact. In the orthogonal case let G ◦ n = G n ∩ SL n +2 n ( E ) be thespecial orthogonal group. In order to maintain a unified notation, we set G ◦ n = G n in theunitary and symplectic cases. Thus G ◦ n is a connected reductive p -adic group.Fix n ∈ N and let G = G n , X = X n and H = H n . For any subset S of G let S ◦ = S ∩ G ◦ n .Let N = n + 2 n so that [ n ] ∈ Y N ( F ) and G is a subgroup of GL N ( E ′ ).5.1. The standard parabolic subgroups of G ◦ . For a decomposition α = ( N , . . . , N k )of N let Q α = L α ⋉ V α be the standard parabolic subgroup of GL N ( E ′ ) of type α withstandard Levi subgroup L α and unipotent radical V α . That is, Q α is the unique parabolicsubgroup containing all upper-triangular matrices in GL N ( E ′ ) with Levi subgroup L α = { diag( g , . . . , g k ) : g i ∈ GL N i ( E ′ ) , i = 1 , . . . , k } . For the definition of a parabolic subgroup of G and the following analysis (particularlyin the orthogonal case when G is not connected) we refer to [MgVW87, CHAPITRE 1, III2].The standard minimal parabolic subgroup of G is P = M ⋉ U with Levi part M = { diag( a , . . . , a n , h, a − τn , . . . , a − τ ) : a , . . . , a n ∈ ( E ′ ) ∗ , h ∈ G } and unipotent radical U = V (1 ( n ) ,n , ( n ) ) ∩ G consisting of block upper triangular unipotentmatrices. (Here 1 ( n ) is the n -tuple (1 , . . . , A be the set of all tuples α of non-negative integers of the form α = ( n , . . . , n k ; r )where k, r ∈ Z ≥ , n i ∈ N , 1 ≤ i ≤ k and n = n + · · · + n k + r . When r = 0 we will simplywrite α = ( n , . . . , n k ) ∈ A and denote by A the subset of all tuples in A with r = 0.For m ∈ N and g ∈ GL m ( E ′ ) let g ∗ = w mt g − τ w m . For α = ( n , . . . , n k ; r ) ∈ A let ι = ι α : GL n ( E ′ ) × · · · × GL n k ( E ′ ) × G r → G be the imbedding ι ( g , . . . , g k ; h ) = diag( g , . . . , g k , h, g ∗ k , . . . , g ∗ ) . When α ∈ A we simply write ι ( g , . . . , g k ) = diag( g , . . . , g k , g ∗ k , . . . , g ∗ ). Denote by M α the image of ι α and let U α = V ( n ,...,n k ,n +2 r,n k ,...,n ) ∩ G and P α = M α ⋉ U α . The groups P α , α ∈ A are parabolic subgroups of G containing P . Themap α P α is a bijection from A to a complete set of representatives of the G -conjugacyclasses of parabolic subgroups of G . A parabolic subgroup of G is called standard if it isof the form P α for some α ∈ A . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 27 Excluding the split even orthogonal case (that is, when F ′ = F , ε = 1 and n = 0) themap α P ◦ α is a bijection from A to the parabolic subgroups of G ◦ containing P ◦ . Inorder to describe the standard parabolic subgroups of G ◦ in the split even orthogonal casewe introduce some further notation.Consider the split even orthogonal case and let κ = κ n = ι ( I n − , w ). We set ι ′ α =Ad( κ ) ◦ ι α and let M ′ α = κM α κ be the image of ι ′ α . When α is clear from the context wealso write ι ′ = ι ′ α . We further write U ′ α = κU α κ and P ′ α = κP α κ = M ′ α ⋉ U ′ α . Then P ′ α isa parabolic subgroup of G containing P . Note that for α = ( n , . . . , n k ; r ) ∈ A we have P α = P ′ α except for α such that r = 0 and n k = 1. Furthermore, for α, β ∈ A we have thatthe two sets { P β , P ′ β } and { P α , P ′ α } intersect if and only if β = α .A parabolic subgroup of G ◦ is called standard if it contains P . Every standard parabolicsubgroup of G ◦ is either of the form P ◦ α or ( P ′ α ) ◦ . However, there are some repetitions. Wehave P ◦ α = ( P ′ α ) ◦ except for α such that r = 0 and n k = 1. Furthermore, if n + · · · + n m +1 = n we also have P ◦ ( n ,...,n m , = P ◦ ( n ,...,n m ;1) . These cases provide the only possible repetitions.In order to unify notation, back to the general case, let A ◦ equal A unless we are in thesplit even orthogonal case in which case we define A ◦ to be the disjoint union of { α = ( n , . . . , n k ; r ) ∈ A : r > r = 0 , n k = 1 } and { [ α, i ] : α = ( n , . . . , n k ) ∈ A , n k = 1 , i ∈ {± }} . For such [ α, i ] we write P [ α,i ] = ( P α i = 1 P ′ α i = − . , M [ α,i ] = ( M α i = 1 M ′ α i = − . and U [ α,i ] = ( U α i = 1 U ′ α i = − . The map a P ◦ a is a bijection from A ◦ to the set of standard parabolic subgroups of G ◦ .5.2. Explication of Corollary 2 for classical symmetric pairs. Let a ∈ A ◦ and P = M ⋉ U with M = M a and U = U a . That is, P ◦ = M ◦ ⋉ U is a standard parabolic subgroupof G ◦ . If a ∈ A ∩ A ◦ let α = a . Otherwise, write a = [ α, ǫ ] where α = ( n , . . . , n k ) ∈ A and ǫ ∈ {± } and set r = 0. If either a ∈ A ∩ A ◦ or ǫ = 1 denote by ι ◦ a the restriction of ι α to GL n ( E ′ ) × · · · × GL n k ( E ′ ) × G ◦ r . Otherwise, let ι ◦ a = ι ′ α . Thus, ι ◦ a : GL n ( E ′ ) × · · · × GL n k ( E ′ ) × G ◦ r → M ◦ is an isomorphism.Let π i be a representation of GL n i ( E ′ ), i ∈ [1 , k ] and π a representation of G ◦ r . (If n = r = 0, by π we will always mean the trivial representaion of the trivial group). Let π = ( π ⊗ · · · ⊗ π k ⊗ π ) ◦ ι ◦ a be the associated representation of M ◦ . We apply the followingstandard notation for parabolic induction π × · · · π k ⋉ π = I G ◦ P ◦ ( π ) . Our main result in this section is the characterization of distinction for classical sym-metric pairs of representations parabolically induced from cuspidal.For ρ ∈ S k and a subset c of [1 , k ] let • I ( ρ c ) = { i ∈ c : ρ ( i ) = i } ; • o( c ) = o M ( c ) be the number of i ∈ c such that n i is odd; • N ( ρ c ) = N M ( ρ c ) = P i ∈ I ( ρ c ) n i . Theorem 2. With the above notation, let x ∈ X ◦ , π i an irreducible cuspidal representationof GL n i ( E ′ ) , i ∈ [1 , k ] and π a cuspidal representation of G ◦ r . The representation π × · · · π k ⋉ π of G ◦ is G ◦ x -distinguished if and only if there exist an involution ρ ∈ S k and a subset c of [1 , k ] such that ρ ( c ) = c and n ρ ( i ) = n i , i ∈ [1 , k ] , y i ∈ Y n i ( E ′ / ( E ′ ) στ , ε ) , i ∈ I ( ρ c ) and z ∈ X r such that π ρ ( i ) ≃ ¯ π ∨ i i c , ρ ( i ) = iπ i is GL n i ( F ′ ) -distinguished i c , ρ ( i ) = iπ ρ ( i ) ≃ π τσi i ∈ c , ρ ( i ) = iπ i is U( y i , E ′ / ( E ′ ) στ ) − distinguished i ∈ c , ρ ( i ) = iπ is ( G ◦ r ) z − distinguishedand furthermore • in the unitary case ( − o( c ) det z Q i ∈ I ( ρ c ) det y τ − i ∈ det x Γ (see (4) ). (Here if n = r = 0 replace det z by one.) • in the orthogonal case the following condition is satisfied: ( − r o( c )+ ( N ( ρ c )2 )(2 det ) o( c ) Y j ∈ I ( w ) det y j ∈ N ( E/F ) if and only if ~ [ r ] ( z ) ~ [ r ] ( I n +2 r ) = ~ [ n ] ( x ) ~ [ n ] ( I N ) . (See § ~ [ r ] . Here if n = 0 replace det byone and if n + 2 r = 0 replace the left hand side of the equality by one.) We furtherremark that ~ [ n ] ( I N ) = Hasse F ( [ n ]) = (det , − nF ( − , − n ) F Hasse F ( ) . A necessary condition for distinction. As a consequence of Theorem 2 we obtaina necessary condition for distinction that is easier to formulate and is useful for applications.We continue to assume that n + · · · + n k + r = n . For w = ρ c ∈ W k and a tuple π = ( π , . . . , π k ; π ) such that π is a representation of G ◦ r and π i is a representation ofGL n i ( E ′ ) let(5) wπ = ( π ′ , . . . , π ′ k ; π ) where π ′ i = ( π ρ − ( i ) i c π ∗ ρ − ( i ) i ∈ c . Corollary 5. Let x ∈ X ◦ , π i an irreducible cuspidal representation of GL n i ( E ′ ) , i ∈ [1 , k ] and π a cuspidal representation of G ◦ r . If the representation π × · · · π k ⋉ π NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 29 of G ◦ is G ◦ x -distinguished and w = ρ c ∈ W ◦ k [2 , M ] is given by Theorem 2 then w ( π , . . . , π k ; π ) = (¯ π ∨ , . . . , ¯ π ∨ k ; π ) . Proof. Write wπ = ( π ′ , . . . , π ′ k ; π ) and recall that for an irreducible representation υ ofGL m ( E ′ ) we have υ ∗ ≃ ( υ ∨ ) τ by [GK75]. The identity π ′ i = ¯ π ∨ i is therefore clear whenever ρ ( i ) = i . For i c such that ρ ( i ) = i it further follows from [Fli91, Proposition 12] and for i ∈ c such that ρ ( i ) = i it further follows from [FLO12, Theorem 6.1(1)]. (cid:3) Distinction for classical Galois pairs-proof of the main result A reduction step. We begin the proof of the theorem with a reduction to the casewhere P is a standard parabolic subgroup of G , that is, either a ∈ A or ǫ = 1 in the abovenotation. This step is only relevant to the split even orthogonal case.Consider the split even orthogonal case and let α = ( n , . . . , n k ) ∈ A . Recall that κ = ι ( I n − ; w ) and for a representation π of G ◦ let π κ be the representation of G ◦ on thespace of π defined by π κ ( g ) = π ( κgκ − ). More generally, for a representation π of M α denote by π κ the representation of M ′ α = κM α κ − defined similarly. Lemma 7. (1) For a representation π of G ◦ we have Hom H ◦ ( π, ) = Hom H ◦ ( π κ , ) . (2) Let α ∈ A and π a representation of M α . Then Hom H ◦ ( I G ◦ P α ( π ) , ) ≃ Hom H ◦ ( I G ◦ P ′ α ( π κ ) , ) . Proof. The first part is immediate from the fact that κ normalizes H ◦ . The second part fol-lows from the first part and the observation that I G ◦ P ′ α ( π κ ) ≃ I G ◦ P α ( π ) κ . Indeed, for ϕ ∈ I G ◦ P α ( π )let ϕ κ ( g ) = ϕ ( κgκ − ), g ∈ G ◦ . Then ϕ ϕ κ realizes this equivalence of representa-tions. (cid:3) We continue with some further notation and preliminaries.6.2. Admissible orbits. For a standard parabolic subgroup P = M ⋉ U of G we recallsome facts on the P ◦ -orbits in X ◦ following [Off17, Section 3]. For x ∈ X ◦ the P ◦ -orbit P ◦ · x is contained in the Bruhat cell P ◦ xP ◦ . By the Bruhat decomposition it correspondsto a double coset in W M ◦ \ W G ◦ /W M ◦ . Let w be the element of minimal length in thisdouble coset, then w is an involution. We say that P ◦ · x is M ◦ -admissible if w normalizes M ◦ . Intersection with N G ◦ ( M ◦ ) defines a bijection from M ◦ -admissible P ◦ -orbits in X ◦ to M ◦ -orbits in X ∩ N G ◦ ( M ◦ ). In the sequel we study M ◦ -orbits in X ∩ N G ◦ ( M ◦ ).6.3. The signed permutation group. Let W k be the signed permutation group on k elements realized as W k = S k ⋉ Ξ k where Ξ k ≃ ( Z / Z ) k is the group of subsets of the integerinterval [1 , k ] with the operation of symmetric difference and the group S k of permutationsof [1 , k ] acts naturally on Ξ k (In W k we have ρ c ρ − = ρ ( c ), ρ ∈ S k , c ∈ Ξ k . That is,conjugation of an element c in Ξ k by the permutation ρ is the image of the set c under the permutation ρ ). Denote by S k [2] the set of involutions in S k and by W k [2] the set ofinvolutions in W k . Note that W k [2] = { ρ c : ρ ∈ S k [2] , c ∈ Ξ k , ρ ( c ) = c } . When we write w = ρ c ∈ W k we always mean that ρ ∈ S k and c ∈ Ξ k .6.4. The split even orthogonal case. Some of the arguments in the sequel will requirefurther justification in the case where F ′ = F , n = 0 and ε = 1 henceforth the split evenorthogonal case.In the split even orthogonal case we define the Weyl group W G of the non-connectedgroup G as W G = N G ( T ) /T . Clearly W G ◦ is a subgroup of W G . Note that T = { ι ( a , . . . , a n ) : a i ∈ F ∗ , i ∈ [1 , n ] } is a maximal split torus of G ◦ contained in M and let e i ∈ X ∗ ( T ) be defined by e i ( ι ( a , . . . , a n )) = a i , i ∈ [1 , n ]. The split connected group G ◦ has a root system R ( T, G ◦ ) = {± ( e i ± e j ) : 1 ≤ i = j ≤ n } of type D n with a basis of simple roots∆ ◦ = { e i − e i +1 : i ∈ [1 , n − } ⊔ { e n − + e n } . The set R ( T, G ) = R ( T, G ◦ ) ⊔ {± e i : i ∈ [1 , n ] } is a root system of type C n with basis∆ = (∆ ◦ \ { e n − + e n } ) ⊔ { e n } . It is easy to see that the action of N G ( T ) on T by conjugation identifies W G as the Weylgroup of the root system R ( T, G ) and in particular, defines an isomorphism W G ≃ W n . In particular, W G admits a length function with respect to the elementary reflectionsassociated with the simple roots in ∆ .For the rest of this section let α = ( n , . . . , n k ; r ) ∈ A ∩ A ◦ , M = M α , U = U α and P = M ⋉ U .6.5. The set W G ( M ) and elementary symmetries. Let { s α : α ∈ ∆ M ◦ } be the set ofelementary symmetries in W G ◦ ( M ◦ ) and ℓ = ℓ M ◦ : W G ◦ ( M ◦ ) → Z ≥ the length functiondefined in [MW95, I.1.7].For the sake of unified notation set W G ( M ) = W G ◦ ( M ◦ ), ℓ M = ℓ M ◦ , ∆ P = ∆ P ◦ and R ( A M , G ) = R ( A M ◦ , G ◦ ) except if r = 0 in the split even orthogonal case.Assume that r = 0 in the split even orthogonal case and note that M = M ◦ . Let W G ( M ) be the subset of elements w ∈ W G such that w has minimal length in wW M and wM w − is a standard Levi subgroup of G (that is, of the form M β for some β ∈ A ). Let R ( A M , G ) be the set of non-zero restrictions to A M of the roots in R ( T, G ) and let ∆ P bethe non-zero restrictions to A M of the roots in ∆ . For α ∈ R ( A M , G ) we write α > the notation R ( T, G ) is somewhat artificial, note that there are no root subgroups for the roots 2 e i NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 31 it is the restriction of a positive root in R ( T, G ) with respect to ∆ and α < A M = { ι ( a I n , . . . , a k I n k ) : a i ∈ F ∗ , i ∈ [1 , k ] } and ∆ P = { α , . . . , α k } where α i ( ι ( a I n , . . . , a k I n k )) = a i a − i +1 , i ∈ [1 , k − α k ( ι ( a I n , . . . , a k I n k )) = a k . Let s i = s i,M = ι ( I n + ··· + n i − , (cid:0) I ni +1 I ni (cid:1) , I n i +2 + ··· + n k ) , i ∈ [1 , k − s k = s k,M = ι ( I n + ··· + n k − ; (cid:0) I nk I nk (cid:1) ) . Note that s i T ∈ W G ( M ) and set