aa r X i v : . [ m a t h . R T ] J u l LINEAR PERIODS FOR UNITARY REPRESENTATIONS
Chang Yang A bstract . Let F be a local non-Archimedean field of characteristic zero with a finiteresidue field. Based on Tadi´c’s classification of the unitary dual of GL n p F q , we clas-sify irreducible unitary representations of GL n p F q that have nonzero linear periods, interms of Speh representations that have nonzero periods. We also give a necessary andsu ffi cient condition for the existence of a nonzero linear period for a Speh representation.
1. Introduction1.1. Main results.
Let F be a local non-Archimedean field of characteristic zero witha finite residue field. Denote the group G n “ GL n p F q . Let p and q be two nonnegativeintegers with p ` q “ n , we denote by H “ H p , q the subgroup of G n of matrices of theform: ˜ g g ¸ with g P G p , g P G q . Let π be a smooth representation of G n on a complex vector space V and χ a character of H , denote by Hom H p π, χ q the space of linear forms l on V such that l p π p h q v q “ χ p h q l p v q for all v P V and h P H . Smooth representations π of G n with Hom H p π, χ q ‰ p H , χ q -distinguished , or simply H -distinguished if χ is the trivial character of H .Elements of Hom H p π, q are called (local) linear periods of π . Linear periods havebeen studied by many authors. The uniqueness of linear periods was proved by Jacquetand Rallis in [ JR ]; the uniqueness of twisted linear periods, with respect to almost all char-acters χ of H and in the case p “ q , was proved by Chen and Sun in [ CS ]. It thus remainsan interesting question of characterizing irreducible representations that have nonzero lin-ear periods. It is known that a tempered representation of GL n p F q has nonzero linearperiods with respect to H n , n if and only if it is a functorial transfer of a generic temperedrepresentation of SO n ` p F q , see [ JS ], [ Mat2 ] and [
Mat4 ]. Another closely related char-acterization of the existence of nonzero linear periods for an essentially square-integrablerepresentation is through poles of the local exterior square L -functions associated with therepresentation, see [ Mat2 ] and references therein. A recent preprint by S´echerre [
S´ec ]studied supercuspidal representations with nonzero linear periods from the point of view of type theory. However, all of these characterizations are for generic representations.Motivated by the recent work of Gan-Gross-Prasad [
GGP ] on branching laws in the non-tempered case, we are led to consider in this work the existence of nonzero linear periodsfor irreducible unitary representations.Our main results are as follows. We refer the reader to Section 2 for unexplainednotation in the following two theorems.T heorem
Let Sp p δ, k q be a Speh representation of G n , where δ is a square-integrable representation of G d with d ą , and k is a positive integer ( n “ dk). ThenSp p δ, k q is H n , n -distinguished if and only if d is even and δ is H d { , d { -distinguished. T heorem An irreducible unitary representation π of G n is H n , n -distinguished ifand only if it is self-dual and its Arthur part π Ar is of the form p σ ˆ σ _ q ˆ ¨ ¨ ¨ ˆ p σ r ˆ σ _ r q ˆ σ r ` ˆ ¨ ¨ ¨ ˆ σ s . where each σ i is a Speh representation for i “ , ¨ ¨ ¨ , s, and each representation σ j isH m j , m j -distinguished for some positive integer m j , j “ r ` , ¨ ¨ ¨ , s. Distinction problem for unitary representation has already been considered by Ma-tringe for local Galois periods in [
Mat3 ] and by O ff en and Sayag for local Symplecticperiods in [ OS1, OS2 ]. We remark that the special case of Theorem 1.2 for representationsof Arthur type (see Theorem 7.3) is similar to [
Mat4 , Theorem 3.13] about local linear pe-riods for generic representations and the main result in [
Mat3 ] about local Galois periodsfor unitary representations. A global analogue of our result is to find the H n , n -distinguishedrepresentation in the automorphic dual of G n , which we will pursue in future works. Wealso refer the reader to [ FJ, JR ] for the role of local linear periods and their global ana-logues in the study of standard L -functions. Most of our work deals with distinctionof parabolically induced representations of G n . The main tool to study distinction of in-duced representations is the geometric lemma of Bernstein-Zelevinsky [ BZ ], which relatesdistinction of an induced representation to distinction of some Jacquet module of the in-ducing data. It was shown by Tadi´c in [ Tad1 ] that every irreducible unitary representationis isomorphic to the parabolic induction of Speh representations or their twists. The ob-servation is that Jacquet modules of Speh representations have a convenient combinatorialdescription similar to that of Jacquet modules of essentially square-integrable representa-tions ([ KL ]). As hinted by the geometric lemma, to classify H n , n -distinguished irreducibleunitary representations, it is necessary to consider H p , q -distinction with respect to a partic-ular family of characters in (2.1), not only of Speh representations, but also of a larger classof representations, ladder representations. The class of ladder representations was intro-duced by Lapid and Minguez in [ LM1 ], and has many remarkable properties which makethem an ideal testing ground for distinction of non-generic representations and some otherquestions in the representation theory of general linear groups, see for example [
FLO ], INEAR PERIODS FOR UNITARY REPRESENTATIONS 3 [ Gur ], [
MOS ] and [
LM2 ]. The most complicated part of the paper, Section 6, is devotedto the study of distinction of ladder representations. Our treatment, although not simple,is largely elementary and combinatorial based on detailed analysis using the geometriclemma. We refer the reader to [
Mat1 ] and [
Mat4 ] for a similar approach to the classifi-cation of distinguished generic representations in Galois symmetric space and our settingrespectively.We next outline the proof of Theorem 1.1. For the ‘if’ part, the existence of non-zerolinear periods for the standard module of a Speh representation Sp p ∆ , k q is guaranteed bythe work of Blanc and Delorme [ BD ] when ∆ is H d { , d { -distinguished. Thus it su ffi cesto show that the maximal proper subrepresentation of the standard module associated withSp p ∆ , k q is not H n , n -distinguished. The explicit structure of this maximal proper subrep-resentation is well known by the work of Tadi´c [ Tad3 ] (see also [
LM1 ]). For the ‘onlyif’ part of Theorem 1.1, however, we cannot expect to get any information on the distin-guishedness of ∆ from that of the standard module of Sp p ∆ , k q when k is an even, as inthis case, the standard module of Sp p ∆ , k q is H n , n -distinguished for any self-dual ∆ by thework of Blanc and Delorme [ BD ]. We instead use the idea of ‘restricting to the mirabolicsubgroup’, and relate linear periods on Sp p ∆ , k q with those on its highest shifted deriva-tive, which is exactly Sp p ∆ , k ´ q . The ‘only if’ part is then proved by induction on k .We remark that the idea of exploiting the theory of derivatives in distinction problems hasalready appeared many times in the literature, see for example [ CPS ], [
Kab ], [
Mat4 ] and[
Mat3 ].The paper is organized as follows. In Section 2 we introduce notations and somepreliminaries on the representation theory of general linear groups. In Section 3 we presentsome general facts on p H p , q , µ a q -distinguished representations, where µ a is the character in(2.1). In this section, we recall a result of Gan which is crucial for our combinatorialstudy of twisted linear periods. In Section 4 we give a detailed analysis of the parabolicorbits of the symmetric space involved and in Section 5 we draw some consequences of thegeometric lemma. Section 6 is devoted to the study of distinction of ladder representations.We then complete the classification in Section 7.
2. Preliminaries on representations of GL n p F q Throughout the paper let F be a local non-Archimedean field of characteristic zerowith a finite residue field.For any n P Z ě , let G n “ GL n p F q and let R p G n q be the category of smooth complexrepresentations of G n of finite length. Denote by Irr p G n q the set of equivalence classes of ir-reducible objects of R p G n q and by C p G n q the subset consisting of supercuspidal represen-tations. (By convention we define G as the trivial group and Irr p G q consists of the trivialrepresentation of G .) Let Irr and C be the disjoint union of Irr p G n q and C p G n q , n ě π P R p G n q , we call n the degree of π . CHANG YANG
Let R n be the Grothendieck group of R p G n q and R “ ‘ n ě R n . The canonical mapfrom the objects of R p G n q to R n will be denoted by π ÞÑ r π s .Denote by ν the character ν p g q “ | det g | on any G n . (The n will be implicit andhopefully clear from the context.) For any π P R p G n q and a P R , denote by ν a π therepresentation obtained from π by twisting it by the character ν a , and denote by π _ thecontragredient of π . The sets Irr and C are invariant under taking contragredient. Fora character χ of F ˆ , define the real part Re p χ q of χ to be the real number a such that | χ p z q| C “ | z | a , z P F ˆ , where | ¨ | C is the absolute value on C .For a subgroup Q of G n , denote by δ Q the modular character of Q .Let p and q be two nonnegative integers with p ` q “ n . Denote by w p , q the matrix w p , q “ ˜ I q I p ¸ . Let H p , q be the subgroup of G n as in the introduction. For a P R , define the character µ a of H p , q by µ a ˜˜ g g ¸¸ “ ν a p g q ν ´ a p g q , g P G p , g P G q . (2.1)(By convention we allow the case where p or q is zero.) The standard parabolic subgroupsof G n are in bijection with compositions p n , ¨ ¨ ¨ , n t q of n . The corresponding standard Levisubgroup is the gorup of block diagonal invertible matrices with block sizes n , ¨ ¨ ¨ , n t . Itis isomorphic to G n ˆ ¨ ¨ ¨ ˆ G n t .Let P “ M ˙ U be a standard parabolic subgroup of G n and σ a smooth, complexrepresentation of M . We denote by Ind G n P p σ q its normalized parabolic induction; for anystandard Levi subgroup L Ă M , we denote by r L , M p σ q the normalized Jacquet module (see[ BZ , § ρ , ¨ ¨ ¨ , ρ t are representations of G n , ¨ ¨ ¨ , G n t respectively, we denote by ρ ˆ ¨ ¨ ¨ ˆ ρ t the representation Ind G n P σ where σ is the representation ρ b ¨ ¨ ¨ b ρ t of M , where M is thestandard Levi subgroup of the parabolic subgroup P corresponding to p n , ¨ ¨ ¨ , n t q .Next we brief review the Jacquet module of a product of representations of finite length[ Zel , § α “ p n , ¨ ¨ ¨ , n t q and β “p m , ¨ ¨ ¨ , m s q be two compositions of n . For every i P t , ¨ ¨ ¨ , t u , let ρ i P R p G n i q . Denoteby Mat α,β the set of t ˆ s matrices B “ p b i , j q with nonnegative integer entries such that s ÿ j “ b i , j “ n i , i P t , ¨ ¨ ¨ , t u , t ÿ i “ b i , j “ m j , j P t , ¨ ¨ ¨ , s u . Fix B P Mat α,β . For any i P t , ¨ ¨ ¨ , t u , α i “ p b i , , ¨ ¨ ¨ , b i , s q is a composition of n i and wewrite the compostion factors of r α i p ρ i q as σ ki “ σ ki , b ¨ ¨ ¨ b σ ki , s , σ ki , j P Irr p G b i , j q , k P t , ¨ ¨ ¨ , l i u , INEAR PERIODS FOR UNITARY REPRESENTATIONS 5 where l i is the length of r α i p ρ i q . For any j P t , ¨ ¨ ¨ , s u and a sequence k “ p k , ¨ ¨ ¨ , k r q ofintegers such that 1 ď k i ď l i , define Σ B , kj “ σ k , j ˆ ¨ ¨ ¨ ˆ σ kt , j P R p G m j q . Then we have r r β p ρ ˆ ¨ ¨ ¨ ˆ ρ t s “ ÿ B P Mat α,β , k r Σ b , k b ¨ ¨ ¨ b Σ B , ks s . By a segment of cuspidal representations we mean aset r a , b s ρ “ t ν a ρ, ν a ` ρ, ¨ ¨ ¨ , ν b ρ u , where ρ P C and a , b P R , b ´ a P Z ě . The representation ν a ρ ˆ ν a ` ρ ˆ ¨ ¨ ¨ ˆ ν b ρ hasa unique irreducible quotient, which is an essentially square-integrable representaton andis denoted by ∆ pr a , b s ρ q . The map r a , b s ρ ÞÑ ∆ pr a , b s ρ q gives a bijection between the set ofsegments of cuspidal representations and the subset of essentially square-integrable repre-sentations in Irr. (In what follows, for simplicity of notation, we shall use ∆ to denote eithera segment of cuspidal representations or the essentially square-integrable representationscorresponding to it; we hope this will not cause any confusion.) We use the conventionthat ∆ pr a , b s ρ q “ b ă a ´ ∆ pr a , a ´ s ρ q “
1, the trivial represntation of G .We denote the extremities of ∆ “ ∆ pr a , b s ρ q by b p ∆ q “ ν a ρ P C and e p ∆ q “ ν b ρ P C respectively. We also write l p ∆ q “ b ´ a ` ∆ .For ρ P C , we denote by Z ρ the set t ν a ρ | a P Z u and call it the cuspidal line of ρ . Wethen trasport the order and additive structure of Z to the cuspidal line Z ρ . Thus we shallsometimes write ν a ρ ` b “ ν a ` b ρ and ν a ρ ď ν b ρ if a ď b , where a , b are integers. By thecontragredient of Z ρ we mean the cuspidal line Z ρ _ .Let ∆ and ∆ be two segments. We say that ∆ and ∆ are linked if ∆ Y ∆ forms asegment but neither ∆ Ă ∆ nor ∆ Ă ∆ . If ∆ and ∆ are linked and b p ∆ q “ b p ∆ q ν j with j ą
0, then we say that ∆ precedes ∆ and write ∆ ă ∆ .A multisegment is a multiset (that is, set with multiplicities) of segments. Denote by O the set of multisegements. For ρ P C , let O ρ denote the multisegements such that all ofits segements are contained in the cuspidal line Z ρ . An order m “ t ∆ , ¨ ¨ ¨ , ∆ t u P O on amultisegments m is of standard form if ∆ i ć ∆ j for all i ă j . Every m P O admits at leastone standard order.Let m “ t ∆ , ¨ ¨ ¨ , ∆ t u P O be ordered in standard form. The representation λ p m q “ ∆ ˆ ¨ ¨ ¨ ˆ ∆ t is independent of the choice of order of standard form. It has a unique irreducible quotientthat we denote by L p m q . The Langlands classification says that the map m ÞÑ L p m q is abijection between O and Irr. G n . We briefly recall the classification of the unitary dual of G n by Tadi´c [ Tad1 , Theorem D]. Let Irr u be the subset of unitarizable representations in Irr, CHANG YANG and D u the subset of all square-integrable classes in Irr u . Let k be a positive integer, andlet δ P D u . The repersentation ν p k ´ q{ δ ˆ ν p k ´ q{ δ ˆ ¨ ¨ ¨ ˆ ν ´p k ´ q{ δ has a unique irreducible unitarizable quotient Sp p δ, k q , called a Speh representation.Suppose 0 ă α ă {
2. The representation ν α Sp p δ, k q ˆ ν ´ α Sp p δ, k q is irreducible andunitarizable; we denote it by Sp p δ, k qr α, ´ α s . Let B be the set of all Sp p δ, k q , Sp p δ, k qr α, ´ α s , where δ P D u , k is a positive integer and 0 ă α ă {
2. By [
Tad1 , Theorem D], anirreducible representation π is unitarizable if and only if it is of the form π ˆ ¨ ¨ ¨ ˆ π t , π i P B , i “ , ¨ ¨ ¨ , t . Moreover, this expresssion is unique up to permutation. We call it a Tadi´c decompositionof π .By an irreducible representation of Arthur type , we mean an irreducible unitary repre-sentation whose Tadi´c decomposition does not involve any Sp p δ, k qr α, ´ α s . For π P Irr u ,we then have a decomposition π “ π Ar ˆ π c , where π Ar is a representation of Arthur typeand is called the Arthur part of π .
3. Preliminaries on p H p , q , µ a q -distinguished representations3.1. Basic facts. L emma (1) Let π be a smooth representation of G n . If π is p H p , q , µ a q -distinguishedfor two nonnegative integers p, q with p ` q “ n and a P R , then π is also p H q , p , µ ´ a q -distinguished;(2) Let π , ¨ ¨ ¨ , π t P Irr p G n q . If π ˆ ¨ ¨ ¨ˆ π t is p H p , q , µ a q -distinguished for two nonneg-ative integers p, q with p ` q “ n and a P R , then π _ ˆ¨ ¨ ¨ˆ π _ t is p H p , q , µ ´ a q -distinguished. P roof . The statement (1) follows from the fact that π – π w q , p . Let ι denote the invo-lution ι p g q “ t g ´ of transpose inversion. Then (2) follows from the fact that p π ˆ ¨ ¨ ¨ ˆ π t q ˝ ι – π _ ˆ ¨ ¨ ¨ ˆ π _ t . Q.E.D.For representations of dimension one, we have the following simple lemma, whoseproof we omit.L emma Let χ be a character of G n . Assume that χ is p H p , q , µ a q -distinguished fornonnegative integers p, q with p ` q “ n and a P R . If q “ (resp. p “ ), then χ is thecharacter ν a (resp. ν ´ a ) of G n ; If p, q ą , then a “ and χ “ , the trivial character ofG n . For untwisted linear periods, we have the following fundamental result due to Jacquetand Rallis [ JR ]. INEAR PERIODS FOR UNITARY REPRESENTATIONS 7 L emma Let p, q be two positive integers with p ` q “ n. If π P Irr p G n q , then dim Hom H p , q p π, q ď . Furthermore, if dim Hom H p , q p π, q “ , then π – π _ . R emark In this work we will not need multiplicity one results about (twisted)linear periods. However, the self-dualness property of distinguished representations is im-portant for applications of the geometric lemma. The uniqueness of twisted linear periods,in the case p “ q, is studied by Chen and Sun in [ CS ] . Their result shows that, for all butfinitely many a, dim Hom H p , p p π, µ a q ď for all π P Irr p G p q . However, due to the author’slimited knowledge, we can not deduce self-dualness property from their arguments whenthe multiplicity is nonzero as in the untwisted case. When n “ m is an even integer, the Shalikasubgroup of G n is defined to be S n “ a b a ¸ ˇˇˇˇˇ a P G m , b P M m + “ G m ˙ N m , m , where M m indicates the set of m ˆ m matrices with entries in F . Define a character ψ S n on S n by ψ S n ˜˜ a a ¸ ˜ b ¸¸ “ ψ F p Tr p b qq , (3.1)where ψ F is a non-trivial character of F . For a smooth representation π of G n , an elementin Hom S n p π, ψ S n q is called a local Shalika period of π .The relation between untwisted linear periods and untwisted Shalika periods is wellknown (see [ JNQ ] for their equivalence in the case of supercuspidal representations; seealso a discussion for relatively square-integrable representations in [
Mat2 , § G n (see [ Gan , Theorem 3.1]).P roposition
Let n “ m be an even integer. For any π P Irr p G n q and σ P Irr p G m q ,one has Hom S n p Θ p π q , σ b ψ S n q – Hom H m , m p π _ , σ b C q , (3.2) where Θ p π q is the big theta lift of π and σ b ψ S n is viewed as a representation of S n “ G m ˙ N m , m . C orollary Let π be a generic representation of G n . The followings are equiva-lent: (1) π is p H m , m , µ a q -distinguished for some a P R ; (2) π is p H m , m , µ a q -distinguished for all a P R ; (3) π is p S n , ψ S n q -distinguished.In particular, if one of the three equivalent conditions holds, then π is self-dual. CHANG YANG P roof . For any a P R , the representation ν a π is still generic. Thus Θ p ν a π q “ p ν a π q _ by a result of Minguez [ M´ın , § H m , m p π, µ a q “ Hom H m , m p π, ν a b ν ´ a q – Hom H m , m p ν a π, ν a b C q– Hom S n p ν a π, ν a b ψ S n q – Hom S n p π, ψ S n q . Q.E.D.R emark
The result of Gan, Proposition 3.5, highlights the importance of deter-mining whether Θ p π q “ π _ for an irreducible representation π of G n . This is true forgeneric representations by a result of Minguez [ M´ın ] , see also [ Gan , Theorem 4.1] . Fur-thermore, in [ FSX ] , the authors prove that if the Godement-Jacquet L function L p s , π q orL p s , π _ q has no pole at s “ { , then Θ p π q – π _ . R emark When n “ , this corollary can be deduced from a result of Wald-spurger on toric periods, which shows that a generic representation of GL p F q is p T , µ a q -distinguished if and only if its central character is trivial, where T is the diagonal torus in GL p F q . Let P n Ă G n be the mirabolicsubgroup of G n consisting of matrices with the last row p , , ¨ ¨ ¨ , , q . We refer the readerto [ BZ , 3.2] for the definition of the following functors Ψ ´ : Alg P n Ñ Alg G n ´ , Ψ ` : Alg G n ´ Ñ Alg P n , Φ ´ : Alg P n Ñ Alg P n ´ , Φ ` : Alg P n ´ Ñ Alg P n . Define π p k q “ Ψ ´ p Φ ´ q k ´ p π | P n q to be the k -th derivative of a representation π of G n .The following proposition can be proved by the same argument as that in [ Kab , Propo-sition 1] (see also [
Mat2 , Proposition 3.1], where the linear subgroups H p , q take di ff erentforms.)P roposition If σ is a representation of P n ´ and χ is a character of H p , q , then Hom P n X H p , q p Φ ` σ, χ q – Hom P n ´ X H q ´ , p p σ, χ w q ´ , p µ ´ { q as complex vector spaces, where χ w q ´ , p is the character of H q ´ , p defined by χ w q ´ , p p g q “ χ p w q ´ , p gw ´ q ´ , p q . In particular, for all a P R , one has Hom P n X H p , q p Φ ` σ, µ a q – Hom P n ´ X H q ´ , p p σ, µ ´ a ´ { q . (3.3)As a corollary, we have the following result due to Matringe [ Mat2 , Theorem 3.1].C orollary
Let ∆ be an essentially square-integrable representation of G n . Letp, q be two positive integers with p ` q “ n, and χ a character of H p , q . Assume that π is p H p , q , χ q -distinguished. Then p “ q. Another application of Proposition 3.9 will generalize Corollary 3.10 to essentiallySpeh representations in Theorem 6.11 of Section 6.3.The following proposition, a direct consequence of Corollary 3.10 and Corollary 3.6,is the starting point of the combinatorial arguments in this work.
INEAR PERIODS FOR UNITARY REPRESENTATIONS 9 P roposition Let ∆ be an essentially square-integrable representation of G n . If ∆ is p H p , q , µ a q -distinguished for two positive integers p, q with p ` q “ n and some a P R ,then p “ q, and ∆ is self-dual.