s α i = s i T . The elements { s α : α ∈ ∆ P } serve the roleof elementary symmetries for W G ( M ). We define the length function ℓ M on W G ( M ) byletting ℓ M ( w ) be the cardinality of { α ∈ R ( A M , G ) : α > , wα < } . With this set up theresults of [MW95, I.1.7 and I.1.8] extend to W G ( M ) without modification.6.6. A choice of representatives in G for W G ( M ) . If n > η ∈ H as follows. In the unitary case set η = I n and in the orthogonal case let η ∈ H \ H ◦ be such that η = I n (see Observation 1).For m ∈ N we define an involution η m ∈ G m as follows. If n > η m = ι ( I m ; η ). If n = 0, in the (split even) orthogonal case set η m = ι ( I m − ; w ) otherwise set η m = I m .For every w ∈ W k we define an element t w = t w,M ∈ G as follows. For ρ ∈ S k let w ρ ∈ GL n ( F ) be the permutation matrix with n i × n j blocks w ρ = ( δ i,ρ ( j ) I n i ) i, j ∈ [1 ,k ] and let t ρ,M = ι ( w ρ ; I n +2 r ) . For r ≤ m ∈ N let η m,M = η m except if r = 0 in the split even orthogonal case where weset η m,M = I m . Note that η m,M = ι ( I m − r ; η r,M ). For i ∈ [1 , k ] let t i,M = I n + ··· + n i − I n i η n i n i +1 + ··· + n k + r,M ε I n i I n + ··· + n i − and note that t i,M ∈ G ◦ except if n i is odd and r = 0 in the split even orthogonal case inwhich case t i,M ∈ G \ G ◦ . Note further that t ,M , . . . , t k,M commute with each other. For c ∈ Ξ k let t c ,M = Y i ∈ c t i,M and recall that o( c ) = o M ( c ) is the number of i ∈ c such that n i is odd. Thus, t c ,M ∈ G ◦ except if o M ( c ) is odd and r = 0 in the split even orthogonal case in which case t c ,M ∈ G \ G ◦ .Finally, for w = ρ c ∈ W k let t w = t w,M = t ρ,M t c ,M . Note that t w ∈ N G ( T ) ∩ H and in fact t w ∈ G ◦ except if o M ( c ) is odd and r = 0 in thesplit even orthogonal case. Furthermore, t w represents an element in W G ( M ) (that is, t w M ∈ W G ( M )) and for g i ∈ GL n i ( E ′ ), i ∈ [1 , k ], h ∈ G r and m = ι ( g , . . . , g k ; h ) ∈ M we have(6) t w mt − w = ı ( g ′ , . . . , g ′ k ; h ′ ) where g ′ i = ( g ρ − ( i ) i c g ∗ ρ − ( i ) i ∈ c and h ′ = η o( c ) r hη − o( c ) r . In particular, t w M t − w = M wα where wα = ( n ρ − (1) , . . . , n ρ − ( k ) ; r ). Note further that for w , w ∈ W k , t w = t w ,M w α , t w = t w ,M and t w w = t w w ,M we have t w t w m ( t w t w ) − = t w w mt − w w , m ∈ M. That is, t − w w t w t w lies in the center of M and in particular in M .Let s , . . . , s k be the standard simple reflections in W k , that is, s i = ( i, i + 1) ∈ S k , i ∈ [1 , k − 1] and s k = { k } ∈ Ξ k . Note that t s , . . . , t s k are representatives of the elementarysymmetries in W G ( M ). It is now a simple consequence of the results in [MW95, I.1.7 andI.1.8] and the discussion in § w t w M : W k → W G ( M ) is bijective.We denote by j M : W G ( M ) → W k its inverse. For w ∈ W G ( M ) and w ′ ∈ W G ( M ′ ) where M ′ = wM w − we have j M ( w ′ w ) = j M ′ ( w ′ ) j M ( w ) . r = 0 in the even orthogonal case. Note that neitherof W G ( M ) and W G ◦ ( M ◦ ) is necessarily contained in the other. In fact, it is easy to seethat W G ( M ) ∩ W G ◦ ( M ◦ ) = { w ∈ W G ( M ) : j M ( w ) = ρ c where o( c ) is even } . Furthermore, clearly W G ◦ ( M ◦ , M ◦ ) ⊆ W G ( M ) ∩ W G ◦ ( M ◦ ) and we conclude that W G ◦ ( M ◦ , M ◦ ) = { w ∈ W G ( M ) : j M ( w ) = ρ c where o( c ) is even and n ρ ( i ) = n i , i ∈ [1 , k ] } . In particular, t j M ( w ) ∈ G ◦ , w ∈ W G ◦ ( M ◦ , M ◦ ).6.7. Orbits and stabilizers in X ∩ N G ◦ ( M ◦ ) . Let W k [2 , M ] = { w = ρ c ∈ W k [2] : n ρ ( i ) = n i , i ∈ [1 , k ] } . We begin with two lemmas that consist of simple computations that allow us to compute M ◦ -orbits and stabilizers in X ∩ N G ◦ ( M ◦ ). Lemma 8. Let w = ρ c ∈ W k [2 , M ] and m = ι ( g , . . . , g k ; h ) ∈ M where g i ∈ GL n i ( E ′ ) , i ∈ [1 , k ] and h ∈ G r . Then t w mt w m = ι ( a , . . . , a k ; b ) where a i = ( g ρ ( i ) ¯ g i i c ε g ∗ ρ ( i ) ¯ g i i ∈ c and b = η o( c ) r hη o( c ) r ¯ h. NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 33 In particular, t w m ∈ X if and only if g ρ ( i ) ¯ g i = I n i i c , ρ ( i ) = ig i ∈ Z n i ( E ) i c , ρ ( i ) = ig ∗ ρ ( i ) ¯ g i = ε I n i i ∈ c , ρ ( i ) = iw n i ¯ g i ∈ Y n i ( E ′ / ( E ′ ) στ , ε ) i ∈ c , ρ ( i ) = iη o( c ) r h ∈ X r . Proof. Note that t w ∈ N G ( M ) and recall that ¯ t w = t w . It follows from the above discussionthat t w lies in the center of M and therefore t w mt w m = t w ( t w mt − w ) ¯ m . In fact, it is easyto compute explicitly that t w = ι ( u , . . . , u k ; I n +2 r ) where u i = ( I n i i c ε I n i i ∈ c . Combined with (6) the lemma follows. (cid:3) Lemma 9. Let w = ρ c ∈ W k [2 , M ] and m = ι ( g , . . . , g k ; h ) , x = ι ( x , . . . , x k , y ) ∈ M where g i , x i ∈ GL n i ( E ′ ) , i ∈ [1 , k ] and h, y ∈ G r . Then mt w xm − = t w ι ( a , . . . , a k ; b ) where a i = ( g ρ ( i ) x i ¯ g − i i c g ∗ ρ ( i ) x i ¯ g − i i ∈ c and b = η o( c ) r hη − o( c ) r y ¯ h − . Proof. Since t w is in the center of M we have mt w = t w mt − w and therefore mt w xm − = t w ( t w mt − w ) xm − . The lemma is therefore straightforward from (6). (cid:3) If r = 0 in the split even orthogonal case let W ◦ k [2 , M ] = { w = ρ c ∈ W k [2 , M ] : o( c ) is even } . For the sake of unified notation set W ◦ k [2 , M ] = W k [2 , M ] otherwise.Recall that for w = ρ c ∈ W k [2] we have I ( w ) = { i ∈ c : ρ ( i ) = i } . For w = ρ c ∈ W ◦ k [2 , M ], y i ∈ Y n i ( E ′ / ( E ′ ) στ , ε ), i ∈ I ( w ) and z ∈ X r let(7) x w ( { y i } i ∈ I ( w ) , z ) = t w ι ( u , . . . , u k ; η o( c ) r z ) where u i = ( I n i i I ( w ) w n i ¯ y i i ∈ I ( w ) . Proposition 6. (1) The disjoint M ◦ -orbit decomposition of X ∩ N G ◦ ( M ◦ ) is given by X ∩ N G ◦ ( M ◦ ) = ⊔ w ∈ W ◦ k [2 ,M ] ⊔ { y i } i ∈ I ( w ) , z M ◦ · x w ( { y i } i ∈ I ( w ) , z ) where y i ranges over a choice of representatives for the two GL n i ( E ′ ) -orbits in Y n i ( E ′ / ( E ′ ) στ , ε ) , i ∈ I ( w ) and z ranges over a set of representatives for the (atmost two) G ◦ r -orbits in X r ∩ η o( c ) r G ◦ r = ( X r \ X ◦ r o( c ) is odd and either n > or r > in the orthogonal case X ◦ r otherwise. (2) In particular, for w ∈ W ◦ k [2 , M ] the number of M ◦ -orbits in X ∩ t w M ◦ is | I ( w ) | + δ w,M where δ w,M ∈ { , } is determined as follows: • In the symplectic case δ w,M = 0 . • In the unitary case δ w,M = ( n + 2 r > n = r = 0 . • In the orthogonal case δ w,M = 0 if – n ∈ { , } and r = 0 or – n = 2 , r = 0 and det is in − F ∗ if o( c ) is even and in − ı F ∗ if o( c ) is odd.Otherwise, δ w,M = 1 .(3) The stabilizer of x w ( { y i } i ∈ I ( w ) , z ) in M ◦ consists of elements ι ( g , . . . , g k ; h ) suchthat g ρ ( i ) = ¯ g i ∈ GL n i ( E ′ ) i c , ρ ( i ) = ig i ∈ GL n i ( F ′ ) i c , ρ ( i ) = ig ∗ ρ ( i ) = ¯ g i ∈ GL n i ( E ′ ) i ∈ c , ρ ( i ) = i ¯ g i ∈ U( y i , ( E ′ / ( E ′ ) στ ) i ∈ c , ρ ( i ) = ih ∈ ( G ◦ r ) z . Proof. Recall that the map w wM ◦ : W G ◦ ( M ◦ , M ◦ ) → N G ◦ ( M ◦ ) /M ◦ is a group isomor-phism. It follows from the discussion in § x ∈ X ∩ N G ◦ ( M ◦ ) then xM = wM where w ∈ W G ◦ ( M ◦ , M ◦ ) is an involution. Furthermore, W G ◦ ( M ◦ , M ◦ ) ⊆ W G ( M ) ∩ W G ◦ ( M ◦ )and t j M ( w ) ∈ G ◦ for w ∈ W G ◦ ( M ◦ , M ◦ ) (see § r = 0 in the split orthogonal case).Note that j M maps the set of involutions in W G ◦ ( M ◦ , M ◦ ) to W ◦ k [2 , M ]. We therefore havethe disjoint union X ∩ N G ◦ ( M ◦ ) = ⊔ w ∈ W ◦ k [2 ,M ] X ∩ t w M ◦ . The disjoint decomposition X ∩ t w M ◦ = ⊔ { y i } i ∈ I ( w ) , z M ◦ · x w ( { y i } i ∈ I ( w ) , z )as well as the explication of the stabilizer M ◦ x w ( { y i } i ∈ I ( w ) , z ) are immediate from Lemmas 8and 9 and the facts that Y m ( E ′ / ( E ′ ) στ , ε ) consists of two GL m ( E ′ )-orbits (see the unitarycase in § Z m ( E ) = GL m ( E ′ ) · I m (see § M ◦ -orbits in X ∩ t w M ◦ further follows from the classification of G ◦ r -orbits in X ◦ r and in X r \ X ◦ r (see Corollary 3 in the symplectic case, Corollary 4 in the unitary case and Proposition 5in the orthogonal case). (cid:3) Finally it will be relevant for us to determine, for x ∈ X ◦ , when is the representative (7)in G ◦ · x . Lemma 10. Let x ∈ X ◦ , w = ρ c ∈ W ◦ k [2 , M ] , y i ∈ Y n i ( E ′ / ( E ′ ) στ , ε ) , i ∈ I ( w ) and z ∈ X r ∩ η o( c ) r G ◦ r . Set x w = x w ( { y i } i ∈ I ( w ) , z ) ∈ X ◦ .(1) In the symplectic case x w ∈ G ◦ · x . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 35 (2) In the unitary case x w ∈ G ◦ · x if and only if ( − o( c ) det z Y i ∈ I ( w ) det y τ − i ∈ det x Γ . (Here, if n = r = 0 replace det z by one. See (4) for the definition of Γ .)(3) In the orthogonal case • for x , x ∈ X ◦ we have G ◦ · x = G ◦ · x if and only if ~ [ n ] ( x ) = ~ [ n ] ( x ) ; • x w ∈ G ◦ · x if and only if the following condition is satisfied: ( − r o( c )+ ( N ( ρ c )2 )(2 det ) o( c ) Y j ∈ I ( w ) det y j ∈ N ( E/F ) if and only if ~ [ r ] ( z ) ~ [ r ] ( I n +2 r ) = ~ [ n ] ( x ) ~ [ n ] ( I N ) . (See § ~ [ r ] . Here if n = 0 replace det by one and if n + 2 r = 0 replace the left hand side of the equality by one.)Proof. In the symplectic case the result of the lemma is immediate from Corollary 3. Inthe unitary case it is immediate from Corollary 4 and the observation thatdet x w ( { y i } i ∈ I ( w ) ; z ) = ( − o( c ) det z Y i ∈ I ( w ) det y τ − i . The orthogonal case is more elaborate and requires a careful computation of invariants.We point out all the ingredients of our computation.It follows from Lemma 6 (1) that ∂ [ n ] ( x ) = ∂ [ n ] ( x ). The first part immediately follows(see § x w ∈ G ◦ · x if and only if ~ [ n ] ( x ) = ~ [ n ] ( x w ).Let z w ∈ GL N ( E ) be such that x w = z w · I N and y w = [ n ] ⋆ z w . By definition ~ ( x w ) =Hasse F ( y w ).We make the following observations. Let ζ m = (cid:0) I m ıI m I m − ıI m (cid:1) and υ m = w m ⋆ζ m . We observethat ζ m · I m = (cid:0) I m I m (cid:1) and υ m = diag(2 w m , − ı w m ) . Note further that w m ⋆ diag( ζ m , ζ m ) = (cid:0) η m η m (cid:1) and that η m , (cid:0) η m η m (cid:1) ∈ w m ⋆ GL m ( F ) = I m ⋆ GL m ( F ).For y ∈ Y m ( E/F, 1) let x ( y ) = (cid:0) y − w m w m ¯ y (cid:1) ∈ X w m , z ( y ) = (cid:0) I m ıw m w m ¯ y − ıw m ¯ yw m (cid:1) ∈ GL m ( E )and η ( y ) = w m ⋆ z ( y ). Note that z ( y ) · I m = x ( y ) and therefore η ( y ) ∈ Y m ( F/F, y ∈ Y m ( F/F, 1) = Y m ( E/F, ∩ GL m ( F ) then η ( y ) = diag(2 y, − ı w m yw m ) . We can deduce that det η ( y )det w m ∈ ı m F ∗ and Hasse F ( η ( y ))Hasse F ( w m ) = (det y, ı ) F (2 , ı ) mF ( − , ı )( m ) F . Indeed, note that η ( y ) ⋆ GL m ( F ) (hence det η ( y ) F ∗ and Hasse F ( η ( y ))) depends only on y ⋆ GL m ( E ) and therefore we can replace y with diag( I m − , det y ). In this case η ( y ) isdiagonal and its invariants are easy to compute. We further recall that det w m = ( − m .Based on the above observations we have that diag( u , . . . , u k , u ) ∈ y w ⋆ GL N ( F ) where u i = ( η ( y i ) i ∈ I ( w ) I n i otherwise and u = [ r ] ⋆ ζ where z = ζ · I n +2 r . It therefore follows thatHasse F ( y w ))Hasse F ( [ n ]) = ~ [ r ] ( z )Hasse F ( [ r ]) Y i ∈ I ( w ) ( ∂ [ r ] ( z ) , det η ( y i )) F (det [ r ] , det w n i ) F Hasse F ( η ( y i ))Hasse F ( w n i ) × Y { i, j ∈ I ( w ): i We note further that o( w ) ≡ o( c )(mod2). We conclude thatHasse F ( y w ))Hasse F ( [ n ]) = (( − r , ı ) o( c ) F ( − , ı )( N ( w )2 ) F ( Y j ∈ I ( w ) det y j , ı ) F ~ [ r ] ( z )Hasse F ( [ r ]) . Recall that from the definition of the quadratic Hilbert symbol we have ( a, ı ) F = 1 if andonly if a ∈ N ( E/F ). The lemma follows. (cid:3) The contribution of an admissible orbit. Let∆(GL m ( E ′ )) = { ( g, g ) : g ∈ GL m ( E ′ ) } be the image of the diagonal imbedding of GL m ( E ′ ) in GL m ( E ′ ) × GL m ( E ′ ). The followingis an immediate consequence of the explication of the stabilizers in Proposition 6 (3). Corollary 6. Let α = ( n , . . . , n k ; r ) ∈ A and M = M α . Let π i be a representationof GL n i ( E ′ ) , i ∈ [1 , k ] and π a representation of G ◦ r . Let w = ρ c ∈ W ◦ k [2 , M ] , y i ∈ Y n i ( E ′ / ( E ′ ) στ , ε ) , i ∈ I ( w ) , z ∈ X r ∩ η o( c ) r G ◦ r and x w ( { y i } i ∈ I ( w ) , z ) ∈ X ◦ be defined as in (7) . The representation ( π ⊗ · · · ⊗ π k ⊗ π ) ◦ ι α of M ◦ is M ◦ x w ( { y i } i ∈ I ( w ) ,z ) -distinguished ifand only if π i ⊗ ¯ π ρ ( i ) is ∆(GL n i ( E ′ )) -distinguished i c , ρ ( i ) = iπ i is GL n i ( F ′ ) -distinguished i c , ρ ( i ) = iπ i ⊗ ¯ π ∗ ρ ( i ) is ∆(GL n i ( E ′ )) -distinguished i ∈ c , ρ ( i ) = iπ i is U( y i , E ′ / ( E ′ ) στ ) − distinguished i ∈ c , ρ ( i ) = iπ is ( G ◦ r ) z − distinguished. (cid:3) Remark . For irreducible representations π and π of GL m ( E ′ ) we clearly have that π ⊗ π is ∆(GL n i ( E ′ ))-distinguished if and only if π ≃ π ∨ . Furthermore, for an irreduciblerepresentation π of GL m ( E ′ ) we have π ∗ ≃ ( π τ ) ∨ (see [GK75]). As a consequence, inthe above lemma (and in its notation) if ρ ( i ) = i and π i and π ρ ( i ) are irreducible then π i ⊗ ¯ π ρ ( i ) is ∆(GL n i ( E ′ ))-distinguished if and only if π ρ ( i ) ≃ ¯ π ∨ i and π i ⊗ ¯ π ∗ ρ ( i ) is ∆(GL n i ( E ′ ))-distinguished if and only if π ρ ( i ) ≃ π στi .6.9. Completion of the proof of Theorem 2. It follows from Lemma 7 that we mayassume without loss of generality that P = P α for some α = ( n , . . . , n k ; r ) ∈ A (with r = 1 in the split even orthogonal case). As a consequence of Corollary 2 we have that π × · · · π k ⋉ π is G ◦ x -distinguished if and only if there exists x ′ ∈ G · x ∩ N G ◦ ( M ◦ ) such that( π ⊗ · · · ⊗ π k ⊗ π ) ◦ ι α is M ◦ x ′ -distinguished. Note that for a representation υ of M ◦ and m ∈ M ◦ we have that ℓ ℓ ◦ υ ( m ) is an isomorphism from Hom M ◦ m · x ′ ( υ, ) to Hom M ◦ x ′ ( υ, )and in particular, υ is M ◦ x ′ -distinguished if and only if it is M ◦ m · x ′ -distinguished. Thetheorem is therefore now immediate from Proposition 6(1), Lemma 10, Corollary 6 andRemark 5. (cid:3) Distinction and conjugate-selfduality in the split odd orthogonal case It was recently proved by Beuzart-Plessis [BP] that for any quasi-split reductive group H defined over F and irreducible, generic representation π of H ( E ) that is H ( F )-distinguishedwe have π σ = π ∨ .When H is the split odd special orthogonal group we apply our previous results in order toprove a global analogue as well as several local generalizations of this result. In particular,we show that in the above statement H ( F )-distinction can be replaced by H ′ ( F )-distinctionwhere H ′ is a non-quasi-split inner form of H such that H ′ ( E ) = H ( E ).Let SO n +1 = SO( w n +1 ) be the split odd special orthogonal group. Over any field k wehave SO n +1 ( k ) = { g ∈ SL n +1 ( k ) : t gw n +1 g = w n +1 } . The following is well known. Since we do not know a proper reference we include theargument. Observation 3. (1) For a p -adic field F and an irreducible, generic representation π of SO n +1 ( F ) we have π ∨ ≃ π .