4. Symmetric spaces and parabolic orbits
The main tool we use to classify distinguished unitary representations is the geometriclemma of Bernstein and Zelevinsky [ BZ , Theorem 5.2]. Applying it requires a detailedanalysis of the double coset space P z G n { H p , q , where P is a parabolic subgroup of G n . As H p , q is a symmetric subgroup of G n , we follow the framework given by O ff en in [ O ff ]. Let G “ G n , H “ H p , q be the subgroup of G n as in theintroduction. Let ε “ ε p , q “ ˜ I p ´ I q ¸ , and θ “ θ p , q be the involution on G n defined by θ p g q “ ε g ε ´ . The symmetric spaceassociated to p G , θ q is X “ t g P G | θ p g q “ g ´ u , equipped with the G -action g ¨ x “ gx θ p g q ´ . The map g ÞÑ g ¨ e gives a bijection of thecoset space G { H onto the orbit G ¨ e Ă X , and thus a bijection of the double coset space P z G { H onto the P -orbits in G ¨ e , where P denotes a parabolic subgroup of G . For any g P G , denote by r g s G the conjugacy class of g in G . Note that the map g ÞÑ g ε gives abijection of G ¨ e onto r ε s G and that the G -action on G ¨ e is transformed to the conjugationaction of G on r ε s G .For any subgroup Q of G and x P X , let Q x “ t g P Q | g ¨ x “ x u be the stabilizer of x in Q . Note that Q x is just the centralizer of x ε in Q . A first coarse classification of the doublecosets in P z G { H is given by certain Weyl elements. Let W be the Weyl group of G . Let W r s “ t w P W | w θ p w q “ e u “ t w P W | w “ e u be the set of twisted involutions in W . For two standard Levi subgroups M and M of G ,let W M M be the set of all w P W that are left W M -reduced and right W M -reduced.Given a standard parabolic subgroup P “ M ˙ U , define a map ι M : P z X Ñ W r s X W M M (4.1)by the relation
PxP “ P ι M p P ¨ x q P . (4.2)For x P X , let w “ ι M p P ¨ x q and L “ M p w q “ M X wMw ´ . Then L is a standard Levi subgroup of M satisfying L “ wLw ´ . It is noted in [ O ff ] that, to apply the geometric lemma inparticular cases, it is necessary to first understand the admissible orbits. Recall that x P X (or a P -orbit P ¨ x in X) is said to be M -admissible if M “ wMw ´ where w “ ι M p P ¨ x q .We now describe the relevant data for M -admissible P -orbits in G ¨ e .By [ O ff , Corollary 6.2], M -admissible P -orbits in G ¨ e is in bijection with M -orbits in G ¨ e X N G p M q , or equivalently M -conjugacy classes in r ε s G X N G p M q .Fix a composition ¯ n “ p n , ¨ ¨ ¨ , n t q of n . Let P “ M ˙ U be the standard para-bolic subgroup of G n associated to ¯ n . Denote by S p ¯ n q t the set of permutations τ on the set t , , ¨ ¨ ¨ , t u such that n i “ n τ p i q for all i P t , ¨ ¨ ¨ , t u . To each τ in S p ¯ n q t , we associate ablock matrix w τ which has I n i on its p τ p i q , i q -block for each i and has 0 elsewhere. Thenthe map τ ÞÑ w τ M defines an isomorphism of groups from S p ¯ n q t to N G p M q{ M . Write an element of M asdiag t A , ¨ ¨ ¨ , A t u . Note that an element w τ diag t A , ¨ ¨ ¨ , A t u of N G p M q has order 2 if andonly if τ “ A i A τ p i q “ I n i for all i P t , ¨ ¨ ¨ , t u . One sees that the M -conjugacy classes in r ε s G X N G p M q are parameterized by the set ofpairs p c τ , τ q where τ P S p ¯ n q t , τ “
1, and c τ is a set of the form tp n k , ` , n k , ´ q | for all k such that τ p k q “ k u such that $’’&’’% n k “ n k , ` ` n k , ´ , n k , ` , n k , ´ ě ř k ,τ p k q“ k n k , ` ` ř p i ,τ p i qq , i ă τ p i q n i “ p ; ř k ,τ p k q“ k n k , ´ ` ř p i ,τ p i qq , i ă τ p i q n i “ q . (4.3)Denote by I p , q p ¯ n q the set of all such pairs.For the M -admissible P -orbit O corresponding to p c τ , τ q in I p , q p ¯ n q , we can choose anatural orbit representative x “ x p c τ ,τ q P O X N G p M q as follows: The matrix x ε has I n i onits p τ p i q , i q -block when τ p i q ‰ i , diag p I n i , ` , ´ I n i , ´ q on its p i , i q -block when τ p i q “ i , and 0elsewhere. One sees easily that M x consists of elements diag t A , ¨ ¨ ¨ , A t u such that $&% A i “ A τ p i q , τ p i q ‰ i ; A i I n i , ` , n i , ´ “ I n i , ` , n i , ´ A i τ p i q “ i . (4.4)Here and in what follows, we denote by I n , n for the diagonal matrix diag t I n , ´ I n u . Thus,when τ p i q “ i , we may further write A i as diag t A i , ` , A i , ´ u . One also has P x “ M x ˙ U x .The following computation of modular characters is indispensable for applications ofthe geometric lemma, see [ O ff , Theorem 4.2]. We omit the proof here as it is obtained bya routine calculation. INEAR PERIODS FOR UNITARY REPRESENTATIONS 11 L emma Let x P G ¨ e X N G p M q be the representative as above of the P-orbitcorresponding to p c , τ q P I p , q p ¯ n q . Then, for m “ diag t A , ¨ ¨ ¨ , A t u P M x , we have δ P x δ ´ { P p m q “ ź i ă j τ p i q“ i ,τ p j q“ j ν p A i , ` q p n j , ` ´ n j , ´ q{ ν p A i , ´ q p n j , ´ ´ n j , ` q{ ν p A j , ` q p n i , ´ ´ n i , ` q{ ν p A j , ´ q p n i , ` ´ n i , ´ q{ (4.5) ¨ ź i ă j τ p i qą τ p j q ν p A i q ´ n j { ν p A j q n i { . For our purposes, we consider only P -orbits in G ¨ e Ă X where P is a maximal parabolic subgroup. Let P “ P k , n ´ k be the standard parabolic subgroupassociated to p k , n ´ k q with M its Levi subgroup. We follow the geometric method asin [ Mat4 ]. The case where | p ´ q | ď H there takes a di ff erent form and thetreatment here is independent.Let V be a n -dimensional F -vector space with a basis t e , ¨ ¨ ¨ , e n u . Let V ` (resp. V ´ )be the subspace of V of dimension p (resp. q ) which is generated by t e , ¨ ¨ ¨ , e p u (resp. t e p ` , ¨ ¨ ¨ , e n u ). The coset space G { P can be identified with the set of subspaces of V ofdimension k . For such a subspace W , set r W “ dim F p W X V ` q , s W “ dim F p W X V ´ q . L emma Let W and W be two subspaces of V of dimension k. Then they are in thesame H-orbit if and only if r W “ r W and s W “ s W . For a pair of nonnegative integers p r , s q , there is a subspace W of V such that r “ r W and s “ s W if and only if r ` s ď k , k ´ s ď p , k ´ r ď q . (4.6)Denote by I kp , q the set of pairs of nonnegative integers p r , s q that satisfying (4.6). Then,by Lemma 4.2, the double cosets in H z G { P can be parameterized by I kp , q . For p r , s q P I kp , q ,call d “ k ´ r ´ s the defect of p r , s q .We first seek a complete set of representatives of P z G { H ; and we split the discussionsinto two cases.Case k ě p . Let W p r , s q be the subspace of V generated by t e , ¨ ¨ ¨ , e r ; e r ` ` e q ` r ` , ¨ ¨ ¨ , e k ´ s ` e q ` k ´ s ; e q ` k ´ s ` , ¨ ¨ ¨ e n ; e p ` , ¨ ¨ ¨ , e k u . Then dim F W r , s “ k , dim F p W p r , s q X V ` q “ r and dim F p W p r , s q X V ´ q “ s . Let ˜ η ´ p r , s q be theblock matrix ˜ C C C C ¸ where C and C are matrices of size p ˆ p and q ˆ q respectively, and C “ ˜ I k ´ s ¸ , C “ ˜ I k ´ s ` q ´ p ¸ C “ ˜ I s ` p ´ k ¸ , C “ ˜ I p ´ r ¸ . Then t ˜ η ´ p r , s q u is a complete set of representatives of the double coset space H z G { P . Takinginverse, we thus get a complete set of representatives t ˜ η p r , s q u of P z G { H .Case k ď p . Let W p r , s q be the subspace of V of dimension k generated by t e , e , ¨ ¨ ¨ , e r ; e r ` ` e n ´ k ` r ` , ¨ ¨ ¨ , e k ´ s ` e n ´ s ; e n ´ s ` , ¨ ¨ ¨ , e n u . Then dim F W r , s “ k , dim F p W r , s X V ` q “ r and dim F p W p r , s q X V ´ q “ s . Let ˜ η ´ p r , s q be theblock matrix ˜ D D D D ¸ where D and D are matrices of size k ˆ k and p n ´ k q ˆ p n ´ k q respectively, and D “ ˜ I k ´ s ¸ , D “ ˜ I n ´ k ´ s ¸ D “ ˜ I s ¸ , D “ ˜ I k ´ r ¸ . Then t ˜ η ´ p r , s q u is a complete set of representatives of the double coset space H z G { P . Takinginverse, we thus get a complete set of representatives t ˜ η p r , s q u of P z G { H .We then describe the relevant data for these general P -orbits in G ¨ e . For p r , s q P I kp , q ,let ˜ x p r , s q “ ˜ η p r , s q θ p ˜ η p r , s q q ´ P G ¨ e . Thus t ˜ x p r , s q u is a complete set of representatives of P -orbits in G ¨ e . Write w p r , s q “ ι M p P ¨ ˜ x p r , s q q . Recall that w p r , s q is left and right W M -reduced.In either case, we have that w p r , s q “ ¨˚˚˚˝ I k ´ d I d I d I n ´ k ´ d ˛‹‹‹‚ . (4.7)Thus L “ L p r , s q “ M X w p r , s q Mw ´ p r , s q is the standard Levi subgroup associated to thecomposition p k ´ d , d , d , n ´ k ´ d q of n . Denote by Q the parabolic subgroup of G n with L its Levi subgroup. We can choose, in either case, an orbit representative x p r , s q P P ¨ ˜ x p r , s q X Lw p r , s q such that x p r , s q ε “ ¨˚˚˚˝ I r , s I d I d I p ` s ´ k , q ` r ´ k ˛‹‹‹‚ . (4.8)So the group L x p r , s q consists of elements diag t A , ` , A , ´ , A , A , A , ` , A , ´ u such that $&% A , ` P G r , A , ´ P G s , A , ` P G p ` s ´ k , A , ´ P G q ` r ´ k ; A “ A P G d . INEAR PERIODS FOR UNITARY REPRESENTATIONS 13
We can also choose η p r , s q P G n such that η p r , s q θ p η p r , s q q ´ “ x p r , s q and that η ´ p r , s q ¨˚˝ A , ` A , ´ A A A , ` A , ´ ˛‹‚ η p r , s q “ ¨˚˝ A , ` A A , ` A , ´ A A , ´ ˛‹‚ P H p , q (4.9)Note that x p r , s q is the natural representative for an L -admissible Q -orbit in G ¨ e chosenin Section 4.3. Thus, by Lemma 4.1, we haveL emma For p r , s q P I kp , q , let x “ x p r , s q and η “ η p r , s q as given above. For a P R ,let µ a be the character of H “ H p , q defined in (2.1) . Form “ diag t A , ` , A , ´ , A , A , A , ` , A , ´ u P L x , we have δ Q x δ ´ { Q p m q “ ν p A , ` q p p ´ q ` s ´ r q{ ν p A , ´ q p q ´ p ` r ´ s q{ ν p A , ` q p s ´ r q{ ν p A , ´ q p r ´ s q{ , (4.10) µ η ´ a p m q “ ν p A , ` q a ν p A , ´ q ´ a ν p A , ` q a ν p A , ´ q ´ a . C orollary Let ρ “ ρ b ρ b ρ b ρ be a pure tensor representation of L. Then ρ is p L x , δ Q x δ ´ { Q µ η ´ a q -distinguished if and only if $’’&’’% ρ – ρ _ ,ρ is p G r ˆ G s , µ a `p p ` s ´ q ´ r q{ q -distinguished ,ρ is p G p ` s ´ k ˆ G q ` r ´ k , µ a `p s ´ r q{ q distinguished . (4.11)
5. Consequences of the geometric lemma5.1. The geometric lemma.
We first recall the formulation of the geometric lemmaof Bernstein and Zelevinsky in [ O ff , Theorem 4.2], and we refer the reader to loc.cit forunexplained notation.P roposition Let P “ M ˙ U be a standard parabolic subgroup of G. Let σ bea representation of M, and χ a character of H. If the representation Ind GP p σ q is p H , χ q -distinguished, then there exist a P-orbit O in P zp G ¨ e q and η P G satisfying x “ η ¨ e P O X Lw (where w “ ι M p P ¨ x q and L “ M p w q ) such that the Jacquet module r L , M p σ q is p L x , δ Q x δ ´ { Q χ η ´ q -distinguished. Here Q “ L ˙ V is the standard parabolic subgroup ofG with standard Levi subgroup L.
We retain the notation of Section 4. As a consequence of the analysis there, we for-mulate the following corollary.C orollary
Let σ resp. σ be a representation of G k resp. G n ´ k . If the repre-sentation σ ˆ σ is p H p , q , µ a q -distinguished for some p, q ě , p ` q “ n and a P R ,then there exists a pair p r , s q P I kp , q with defect d “ k ´ r ´ s such that the representationr p k ´ d , d q σ b r p d , n ´ k ´ d q σ of L is p L x , δ Q x δ ´ { Q µ η ´ a q -distinguished, where L is the standardLevi subgroup of G n associated to p k ´ d , d , d , n ´ k ´ d q , Q is the standard parabolic subgroup with L its Levi part, x “ x p r , s q is given in (4.8) and η “ η p r , s q P G n such thatx “ η ¨ e and (4.9) holds. Wenow apply Corollary 5.2 to distinction of products of essentially square-integrable repre-sentations.L emma
Let m “ t ∆ , ¨ ¨ ¨ , ∆ r u and m “ t ∆ , ¨ ¨ ¨ , ∆ s u be two multisegments.(not necessarily ordered in standard form). If ∆ ˆ ¨ ¨ ¨ ˆ ∆ r – ∆ ˆ ¨ ¨ ¨ ˆ ∆ s , then m “ m . P roof . By our assumption and the commutativity of R , we know that L p m q is a sub-quotient of λ p m q . Similarly, L p m q is a subquotient of λ p m q . That m “ m then followsfrom [ Tad2 , Theorem 5.3] (or the Bernstein-Zelevinsky theory, [
Zel , 7.1 Theorem], forLanglands classification after applying the Zelevinsky involution). Q.E.D.P roposition
Let π “ ∆ ˆ ¨ ¨ ¨ ˆ ∆ t be a representation of G n , where ∆ i “ ∆ pr a i , b i s ρ i q is an essentially square-integrable representation of G n i , i “ , ¨ ¨ ¨ , t. (Herewe assume all a i , b i are integers.) Suppose that π is p H p , q , µ a q -distinguished with p, q twononnegative integers, p ` q “ n, and a P R . Then there exist an integer c t satisfyinga t ´ ď c t ď b t such that one of the following cases must hold: Case A1.
One has a t “ c t ă b t . The representation ∆ pr a t , c t s ρ t q “ b p ∆ t q is eitherthe character ν a `p q ´ p ` q{ or the character ν ´ a `p p ´ q ` q{ of G ; and there exists i Pt , , ¨ ¨ ¨ , t ´ u , and an integer c i , a i ď c i ď b i , such that (i) one has ∆ pr a t ` , b t s ρ t q _ – ∆ pr a i , c i s ρ i q ; (ii) the representation ∆ ˆ ¨ ¨ ¨ ˆ ∆ pr c i ` , b i s ρ i q ˆ ¨ ¨ ¨ ˆ ∆ t ´ is p H p ´ n t , ` q ´ n t , µ a ` { q or p H ` p ´ n t , q ´ n t , µ a ´ { q -distinguished, depending onb p ∆ t q . Case A2.
One has a t ď c t ă b t . The representation ∆ pr a t , c t s ρ t q , with its degree n t aneven integer, is H n t { , n t { -distinguished; and there exists i P t , , ¨ ¨ ¨ , t ´ u and an integerc i , a i ď c i ď b i , such that (i) one has ∆ pr c t ` , b t s ρ t q _ – ∆ pr a i , c i s ρ i q ; (ii) the representation ∆ ˆ ¨ ¨ ¨ ˆ ∆ pr c i ` , b i s ρ i q ˆ ¨ ¨ ¨ ˆ ∆ t ´ is p H p , q , µ a q -distinguished with p “ p ´ n t ` n t { and q “ q ´ n t ` n t { . Case B1.
One has c t “ b t . The representation ∆ pr a t , c t s ρ t q “ ∆ t is either the character ν a `p q ´ p ` q{ or the character ν ´ a `p p ´ q ` q{ of G ; and the representation ∆ ˆ ¨ ¨ ¨ ˆ ∆ t ´ is p H p ´ , q , µ a ` { q or p H p , q ´ , µ a ´ { q -distinguished, depending on ∆ t . Case B2.
One has c t “ b t . The representation ∆ t is H n t { , n t { -distinguished, where n t is even; and the representation ∆ ˆ ¨ ¨ ¨ ˆ ∆ t ´ is p H p ´ n t { , q ´ n t { , µ a q -distinguished. Case C.
One has c t “ a t ´ . There exists i P t , , ¨ ¨ ¨ , t ´ u and an integer c i ,a i ď c i ď b i , such that (i) one has ∆ _ t – ∆ pr a i , c i s ρ i q ; (ii) the representation ∆ ˆ ¨ ¨ ¨ ˆ ∆ pr c i ` , b i s ρ i q ˆ ¨ ¨ ¨ ˆ ∆ t ´ is p H p ´ n t , q ´ n t , µ a q -distinguished. P roof . Write σ “ ∆ ˆ ¨ ¨ ¨ ˆ ∆ t ´ and σ “ ∆ t , and k “ n ´ n t . By Corollary 5.2, inits notation, there exists p r , s q P I kp , q with defect d “ k ´ r ´ s such that the representation r p k ´ d , d q σ b r p d , n ´ k ´ d q σ of L is p L x , δ Q x δ ´ { Q µ η a q -distinguished. By [ Zel , 9.5], the Jacquetmodule r p d , n ´ k ´ d q σ of σ is either zero or of the form ∆ pr c t ` , b t s ρ t q b ∆ pr a t , c t s ρ t q for certain integer c t with a t ´ ď c t ď b t . By [ Zel , 1.2, 1.6], there exists a filtration0 Ă V Ă ¨ ¨ ¨ Ă V “ r p k ´ d , d q σ such that each successive factor is equivalent to arepresentation of the form ∆ pr c ` , b s ρ q ˆ ¨ ¨ ¨ ˆ ∆ pr c t ´ ` , b t ´ s ρ t ´ q b ∆ pr a , c s ρ q ˆ ¨ ¨ ¨ ˆ ∆ pr a t ´ , c t ´ s ρ t ´ q , for certain integers c i such that a i ´ ď c i ď b i , i “ , ¨ ¨ ¨ , t ´
1. Therefore, there existsintegers c i , i “ , , ¨ ¨ ¨ , t , such that the pure tensor representation t ´ ź i “ ∆ pr c i ` , b i s ρ i q b t ´ ź i “ ∆ pr a i , c i s ρ i q b ∆ pr c t ` , b t s ρ t q b ∆ pr a t , c t s ρ t q is p L x , δ Q x δ ´ { Q µ η ´ a q -distinguished. By Corollary 4.4, we have ∆ pr c t ` , b t s ρ i q _ – t ´ ź i “ ∆ pr a i , c i s ρ i q . By Lemma 5.3, c i “ a i ´ i between 1 and t ´
1. So, for this i , we have ∆ pr c t ` , b i s ρ i q _ – ∆ pr a i , c i s ρ i q . (5.1)Corollary 4.4 also implies that ∆ pr a t , c t s ρ t q is p G p ` s ´ k ˆ G q ` r ´ k , µ a `p s ´ r q{ q -distinguished,(5.2)and that ∆ ˆ ¨ ¨ ¨ ˆ ∆ pr c i ` , b i s ρ i q ˆ ¨ ¨ ¨ ˆ ∆ t ´ is p G r ˆ G s , µ a `p p ´ q ` s ´ r q{ q -distinguished.(5.3)When a t ď c t ă b t , we have two subcases. If c t “ a t and the degree of ρ t equals to 1,it follows from (5.2) that p p ` s ´ k , q ` r ´ k q “ p , q or p , q . By (5.3), (5.1) and simple calculations, we then have Case A1; Otherwise, the representation ∆ pr a t , c t s ρ t q is not onedimensional. Thus, in (5.2) we have p ` s ´ k ą q ` r ´ k ą
0. By Proposition 3.11,we get that ∆ pr a t , c t s ρ t q is H n t { , n t { -distinguished with n t its degree. The rest statements ofCase A2 follow from simple calculations. Thus we have Case A2.When c t “ b t , we have two subcases. If ∆ t is a character of G , then by similararguments as in Case A1, we have Case B1. Otherwise, by similar arguments as in CaseA2, we have Case B2. In these two cases, we have d “ c i “ a i ´ c t “ a t ´
1, by (5.2), we have p ` s ´ k “ q ` r ´ k “
0. The statements ofCase C follow from (5.3), (5.1) and simple calculations. So we are done. Q.E.D.We can get a simplified version of the above proposition that is less precise but stilluseful in many applications of the geometric lemma.C orollary
Let π “ ∆ ˆ ¨ ¨ ¨ ˆ ∆ t be as above. If π is p H p , q , µ a q -distinguishedwith p, q and a as above, then either the representation ∆ t is the character ν a `p q ´ p ` q{ orthe character ν ´ a `p p ´ q ` q{ of G , or there is i P t , , ¨ ¨ ¨ , t u such that e p ∆ t q _ – b p ∆ i q . P roof . Note that in all cases other than Case B1, we have a duality relation. Q.E.D.With regard to the duality relation between extremities of segments, we have a slightlymore general proposition, whose proof is similar to that of Proposition 5.4 and is omittedhere.P roposition Let π “ ∆ ˆ ¨ ¨ ¨ ˆ ∆ t and π “ ∆ ˆ ¨ ¨ ¨ ˆ ∆ s be two produts ofessentially square-integrable representations. If π ˆ π is p H p , q , µ a q -distinguished for twononnegative integers p, q, p ` q “ n and a P R , then there are two possibilities here:(1). One has π is p H p , q , µ a q -distinguished and π is p H p , q , µ a q -distinguished forsome p i , q i and a i , i “ , ;(2). There exist i between and t, and j between and s such that e p ∆ j q _ – b p ∆ i q . Finally, we have the following result on su ffi cient conditions for distinction of inducedrepresentations due to Matringe [ Mat4 , Proposition 3.8]. It also follows directly from [ O ff ,Proposition 7.1] and Corollary 4.4.L emma Let n “ m and n “ m be even integers, let a P R . Assumethat π is p H m , m , µ a q -distinguished and π is p H m , m , µ a q -distinguished. Then π ˆ π is p H m ` m , m ` m , µ a q -distinguished.