(2) For a number field k and an irreducible, generic, cuspidal automorphic representa-tion Π of SO n +1 ( A k ) we have Π ∨ = Π .Proof. Consider first the local statement and assume further that π is unramified. Then, π is fully induced from a character χ of the standard Borel subgroup. Since χ − is in theWeyl orbit of χ it follows that π is selfdual.In the global setting, this implies that Π v is selfdual for almost all places v of k . Theglobal statement is therefore immediate from the rigidity theorem of Jiang and Soudry[JS03, Theorem 5.3].Now a simple globalization argument such as [Sha91, Proposition 5.1] implies that everyirreducible, generic and cuspidal representation π of SO n +1 ( F ) is selfdual.For the general local case, in terms of its cuspidal support, π ∨ is a subquotient ofa representation induced from an irreducible, cuspidal and generic representation of astandard Levi subgroup. That is, there exists a decomposition n = n + · · · + n k + r ,irreducible cuspidal representations π i of GL n i ( F ), i ∈ [1 , k ] and an irreducible, genericand cuspidal representation π of G ◦ r such that π ∨ is a subquotient of π × · · · × π k ⋉ π . Itfollows from the cuspidal case that π is selfdual. Thus both π and π ∨ are subquotients of π ∨ × · · · × π ∨ k ⋉ π ∨ . However, by a well known result of Rodier [Rod73], π ∨ × · · · × π ∨ k ⋉ π ∨ has a unique irreducible, generic subquotient and it follows that π is selfdual. (cid:3) Let G be a semi-simple algebraic group and H a reductive subgroup of G both definedover a number field k . It follows from [AGR93, Proposition 1] that the period integral Z H ( k ) \ H ( A k ) φ ( h ) dh is absolutely convergent for every cuspidal automorphic form φ on G ( A k ). A cuspidalautomorphic representation Π of G ( A k ) is called H -distinguished if this period integral isnon-vanishing for some cusp form φ in the space of Π. NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 39 Theorem 3. Let K/k be a quadratic extension of number fields with Galois action σ , G =Res K/k (SO n +1 ) , X = { x ∈ G : x σ = x − } and g · x = gxg − σ the action of G on X by twistedconjugation. Denote by G x the stabilizer of x ∈ X ( k ) in G , an algebraic group defined over k . Let Π be an irreducible cuspidal generic representation of G ( A k ) = SO n +1 ( A K ) that is G x -distinugished. Then Π σ ≃ Π ∨ .Proof. Write Π = ⊗ v Π v where the restricted tensor product is over all places v of k . ThenΠ v is G x ( k v )-distinguished for all places v of F .If v is split in K (that is K v ≃ k v × k v ) then we may identify G ( K v ) with SO n +1 ( k v ) × SO n +1 ( k v ) and σ acts on it by interchanging the coordinates. Then x v = ( x , x − ) forsome x ∈ SO n +1 ( k v ) and G x ( k v ) = { ( g, x − gx ) : g ∈ SO n +1 ( k v ) } . Write Π v = π ⊗ π for irreducible representations π and π of SO n +1 ( k v ). Since Π v is G x ( k v )-distinguished we see that π ≃ π ∨ , that is, Π σv ≃ Π ∨ v .Since GL n +1 has trivial Galois cohomology, there exists z ∈ GL n +1 ( K ) such that x = zz − σ . Note that y = t zw n +1 z is a symmetric matrix in GL n +1 ( k ). Since det x = 1we have det y k ∗ = det w n +1 k ∗ . Since the Hasse invariant of y is one at almost all placesit follows that y and w n +1 are in the same GL n +1 ( k v )-orbit for almost all places v of k .It follows from Lemma 3 that G x ( k v ) is SO n +1 ( K v )-conjugate to SO n +1 ( k v ) for almostall places v of k that are inert in K . Let v be a finite and inert place of k such thatΠ v is unramified. Since Π is generic we can write Π v ≃ χ × · · · χ n ⋉ as irreduciblyinduced from a character of the Borel subgroup. Thus, Π v ≃ χ ′ ρ (1) × · · · χ ′ ρ ( n ) ⋉ whenever χ ′ i ∈ { χ i , χ − i } , i ∈ [1 , n ] and ρ ∈ S n . It now follows from Corollary 5 that Π σv = Π ∨ v .We conclude that Π σv = Π ∨ v for almost all places v of k and the theorem now followsfrom the rigidity theorem of Jiang and Soudry [JS03, Theorem 5.3]. (cid:3) For the rest of this section we go back to the notation of § G ◦ = SO n +1 ( E ) and X ◦ = { x ∈ G ◦ : x − = x σ } . Since X = X ◦ ⊔ ( − X ◦ )we will use without further mention the fact that G ◦ x = G ◦− x , x ∈ X .A representation π of GL m ( E ) is called essentially tempered if | det | − e ⊗ π is temperedfor some real number e . The number e is uniquely determined by π and we write e ( π ) = e .Every irreducible and cuspidal representation of GL m ( E ) is essentially tempered. Theorem 4. Let π be an irreducible, generic representation of G ◦ . If π is G x -distinguishedfor some x ∈ X then π σ ≃ π ∨ .Proof. If π is cuspidal this is immediate from Theorem 3 and the globalization of distin-guished, generic representations in [BP]. We apply the cuspidal case in order to completethe proof of the theorem for a general π .In terms of its cuspidal support we realize π as a quotient of a representation induced froman irreducible, generic and cuspidal representation of a standard Levi subgroup of G ◦ . Thatis, there exists a decomposition n = n + · · · + n k + r , irreducible cuspidal representations π i of GL n i ( E ), i ∈ [1 , k ] and an irreducible, generic and cuspidal representation π of G ◦ r such that π ∨ is a quotient of I = π × · · · × π k ⋉ π . Hence I is also G ◦ x -distinguished.Furthermore, π is the unique generic irreducible subquotient of I by [Rod73].By Theorem 2 the representation π is ( G ◦ r ) z -distinguished for some z ∈ X r and by thecuspidal case of this theorem we conclude that π σ ≃ π ∨ . It further follows from Corollary5 that ( π , . . . , π k ) = (( π ′ ρ (1) ) σ , . . . , ( π ′ ρ ( k ) ) σ ) for some involution ρ ∈ S k and some choice π ′ i ∈ { π i , π ∨ i } , i ∈ [1 , k ]. Since ( π ′ ρ (1) ) σ × · · · × ( π ′ ρ ( k ) ) σ ⋉ ( π ∨ ) σ has the same decompositionseries as ( I ∨ ) σ we conclude that I and ( I ∨ ) σ have the same decomposition series. Thus,( π ∨ ) σ is a generic subquotient of I and therefore π σ = π ∨ . (cid:3) Recall that by generalized injectivity [Han20], if a standard module is generic and re-ducible then its Langlands quotient is not generic. We now extend our result and showthat for a certain family of irreducible, degenerate representations distinction implies Galoisinvariance. Proposition 7. Let I be a standard module of G ◦ of the form I = π × · · · × π s ⋉ T where π i is irreducible cuspidal for i ∈ [1 , s ] and such that e ( π ) ≥ · · · ≥ e ( π s ) > and T is irreducible, generic and has unitary cuspidal support (in particular, T is tempered).If either I or I ∨ is G ◦ x -distinguished for some x ∈ X then I σ = I . Consequently, if theLanglands quotient of I is G ◦ x -distinguished for some x ∈ X then it is Galois invariant.Proof. The representation T is the unique generic direct summand of a semi-simple repre-sentation of the form π s +1 × · · · × π k ⋉ π where π is an irreducible, generic and cuspidalrepresentation of G ◦ r for some r and π i is irreducible cuspidal for i ∈ [ s + 1 , k ]. Therefore I is a quotient of Π = π × · · · × π k ⋉ π and I ∨ is a quotient of Π ∨ .If I (resp. I ∨ ) is G ◦ x -distinguished then so is Π (resp. Π ∨ ). It follows from Theorem 2 that π (resp. π ∨ ) is ( G ◦ r ) z -distinguished for some z ∈ X r and from Theorem 4 and Observation3 that π σ = π . It further follows from Corollary 5, the inequalities e ( π ) ≥ · · · ≥ e ( π s ) > e ( π s +1 ) = · · · = e ( π k ) = 0 that we have the following equalities of multi-sets of cuspidal representations { π , . . . , π s } = { π σ , . . . , π σs } and { π ′ s +1 , . . . , π ′ k } = { π σs +1 , . . . , π σk } for some choice of π ′ i ∈ { π i , π ∨ i } , i ∈ [ s + 1 , k ]. By well known results on standard modulesfor the general linear group over E the first identity implies the equality of the inducedrepresentations π × · · · × π s = π σ × · · · × π σs while the second identity combined with the fact that π is Galois invariant imply theequality of the semi-simple representations π s +1 × · · · × π k ⋉ π = π σs +1 × · · · × π σk ⋉ π σ . Since T σ is a generic summand of the right hand side we conclude that T = T σ . Alltogether, this implies that I is Galois invariant.The second part of the proposition follows from the first part and the facts that Galoisinvariance of I implies Galois invariance of its Langlands quotient and G ◦ x -distinction ofthe Langlands quotient implies G ◦ x -distinction of I . (cid:3) NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 41 Remark . The proof of Proposition 7 uses Theorem 4 only in the case where π is cuspidal.In fact, the proposition can be used as an alternative way to reduce Theorem 4 to thecuspidal case as follows. Let π be as in Theorem 4. We freely use the notation of the proofof the proposition. We may realize π ∨ as the unique generic irreducible subquotient of arepresentation of the form Π. We may now define T to be the unique generic summandof the semi-simple representation π s +1 × · · · × π k ⋉ π so the standard module I is definedas in the proposition. Thus, π ∨ is the unique generic irreducible subquotient of I and bygeneralized injectivity [Han20], π ∨ is a subrepresentation of I . Consequently, π is a quotientof I ∨ and therefore I ∨ is G ◦ x -distinguished. It therefore follows from the proposition that I is Galois invariant. Hence( π ∨ ) σ is an irreducible generic subquotients of I and thereforeequals π ∨ . It follows that π is Galois invariant.8. On distinction and local Langlands correspondence In this section we identify a certain family of irreducible representations π of the classicalgroup G ◦ (in the notation of § N ( E ′ ) is also irreducibly induced from cuspidal and Galois distinctionof π implies Galois distinction of its transfer. We only consider quasi-split classical groupsand the family of representations will contain, in particular, all generic principal seriesrepresentations (that is, irreducibly induced from a character of the Borel subgroup).The results are simple applications of Theorem 2, however, we make the relation toPrasad’s conjectures in [Pra] explicit and this requires lengthy and rather technical com-putations.8.1. On Galois distinction for GL N . For m = m + · · · + m t and representations π i ofGL m i ( E ′ ), i ∈ [1 , t ] we denote by π × · · · × π t the representation of GL m ( E ′ ) parabolicallyinduced from π ⊗ · · · ⊗ π t (with the usual normalized induction). If π i is GL m i ( F ′ )-distinguished for all i ∈ [1 , t ] then π × · · · × π t is GL m ( F ′ )-distinguished by a closedorbit argument (see e.g. [Fli92, Proposition 26]). Furthermore, for any representation π ofGL m ( E ′ ) of finite length, the representation π ∨ × π σ of GL m ( E ′ ) is GL m ( F ′ )-distinguishedby an open orbit argument (see e.g. [Gur15, Proposition 2.3]).For a representation π of GL m ( E ′ ) and a character χ of ( E ′ ) ∗ let πχ = π ⊗ ( χ ◦ det). Let η E ′ /F ′ be the unique character of ( F ′ ) ∗ with kernel, the index two subgroup N ( E ′ /F ′ ), andlet η be any extension of η E ′ /F ′ to a character of ( E ′ ) ∗ . In particular, η σ = η − . Clearly, π is GL m ( F ′ )-distinguished if and only if πη is (GL m ( F ′ ) , η E ′ /F ′ ◦ det)-distinguished. Since,with the above notation, ( π ×· · ·× π t ) η ≃ ( π η ) ×· · ·× ( π t η ), the closed orbit argument alsoimplies that if π i is (GL m i ( F ′ ) , η E ′ /F ′ ◦ det)-distinguished for all i ∈ [1 , t ] then π × · · · × π t is (GL m ( F ′ ) , η E ′ /F ′ ◦ det)-distinguished. Similarly, ( π ∨ × π σ ) η ≃ ( πη − ) ∨ × ( πη − ) σ andtherefore, the open orbit argument also implies that for any representation π of GL m ( E ′ ) offinite length, the representation π ∨ × π σ of GL m ( E ′ ) is (GL m ( F ′ ) , η E ′ /F ′ )-distinguished.In the next lemma we freely use both the open and the closed orbit argument.We make the following simple