6. Distinction of ladder representations6.1. Notations and basic facts.
The class of ladder representations was first intro-duced by Lapid and Minguez in [
LM1 ], and was further studied by Lapid and his collabo-rators in [ KL ] and [ LM2 ]. We start by review some basic facts of these representations.
INEAR PERIODS FOR UNITARY REPRESENTATIONS 17
Definitions.
Let ρ P C . By a ladder we mean a set t ∆ , ¨ ¨ ¨ , ∆ t u P O ρ suchthat b p ∆ q ą ¨ ¨ ¨ ą b p ∆ t q and e p ∆ q ą ¨ ¨ ¨ ą e p ∆ t q . (6.1)A representation π P Irr is called a ladder representation if π “ L p m q where m P O ρ is aladder. Whenever we say that m “ t ∆ , ¨ ¨ ¨ , ∆ t u P O ρ is a ladder, we implicitly assumethat m is already ordered as in (6.1).L emma If m P O ρ is a ladder, so is m _ P O ρ _ . We have L p m q _ “ L p m _ q . We introduce some more notation. For a ladder m “ t ∆ , ¨ ¨ ¨ , ∆ t u P O ρ ordered asin (6.1) and π “ L p m q , we shall denote b p ∆ q by b p π q , called the beginning of the ladderrepresentaion π ; denote e p ∆ t q by e p π q , called the end of π . We shall denote ht p π q “ t ,called the height of π .We say that π is a decreasing (resp. increasing ) ladder representation if l p ∆ q ě ¨ ¨ ¨ ě l p ∆ t q p resp. l p ∆ q ď ¨ ¨ ¨ ď l p ∆ t qq . We say that π is a left aligned (resp. right aligned ) representation if b p ∆ i q “ b p ∆ i ` q ` e p ∆ i q “ e p ∆ i ` q ` i “ , ¨ ¨ ¨ , t ´
1. Note that left aligned repreesentations aredecreasing ladder representations and right aligned representations are increasing ladderrepresentations.A ladder representation is called an essentially Speh representation if it is both leftaligned and right aligned. Let ∆ be an essentially square-integrable representation of G d and k a positive integer. Then m “ t ν p k ´ q{ ∆ , ν p k ´ q{ ∆ , ¨ ¨ ¨ , ν p ´ k q{ ∆ u is a ladder, andthe ladder representation L p m q is an essentially Speh representation, which we denote bySp p ∆ , k q . All essentially Speh representations can be obtained in this manner.Let us further write ∆ i “ ∆ pr a i , b i s ρ q . (The a i ’s are integers by our convention.) By a division of π as two ladder representations π and π , denoted by π “ π \ π , we meanthat there exist integers c i with a i ´ ď c i ď b i , i “ , ¨ ¨ ¨ , t , such that c ą c ą ¨ ¨ ¨ ą c t and that π “ L p ∆ pr a , c s ρ q , ¨ ¨ ¨ , ∆ pr a t , c t s ρ qq ,π “ L p ∆ pr c ` , b s ρ q , ¨ ¨ ¨ , ∆ pr c t ` , b t s ρ qq . Note that if π is an essentially Speh representation and π “ π \ π with neither π nor π the trivial representation of G , then we have b p π q “ b p π q and e p π q “ e p π q .6.1.2. Standard module.
One useful property of ladder representations is that the re-lation between them and their standard modules is explicit. Let m “ t ∆ , ¨ ¨ ¨ , ∆ t u P O ρ bea ladder with ∆ i “ ∆ pr a i , b i sq ρ . Set K i “ ∆ ˆ ¨ ¨ ¨ ˆ ∆ i ´ ˆ ∆ pr a i ` , b i s ρ q ˆ ∆ pr a i , b i ` s ρ q ˆ ∆ i ` ˆ ¨ ¨ ¨ ˆ ∆ t , for i “ , ¨ ¨ ¨ , t ´
1. (By our convention, K i “ a i ą b i ` ` LM1 , Theorem 1]we have P roposition With the above notation let K be the kernel of the projection λ p m q Ñ L p m q . Then K “ ř t ´ i “ K i . Jacquet modules.
The Jacquet modules of ladder representations were com-puted in [ KL , Corollary 2.2], where it is shown that the Jacquet module of a ladder rep-resentation is semisimple, multiplicity free, and that its irreducible constituents are them-selves tensor products of ladder representations. For us, we need only the Jacquet modulewith respect to a maximal parabolic subgroup. We record the result in [ KL ] here. Let P “ M ˙ U be the standard parabolic subgroup of G n associated to p k , n ´ k q .P roposition Let m “ t ∆ , ¨ ¨ ¨ , ∆ t u P O ρ be a ladder with ∆ i “ r a i , b i s ρ , and π “ L p m q . Then r M , G p π q “ ÿ π “ π \ π π b π , where the summation takes over all divisions of π such that the degree of π is n ´ k andthat the degree of π is k. Bernstein-Zelevinsky derivatives.
The full derivative of a ladder representationwas computed in [
LM1 , Theorem 14], where it is shown that the semisimplification of allof the derivatives of a ladder representation consists of ladder representations of smallergroups. In particular, the derivatives of a left aligned representation take simple forms,which we recall here.L emma
Let ρ P C p G d q , and m “ t ∆ , ¨ ¨ ¨ , ∆ t u P O ρ be a ladder with ∆ i “ ∆ pr a i , b i s ρ q . Suppose that π “ L p m q is a left aligned representation. If k is not divided byd, then π p k q “ . If k “ rd, then π p k q “ L p ∆ pr a ` r , b s ρ q , ∆ , ¨ ¨ ¨ , ∆ t q . In this section wedraw another consequence of Corollary 5.2 when applied to products of essentially Spehrepresentations. Note first that, as a consequence of Lemma 6.4, we haveL emma
Let σ and π i be left aligned representations of G n and G n i , i “ , ¨ ¨ ¨ , k.If σ – π ˆ ¨ ¨ ¨ ˆ π k , then we have k “ . In view of Lemma 6.5 and the description of Jacquet modules of a ladder representa-tion in Proposition 6.3, we can formulate the following proposition, whose proof is similarto that of Proposition 5.4 and is omitted here.P roposition
Let π “ π ˆ ¨ ¨ ¨ ˆ π t be a representation of G n , where π i is a Spehrepresentation of G n i , i “ , ¨ ¨ ¨ , t. Assume that π is p H p , q , µ a q -distinguished with p, q twononnegative integers, p ` q “ n and a P R . Then there exist a division of π t as two ladderrepresentations π t and π t , π t “ π t \ π t , with degrees n t and n t respectively, such that oneof the following cases must hold: INEAR PERIODS FOR UNITARY REPRESENTATIONS 19
Case A . The representation π t is neither π t nor the trivial representation of G . Thereexists i , ď i ď t ´ , and a division of π i as two ladder representations π i and π i , π i “ π i \ π i , such that (i) π t is p H r , s , µ a `p r ´ s ` q ´ p q{ q -distinguished, for two nonnegative integers r , s ě ,r ` s “ n t ; (ii) π t – π i ; (iii) the representation π ˆ ¨ ¨ ¨ ˆ π i ´ ˆ π i ˆ π i ` ˆ ¨ ¨ ¨ ˆ π t ´ is p H r , s , µ a `p s ´ r ` p ´ q q{ q -distinguished, for two nonnegative integers r , s ě ,r ` s “ n ´ n t ´ n t . Case B . One has π t “ π t is p H r , s , µ a `p r ´ s ` q ´ p q{ q -distinguished, for two nonnegativeintegers r, s ě , r ` s “ n t , and the representation π ˆ ¨ ¨ ¨ ˆ π t ´ is p H r , s , µ a `p s ´ r ` p ´ q q{ q -distinguished, for two nonnegative integers r , s ě , r ` s “ n ´ n t . Case C . The representation π t is the trivial representation of G , so π t “ π t . Thereexists i , ď i ď t ´ , and a division of π i as two ladder representations π i and π i , π i “ π i \ π i , such that (i) π _ t – π i ; (ii) the representation π ˆ ¨ ¨ ¨ ˆ π i ´ ˆ π i ˆ π i ` ˆ ¨ ¨ ¨ ˆ π t ´ is p H r , s , µ a `p s ´ r ` p ´ q q{ q -distinguished, for two nonnegative integers r , s ě , r ` s “ n ´ n t . To apply this proposition, it is desirable to change the ordering of the representationsin the product. The commutativity of a product of two ladder representations was studiedby Lapid and Minguez in [
LM2 ]. Here we present a special case of their results that issu ffi cient for our purpose.L emma Let ρ P C . Let m , m P O ρ be two ladders, with m “ t ∆ , , ¨ ¨ ¨ , ∆ , t u and m “ t ∆ , , ¨ ¨ ¨ , ∆ , t u . Suppose that L p m q is an essentially Speh representation and L p m q is a right aligned representation. If e p ∆ , t q “ e p ∆ , t q and t ď t , or e p ∆ , t q “ e p ∆ , t q and b p ∆ , t q ď b p ∆ , t q , then L p m q ˆ L p m q is irreducible and L p m q ˆ L p m q “ L p m q ˆ L p m q . P roof . Note that the results in [ LM2 ] are expressed in terms of Zelevinsky classifica-tion. By the combinatorial description of Zelevinsky involution by Moeglin-Waldspurger[
MeJLW ] (see also [
LM1 , § m t and m t of m and m . The assertion then follows fromProposition 6.20 and Lemma 6.21 in [ LM2 ]. Q.E.D.