observation that, in conjunction with Theorem 2, willallow us to relate between Galois distinction for a classical group and a general lineargroup. Lemma 11. Let N = 2( n + · · · + n k )+ r , Π a representation of GL r ( E ′ ) , π i an irreduciblerepresentation of GL n i ( E ′ ) , i ∈ [1 , k ] and χ be either the trivial character of ( F ′ ) ∗ or η E ′ /F ′ .Assume that(1) the representation Π = π × · · · × π k × Π × ( π ∨ k ) τ × · · · × ( π ∨ ) τ of GL N ( E ′ ) is irreducible,(2) Π is (GL r ( F ′ ) , χ ◦ det) -distinguished and(3) there exist an involution ρ ∈ S k and a subset c of [1 , k ] such that ρ ( c ) = c and n ρ ( i ) = n i , i ∈ [1 , k ] and y i ∈ Y n i ( E ′ / ( E ′ ) στ , ε ) for any i ∈ c such that ρ ( i ) = i suchthat π ρ ( i ) ≃ ( π ∨ i ) σ i c , ρ ( i ) = iπ i is (GL n i ( F ′ ) , χ ◦ det) -distinguished i c , ρ ( i ) = iπ ρ ( i ) ≃ π τσi i ∈ c , ρ ( i ) = iπ i is U( y i , E ′ / ( E ′ ) στ ) − distinguished i ∈ c , ρ ( i ) = i. Then Π is (GL N ( F ′ ) , χ ◦ det) -distinguished.Proof. Since Π is irreducible we may freely permute its factors. For i c such that ρ ( i ) = i we have that π i × π ρ ( i ) ≃ π i × ( π ∨ i ) σ and ( π ∨ i ) τ × ( π ∨ ρ ( i ) ) τ ≃ ( π ∨ i ) τ × π στi . For i ∈ c such that ρ ( i ) = i we have π i × ( π ∨ ρ ( i ) ) τ ≃ π i × ( π ∨ i ) σ . For i ∈ c such that ρ ( i ) = i it follows from [FLO12, Theorem 6.1(1)] that π i ≃ π στi andtherefore that π i × ( π ∨ i ) τ ≃ π στi × ( π ∨ i ) τ . By the open orbit argument, it follows that Π can be expressed as a product of representa-tions each of the appropriate GL m ( E ′ ) that is (GL m ( F ′ ) , χ ◦ det)-distinguished. Therefore,Π is (GL N ( F ′ ) , χ ◦ det)-distinguished by the closed orbit argument. (cid:3) L-groups and transfer maps. Recall (see § 5) that G ◦ = H ◦ ( E ) where H ◦ is theconnected component of the algebraic group H = { g ∈ GL N : t g τ [ n ] g = [ n ] } definedover F . We denote by W k the Weil group of the non-archimedean local field k and by W ′ k = W k × SL ( C ) the Weil-Deligne group.Let G ◦ = H ◦ E be the base change of H ◦ to E . Assume for the rest of this work that G ◦ is quasi-split. The Langlands dual group (L-group) of G ◦ is c G ◦ ( C ) ⋉ W E with connectedcomponent c G ◦ ( C ). For many applications it suffices to keep track of the action of W E on c G ◦ ( C ) and it is convenient to consider a simplified version of the L -group. For instance,if G ◦ is split then W E acts trivially on c G ◦ ( C ) and we only keep track of the connectedcomponent.We denote by L G ◦ the L -group of G ◦ defined as follows. In the symplectic, the oddorthogonal and the split even orthogonal cases, we take L G ◦ = c G ◦ ( C ). Explicitly, c G ◦ ( C ) NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 43 equals SO n +1 ( C ), Sp n ( C ) or SO n ( C ) in the symplectic, odd orthogonal or split evenorthogonal case respectively. In the non-split but quasi-split even orthogonal case, G ◦ splits over a quadratic extension Q/E and we set L G ◦ = SO n +2 ( C ) ⋉ Gal( Q/E ) . Sending the Galois involution θ Q/E of Q/E to s = diag( I n , w , I n ) we obtain an identifica-tion L G ◦ = O n +2 ( C )with the split orthogonal group defined with respect to w n +2 . Hence in all these caseswe have a natural imbedding of L G ◦ into GL m ( C ) where m = m ( n ) equals 2 n + 1 in thesymplectic case, 2 n in the split orthogonal case and 2 n + 2 in the quasi-split non-splitorthogonal case.In the unitary case L G ◦ = GL N ( C ) ⋉ Gal( E ′ /E )where the Galois involution τ of E ′ /E acts on GL N ( C ) by τ gτ − = J N t g − J − N where J N = diag(1 , − , . . . , ( − N − ) w N . In this case we set m = m ( n ) = N .We denote by Φ( G ◦ ) the set of L -parameters for G ◦ . These are the c G ◦ ( C )-conjugacyclasses [ φ ] of (admissible) L -homomorphisms φ : W ′ E → L G ◦ . Similarly, we denote byΦ(GL m ( E ′ )) the set of L -parameters of GL m ( E ′ ). That is, the GL m ( C )-conjugacy classes[ φ ] of L -homomorphisms φ : W ′ E ′ → GL m ( C ).Note that in all cases an L -homomorphism φ : W ′ E → L G ◦ for G ◦ maps W ′ E ′ intoGL m ( C ) and therefore defines an L -homomorphism for GL m ( E ′ ). Denote by I : Φ( G ◦ ) → Φ(GL m ( E ′ )) the map defined by I ([ φ ]) = [ φ | W ′ E ′ ] . The local Langlands correspondence is a conjectural classification of the irreducible rep-resentations of a local reductive group in terms of, so called, enhanced L -parameters. Thework of Arthur [Art13] followed by that of Mok [Mok15] establish the local Langlandscorrespondence for tempered representations of the quasi-split classical groups G ◦ that weconsider. In fact, we will need a much weaker version of their result as formulated in[Kal16, Conjecture A]. In conjunction with [ABPS14] or [SZ18] it provides a finite to onemap from the set of isomorphism classes of irreducible representations of G ◦ to the setΦ( G ◦ ) of L-parameters of G ◦ with certain natural properties. The fibers of this map formthe L -packets for G ◦ . Together with the local Langlands correspondence for general lineargroups ([Hen00],[HT01]) this defines a transfer of irreducible representations from G ◦ toGL m ( E ′ ) along the map I . We denote by T ( π ) the transfer of an irreducible representa-tion π of G ◦ . It is the representation of GL m ( E ′ ) with parameter I ([ φ ]) where [ φ ] is theparameter of π .It is known by the work of Arthur, Mok and others (see [Ato17]), that whenever onefixes a non-degenerate character ψ of the unipotent radical of the Borel subgroup of G ◦ ,each tempered L-packet contains a unique ψ -generic representation. Transfer of induced generic representations. Throughout this section fix a de-composition n = n + · · · + n k + r and representations π of G ◦ r and π i of GL n i ( E ′ ), i ∈ [1 , k ].In the split even orthogonal case, if r = 0 we also fix a sign in order to choose one of the twostandard parabolic subgroups of type ( n , . . . , n k ) (see § π = π × · · · π k ⋊ π of G ◦ and Π = π × · · · π k × T ( π ) × ( π ∨ k ) τ × · · · ( π ∨ ) τ . of GL m ( E ′ ) where m = m ( n ) and the representation T ( π ) of GL m ( r ) ( E ′ ) is the transfer of π defined by the local Langlands correspondence.We will say that an irreducible representation of G ◦ is generic if it has a Whittakermodel with respect to a non-degenerate character of the standard maximal unipotentsubgroup of G ◦ . By the standard module theorem [Mui01, Theorem 1.1] every irreduciblegeneric representation of G ◦ is of the form π above where π is an irreducible generictempered representation of G ◦ r and π , . . . , π k are irreducible essentially square-integrablerepresentations such that e ( π ) ≥ · · · ≥ e ( π k ) > 0. The standard parabolic subgroup of G ◦ and the data ( π , . . . , π k ; π ) are uniquely determined by π . When this is the case, wewill show that Π is also irreducible and generic and in fact Π = T ( π ). Lemma 12. Assume that π is tempered, π , . . . , π k are essentially square-integrable, e ( π ) ≥ · · · ≥ e ( π k ) ≥ and both π and Π are irreducible. Then Π = T ( π ) .Proof. Let M ◦ ≃ GL n ( E ′ ) × · · · × GL n k ( E ′ ) × G ◦ r be the standard Levi subgroup ofthe standard parabolic subgroup of G ◦ fixed in the discussion above from which π is in-duced. Denote by L M ◦ the relevant standard Levi subgroup of L G ◦ (cf. [Bor79, § M ◦ . Let M ′ be the standard Levi subgroup of GL m ( n ) ( C ) of type( n , . . . , n k , m ( r ) , n k , . . . , n ). The connected component of L M ◦ equals M ′ ∩ G ◦ .By compatibility of the local Langlands correspondence with parabolic induction for tem-pered representations (as in [Kal16, Conjecture A]) and with the Langlands classification([ABPS14] or [SZ18]), the L-parameter [ φ ] of π is represented by some L-homomorphism φ : W ′ E → L M ◦ with image in L M ◦ . Then φ maps W ′ E ′ into the connected component of L M ◦ and inparticular into M ′ = diag(GL n ( C ) , . . . , GL n k ( C ) , GL m ( r ) ( C ) , GL n k ( C ) , . . . , GL n ( C )) andfrom the compatibility condition defining an L -homomorphism one checks that I [ φ ] = [ φ | W ′ E ′ ] = [diag( φ , . . . , φ k , ( φ ) | W ′ E ′ , φ ′ k , . . . , φ ′ )]where [ φ i ] is the L -parameter of π i , i ∈ [0 , k ] and [ φ ′ i ] the L -parameter of ( π ∨ i ) τ , i ∈ [1 , k ].That is, [ φ ] is the L -parameter for Π. (cid:3) Lemma 13. Assume that π is generic tempered, π , . . . , π k are essentially square-integrable, e ( π ) ≥ · · · ≥ e ( π k ) ≥ and π is irreducible. Then Π is irreducible and generic. In partic-ular Π = T ( π ) . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 45 Proof. Note that T ( π ) is tempered and in particular unitary. If e ( π ) = 0 then Π is inducedfrom a unitary representation and is therefore irreducible by a theorem of Bernstein [Ber84].Otherwise, let l be maximal such that e ( π l ) > 0. Replacing π by π l +1 × · · · × π k ⋊ π and k by l we may assume without loss of generality that l = k , i.e., that e ( π k ) > π is irreducible, we also have that π ′ i ⋊ π , π ′ i × π ′ j are irreducible for all i, , j ∈ [1 , k ]where π ′ i ∈ { π i , ( π ∨ i ) τ } for i ∈ [1 , k ]. Since T ( π ) is tempered it is of the form T ( π ) ≃ δ × · · · × δ t for a unique multi-set { δ , . . . , δ t } of square-integrable representations. Since the standardmodule π i ⋊ π is irreducible it follows from [Mui01, Lemma 1.1 and Theorem 1.1] that L (1 , π ∨ i , π ) = L (1 , π ∨ i , T ( π )) = t Y j =1 L (1 , π ∨ i , δ j ) = ∞ where L ( s, σ , σ ) denotes the local Rankin-Selberg L -function of irreducible representa-tions σ i of GL m i ( E ′ ), i = 1 , 2. The first equality can be viewed as a definition of the lefthand side and the second follows from [JPSS83, Proposition 8.4]. It follows from [MO18,Lemma 2.3], in its terminology, that δ j does not precede π i and since e ( π i ) > e ( δ j ) itis also the case that π i does not precede δ j . It therefore follows from [Zel80, Theorem 9.7]that π i × δ j is irreducible for every i ∈ [1 , k ] and j ∈ [1 , t ]. It follows from [GGP12, Theo-rem 8.1] that ( T ( π ) ∨ ) τ ≃ T ( π ) and consequently { δ , . . . , δ t } is stable under δ ( δ ∨ ) τ .Hence π i × ( δ ∨ j ) τ and therefore also its conjugate dual ( π ∨ i ) τ × δ j are irreducible. It nowfollows from [Zel80, Theorem 9.7] that Π is irreducible and generic. The equality Π = T ( π )follows from Lemma 12. (cid:3) A consequence of a conjecture of Prasad. We state a consequence of a conjectureof Prasad for distinguished representations in the context of a Galois pair. The conjectureis formulated in terms of the opposition group Y op = Y op ,E/F defined in [Pra, § 5] and acharacter ω Y = ω Y ,E/F of Y ( F ) of order at most two constructed in [Pra, Proposition 6.4]for any connected reductive group Y defined and quasi-split over F with respect to thequadratic extension E/F .We begin by explicating Y op and ω Y for groups Y relevant to this work.8.4.1. A table for Y op and ω Y ,E/F . We denote by U m,K/F the quasi-split unitary groupdefined over F associated with a quadratic extension K/F and defined by w m . Let wsn :U m,K/F ( F ) → K ∗ /F ∗ be the composition of det : U m,K/F ( F ) → ( K/F ) with the inverseof the isomorphism zF ∗ zz − τ from K ∗ /F ∗ to U ,K/F ( F ). (In [Wal59], Wall interpretedthis map as a spinor norm analogue.) For the purpose of the table below, K/F will be aquadratic extension such that K ∩ E = F so that KE/F is a Klein extension and containsa third quadratic extension of F , that we denote by K ′ , different than K and E .We denote by SO m any special orthogonal group defined and quasi-split over F of theform { g ∈ SL m : t gyg = y } for some y = t y ∈ GL m ( F ). ( With this convention SO stands for a rank one torus that is either split over F (isomorphic to GL ) or splits over a quadratic extension L of F (isomorphic to U ,L/F ). We denote by sn : SO m ( F ) → F ∗ /F ∗ the spinor norm map.We also denote by η E/F the quadratic character of F ∗ with kernel N ( E/F ). We have Y ω Y Y op GL m η m − E/F ◦ det U m,E/F U m,E/F GL n Sp m Sp m SO m η mE/F ◦ sn SO m U m,K/F η m − EK/K ◦ wsn U m,K ′ /F In the sequel, this table is sometimes used without further mention. For the sake ofcompleteness we provide the details of the computation in Appendix A.8.4.2. Prasad’s conjecture and transfer of distinction for quasi-split classical groups. Let Y be an F -quasi-split group. Since ( Y op ) E ≃ Y E the L -group L Y ( E ) of Y E is a sub-group of the L -group L Y op ( F ) of Y op with the same connected component and for any L -homomorphism φ : W ′ F → L Y op ( F ) we have φ ( W ′ E ) ⊆ L Y ( E ). This defines a basechangemap BC EF : Φ( Y op ( F )) Φ( Y ( E )) , BC EF ([ φ ]) = [ φ | W ′ E ]from L -parameters of Y op ( F ) to L -parameters of Y ( E ).We say that a representation π of Y ( E ) is E/F -generic if it is ( U, ψ )-generic for a σ -stable maximal unipotent subgroup U of Y ( E ) and a non-degenerate charactacter ψ of U satisfying ψ σ = ψ − (which is the same as ψ | U σ ≡ ). The following is a part of [Pra,Conjecture 13.3]. Conjecture 1 (Dipendra Prasad) . Assume that the Langlands correspondence is knownfor Y ( E ) and Y op ( F ) .(1) Let π be a ( Y ( F ) , ω Y ,E/F ) -distinguished irreducible representation of Y ( E ) . Thenthe parameter of the L-packet of π belongs to BC EF (Φ( Y op ( F ))) .(2) Conversely, if the L -packet of an E/F -generic representation π of Y ( E ) corre-sponds to a parameter in BC EF (Φ( Y op ( F ))) then π is ( Y ( F ) , ω Y ,E/F ) -distinguished.Remark . This conjecture is now a theorem when Y = GL n . It follows from the resultsin [Kab04, AKT04, AR05, Hen10, Mat11, Gur15].Fix a quadratic extension K of F contained in E ′ (so that E ′ /K is also quadratic) andlet U n ( E ′ /K ) = U n,E ′ /K ( K ). We will need the following characterization of the image of BC E ′ K : Φ(U n ( E ′ /K )) → Φ(GL n ( E ′ )) . We say that an L-homomorphism φ E ′ : W ′ E ′ → GL n ( C ) NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 47 (or the L -parameter [ φ ]) is E ′ /K -conjugate of sign ǫ ∈ {± } (we also say E ′ /K -conjugateorthogonal if the sign is 1 and E ′ /K -conjugate symplectic if the sign is − 1) if there existsa non-degenerate bilinear form B : C n × C n → C such that(8) B ( φ E ′ ( w ) x, φ E ′ ( sws − ) y ) = B ( x, y ) and B ( x, φ E ′ ( s ) y ) = ǫB ( y, x )for w ∈ W ′ E ′ , x, y ∈ C n where s is a choice of an element of W ′ K \ W ′ E ′ . Theorem 5. ([GGP12, Theorem 8.1]) An L-homomorphism φ E ′ as above is E ′ /K -conjugateof sign ( − n − if and only if [ φ E ′ ] ∈ BC E ′ K (Φ(U n ( E ′ /K ))) . We will also need the following slightly more precise result. Theorem 6. ([Kab04, AKT04, AR05, Hen10, Mat11]) Let π be a generic irreducible rep-resentation of GL n ( E ′ ) with L-parameter [ φ π ] . Then φ π is E ′ /K -conjugate symplectic ifand only if π is (GL n ( K ) , η E ′ /K ◦ det) -distinguished and E ′ /K -conjugate orthogonal if andonly if π is GL n ( K ) -distinguished. We are now in a position to show that the first part of Conjecture 1 for the genericspectrum implies a relation between distinction in the classical group and distinction ofthe transfer. Proposition 8. Let Y = H ◦ be a quasi-split classical F -group of F -rank n , defined asin § (1) holds for Y . If π is an irreducible, genericand ( Y ( F ) , ω Y ,E/F ) -distinguished representation of Y ( E ) then the representation T ( π ) of GL m ( E ′ ) (for the appropriate m = m ( n ) ) is (GL m ( F ′ ) , χ ) -distinguished where χ is thetrivial character in the even orthogonal case and equals ω GL m ,E ′ /F ′ otherwise.Proof. Let [ φ ] ∈ Φ( Y ( E )) be the parameter of π so that I ([ φ ]) ∈ Φ(GL m ( E ′ )) is theparameter of T ( π ). It follows from Lemma 13 that T ( π ) is generic and therefore, byTheorem 6, it suffices to show that [ φ ′ ] is E ′ /F ′ -conjugate of sign ǫ where ǫ equals one inthe even orthogonal case and ( − m − otherwise.In the unitary case m = n , Y ( E ) = U n ( E ′ /E ), Y op ( F ) = U n ( F ′′ /F ′ ) where F ′′ = ( E ′ ) στ and I = BC E ′ E : Φ(U n ( E ′ /E )) → Φ(GL n ( E ′ ))is the corresponding basechange map. This implies the commutativity of the diagramΦ(U n ( E ′ /E )) I / / Φ(GL n ( E ′ ))Φ(U n ( F ′′ /F )) BC F ′ F / / BC EF O O Φ(U n ( E ′ /F ′ )) . BC E ′ F ′ O O (Both compositions from Φ(U n ( F ′′ /F )) to Φ(GL n ( E ′ )) are defined by restriction to W ′ E ′ .)By assumption, [ φ ] ∈ Φ(U n ( E ′ /E )) lies in the image of BC EF and therefore I ([ φ ]) lies inthe image of BC E ′ F ′ and by Theorem 5 it is E ′ /F ′ -conjugate of sign ( − n − . Consider now the symplectic or orthogonal case. Again, by assumption [ φ ] lies in theimage of BC EF : Φ( Y op ( F )) → Φ( Y ( E )). Let φ ′ : W ′ F → L Y op ( F ) be such that[ φ ] = I ( BC EF ([ φ ] ′ )) = [ φ ′| W ′ E ] . Note that there exists a non-degenerate bilinear form B : C N × C N → C N such that B ( y, x ) = ǫB ( x, y ), x, y ∈ C N and the image of φ ′ preserves B . Let s ∈ W F \ W E anddefine the bilinear form B ′ : C N × C N → C N by B ′ ( x, y ) = B ( φ ′ ( s ) x, y ) . It is now a straightforward verification that B ′ and φ satisfy the conditions (8) and since I is defined via the inclusion L Y ( E ) ⊆ GL N ( C ) it follows that I ([ φ ]) is E/F -conjugate ofsign ǫ . (cid:3) Remark . In the odd orthogonal case we could alternatively argue as in the unitary caseusing the commutative diagramΦ(SO n +1 ( E )) I / / Φ(GL n ( E ))Φ(SO n +1 ( F )) I F / / BC EF O O Φ(U n ( E/F )) BC EF O O where I F is induced by the inclusion GL n ( C ) σ ⊆ L U n ( E/F ) as GL n ( C ) σ is conjugate toSp n ( C ) in GL n ( C ). Corollary 7. Assume that − ∈ N ( E/F ) . In the notation of Proposition 8 assume thatConjecture 1 (1) holds for Y . If π is a generic irreducible representation of Y ( E ) that is Y ( F ) -distinguished then T ( π ) is GL m ( n ) ( F ′ ) -distinguished.Proof. Taking the formula for ω Y ,E/F into account, the corollary is exactly the statementof Proposition 8 except in the even unitary and odd orthogonal cases. Consider these twocases now. Let ω be the quadratic character of Y ( E ) that extends the character ω Y ,E/F of Y ( F ) given by Lemma 15 (see Remark 10). The representation π ⊗ ω is ( Y ( F ) , ω Y ,E/F )-distinguished and since in both cases m ( n ) is even it follows from Proposition 8 that T ( π ⊗ ω ) is (GL m ( n ) ( F ′ ) , η E ′ /F ′ ◦ det)-distinguished. By Lemma 16 we have T ( π ⊗ ω ) = T ( π ) ⊗ ( η E ′ /F ′ ◦ det) and it therefore follows that T ( π ) is GL m ( n ) ( F ′ )-distinguished asrequired. (cid:3) Transfer of distinction for principal series. Let H ◦ = H ◦ n be a quasi-split classical F -group of F -rank n , defined as in § H ◦ = H ◦ n = H ◦ ( F ) and G ◦ = G ◦ n = H ◦ ( E ).Our goal in this section is to show that for certain irreducible representations π of G ◦ induced from a cuspidal representation of a Levi subgroup of the form M ◦ α with α ∈ A (see § ∈ Y ( F/F, 1) is such that SO( ) is anisotropic and NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 49 splits over a unique quadratic extension K = F [ √− det ] of F . Let α be a character ofSO( , E ).If K = E then SO( , E ) ≃ E ∗ . Let λ α be the character of E ∗ corresponding to α underthis identification. It is a straightforward observation that T ( α ) = λ α × λ − α . Otherwise, let L = KE and SO ( L/E ) = SO( , E ). In particular, SO ( L/E ) ≃ ( L/E ) ≃ L ∗ /E ∗ where the right isomorphism is via Hilbert 90. We denote by λ α the correspondingcharacter of L ∗ trivial on E ∗ . Let φ : W ′ L → C ∗ be the L -parameter of λ α . The twodimensional representation Ind W ′ E W ′ L ( φ ) is therefore a L -homomorphism corresponding to arepresentation π ( α ) of GL ( E ). In fact, the representation π ( α ) has a model as the spaceof α -coinvariants in the Weil representation (see [Mat08, D´efinition 2.1.1]). According toour convention, L SO ( L/E ) = O ( C ) and for the L -homomorphism φ α : W ′ E → O ( C )corresponding to α it is easy to see that composing with the inclusion O ( C ) ⊆ GL ( C )we get that I ( φ α ) ≃ Ind W ′ E W ′ L ( φ ). That is, T ( α ) = π ( α ). Proposition 9. Let α be a character of G ◦ that is ( G ◦ ) z -distinguished for some z ∈ X .Then the representation T ( α ) of GL m (0) ( E ′ ) is GL m (0) ( F ′ ) -distinguished.Proof. The lemma is trivially true if m (0) = 0 (both α and T ( α ) equal the trivial characterof the trivial group). Otherwise, either m (0) = 1 in the odd unitary case or m (0) = 2 inthe quasi-split non split even orthogonal case.In the first case G ◦ = ( E ′ /E ) and ( G ◦ ) z = ( F ′ /F ) is independent of z ∈ X . Thus α is a character of ( E ′ /E ) that is trivial on ( F ′ /F ) and T ( α )( a ) = α ( a − τ ), a ∈ ( E ′ ) ∗ .Since a − τ ∈ ( F ′ /F ) for a ∈ ( F ′ ) ∗ we conclude that T ( α ) is indeed ( G ◦ ) z -distinguished.In the second case H ◦ = SO( ). If − det j ∈ E ∗ then as discussed above T ( α ) = λ α × λ − α and G ◦ ≃ E ∗ . Under this isomorphism the group ( G ◦ ) z identifies with F ∗ if z ∈ X ◦ andwith ( E/F ) otherwise. In the first case this means that λ α (and hence also λ − α ) is trivialon F ∗ and in the second case that λ σα = λ α . Consequently, T ( α ) is GL ( F )-distinguishedby a closed orbit argument in the first case and by an open orbit argument in the second(see § − det j E ∗ set K = F [ √− det ] and L = KE and let K ′ be the thirdquadratic extension of F contained in L that is different from E and K . As observed above T ( α ) = π ( α ) and G ◦ ≃ ( L/E ) . In fact, the full orthogonal group G is isomorphic to thesemi-direct product ( L/E ) ⋊ Z where the non-trivial element of Z acts on ( L/E ) by theGalois action of L/E . Under this isomorphism ( G ◦ ) z identifies with ( K/F ) if z ∈ X ◦ andwith ( K ′ /F ) otherwise. In other words, with the above notation, λ α is trivial on either K ∗ or ( K ′ ) ∗ . The fact that for such α the representation π ( α ) is GL ( F )-distinguished isproved in [Mat08, Th´eor`eme 2.3.3]. (cid:3) Let n = n + · · · + n k and M ◦ = M ◦ ( n ,...,n k ) be the corresponding standard Levi subgroupof G ◦ . For x = x w ( { y i } i ∈ I ( w ) , z ) ∈ G ◦ · I N defined by (7) let η ∈ G ◦ be such that η · I N = x .Note that M ◦ x = M ◦ ∩ ηH ◦ η − and therefore the character ω H ◦ ,E/F ◦ Ad( η − ) of ηH ◦ η − restricts to a character of M ◦ x . Lemma 14. Consider either the even unitary or odd orthogonal case. With the abovenotation let g i ∈ GL n i ( E ′ ) , i = 1 , . . . , k be such that m = ι ( g , . . . , g k ) ∈ M x . We have ω H ◦ ,E/F ( η − mη ) = Y i c , ρ ( i )= i η E ′ /F ′ (det g i ) . Proof. Note that in both cases G ◦ is the trivial group. In the even unitary case we have ω U n,F ′ /F ,E/F = η E ′ /F ′ ◦ α − ◦ detwhere α : ( F ′ ) ∗ /F ∗ → ( F ′ /F ) is the isomorphism aF ∗ a − τ . It follows from Proposition6 (3) that det( η − mη ) = det m = ( aa σ ) − τ b − τ c − τ where a = Q i<ρ ( i ) det g i ∈ ( E ′ ) ∗ , b = Q i ∈ c , ρ ( i )= i det g i ∈ ( E ′ / ( E ′ ) στ ) and c = Q i c , ρ ( i )= i det g i ∈ ( F ′ ) ∗ . We have η E ′ /F ′ ( α − (( aa σ ) − τ )) = η E ′ /F ′ ( N E ′ /F ′ ( a )) = 1and η E ′ /F ′ ( α − ( c − τ )) = η E ′ /F ′ ( c ) . It remains to see that η E ′ /F ′ ( α − ( b − τ )) = 1. Let d ∈ ( E ′ ) ∗ be such that b = d − στ (byHilbert 90 ). Then b − τ = ( dd σ ) − τ and therefore η E ′ /F ′ ( α − ( b − τ )) = η E ′ /F ′ ( N E ′ /F ′ ( d )) = 1 . The even unitary case follows. In the odd orthogonal case ω SO n +1 ,E/F = η E/F ◦ sn . It follows from Proposition 6 (3) that m = ι ( g , . . . , g k ) ( g i ) i ≤ ρ ( i ) defines an isomorphism M x ≃ [ × i<ρ ( i ) GL n i ( E )] × [ × i c , ρ ( i )= i GL n i ( F )] × [ × i ∈ c , ρ ( i )= i U( y i , E/F )] . Let m i = ι ( h , . . . , h k ) where h j = I n j unless j ∈ { i, ρ ( i ) } in which case h j = g j . Thenclearly m = Q i ≤ ρ ( i ) m i with all m i ’s commuting. Set also m ◦ = Q i ∈ c , ρ ( i )= i m i . We computeseparately η E/F ◦ sn( η − m i η ) for each i such that either i c or ρ ( i ) = i as well as η E/F ◦ sn( η − m ◦ η ).The computation for either i c or ρ ( i ) = i is implicitly based on finding η explicitly.More precisely, let x i = I n i i c , ρ ( i ) = iι (cid:0) I ni I ni (cid:1) i c , ρ ( i ) = i (cid:0) I ni I ni (cid:1) i ∈ c , ρ ( i ) = i and ℓ i = ι ( g i ) i c , ρ ( i ) = iι ( g i , g σi ) i c , ρ ( i ) = iι ( g i , ¯ g ∗ i ) i ∈ c , ρ ( i ) = i. Then η E/F ◦ sn( η − m i η ) = η E/F ◦ sn( η − i ℓ i η i ) where η i ∈ SO k i ( E ) is such that η i · I k i = x i and k i equals n i if ρ ( i ) = i and 2 n i otherwise. Below we provide an explicit such η i whenevereither i c or ρ ( i ) = i . This shows that x ∈ SO n +1 ( E ) · x ′ where x ′ = x I ( w ) ( { y i } i ∈ I ( w ) , z ) NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 51 (here we think of the set of indices I ( w ) as an element of W k ), that is, x ′ ∈ SO n +1 ( E ) · I n +1 .Similarly, set x ◦ = I t ( − t I t ι ( w t ¯ y ) ∈ X ◦ t +1 where i < · · · < i j are such that I ( w ) = { i , . . . , i j } , t = n i + · · · , + n i j and y =diag( y i , . . . , y i j ). Applying Lemma 10 to both x ′ and x ◦ we conclude that x ◦ = η ◦ · I t +1 for some η ◦ ∈ SO t +1 ( E ) and we then have η E/F ◦ sn( η − m ◦ η ) = η E/F ◦ sn(( η ◦ ) − ι ( g i , . . . , g i j ) η ◦ ) . We also note that ι ( g i , . . . , g i j ) ∈ U ( y, E/F ).Furthermore, our explicit computations of the spinor norm are based on the fact thatany h ∈ SO m ( F ) can be written as an even product h = s v . . . s v k of orthogonal reflectionswith respect to anisotropic vectors v i ∈ F m for the quadratic form q ( v ) = t vw m v , v ∈ F m and that in such a situationsn( h ) = q ( v ) . . . q ( v k ) F ∗ ∈ F ∗ /F ∗ . Based on this we have(9) sn( ι ( h )) = det h F ∗ , h ∈ GL m ( F ) . Indeed, any homomorphism from GL m ( F ) to F ∗ /F ∗ factors through determinant andtherefore sn( ι ( h )) = χ (det h ) for some homomorphism χ : F ∗ → F ∗ /F ∗ . In order to showthat χ is the natural projection it suffices to show that sn( ι (diag( I m − , t ))) = tF ∗ , t ∈ F ∗ and this reduces to an SO ( F ) computation of sn(diag( t, t − )). Indeed,(10) diag( t, t − ) = s v ( t ) s v (1) where t v ( t ) = ( t, 1) so that q ( v ( t )) = 2 t .If i c and ρ ( i ) = i then we may choose η i = I n i and η E/F (sn( η − m i η )) = η E/F (sn( ι ( g i ))) = η E/F (det g i )by (9).If i c and i < ρ ( i ) then η E/F (sn( η − m i η )) = η E/F (sn( ι ( ζ − n i diag( g i , g σi ) ζ n i ))where ζ m = (cid:0) I m ıI m I m − ıI m (cid:1) and the right hand side is computed in SO n i ( F ). (Recall from theproof of Lemma 10 that ζ m · I m = (cid:0) I m I m (cid:1) .) Note that ζ − n i diag( g i , g σi ) ζ n i ∈ GL n i ( F ) andby (9) we see that η E/F (sn( η − m i η )) = η E/F ( N E/F (det g i )) = 1 . For the next step we introduce the matrix ξ m = I m ıI m − ı I m I m I m − ıI m ı I m I m ∈ SO m ( E ) that