The aim of this section is toprove Theorem 6.11 that generalizes Corollary 3.10 to the case of essentially Speh repre-sentations. A key ingredient in its proof is Proposition 6.10 which also gives a proof of the‘only if’ part of Theorem 1.1.P roposition
Let π be an essentially Speh representation of G n which is not onedimensional. Assume that π is p H p , q , µ a q -distinguished for two positive integers p, q withp ` q “ n and some a P R . Then π is self-dual. P roof . Write π “ L p m q with m “ t ∆ , ¨ ¨ ¨ , ∆ t u a ladder. Then ∆ ˆ ¨ ¨ ¨ ˆ ∆ t is p H p , q , µ a q -distinguished by our assumption. As π is not one dimensional, the degree of ∆ t is not 1. Thus by Corollary 5.5, there exists i P t , , ¨ ¨ ¨ , t u such that b p ∆ i q – e p ∆ t q _ .We claim that i “
1. If so, by Lemma 6.1, we see easily that π is self-dual. In fact, ifotherwise i ą
1, we then apply Proposition 5.6 to π ˆ π , where π “ ∆ ˆ ¨ ¨ ¨ ˆ ∆ i ´ and π “ ∆ i ˆ ¨ ¨ ¨ ˆ ∆ t . We get either that b p ∆ j q – e p ∆ i ´ q _ for some j P t , , ¨ ¨ ¨ , i ´ u ,or that b p ∆ j q – e p ∆ k q _ for some j P t , , ¨ ¨ ¨ , i ´ u and some k P t i , i ` , ¨ ¨ ¨ , t u . Ineither case, this will contradict with the relation b p ∆ i q – e p ∆ t q _ . Q.E.D.P roposition Let π be an essentially Speh representation of G n which is not onedimensional. If for some a P R , a ‰ , the representation π is p H p , q , µ a q -distinguished fortwo positive integers p, q with p ` q “ n. Then we have p “ q. P roof . This follows easily from Proposition 6.8 and consideration of the central char-acter of π . Q.E.D.P roposition Let π “ Sp p ∆ , l q be an essentially Speh representation of G n , where ∆ is an essentially square-integrable representaion of G d , d ą , and l is a positive integer.Assume that π is H p , q -distinguished or p H p , q , µ ´ { q for two positive integers p, q, p ` q “ n. Then the degree d of ∆ is even, and ∆ is H d { , d { -distinguished; also we have p “ q. P roof . We prove this by induction on l . The case l “ π is p H p , q , µ a q -distinguished with a “ ´ {
2. By Proposition6.8 we know that π is self-dual, hence ∆ is also self-dual. In view of Lemma 3.1, we mayassume that p ě q . Now we have Hom P n X H p , q p π, q ‰
0. By [ BZ , § π | P n of π to P n has a filtration which has composition factors p Φ ` q i ´ Ψ ` p π p i q q , i “ , ¨ ¨ ¨ , h ,where π p h q is the highest derivative of π . We first analyze linear functionals on these factorsapces using the theory of Bernstein-Zelevinsky derivatives.(1) When i “ k is even. If q ą k and p ą k ´
1, by applying (3.3) repeatly, we haveHom P n X H p , q pp Φ ` q i ´ Ψ ` p π p i q q , µ a q – Hom P n ´ i ` X H q ´ k , p ´ k ` p Ψ ` p π p i q q , µ ´ a ´ { q (6.2) – Hom H q ´ k , p ´ k p ν { π p i q , µ ´ a ´ { q . Otherwise, there exists i ě P n X H p , q pp Φ ` q i ´ Ψ ` p π p i q q , µ a q – Hom P n ´ i ` i ` pp Φ ` q i Ψ ` p π p i q q , µ a q , (6.3)where a “ a or ´ a ´ { i odd or even. INEAR PERIODS FOR UNITARY REPRESENTATIONS 21 (2) When i “ k ` q ą k and p ą k , by applying (3.3) repeatly, we haveHom P n X H p , q pp Φ ` q i ´ Ψ ` p π p i q q , µ a q – Hom P n ´ i ` X H p ´ k , q ´ k p Ψ ` p π p i q q , µ a q (6.4) – Hom H p ´ k , q ´ k ´ p ν { π p i q , µ a q . Otherwise, there exists i ě P n X H p , q pp Φ ` q i ´ Ψ ` p π p i q q , µ a q – Hom P n ´ i ` i ` pp Φ ` q i Ψ ` p π p i q q , µ a q , (6.5)where a “ a or ´ a ´ { i even or odd.We claim that the factor spaces corresponding to non-highest derivatives contributenothing, that is, we haveHom P n X H p , q pp Φ ` q i ´ Ψ ` p π p i q q , µ a q “ , for all 1 ď i ă h . (6.6)We shall discuss separately according to i is even or odd, a “ ´ {
2. Notefirst that, by Lemma 6.4, when 1 ď i ă h , the i -th derivative π p i q is either 0 or a ladderrepresentation of the form L p ∆ ˆ ν p l ´ q{ ∆ ˆ ¨ ¨ ¨ ˆ ν p ´ l q{ ∆ q , (6.7)where the segement of ∆ is a proper subsegment of that of ν p l ´ q{ ∆ with the same end.In particular, π p i q is either 0 or an irreducible representation. Thus, if we are in the casewhere (6.3) or (6.5) is true, thenHom P n X H p , q pp Φ ` q i ´ Ψ ` p π p i q q , µ a q – Hom P n ´ i ` i ` pp Φ ` q i Ψ ` p π p i q q , µ a q“ , as the representation p Φ ` q i Ψ ` p π p i q q is either 0 or an irreducible representation of P n ´ i ` i ` that is not one dimensional by [ BZ , 3.3 Remarks].Now we deal with the case where (6.2) or (6.4) is true. Note that, from (6.7), we knowthat ν { π p i q is either 0 or the unique irreducible quotient of ν { ∆ ˆ Sp p ∆ , l ´ q . Wediscuss as follows.Case (1) where a “ i “ k is even. By (6.2), it su ffi ces to show thatHom H q ´ k , p ´ k p ν { ∆ ˆ Sp p ∆ , l ´ q , µ ´ { q “ . (6.8)Assume, on the contrary, that ν { ∆ ˆ Sp p ∆ , l ´ q is p H q ´ k , p ´ k , µ ´ { q -distinguished. Notethat none of the ends of the segments of Sp p ∆ , l ´ q is dual to the beginning of ν { ∆ .Thus, by Proposition 6.6, we have that ν { ∆ is p H r , s , µ p s ´ p ´ r ` q ´ q{ q -distinguished andSp p ∆ , l ´ q is p H q ´ k ´ r , p ´ k ´ s , µ p s ´ r ´ q{ q -distinguished. (Case A and Case C are elimi-nated.) If the degree of ν { ∆ is greater than 1, then ν { ∆ is self-dual by Proposition3.11. This is absurd because the central character of ν { ∆ has positive real part; If thedegree of ν { ∆ is 1, then p r , s q “ p , q or p , q . If r “ s “
0, then Sp p ∆ , l ´ q is p H q ´ k ´ , p ´ k , µ ´ q -distinguished. Thus we have p “ q ´ p ě q ; If r “ s “
1, then Sp p ∆ , l ´ q is p H q ´ k , p ´ k ´ , q -distinguished. So, by induction hypothesis, we have p ´ “ q . Thisforces ν { ∆ “ ν { , that is e p ν p l ´ q{ ∆ q “ , the trivial character of G . This is impossi-ble as ∆ is self-dual and its degree d is greater than 1. Case (2) where a “ i “ k ` H p ´ k , q ´ k ´ p ν { π p i q , q “ , (6.9)as the central character of ν { π p i q has positive real part when i ă h .The arguments for the remaining two cases where a “ ´ { i is even or odd aresimilar to that of the above two cases; and we omit here. So we have proved (6.6).By Lemma 6.4, we know that the highest derivative of π is π p d q and ν { π p d q “ Sp p ∆ , l ´ q . Now we haveHom P n X H p , q pp Φ ` q d ´ Ψ ` p π p d q q , µ a q ‰ , (6.10)where a “ ´ {
2. We use once again the theory of derivatives to analyze the left handside of (6.10). Recall that we have already assumed p ě q . By similar analysis as above,the only possible case is when (6.2) is true, that is, d is even andHom P n X H p , q pp Φ ` q i ´ Ψ ` p π p d q q , µ a q – Hom H q ´ k , p ´ k p Sp p ∆ , l ´ q , µ ´ a ´ { q . Note that when a “ ´ { ´ a ´ { “ ´ { heorem Let π be an essentially Speh representation of G n that is not one di-mensional. If π is p H p , q , µ a q -distinguished for two positive integers p, q with p ` q “ nand a P R , then we have p “ q. The following result, which is a consequence of the result of Gan, relates the study oflinear periods and Shalika periods for essentially Speh representations.P roposition
Let π be an essentially Speh representation of G m that is not onedimensional. Write π “ L p m q with m “ t ∆ , ¨ ¨ ¨ , ∆ t u P O ρ and ρ P C . For a P R , assumethat the character ν a or ν ´ a of G does not belong to the setS “ t ν ´ { b p ∆ t q , ν ´ { b p ∆ t ´ q , ¨ ¨ ¨ , ν ´ { b p ∆ qu . Then the followings are equivalent: (1) π is p H m , m , µ a q -disintuished; (2) π is p S n , ψ S n q -distinguished.In particular, if π is p H m , m , µ a q -distinguished, then π is H m , m -distinguished. P roof . In either (1) or (2), the representation π is self-dual by Proposition 6.8 andwork of Jacquet-Rallis [ JR ], which we assume is the case. By the argument in the proofof Corollary 3.6, it su ffi ces to show that Θ pp ν a π q _ q “ ν a π . By [ FSX , Corollary 1.5], itsu ffi ces to show that the Godement-Jacquet L -function L p s , ν a π q or L p s , ν ´ a π q does not has INEAR PERIODS FOR UNITARY REPRESENTATIONS 23 a pole at s “ {
2. By the computations in [
JPSS ], we have L p s , ν ˘ a π q “ t ź i “ L p s , ν ˘ a e p ∆ i qq “ t ź i “ L p s , ν ˘ a b p ∆ i q _ q . (6.11)The last equality in (6.11) comes from the self-dualness property of π . We know that, for acuspidal representation ρ of G d , the L -function L p s , ρ q has a pole at s “ { d “ ρ “ ν ´ { . It follows easily that the L -function L p s , ν a π q (resp. L p s , ν ´ a π q ) hasa pole at s “ { ν a (resp. ν ´ a ) of G belongs to S . Q.E.D. The main result of this section isTheorem 6.18, which asserts that under certain restrictions on p , q and a , p H p , q , µ a q -distinguished left aligned representations (or right aligned representations by symmetry)are essentially Speh. The reader is advised to skip this section for the first reading, godirectly to the proof of the classification of distinguished unitary representations of Arthurtype in Theorem 7.3, and then read this section when necessary.P roposition Let ρ P C p G d q , d ą , and m “ t ∆ , ¨ ¨ ¨ , ∆ t u P O ρ be a ladder.Assume that π “ L p m q is a decreasing or an increasing ladder representation of G n . If π is p H p , q , µ a q -distinguished for two nonnegative integers p, q, p ` q “ n and some a P R ,then all the l p ∆ i q ’s are the same. Moreover, π is self-dual. P roof . In view of Lemma 3.1, we may assume that l p ∆ q ď l p ∆ q ď ¨ ¨ ¨ ď l p ∆ t q . If p or q is zero, then π is a one dimensional representation of G n , hence all l p ∆ i q are 1. So weassume that p , q ą
0. By our assumption, the representation ∆ ˆ ¨ ¨ ¨ ˆ ∆ t is p H p , q , µ a q -distinguished. We now appeal to Proposition 5.4. Write ∆ i “ ∆ pr a i , b i s ρ q , i “ , , ¨ ¨ ¨ , t .Note that by our assumption that d ą
1, Case A1 and Case B1 cannot happen.Case A2. In this case We have a t ď c t ă b t , and ∆ pr a t , c t s ρ q is self-dual. Thuswe have ν a t ρ – ν ´ c t ρ _ , and consequently p a t ` c t q d ` p w ρ q “
0. We also have ∆ pr c t ` , b t s ρ q _ – ∆ pr a i , c i s ρ q for some i ă t and c i ě a i . Thus we get ν a i ρ – ν ´ b t ρ _ ,and then p a i ` b t q d ` p w ρ q “
0. But this is absurd because a i ą a t and b t ą c t .Case B2. In this case we have c t “ b t , and ∆ pr a t , b t s ρ q is self-dual. Thus we have ν a t ρ – ν ´ b t ρ _ , and consequently p a t ` b t q d ` p w ρ q “
0. We also have ∆ ˆ ¨ ¨ ¨ ˆ ∆ t ´ is p H p , q , µ a q -distinguished for some p , q and a . If t “
1, there is nothing to be proved.If t ą
1, by Corollay 5.5, we get that p ν b t ´ ρ q _ – ν a i ρ for some 1 ď i ď t ´
1. Thus weget p a i ` b t ´ q d ` p w ρ q “
0. This is absurd because a i ą a t and b t ´ ą b t .So the only possible case is Case C. We then have ∆ pr a t , b t sq _ – ∆ pr a i , c i sq for i ă t and certain a i ď c i ď b i . Note that, by our assumption, we have l p ∆ i q ď l p ∆ t q . Thus wehave l p ∆ i q “ l p ∆ t q . We claim that i “
1. If this is the case, then all l p ∆ i q ’s will be the sameby our assumption. Indeed, if i ą
1, consider the p H p , q , µ a q -distinguished representaion p ∆ ˆ ¨ ¨ ¨ ˆ ∆ i ´ q ˆ p ∆ i ˆ ¨ ¨ ¨ ˆ ∆ t q . By Propostion 5.6, either we have e p ∆ i ´ q _ – b p ∆ a q with 1 ď a ď t ´
1, or we have e p ∆ c q _ – b p ∆ b q with 1 ď b ď t ´ i ď c ď t . We then get a contradiction as in Case A2 or B2. The assertion on the self-dualness property follows from a repeated analysis asabove. Q.E.D.If we drop the assumption that d ą
1, the situation is complicated by the possibleoccurrence of Case A1 or Case B1. However, we still have the following result on theshape of right aligned representations when it is distinguished.P roposition
Let ρ be a character of G , and m P O ρ be a ladder. Assume that π “ L p m q is a right aligned representation of G n . If π is p H p , q , µ a q -distinguished for twononnegative integers p, q, p ` q “ n and some a P R , then either(1) we have m “ t ∆ , ¨ ¨ ¨ , ∆ i , ∆ i ` , ¨ ¨ ¨ , ∆ i ` i , ∆ i ` i ` , ¨ ¨ ¨ , ∆ i ` i ` i u (6.12) with i , i and i ě , such that l p ∆ k q “ when ď k ď i , l p ∆ i ` k q “ l ą when ď k ď i , l p ∆ i ` i ` k qq “ l ` when ď k ď i , and that e p ∆ i ` i ` i q _ – b p ∆ i ` q (SeeFigure 1 for an example),or(2) we have m “ t ∆ , ¨ ¨ ¨ , ∆ i , ∆ i ` , ¨ ¨ ¨ , ∆ i ` i u (6.13) with i and i ą , such that l p ∆ k q “ when ď k ď i , l p ∆ i ` k q “ when ď k ď i ,and that e p ∆ i ` i q _ – b p ∆ q (See Figure 2 for an example). P roof . Write m “ t ∆ , ¨ ¨ ¨ , ∆ t u . If l p ∆ t q “
1, then π is a one dimensional repre-sentation and m is of the form (6.12) with i “ i “
0. If l p ∆ t q “ l p ∆ q “
2, then π isan essentially Speh representation. It follows from Proposition 6.8 that m is of the form(6.12) with i “ i “
0. If l p ∆ t q “ l p ∆ q “
1, then π can be realized as the uniqueirreducible quotient of π ˆ π , where π is a one dimensional representation and π is anessentially Speh representation of length 2. Thus π ˆ π is p H p , q , µ a q -distinguished. ByProposition 6.6, either m is of the form (6.13) (Case A), or of the form (6.12) with i “ i ą i ą l “ π and π . If l p ∆ t q ą
2, then we apply Proposition 5.4 to the product ∆ ˆ ¨ ¨ ¨ ˆ ∆ t and discuss case by case. Note first that Case A2 cannot happen by similararguments as those in Proposition 6.13; Case B1 cannot happen by our assumption on ∆ t .If in the remaining cases, it follows from Corollary 5.5 and arguments similar to those inProposition 6.13 that m is of the form (6.12). Q.E.D.As shown by Proposition 6.14, distinguished left aligned representations need not beessentially Speh in general. Fortunately, after imposing certain conditions on p , q , and a ,we can show that distinguished left aligned representations (or right aligned representationsby symmetry) are essentially Speh.The following lemma is a simple consequence of Lemma 3.2. INEAR PERIODS FOR UNITARY REPRESENTATIONS 25 ˝‚ ˝˝ ˝ ˝˝ ˝ ‚
Figure 1.
An example of a ladder of the form (6.12) with i “ i “ i “ ‚˝˝ ˝˝ ‚ Figure 2.
An example of a ladder of the form (6.13) with i “ i “ emma Let π be a one dimensional representation of G n . If π is p H p , q , µ p q ´ p q{ q -distinguished with p ` q “ n, then π is either the trivial character of G n or the character ν ´ n { of G n . In particular, b p π q is either ν p n ´ q{ or ν ´ { (of G ). L emma Keep the notation as in Proposition 6.14, let π “ L p m q with m of theform (6.13) (see Figure 2). Then π cannot be p H p , q , µ p q ´ p q{ q -distinguished. P roof . We assume on the contrary that π is p H p , q , µ p q ´ p q{ q -distinguished. Note that π can be realized as the unique quotient of π ˆ π , where π is a one dimensional repre-sentation, π is an essentially Speh representation of length 2, and e p π q – b p π q _ . Thus, π ˆ π is p H p , q , µ p q ´ p q{ q -disintuished. By Proposition 6.6, there exist divisions of π and π , π “ π \ π and π “ π \ π respectively, such that, among other things, π is p H r , s , µ p s ´ r q{ q -distinguished for two nonnegative integers r and s . Note that π is not thetrivial representation of G by our assumption and Proposition 6.8. We shall discuss furtheraccording to the value of r and s .(1) If exactly one of r and s is 0, then π is the character ν ´ n { of G n . Thus b p π q “ ν ´ { , e p π q “ ν { . By Proposition 6.6, we also have π – π . Thus b p π q “ e p π q _ “ ν ´ { “ b p π q , which is absurd.(2) If r ą s ą
0, then π is the character of G r . Thus b p π q “ ν r ´ { and e p π q “ ν ´ r ` { . By Proposition 6.6, we have π – π . Thus we have b p π q “ ´ r ´ { π is also a one dimensional representation which is p H r , s , µ p s ´ r q{ q -distinguished for certain nonnegative integers r and s . By Lemma 6.15, we have that b p π q “ ν p n ´ q{ or ν ´ { , which contradicts with the fact that b p π q ă ν ´ r ´ { with r apositive integer.(3) If r “ s “
0, we have two subcases according to whether π is a one dimensionalrepresentaiton or not. If it is, one can easily check that b p π q “ ν ´ { , which will con-tradicts with Lemma 6.15. If it is not, we see that the contragredient of π corresponds toa ladder of the form (6.13). By the proof of Proposition 6.14, we have b p π q _ is either b p π q ` e p π q ´
1. But, note that b p π q “ e p π q ´
2. It follows from the rela-tion π – π that b p π q _ “ b p π q `
2. This is absurd as π is one dimensional and b p π q ą e p π q . Q.E.D.P roposition Let ρ be a character of G , and m P O ρ be a ladder. Assume that π “ L p m q is a left aligned representation of G n . If π is p H p , q , µ p p ´ q q{ q -distinguished fortwo nonnegative integers p, q, p ` q “ n, then π is an essentially Speh representation. P roof . The case where one of p , q is zero is clear, so we assume p and q are twopositive integers. We may further assume that p ď q by Lemma 3.1. By considering thecontragredient π _ “ L p m _ q , we see from Lemma 6.16 that m _ is of the form (6.12). So,we may write m “ t ∆ , ¨ ¨ ¨ , ∆ i , ∆ i ` , ¨ ¨ ¨ , ∆ i ` i , ∆ i ` i ` , ¨ ¨ ¨ , ∆ i ` i ` i u with i , i and i ě
0, such that l p ∆ k q “ l ą ď k ď i , l p ∆ i ` k q “ l ´ ď k ď i , l p ∆ i ` i ` k qq “ ď k ď i , and that e p ∆ i ` i q _ – b p ∆ q .We may as well assume that i and i are not all zero. Our first step is to show that i “
0. If not so, we realize π , in the obvious way, as the unique irreducible quotient of π ˆ π ˆ π with π i an essentially Speh representation for each i , such that π is a characterof G n , n ą
0, and that at least one of π and π is not the trivial representation of G .By our assumption on π , the representation π ˆ π ˆ π is p H p , q , µ p p ´ q q{ q -distinguished.Now we appeal to Propositio 6.6. Note that as i ą e p ∆ i ` i ` i q is not dual to b p ∆ q or b p ∆ i ` q . We can conclude that π is p H r , s , µ p r ´ s q{ q -distinguished with respect to twononnegative integers r and s . (Case A and Case C are eliminated.) By Lemma 6.15, π is either the trivial represntation of G n or the character ν n { of G n . In particular, b p ∆ i ` i ` q “ ν p n ´ q{ or ν n ´ { . But this will contradict with the fact that e p ∆ i ` i q _ – b p ∆ q .Our next step is to show that i “ i “
0. Assume on the contrary that i ą i ą
0. By our assumption on π , the representation π _ “ L p m _ q is p H p , q , µ p q ´ p q{ q -distinguished. Thus, the representation ∆ _ i ` i ˆ ¨ ¨ ¨ ˆ ∆ _ i ` ˆ ∆ _ i ˆ ¨ ¨ ¨ ˆ ∆ _ is p H p , q , µ p q ´ p q{ q -distinguished. By Proposition 5.4, we deduce that b p ∆ _ q – e p ∆ q _ is the character ν q ´ p ` { or ν p ´ q ` { of G . (This is the consequence of Case A1; CaseA2 and Case B2 are eliminated by arguments similar to that in Proposition 6.13; CaseB1 and Case C are eliminated by our assumptions.) It follows easily from the condition INEAR PERIODS FOR UNITARY REPRESENTATIONS 27 b p ∆ q – e p ∆ i ` i q _ and the assumption p ď q that e p ∆ q _ “ ν p ´ q ` { . Hence we have e p ∆ q “ ν q ´ p ´ { .We show that i “ q ´ p and e p ∆ i q “ ν { by consideration on the central character of π . In fact, on the one hand, we see from the assumption on m and the fact e p ∆ q “ ν q ´ p ´ { that the central character w π of π is ν a where a “ p q ´ p q i ´ i {
2; on the othe hand, as π is p H p , q , µ p p ´ q q{ q -distinguished, we have w π “ ν a where a “ p q ´ p q {
2. Thus theassertion follows. Also, from the fact that e p ∆ i ` i q _ – b p ∆ q , we get that b p ∆ q “ ν i ` { .Thus, l p ∆ q “ q ´ p ´ i ą
2, in particular i ą i .Now, as in the first step, we have that π ˆ π is p H p , q , µ p p ´ q q{ q -distinguished, where π “ L p ∆ , ¨ ¨ ¨ , ∆ i q and π “ L p ∆ i ` , ¨ ¨ ¨ , ∆ i ` i q . We appeal to Proposition 6.6 again,and claim that Case A and Case B cannot happen. In fact, if Case A or Case B happens,there will be a division of π as π and π , where π is not the trivial representation of G , such that π is p H r , s , µ p r ´ s q{ q -distinguished for two nonnegative integers r and s . Inparticular, the central character w π of π has nonnegative real part. But this will contradictwith the fact that e p ∆ i ` q “ ν ´ { . So, there exists a division of π as two ladder repre-sentations π and π such that π – π and that π is p H p ´ n , q ´ n , µ p p ´ q q{ q -distinguished.Note that π is a right aligned representation, and is not a one dimensional representationdue to the fact that i ą i . By Proposition 6.14, we then get a contradiction as we cancheck easily that the ladder m of π is not of the form (6.12) or (6.13). Q.E.D.Finally, combining Proposition 6.13 and 6.17, we getT heorem Let π be a left aligned (resp. right aligned) representation of G n . If π is p H p , q , µ p p ´ q q{ q (resp. p H p , q , µ p q ´ p q{ q )-distinguished for two nonnegative integers p, q,p ` q “ n, then π is an essentially Speh representation.