satisfies ξ m · I m = (cid:18) I m I m (cid:19) . If i ∈ c and i < ρ ( i ) then η E/F (sn( η − m i η )) = η E/F (sn( ξ − n i ι ( g i , ¯ g ∗ i ) ξ n i )) . By the argument used in the previous steps, in order to show that this equals to onefor every g i ∈ GL n i ( E ) it suffices to consider g i = diag( I n i − , t ) for all t ∈ E ∗ . Thisreduces us to the computation in SO ( F ) of η E/F (sn( ξ − ι ( t, t − σ ) ξ )). Note that for v = t (0 , , , 0) the orthogonal reflection s v = diag(1 , − w , 1) and clearly sn( ξ − ι ( t, t − σ ) ξ ) =sn( s v ξ − ι ( t, t − σ ) ξ s v ). Furhtermore, s v ξ − ι ( t, t − σ ) ξ s v = ι ( a ( t )) where a ( t ) = (cid:18) ( t + t σ ) ı ( t σ − t ) ı ( t σ − t ) ( t + t σ ) (cid:19) . Since det a ( t ) = N E/F ( t ) it now follows from (9) that η E/F (sn( ι ( a ( t )))) = 1 as required.To conclude the lemma it remains to show that for y ∈ Y m ( E/F, x ( y ) = I m ( − m I m ι ( w m ¯ y ) ∈ SO m +1 ( E ) · I m +1 and g ∈ U( y, E/F ) we have(11) η E/F (sn( η ( y ) − ι ( g ) η ( y )) = 1where η ( y ) ∈ SO m +1 ( E ) is such that η ( y ) · I m +1 = x ( y ). Indeed, we only need to show thisfor y of the form diag( y i , . . . , y i j ) and g of the form diag( g i , . . . , g i j ) as above. However,it is more convenient to show this more general statement.Assume first that − ∈ N ( E/F ). Then there is a quadratic character ω of E ∗ thatextends η E/F (see Remark 10). Now denoting by sn E the spinor norm on SO m +1 ( E ) thecharacter ω ◦ sn E of SO m +1 ( E ) is well defined and extends η E/F ◦ sn. Note further that ω is trivial on ( E/F ) . Indeed, ω is trivial on both N ( E/F ) and E ∗ and by Hilbert 90the group ( E/F ) is contained in the product N ( E/F ) E ∗ ( z σ − = N E/F ( z ) z − , z ∈ E ∗ ).Therefore, η E/F (sn( η ( y ) − ι ( g ) η ( y )) = ω (sn E ( η ( y ) − ι ( g ) η ( y )) = ω (sn E ( ι ( g ))) . By (9) this equals ω (det g ) that equals 1 since det g ∈ ( E/F ) .Assume now that − N ( E/F ). Note that if η ( y ) · I m +1 = x ( y ) and h ∈ GL m ( E )then ( ι ( h ) η ( y )) · I m +1 = x ( t h − σ yh − ) = x ( y ⋆ h − ) and U ( y ⋆ h − , E/F ) = hU ( y, E/F ) h − .With our assumption it therefore suffices to show (11) for y = diag( a, I m − ) with a = ± § a and let x = x ( y ) and η = η ( y ). Let e , . . . , e m +1 be the standardbasis of F m +1 and v i = η − e i , i = 1 , . . . , m + 1. Note that x = I m +1 and xe = e m +1 except when a = − m = 1 where xe = − e m +1 . Consequently, v σ = η − σ e = η − xe = ( − v m +1 a = − m = 1 v m +1 otherwise. NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 53 It follows from Lemma 10 (or by direct computation) that for m = 2 and a = 1 wehave x ( y ) = w SO ( F ) · I . We may therefore exclude this case. In all other cases,we claim that [U( y, E/F ) , U( y, E/F )] = SU( y, E/F ). Indeed, for m = 1 this is clear andfor all other cases (excluding m = 2 and a = 1) y admits an isotropic vector. It thereforefollows from [Die48, Th´eor`eme 5] that every proper normal subgroup of SU( y, E/F ) iscentral. It is easy to see that [U( y, E/F ) , U( y, E/F )] is not central. Consequently, there isa homomorphism χ : ( E/F ) → F ∗ /F ∗ such that sn ◦ Ad( η − ) ◦ ι = χ ◦ det. It thereforeforesuffices to show (11) for g = diag( u, I m − ) ∈ U ( y, E/F ) with u ∈ ( E/F ) . Note that forsuch g as in (10) we have ι ( g ) = s ue + e m +1 s e + e m +1 . Applying Hilbert 90, let z ∈ E ∗ be such that z σ − = u . Then ue + e m +1 and σ ( z ) e + ze m +1 are co-linear in E m +1 . Furthermore we have( v + v m +1 ) σ = ( − ( v + v m +1 ) a = − m = 1 v + v m +1 otherwise.and similarly( σ ( z ) v + zv m +1 ) σ = ( − ( σ ( z ) v + zv m +1 ) a = − m = 1 σ ( z ) v + zv m +1 otherwise.It follows that both w = α ( v + v m +1 ) and w ′ = α ( σ ( z ) v + zv m +1 ) are in F m +1 where α = ( ı a = − m = 11 otherwise.We deduce that η − ι ( g ) η = η − s σ ( z ) e + ze m +1 s e + e m +1 η = s σ ( z ) v + zv m +1 s v + v m +1 = s w ′ s w and since q = q ◦ Ad( η − ) we havesn( η − ι ( g ) η ) = q ( w ) q ( w ′ ) F ∗ = q ( α ( σ ( z ) e + ze m +1 )) q ( α ( e + e m +1 )) F ∗ = 4 α N E/F ( z ) F ∗ . Since 4 α ∈ F ∗ , we obtain η E/F (sn( η − ι ( g ) η )) = η E/F ( N E/F ( z )) = 1as required. (cid:3) For a certain class of generic irreducible representations π of G ◦ we are now in a positionto prove unconditionally the relation between distinction of π and of its transfer T ( π ) thatProposition 8 shows follows more generally from Conjecture 1. In particular, π could beany irreducible principal series. Theorem 7. In the notation of § π i is cuspidal for i = 0 , . . . , k (in partic-ular, π is the trivial character of the trivial group except in the odd unitary or quasi-splitbut non-split even orthogonal cases where π is a character of G ◦ ) and π is irreducible. (Inparticular, π could be any irreducible generic principal series representation of G ◦ .) (1) If π is H ◦ -distinguished then its transfer T ( π ) is GL m ( n ) ( F ′ ) -distinguished.(2) If π is ( H ◦ , ω H ◦ ,E/F ) -distinguished then its transfer T ( π ) is (GL m ( n ) ( F ′ ) , χ ) -distinguishedwhere χ is the trivial character in the even orthogonal case and equals ω GL m ( n ) ,E ′ /F ′ otherwise.Proof. Since π is irreducible, applying an appropriate Weyl element we assume withoutloss of generality that e ( π ) ≥ · · · ≥ e ( π k ) ≥ 0. It follows from Lemma 13 that T ( π ) = π × · · · π k × T ( π ) × ( π ∨ k ) τ × · · · ( π ∨ ) τ . If π is H ◦ -distinguished then it follows from Theorem 2 and (in the odd unitary or quasi-split but non split even orthogonal case) Proposition 9 that T ( π ) satisfies the conditionsfor Lemma 11 and we deduce that T ( π ) is GL m ( n ) ( F ′ )-distinguished. Note that we onlyapplied the necessary condition for distinction of Theorem 2.The second part of the theorem is only different from the first in the odd orthogonal andin the even unitary cases. Assume we are in one of those cases and note that G ◦ is nowthe trivial group. Suppose that π is ( H ◦ , ω H ◦ ,E/F )-distinguished. It follows from [Off17,Corollary 5.2 and Corollary 6.9], the orbits and stabilizers analysis done in Section 6.7,Lemma 14 and the fact that υ ∗ = ( υ τ ) ∨ for an irreducible representation υ of GL m ( E ′ ) (see[GK75]), that there exist an involution ρ ∈ S k and a subset c of [1 , k ] such that ρ ( c ) = c and n ρ ( i ) = n i , i ∈ [1 , k ] and y i ∈ Y n i ( E ′ / ( E ′ ) στ , 1) for all i ∈ c such that ρ ( i ) = i such that π ρ ( i ) ≃ ¯ π ∨ i i c , ρ ( i ) = iπ i is (GL n i ( F ′ ) , η E ′ /F ′ ◦ det)-distinguished i c , ρ ( i ) = iπ ρ ( i ) ≃ π τσi i ∈ c , ρ ( i ) = iπ i is U( y i , E ′ / ( E ′ ) στ ) − distinguished i ∈ c , ρ ( i ) = i. It now follows from Lemma 11 that T ( π ) is (GL m ( n ) ( F ′ ) , η E ′ /F ′ ◦ det)-distinguished. Sincein both cases m ( n ) is even the theorem follows. (cid:3) It is now tempting to suggest that a global analogue of this problem also holds. Let K/k be a quadratic extension of number fields and H ◦ be a k -quasisplit classical group amongstthe ones considered in this paper. Recall from [CPSS11, Theorem 1.1] that for everyglobally generic cuspidal automorphic representation π of H ◦ ( A K ) the functorial lift T ( π )exists and is an automorphic representations of GL m ( n ) ( A K ). Furthermore, the definitionof Prasad’s character ω H ◦ ,K/k makes sense globally (see A.2 in Appendix A). Conjecture 2. Let π be an irreducible globally generic cuspidal automorphic representationof H ◦ ( A K ) such that T ( π ) is a cuspidal automorphic representation of GL m ( n ) ( A K ) .(1) If π is H ◦ ( A k ) -distinguished, then T ( π ) is GL m ( n ) ( A k ) -distinguished.(2) If π is ( H ◦ ( A k ) , ω H ◦ ,K/k ) -distinguished, then T ( π ) is (GL m ( n ) ( A k ) , χ GL m ( n ) ,K/k ) -distinguishedwhere χ GL m ( n ) ,K/k is the trivial character in the even orthogonal case and equals ω GL m ( n ) ,K/k otherwise. NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 55 Appendix A. Explicit computations of Prasad’s quadratic character andopposition group A.1. Prasad’s quadratic character. Let Y be a connected reductive group defined andquasi-split over F . We recall the construction of a quadratic character ω Y ,E/F of Y = Y ( F )in [Pra, Proposition 6.4] associated to Y and the quadratic extension E/F .The character ω Y ,E/F factors through the natural projection Y Y ad = Y ad ( F ) to theadjoint group and it suffices to define it for adjoint groups.Denote by Y sc the simply connected cover of Y ad , by Z its center and by p : Y sc → Y ad the natural projection. Then we have an exact sequence of algebraic F -groups(12) 1 → Z → Y sc → Y ad → → Z ( F ) → Y sc ( F ) p −→ Y ad ( F ) δ p,F −→ H ( F, Z ) . (The Galois cohomology group H ( F, Z ) and the connecting homomorphism δ p,F are de-fined for example in [Ser94].)Let B ad = T ad ⋉ U be a Borel subgroup of Y ad with a maximal torus T ad and unipotentradical U definded over F . Then p − ( B ad ) is a Borel subgroup of Y sc and Z ⊆ T sc = p − ( T ad ). Thus p identifies U with its pre-image in Y sc that we still denote by U . It isknown that the half sum (with multiplicities) ρ of the positive roots of T sc (with respectto U ) is a weight (i.e. an algebraic character) of T sc defined over F and that 2 ρ is a weightof T ad , that is, contains Z in its kernel. By restriction ρ induces a character ρ | Z → {± } .By the functorial properties of Galois cohomology ρ gives rise to a homomorphism[ ρ ] F : H ( F, Z ) → H ( F, {± } ) ≃ F ∗ /F ∗ . Let η E/F be the quadratic character of F ∗ with kernel ( E/F ) , the norm one subgroup.Then ω Y ad ,E/F = η E/F ◦ [ ρ ] F ◦ δ p,F : Y ad → {± } is the character defined by Prasad for Y ad . Composing with the natural map from Y to Y ad we obtain Prasad’s character ω Y ,E/F : Y → {± } . Remark . For semi-simple Y the character ω Y ,E/F can be defined more directly since theprojection p : Y sc → Y ad factors through Y . Let p ′ : Y sc → Y be the correspondingprojection (so that p is the composition of p ′ with the narural projection Y → Y ad ). Thenthe short exact sequence 1 → Z ′ → Y sc p ′ −→ Y → δ p ′ ,F : Y ( F ) → H ( F, Z ′ ). The kernel Z ′ of p ′ isa subgroup of Z so that ρ | Z ′ : Z ′ → {± } defines the map [ ρ ] F : H ( F, Z ′ ) → H ( F, {± } )and we have ω Y ,E/F = η E/F ◦ [ ρ ] F ◦ δ p ′ ,F . Taking both F -points and E -points in Equation (12) and composing with [ ρ ] E and [ ρ ] F respectively we obtain the commutative diagram1 / / Z ( E ) / / Y sc ( E ) / / Y ad ( E ) δ p,E / / H ( E, Z ) [ ρ ] E / / H ( E, {± } ) ∼ / / E ∗ /E ∗ / / Z ( F ) / / i F,E O O Y sc ( F ) / / i F,E O O Y ad ( F ) δ p,F / / i F,E O O H ( F, Z ) [ ρ ] F / / [ i F,E ] O O H ( F, {± } ) [ i F,E ] O O ∼ / / F ∗ /F ∗ i O O where the vertical arrows i F,E are the natural embeddings of F -points into E -points of analgebraic group defined over F and [ i F,E ] are the maps induced by restriction of cocyclesfrom the absolute Galois group of F to that of E . We arrive at the following statement. Lemma 15. We have the following equality of characters of Y ad ( F )[ ρ ] E ◦ δ p,E ◦ i F,E = [ i F,E ] ◦ [ ρ ] F ◦ δ p,F . In particular, if η E/F extends to a quadratic character η of E ∗ then ω ad E,η := η ◦ [ ρ ] E ◦ δ p,E is a character of Y ad ( E ) that extends ω Y ad ,E/F and hence, composition with the naturalprojection Y ( E ) → Y ad ( E ) gives rise to a character ω E,η of Y ( E ) that extends ω Y ,E/F .Proof. The lemma is immediate from the observation that the map [ i F,E ] : H ( F, {± } ) → H ( E, {± } ) identifies with the natural map i : F ∗ /F ∗ → E ∗ /E ∗ induced by the inclusion F ∗ ⊆ E ∗ . (cid:3) Remark . There exists a quadratic character η of E ∗ that extends η E/F if and only if − ∈ N ( E/F ). Indeed, such η exists if and only if η E/F considered as a character of F ∗ /F ∗ is trivial on the kernel of the natural map aF ∗ aE ∗ : F ∗ /F ∗ → E ∗ /E ∗ andif a ∈ E \ F is such that a ∈ F then this kernel is F ∗ ⊔ a F ∗ while − a ∈ N ( E/F ).When this is the case, we can choose a quadratic extension L/E such that η L/E extends η E/F . Note that in this case we have ω E,η L/E = ω Y E ,L/E where Y E is the base change of Y to E and the left hand side is defined in the lemma.We compute ω Y ,E/F when Y is either GL n or the quasi-split classical group H ◦ as in § • Assume that Y = GL n so that Y ad = PGL n and Y sc = SL n . Recall thatEnd( G m ) = Z where we identify n ∈ Z with the n th power endomorphism ofthe multiplicative group G m . Let µ n be the algebraic subgroup of G m defined bythe exact sequence 1 −→ µ n −→ G m n −→ G m −→ . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 57 Consider the following two commutative diagrams1 / / µ n ( n ) (cid:15) (cid:15) / / G m ( n ) (cid:15) (cid:15) n / / G mn − (cid:15) (cid:15) / / / / µ / / G m / / G m / / / / µ n / / SL n / / PGL n / / / / µ n − (cid:15) (cid:15) / / G m × SL np O O p (cid:15) (cid:15) / / GL ns O O det (cid:15) (cid:15) / / / / µ n / / G m n / / G m / / s : GL n → PGL n is the natural projection and p i the projection to the i thcoordinate, i = 1 , 2. Note that Z = µ n is the center of SL n and ρ | Z = (cid:0) n (cid:1) . Bypassing to F -points and applying the long exact sequence of [Ser94, Proposition 43]we, in particular, get from the first diagram the commutative diagram F ∗ n − (cid:15) (cid:15) δ n / / H ( F, µ n ) [ ( n ) ] (cid:15) (cid:15) F ∗ δ / / H ( F, µ ) ∼ / / F ∗ / ( F ∗ ) and from the second diagram the commutative diagramPGL n ( F ) δ / / H ( F, µ n )GL n ( F ) s O O det (cid:15) (cid:15) δ ′ / / H ( F, µ n ) [ − (cid:15) (cid:15) F ∗ δ n / / H ( F, µ n )where δ n , δ , δ and δ ′ are the corresponding connecting homomorphisms. Note that ω GL n ,E/F = η E/F ◦ [ (cid:18) n (cid:19) ] F ◦ δ ◦ s. Putting the two last diagrams together we obtain[ − (cid:18) n (cid:19) ] F ◦ δ ◦ s = δ ◦ ( n − ◦ det . Since δ induces the natural projection F ∗ → F ∗ /F ∗ and [ − F the identity mapon F ∗ /F ∗ we deduce that ω GL n ,E/F = η n − E/F ◦ det . • In the symplectic case we have that ω Sp n ,E/F is trivial. Indeed, if Y = Sp n then Y = Y ad = Y sc and Z = { } . In fact, Sp n ( F ) has no non-trivial cahracters. • Consider the quasi-split orthogonal case. Thus, Y = SO( [ n ]) where in the oddcase we may take [ n ] = w n +1 and in the even case, either [ n ] = w n (the splitcase) or is anisotropic of size two (the quasi-split non-split case). If n = 0 then Y is anisotropic, hence ρ is trivial and therefore so is ω Y ,E/F . Assume now that n ≥ 1. Then Y is semi-simple and Y sc = Spin( [ n ]) is the associated spin group.The short exact sequence1 → µ → Spin( [ n ]) → SO( [ n ]) → δ SO : SO( [ n ] , F ) → H ( F, µ ) ≃ F ∗ / ( F ∗ ) . By Remark 9 we have ω SO( [ n ]) ,E/F = η E/F ◦ [ ρ ] F ◦ δ SO . One verifies that in the evencase ρ is a weight of the diagonal maximal split torus while in the odd case it is not.In particular, in the even case [ ρ ] F is the trivial map so that ω SO( [ n ]) ,E/F is trivialand in the odd case [ ρ ] F is the identity map so that ω SO( [ n ]) ,E/F = η E/F ◦ δ SO .The connecting homomorphism δ SO is known to be the spinor norm sn. Forconvenience we provide the argument that is analogous to the GL n computation.Denote by e sn : GSpin( [ n ]) → G m the spinor norm on GSpin( [ n ]), so that if p : GSpin( [ n ]) → SO( [ n ])is the natural projection with kernel G m , the composed mapsGSpin( [ n ] , F ) p → SO( [ n ]) sn → F ∗ /F ∗ and GSpin( [ n ] , F ) e sn → F ∗ → F ∗ /F ∗ are equal. Consider the commutative diagram1 / / µ / / Spin( [ n ]) / / SO( [ n ]) / / / / µ / / G m × Spin( [ n ]) p O O p (cid:15) (cid:15) / / GSpin( [ n ]) p O O e sn (cid:15) (cid:15) / / / / µ / / G m / / G m / / . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 59 Taking F -points we get the following commutative diagramSO( [ n ] , F ) δ SO / / H ( F, µ ) ≃ F ∗ / ( F ∗ ) GSpin( [ n ] , F ) p O O e sn (cid:15) (cid:15) δ GSpin / / H ( F, µ ) ≃ F ∗ / ( F ∗ ) F ∗ δ / / H ( F, µ ) ≃ F ∗ /F ∗ . Since p is surjective on F -points (see e.g. [Sch85, Chapter 9, Theorem 3.3]) and δ induces the natural projection from F ∗ to F ∗ /F ∗ we deduce that δ SO = sn. Alltogether we conclude that for all n ≥ ω SO( [ n ]) ,E/F = η n E/F ◦ snwhere n ∈ { , , } is the size of . • Let Y = U n,K/F = U K/F ( w n ) be the F -quasi-split unitary group with respect toa quadratic extension K/F and τ the Galois involution associated to K/F . Thus, Y sc = SU n,K/F and Y ad = PU n,K/F . Let m : U n,K/F → PU n,K/F be the naturalprojection. Its restriction to SU n,K/F gives rise to the short exact sequence1 → µ ′ n → SU n,K/F → PU n,K/F → µ ′ n of SU n,K/F is its intersection with the center Res K/F ( µ n )of Res K/F (SL n ). By taking F -points we get the connecting homomorphism δ :PU n,K/F ( F ) → H ( F, µ ′ n ) and ω U n,K/F ,E/F = η E/F ◦ [ ρ ] F ◦ δ ◦ m. Note that if n is odd then the only homomorphism µ ′ n → µ is the trivial one andit follows that [ ρ ] F and therefore also ω U n,K/F ,E/F is trivial.We assume from now on that n is even. Then one verifies that ρ ( z ) = z − n , z ∈ µ ′ n ( F ) . We now interpret the connecting homomorphism δ . Denote bywsn : U n,K/F ( F ) = U n ( K/F ) → K ∗ /F ∗ the composition of the determinantdet : U n,K/F ( F ) → U ,K/F ( F )with the inverse of the isomorphism zF ∗ zz − τ from K ∗ /F ∗ to U ,K/F ( F ). (In[Wal59], Wall interpreted this map as a spinor norm analogue.) We show that(13) [ ρ ] F ◦ δ ◦ m = N K/F ◦ wsn . The character N K/F ◦ wsn naturally appears in another manner, which we nowdescribe. Let δ K ∈ K \ F be such that δ K ∈ F . We denote by O q n the orthogonal group defined over F with respect to the quadratic form q on F n associated to thebilinear form b : ( (cid:18) X X (cid:19) , (cid:18) Y Y (cid:19) ) Tr K/F (( t X − δ K t X ) w n ( Y + δ K Y )) , X i , Y i ∈ F n , i = 1 , . Since we assume that n is even, one checks that the quadratic space ( F n , q ) is asum of hyperbolic planes hence O q n is F -split (for n odd it is quasi-split and splitsover K ). There is now an obvious natural imbedding i : U n,K/F → SO q n , defined over F , and by [Sch85, Chapter 10, Theorem 1.5] we have N K/F ◦ wsn = sn ◦ i. We will make use of this equality to prove equation (13). Consider the commutativediagram 1 / / µ ′ n i / / U ,K/F × SU n,K/F s / / p (cid:15) (cid:15) U n,K/Fm (cid:15) (cid:15) / / / / µ ′ n / / SU n,K/F / / PU n,K/F / / i ( µ ) = ( µ − , µI n ) and s ( z, u ) = zu , that implies the following oneU n,K/F ( F ) m (cid:15) (cid:15) δ s ,F / / H ( F, µ ′ n )PU n,K/F ( F ) δ / / H ( F, µ ′ n )so that δ ◦ m = δ s ,F . Let U n,K/F = { ( z, u ) ∈ U ,K/F × U n,K/F , det( u ) = z } and s : U n,K/F → U n,K/F the projection to the second coordinate s ( z, u ) = u. The map i : µ ( µ, I n ) identifies µ ′ = µ with the kernel of s . Let p n, :U ,K/F × SU n,K/F → U n,K/F be given by p n, ( z, u ) = ( z n/ , zu ) = ( ρ ( z ) − , zu ) . NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 61 This gives the commutative diagram1 / / µ ′ n i / / ρ (cid:15) (cid:15) U ,K/F × SU n,K/F s / / p n, (cid:15) (cid:15) U n,K/F / / / / µ ′ i / / U n,K/F s / / U n,K/F / / . By taking F -points we obtain the commutative diagramU n,K/F ( F ) δ s ,F / / H ( F, µ ′ n ) [ ρ ] F (cid:15) (cid:15) U n,K/F ( F ) δ s ,F / / H ( F, µ ′ )so that δ s ,F = [ ρ ] F ◦ δ s ,F . To prove (13) it remains to show that(14) δ s ,F = sn ◦ i. To this end we recall the exact sequence1 → µ u → Spin q n v → SO q n → j : U n,K/F → Spin q n defined over F such that the following diagram is commutative1 / / µ ′ i / / U n,F ′ /F j (cid:15) (cid:15) s / / U n,F ′ /Fi (cid:15) (cid:15) / / / / µ u / / Spin n v / / SO n / / . Taking F -points and recalling from the orthogonal case that the spinor norm sn is theconnecting homomorphism associated with the bottom row we arrive at (14) and completeour proof.To show that such j exists we first make the following simple observation. There existsat most one map j : U n,K/F → Spin q n such that v ◦ j = i ◦ s . Indeed, if j ′ is another suchmap then u j ′ ( u ) j ( u ) − : U n,K/F → µ must be trivial since U n,K/F is connected. It istherefore enough to find such a j defined over K . Indeed, if the above diagram commutesfor j it also does for j τ . Over K we may identifyU n,K/F ≃ GL n , and U n,K/F ≃ { ( z, g ) ∈ G m × GL n , det( g ) = z } =: GL n and with these identifications s ( z, g ) = g and i : GL n → SO q n is an imbedding definedover K . It now suffices to show that there exists j : GL n → Spin q n defined over K suchthat v ◦ j = s ◦ i and j ( − , I n ) = u ( − j suchthat v ◦ j = s ◦ i then j ( − , I n ) = u ( − 1) is automatic. Indeed, otherwise j defines amap ¯ j : GL n → Spin q n such that v ◦ ¯ j = i . Note that GL n , Spin q n and SO q n are allof rank n . Since i is injective it maps any maximal torus of GL n to a maximal torus T of SO q n . Furthermore, ¯ j must also be injective and therefore ¯ j ( T ) is a maximal torus ofSpin q n . Clearly ¯ j ( T ) is a subgroup of v − ( T ). It is a well known fact on simply connectedcovers that v − ( T ) is a maximal torus of Spin q n and therefore v − ( T ) = ¯ j ( T ). Since u ( µ ) ⊆ v − ( T ) is in the kernel of v this contradicts the injectivity of i .Note that we may further identify GL n as the semidirect product G m ⋉ SL n where theaction of G m on SL n is given by zgz − = (cid:18) z I n − (cid:19) g (cid:18) z − I n − (cid:19) , z ∈ G m , g ∈ SL n . Indeed ( z, g ) ( z, diag( z , I n − ) g ) : G m ⋉ SL n → GL n is an isomorphism. It now sufficesto show that there is a map j : G m ⋉ SL n → Spin q n defined over K such that j ( z, g ) = i (diag( z , I n − ) g ) . Let j = i | SL n : SL n → SO q n and j : G m → SO q n be given by j ( z ) = i (diag( z , I n − )).We claim that it suffices to lift j and j to Spin q n . Indeed, if ¯ j : SL n → Spin q n and¯ j : G m → Spin q n are K -morphisms such that v ◦ ¯ j t = j t , t = 1 , ( z, g ) = ¯ j ( z )¯ j ( g )is automatically a K -morphism. Indeed, we need to verify that¯ j ( z )¯ j ( g )¯ j ( z ) − = ¯ j (cid:18) z I n − (cid:19) g (cid:18) z − I n − (cid:19) . Fix z and let f, f ′ : SL n → Spin q n be defined by the left and right hand sides of the aboveequation respectively. Then v ◦ f = v ◦ f ′ and therefore g f ′ ( g ) f ( g ) − : SL n → µ mustbe trivial as SL n is connected. Thus, f ′ = f as required.To complete the proof we observe that j and j both lift to Spin q n . The map j hasa lifting since SL n is simply connected. Indeed, let Y be the image of j in SO q n and Y ′ = v − ( Y ) ◦ (the connected component of the pre-image of Y in Spin q n ). Then j isan isomorphism of SL n with Y and v : Y ′ → Y is an isogeny (with kernel contained in µ ). Since SL n and therefore also Y is simply connected, v | Y ′ is in fact an isomorphismand therefore ¯ j = ( v | Y ′ ) − ◦ j : SL n → Spin q n is such that j = v ◦ ¯ j . Let Z ≃ G m be the image of j in SO q n and Z ′ = v − ( Z ) ◦ ≃ G m be the connected component of itspre-image in Spin q n . If Z ′ contains the kernel u ( µ ) of v then v | Z ′ = w is the squareof an isomorphism w : Z ′ → Z (any endomorphism of G m with kernel µ is either thesquare map 2 or its composition with the inverse map). We can then define the desiredlifting ¯ j ( z ) = w − ( i (diag( z, I n − ))) such that v ◦ ¯ j = j . Otherwise, v | Z ′ : Z ′ → Z is anisomorphism and we can define ¯ j = ( v | Z ′ ) − ◦ j . This completes our computation. (In NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 63 fact, the second case cannot occur, it would result in a map ¯ j : GL n → Spin q n such that v ◦ ¯ j = i and as observed before this contradicts the fact that i is injective.)All together we proved the formula ω U n,K/F ,E/F = η n − E/F ◦ N K/F ◦ wsn . Remark . In particular, ω U n,E/F ,E/F is always the trivial character since η E/F ◦ N E/F isthe trivial character of E ∗ . For K = E we have ω U n,K/F ,E/F = η n − EK/K ◦ wsn . Indeed, in order to see that η EK/K = η E/F ◦ N K/F we observe that on the one hand theright hand side is clearly trivial on N ( EK/K ) and on the other hand it is not the trivialcharacter since by local class field theory N ( K/F ) and N ( E/F ) are two different subgroupsof F ∗ of index two. Note further that η EK/K ◦ wsn is well defined since F ∗ ⊆ N ( EK/K )by Lemma 5.We describe another interpretation of ω Y ,E/F provided in [Pra] (see also [BP18, § Y is quasi-split then according to [LL15] the Langlands morphism α Y : H ( W F , Z ( b Y ( C ))) → Hom( Y, C × )is bijective. The class χ Y ,E/F := α − Y ( ω Y ,E/F ) : W F → Z ( b Y ( C ))is represented by the unique cocycle χ Y ,E/F that maps W E to the identity and the non-trivial class in W F /W E to the image of − I by any principal SL -morphism from SL ( C )to b Y ( C ). In fact, χ Y ,E/F is a quadratic character of W F and in the sequel we view χ Y ,E/F as a character of W ′ F trivial on SL ( C ) so that it makes sense to twist an L -homomorphismof Y by χ Y ,E/F . The following lemma follows easily. Lemma 16. Suppose that − ∈ N ( E/F ) and let L/E be as in Remark 10, so that L ′ = LE ′ is a quadratic extension of E ′ . Let Y be either SO n +1 or U n,E/F and let N be either n + 1 or n respectively. For [ φ ] ∈ Φ( Y ( E )) we have I ([ χ Y E ,L/E φ ]) = [ χ GL N ,L ′ /E ′ φ ′ ] where [ φ ′ ] = I ([ φ ]) . Proof. In both cases, the principal SL -morphism from SL ( C ) to b Y ( C ) is the irreducible2 n -dimensional representation (it has image in Sp n ( C ) according to [GW09, Lemma3.2.15]).If Y = SO n +1 then E ′ = E and L ′ = L , and χ Y E ,L/E identifies with the character ofGal( L/E ) sending the non-trivial Galois element σ L/E to − I n in the center of Sp n ( C ),whereas χ GL N ,L/E identifies with the character of Gal( L/E ) sending σ L/E to − I n in thecenter of GL n ( C ). As I is induced by composing L-homomorphisms with the naturalinclusion of dual groups the result follows.If Y = U n,E/F then I [ φ ] = [ φ | W E ′ ]. Now χ Y E ,L/E identifies with the character ofGal( L/E ) sending σ L/E to − I n in the center of GL n ( C ), whereas χ GL n ,L ′ /E ′ identifieswith the character of Gal( L ′ /E ′ ) sending σ L ′ /E ′ to − I n in the center of GL n ( C ), and theresult follows. (cid:3) A.2. Prasad’s global character. Let K/k be a quadratic extension of number fields,and let η K/k be the quadratic automorphic character of A × k attached to K/k by globalclass field theory. It decomposes as η K/k = Q v η K v /k v where the product is over all places v of k and η K v /k v is the trivial character if v splits in K .Denote by Y a k -quasi-split group. Thanks to Section A.1 for each place v of k thenatural map Y ( k v ) p v −→ Y ad ( k v ) gives rise to the local homomorphism N v := [ ρ ] k v ◦ δ p,k v ◦ p v : Y ( k v ) → k ∗ v /k ∗ v , and with the obvious notation the exact sequence (12) also gives rise to the homomorphism N k := [ ρ ] k ◦ δ p,k ◦ p k : Y ( k ) → k ∗ /k ∗ that satisfies j k v ◦ N k = N v ◦ i k v where i k v : Y ( k ) → Y ( k v ) is the natural inclusion and j k v : k ∗ /k ∗ → k ∗ v /k ∗ v the natural homomorphism. The local homomorphisms N v allowsone to define the global homomorphism N = Y v N v : Y ( A k ) → A × k / A × k . Note that N | Y ( k ) = N k . We set ω Y ,K/k := η K/k ◦ N. It is a smooth character of Y ( A k ), which is in fact automorphic since η K/k is and N | Y ( k ) equals N k composed with the natural diagonal map from k ∗ /k ∗ to A × k / A × k . Prasad’sglobal character is related to Prasad’s local characters by the following formula: ω Y ,K/k = Y v ω Y ,K v /k v where ω Y ,K v /k v is the trivial character of Y ( k v ) if v splits in K .A.3. The opposition group. Following [Pra, § 5] we define the opposition group Y op forany F -quasi-split reductive group Y . We recall that Y is a form of a unique F -split group Y s , and that any quasi-split form of Y s corresponds to an element in Hom(Gal( ¯ F /F ) , Out( Y s )).Denote by µ Y : Gal( ¯ F /F ) → Out( Y s ) the homomorphism corresponding to Y .The group Out( Y s ) of outer automorphisms of Y s identifies with the group of pinned au-tomorphisms of Y s associated with a quadruple ( Y s , B , T , ( X α ) α ∈ ∆ ) where T is a maximaltorus inside a Borel subgroup B of Y s , ∆ is a basis of simple roots of Y s with respect to( B , T ) and ( X α ) α ∈ ∆ is a pinning, all defined over F (see for example [Bor79]). Denote by w the longest element in the Weyl group of Y s associated to T . Then Aut( H s , B , T , ( X α ) α ∈ ∆ )contains a unique element sending X α to X − w ( α ) , α ∈ ∆, known as the Chevalley invo-lution of Y s , which corresponds to an involution i c in the center Z (Out( Y s )) of Out( Y s )independent of the choice of data ( B , T , ( X α ) α ∈ ∆ ). There is a unique element µ E/F ∈ Hom(Gal( E/F ) , Z (Out( Y s ))) ֒ → Hom(Gal( ¯ F /F ) , Out( Y s ))such that µ E/F ( σ ) = i c . Then µ Y op := µ E/F µ Y ∈ Hom(Gal( ¯ F /F ) , Out( Y s )) NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 65 corresponds to a unique F -quasi-split form of Y s that we denote by Y op . Taking F -pointswe call Y op = Y op ( F ) the opposition group of Y . Note that Y and Y op are isomorphicover E . It further follows from the definition that ( Y op ) op = Y .We now compute the opposition groups in the cases of interest to us. • If Y = GL n then Y = Y s and µ Y is the trivial character. Furthermore, theChevalley involution of GL n corresponds to i c ∈ Z (Out(GL n )) the class of theautomorphism g w nt g − w − n . It is now easily verified that Y op = U n,E/F . • If Y = Sp n or Y = SO( [ n ]) (either the quasi-split orthogonal or symplectic case)then the Chevalley involution of Y s is trivial and therefore Y = Y op . • If Y = U n,F ′ /F then Y s = GL n and µ Y factors through Gal( F ′ /F ) and sends τ to i c . Hence µ Y op = µ E/F µ Y factors through Gal( E ′ /F ) and furthermore maps στ to i c hence lies in the kernel of µ Y op . Thus, µ Y op factors through Gal( F ′′ /F ) where F ′′ = ( E ′ ) στ , and maps the non-trivial involution of F ′′ /F (the restriction of either σ or τ ) to i c . That is, Y op = U n,F ′′ /F . References [ABPS14] Anne-Marie Aubert, Paul Baum, Roger Plymen, and Maarten Solleveld, On the local Langlandscorrespondence for non-tempered representations , M¨unster J. Math. (2014), no. 1, 27–50.MR 3271238 43, 44[AGR93] Avner Ash, David Ginzburg, and Steven Rallis, Vanishing periods of cusp forms over modularsymbols , Math. Ann. (1993), no. 4, 709–723. MR 1233493 (94f:11044) 38[AKT04] U. K. Anandavardhanan, Anthony C. Kable, and R. Tandon, Distinguished representations andpoles of twisted tensor L -functions , Proc. Amer. Math. Soc. (2004), no. 10, 2875–2883.MR 2063106 46, 47[AR05] U. K. Anandavardhanan and C. S. Rajan, Distinguished representations, base change, andreducibility for unitary groups , Int. Math. Res. Not. (2005), no. 14, 841–854. MR 2146859 46,47[Art13] James Arthur, The endoscopic classification of representations , American Mathematical Soci-ety Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013,Orthogonal and symplectic groups. MR 3135650 43[Ato17] Hiraku Atobe, On the uniqueness of generic representations in an L -packet , Int. Math. Res.Not. IMRN (2017), no. 23, 7051–7068. MR 3801418 43[BD08] Philippe Blanc and Patrick Delorme, Vecteurs distributions H -invariants de repr´esentationsinduites, pour un espace sym´etrique r´eductif p -adique G/H , Ann. Inst. Fourier (Grenoble) (2008), no. 1, 213–261. MR 2401221 (2009e:22015) 10, 11[Ber84] Joseph N. Bernstein, P -invariant distributions on GL( N ) and the classification of unitaryrepresentations of GL( N ) (non-Archimedean case) , Lie group representations, II (CollegePark, Md., 1982/1983), Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 50–102. MR 748505 (86b:22028) 45[BM19] Paul Broussous and Nadir Matringe, Multiplicity One for Pairs of Prasad–Takloo-BighashType , International Mathematics Research Notices (2019), rnz254. 18[Bor79] A. Borel, Automorphic L -functions , Automorphic forms, representations and L -functions(Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos.Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61. MR MR546608(81m:10056) 44, 64 [BP] Rapha¨el Beuzart-Plessis, On distinguished generic representations , Preprint. 4, 38, 39[BP18] , On distinguished square-integrable representations for Galois pairs and a conjectureof Prasad , Invent. Math. (2018), no. 1, 437–521. MR 3858402 63[CPSS11] J. W. Cogdell, I. I. Piatetski-Shapiro, and F. Shahidi, Functoriality for the quasisplit classicalgroups , On certain L -functions, Clay Math. Proc., vol. 13, Amer. Math. Soc., Providence, RI,2011, pp. 117–140. MR 2767514 (2012f:22036) 54[Die48] Jean Dieudonn´e, Sur les groupes classiques , Actualit´es Sci. Ind., no. 1040 = Publ. Inst. Math.Univ. Strasbourg (N.S.) no. 1 (1945), Hermann et Cie., Paris, 1948. MR 0024439 53[Fli91] Yuval Z. Flicker, On distinguished representations , J. Reine Angew. Math. (1991), 139–172. MR 1111204 29[Fli92] , Distinguished representations and a Fourier summation formula , Bull. Soc. Math.France (1992), no. 4, 413–465. MR 1194271 41[FLO12] Brooke Feigon, Erez Lapid, and Omer Offen, On representations distinguished by unitarygroups , Publ. Math. Inst. Hautes ´Etudes Sci. (2012), 185–323. MR 2930996 18, 29, 42[GGP12] Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad, Symplectic local root numbers, centralcritical L values, and restriction problems in the representation theory of classical groups ,Ast´erisque (2012), no. 346, 1–109, Sur les conjectures de Gross et Prasad. I. MR 3202556 45,47[GK75] I. M. Gel ′ fand and D. A. Kajdan, Representations of the group GL( n, K ) where K is a localfield , Lie groups and their representations (Proc. Summer School, Bolyai J´anos Math. Soc.,Budapest, 1971), Halsted, New York, 1975, pp. 95–118. MR 0404534 (53 A criterion for integrability of matrix coefficients withrespect to a symmetric space , J. Funct. Anal. (2016), no. 12, 4478–4512. MR 3490774 13[Gur15] Maxim Gurevich, On a local conjecture of Jacquet, ladder representations and standard mod-ules , Math. Z. (2015), no. 3-4, 1111–1127. MR 3421655 18, 41, 46[GW09] Roe Goodman and Nolan R. Wallach, Symmetry, representations, and invariants , GraduateTexts in Mathematics, vol. 255, Springer, Dordrecht, 2009. MR 2522486 (2011a:20119) 63[Han20] Marcela Hanzer, Generalized injectivity conjecture for classical p -adic groups II , J. Pure Appl.Algebra (2020), no. 1, 149–168. MR 3986415 40, 41[Hen00] Guy Henniart, Une preuve simple des conjectures de Langlands pour GL( n ) sur un corps p -adique , Invent. Math. (2000), no. 2, 439–455. MR 1738446 43[Hen10] , Correspondance de Langlands et fonctions L des carr´es ext´erieur et sym´etrique , Int.Math. Res. Not. IMRN (2010), no. 4, 633–673. MR 2595008 46, 47[HT01] Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimuravarieties , Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ,2001, With an appendix by Vladimir G. Berkovich. MR MR1876802 (2002m:11050) 43[HW93] A. G. Helminck and S. P. Wang, On rationality properties of involutions of reductive groups ,Adv. Math. (1993), no. 1, 26–96. MR 1215304 (94d:20051) 7, 11, 14[Jac95] Herv´e Jacquet, The continuous spectrum of the relative trace formula for GL(3) over a qua-dratic extension , Israel J. Math. (1995), no. 1-3, 1–59. MR 1324453 (96a:22029) 3, 10[JLR99] Herv´e Jacquet, Erez Lapid, and Jonathan Rogawski, Periods of automorphic forms , J. Amer.Math. Soc. (1999), no. 1, 173–240. MR 1625060 (99c:11056) 3, 10[JPSS83] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions , Amer. J.Math. (1983), no. 2, 367–464. MR 701565 (85g:11044) 45[JS03] Dihua Jiang and David Soudry, The local converse theorem for SO(2 n + 1) and applications ,Ann. of Math. (2) (2003), no. 3, 743–806. MR 1983781 (2005b:11193) 4, 38, 39[Kab04] Anthony C. Kable, Asai L -functions and Jacquet’s conjecture , Amer. J. Math. (2004),no. 4, 789–820. MR 2075482 (2005g:11083) 46, 47 NTERTWINING PERIODS AND DISTINCTION FOR P-ADIC GALOIS SYMMETRIC PAIRS 67 [Kal16] Tasho Kaletha, The local Langlands conjectures for non-quasi-split groups , Families of auto-morphic forms and the trace formula, Simons Symp., Springer, [Cham], 2016, pp. 217–257.MR 3675168 43, 44[Lag08] Nathalie Lagier, Terme constant de fonctions sur un espace sym´etrique r´eductif p -adique , J.Funct. Anal. (2008), no. 4, 1088–1145. MR MR2381204 (2009d:22013) 11, 16[LL15] Jean-Pierre Labesse and Erez Lapid, Characters of G over local and global fields , 2015, ap-pendix to E. Lapid, Z. Mao, A conjecture on Whittaker-Fourier coefficients of cusp forms, J.Number Theory 146 (2015), 448-505. 63[LR03] Erez M. Lapid and Jonathan D. Rogawski, Periods of Eisenstein series: the Galois case , DukeMath. J. (2003), no. 1, 153–226. MR 2010737 3, 10, 11, 12, 15[Mat08] Nadir Matringe, Distinction et fonctions L d’Asai pour les repr´esentations g´en´eriques desgroupes lin´eaires g´en´eraux sur les corps p-adiques , PhD Thesis, UNIVERSIT´E PARIS.DIDEROT - Paris 7 (2008). 49[Mat11] , Distinguished generic representations of GL( n ) over p -adic fields , Int. Math. Res. Not.IMRN (2011), no. 1, 74–95. MR 2755483 (2012f:22032) 46, 47[Mat17a] , Distinction of the Steinberg representation for inner forms of GL ( n ), Math. Z. (2017), no. 3-4, 881–895. MR 3719517 18[Mat17b] Nadir Matringe, Gamma factors of intertwining periods and distinction for inner forms of gl ( n ), 2017. 18[MgVW87] Colette Mœ glin, Marie-France Vign´eras, and Jean-Loup Waldspurger, Correspondances deHowe sur un corps p -adique , Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin,1987. MR 1041060 26[MO18] Nadir Matringe and Omer Offen, Gamma factors, root numbers, and distinction , Canad. J.Math. (2018), no. 3, 683–701. MR 3785415 45[Mok15] Chung Pang Mok, Endoscopic classification of representations of quasi-split unitary groups ,Mem. Amer. Math. Soc. (2015), no. 1108, vi+248. MR 3338302 43[Mui01] Goran Mui´c, A proof of Casselman-Shahidi’s conjecture for quasi-split classical groups , Canad.Math. Bull. (2001), no. 3, 298–312. MR 1847492 (2002f:22015) 44, 45[MW95] C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series , CambridgeTracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995, Une para-phrase de l’´Ecriture [A paraphrase of Scripture]. MR 1361168 (97d:11083) 5, 6, 30, 31, 32[Off06] Omer Offen, On symplectic periods of the discrete spectrum of GL n , Israel J. Math. (2006), 253–298. MR 2254544 3[Off17] , On parabolic induction associated with a p -adic symmetric space , J. Number Theory (2017), 211–227. MR 3541705 3, 11, 12, 13, 14, 15, 17, 18, 29, 54[Pra] Dipendra Prasad, A relative langlands correspondence and geometry of parameter spaces ,Preprint. 3, 41, 45, 46, 55, 63, 64[Rod73] Fran¸cois Rodier, Whittaker models for admissible representations of reductive p -adic splitgroups , Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI,Williams Coll., Williamstown, Mass., 1972), Amer. Math. Soc., Providence, R.I., 1973,pp. 425–430. MR 0354942 (50 Quadratic and Hermitian forms , Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag,Berlin, 1985. MR MR770063 (86k:11022) 59, 60[Ser94] Jean-Pierre Serre, Cohomologie galoisienne , fifth ed., Lecture Notes in Mathematics, vol. 5,Springer-Verlag, Berlin, 1994. MR 1324577 (96b:12010) 55, 57[Sha91] Freydoon Shahidi, Langlands’ conjecture on Plancherel measures for p -adic groups , Harmonicanalysis on reductive groups (Brunswick, ME, 1989), Progr. Math., vol. 101, Birkh¨auserBoston, Boston, MA, 1991, pp. 277–295. MR 1168488 (93h:22033) 38 [SZ18] Allan J. Silberger and Ernst-Wilhelm Zink, Langlands classification for L -parameters , J. Al-gebra (2018), 299–357. MR 3834776 43, 44[SX] Miyu Suzuki and Hang Xue, Linear intertwining periods and epsilon dichotomy for linearmodels The structure of a unitary factor group , Inst. Hautes ´Etudes Sci. Publ. Math.(1959), no. 1, 23 pp. (1959). MR MR0104764 (21 Induced representations of reductive p -adic groups. II. On irreducible repre-sentations of GL( n ), Ann. Sci. ´Ecole Norm. Sup. (4) (1980), no. 2, 165–210. MR 584084(83g:22012) 45 Email address : [email protected] Universit´e de Poitiers, Poitiers, France Email address : [email protected]@brandeis.edu