7. Distinction in the unitary dual7.1. The case of Speh representations.
We now classify distinguished Speh repre-sentations in terms of distinguished discrete series. In fact, we will do it for essentiallySpeh representations.T heorem
Let n “ m, and Sp p ∆ , k q be an essentially Speh representation of G n ,where ∆ is an essentially square-integrable representation of G d with d ą , and k isa positive integer. Then Sp p ∆ , k q is H m , m -distinguished if and only if d is even and ∆ isH d { , d { -distinguished. P roof . One direction has been proved in Proposition 6.10. We now assume that d iseven and that ∆ is H d { , d { -distinguished. By [ O ff , Proposition 7.2], which is based on thework of Blanc and Delorme [ BD ], the representation ν p k ´ q{ ∆ ˆ ν p k ´ q{ ∆ ˆ ¨ ¨ ¨ ˆ ν p ´ k q{ ∆ (7.1) is H m , m -distinguished. (The distinguishedness of ∆ is unnecessary when k is even).We havethe following exact sequence of representations of G n ,0 Ñ K Ñ ν p k ´ q{ ∆ ˆ ν p k ´ q{ ∆ ˆ ¨ ¨ ¨ ˆ ν p ´ k q{ ∆ Ñ Sp p ∆ , k q Ñ , (7.2)where the kernel K “ ř k ´ i “ K i is given explicitly in Proposition 6.2. To show that Sp p ∆ , k q is H m , m -distinguished, it su ffi ces to show that each K i is not H m , m -distinguished. Write therepresentation (7.1) as ∆ pr a , b s ρ q ˆ ¨ ¨ ¨ ˆ ∆ pr a k , b k s ρ q , here the cuspidal representation ρ is taken to be self-dual and thus a i and b i , i “ , , ¨ ¨ ¨ , k need not be integers. So we have a i ` “ a i ´ , b i ` “ b i ´ , i “ , ¨ ¨ ¨ , k ´ a i ` b k ` ´ i “ , i “ , ¨ ¨ ¨ , k . We further omit the subscript ρ in the sequel. Recall that, by Proposition 6.2, K i “ ∆ pr a , b sq ˆ ¨ ¨ ¨ ˆ ∆ pr a i ` , b i sq ˆ ∆ pr a i , b i ` sq ˆ ¨ ¨ ¨ ˆ ∆ pr a k , b k sq . If i ` ď p k ` q{ K i is H m , m -distinguished, by applying Proposition 5.4 repeatly,we get that ∆ pr a i ` , b i sq ˆ ∆ pr a i , b i ` sq ˆ ¨ ¨ ¨ ˆ ∆ pr a k ` ´ i , b k ` ´ i sq is H m , m -distinguishedfor certain m . (In each step, only Case C is possible.) When we apply Proposition 5.4 onceagain, still, only Case C is possible. But this is absurd as l p ∆ pr a i , b i ` sqq ă l p ∆ q . Similararguments can show that K i is not H m , m -distinguished if i ě p k ` q{ k is even and i “ k {
2. In what follows, to save notation,we sometimes write H -distinguished for H m , m -distinguished when there is no need toaddress m . If K i is H m , m -distinguished, by applying Proposition 5.4 repeatly, we get that ∆ pr a i ` , b i sq ˆ ∆ pr a i , b i ` sq is H -distinguished. This in turn implies that both ∆ pr a i ` , b i sq and ∆ pr a i , b i ` sq are H -distinguished by Proposition 5.6. Let us write ∆ “ St p ρ, l q . Thenby our assumption on i , we have ∆ pr a i , b i ` sq “ St p ρ, l ´ q and ∆ pr a i ` , b i sq “ St p ρ, l ` q .By [ Mat2 , Theorem 6.1], we can conclude that St p ρ, l q is H -distinguished if and only ifSt p ρ, l ´ q (or St p ρ, l ` q ) is not H -distinguished. Actually, as ρ is self-dual, the L -function L p s , φ p ρ q b φ p ρ qq has a simple pole at s “
0, where φ p ρ q is the Langlands parameter of ρ .By the factorization L p s , φ p ρ q b φ p ρ qq “ L p s , Λ ˝ φ p ρ qq ¨ L p s , Sym ˝ φ p ρ qq , we know that exactly one of the symmetric or exterior square L -factors of ρ has a pole at s “
0. The above conclusion then follows from [
Mat2 , Theorem 6.1] where distinction ofSt p ρ, l q is related to the pole of symmetric or exterior square L -facotrs of ρ according to l is even or odd. Thus by our assumption that ∆ is H d { , d { -distingusihed, we get that K i isnot H m , m -distinguished. So we are done. Q.E.D. We start with an auxiliary result, which is needed in one stepof the proof of Theorem 7.3.P roposition
Let π “ π ˆ ¨ ¨ ¨ ˆ π t be an irreducible unitary representation of G m with each π i a Speh representation. Let h be a positive integer. Assume that, for all of those π i such that supp p π i q is contained in the cuspidal line Z ν ´ { , we have b p π i q ď ν h ´ { . If INEAR PERIODS FOR UNITARY REPRESENTATIONS 29 the representation π ˆ ν ´ h { is p H m , m ` h , µ h { q -distinguished, where ν ´ h { is viewed as arepresentation of G h , then π is H m , m -distinguished. P roof . A crucial fact, on which we rely, is that π is a commutative product of Spehrepresentations. We may arrange the ordering of representations in the product and rewrite π “ σ ˆ ¨ ¨ ¨ ˆ σ r ˆ π ˆ ¨ ¨ ¨ ˆ π k , where these π i , i “ , ¨ ¨ ¨ , k , are all the representations appeared in the Tadi´c decomposi-tion of π with b p π i q “ ν h ´ { . We prove the statement by induction on k .As the representation π ˆ ν ´ h { is p H m , m ` h , µ h { q -distinguished, by Proposition 6.6,there exist two representations σ and σ of dimension one, ν ´ h { “ σ \ σ , such that,among other things, σ is p H a , b , µ h `p a ´ b q{ q -distinguished for two nonnegative integers a and b .(1). If σ is not the trivial representation of G , that is, a and b are not all zero, we havethree cases. If a ą b ą
0, then by Lemma 3.2, σ must be the trivial representation of G a ` b . This is absurd as we have b p σ q “ ν ´ { ; If a ą b “
0, then σ is thecharacter ν h ` a { of G a . Thus b p σ q “ ν h ` a ´ { which is absurd; If a “ b ą
0, wesee easily that a “ b “ h , that is, σ is the character ν ´ h { of G h . So, it follows fromProposition 6.6 Case B, that π is H m , m -distinguished.(2). If σ is the trivial representaion of G , then we are in Case C of Proposition 6.6.Note that when k “
0, as none of b p σ j q equals to ν h ´ { by our assumption, Case C cannothappen. Thus, by arguments in (1), we have proved the statement for k “
0. If k ą σ is the trivial representaion of G , then by Proposition 6.6, there exists i , 1 ď i ď k andtwo ladder representations π i and π i , π i “ π i \ π i , such that π i is the character ν h { of G h and the representation σ ˆ ¨ ¨ ¨ ˆ σ r ˆ π ˆ ¨ ¨ ¨ ˆ π i ´ ˆ π i ˆ π i ` ˆ ¨ ¨ ¨ ˆ π k (7.3)is p H m ´ h , m , µ h { q -distinguished. We have two subcases. If π i is one dimensional, then π i must be the trivial representaion of G h as we have b p π i q “ ν h ´ { . Thus π i is thecharacter ν ´ h { of G h . By Lemma 6.7, we know that ν ´ h { ˆ π j “ π j ˆ ν ´ h { for j “ , ¨ ¨ ¨ , k . So we move π to the end of the product (7.3) and get by induction hypothesisthat σ ˆ ¨ ¨ ¨ ˆ σ r ˆ π ˆ ¨ ¨ ¨ ˆ π i ´ ˆ π i ` ˆ ¨ ¨ ¨ ˆ π k is H m ´ h , m ´ h -distinguished. By adding π i “ h , we get that π is H m , m -distingusihed. Ifotherwise π i is not one dimensional, we can also move π i to the end of the product (7.3)by Lemma 6.7. By passing to the contragredient, we get that the representation σ _ ˆ ¨ ¨ ¨ ˆ σ _ r ˆ π _ ˆ ¨ ¨ ¨ ˆ π _ i ´ ˆ π _ i ` ˆ ¨ ¨ ¨ ˆ π _ k ˆ p π i q _ (7.4)is p H m , m ´ h , µ ´ h { q -distinguished. We claim that this is impossible, and thus we are done.Indeed, we realize the representation (7.4) in a natural way as a quotient of products ofessentially square-integrable representations, which is then p H m , m ´ h , µ ´ h { q -distinguished. Note that e pp π i q _ q “ ν ´ h ´ { . This product of essentially square-integrable representa-tion cannot be p H m , m ´ h , µ ´ h { q -distinguished by Corollary 5.5 and our assumption on therepresentation π . Q.E.D.T heorem Let π be an irreducible unitary representation of G m of Arthur type.Then π is H m , m -distinguished if and only if π is of the form p σ ˆ σ _ q ˆ ¨ ¨ ¨ ˆ p σ r ˆ σ _ r q ˆ σ r ` ˆ ¨ ¨ ¨ ˆ σ s . (7.5) where each σ i is a Speh representation for i “ , ¨ ¨ ¨ , r, and each representation σ j isH m j , m j -distinguished for some positive integer m j , j “ r ` , ¨ ¨ ¨ , s. P roof . By the work of Blanc and Delorme [ BD ], we know that σ j ˆ σ _ j is H m j , m j -distinguished with m j the degree of σ j , j “ , ¨ ¨ ¨ , r . One direction then follows fromLemma 5.7. Write π “ π ˆ ¨ ¨ ¨ ˆ π t to be the Tadi´c decomposition of π . We provethe other direction by induction on t . The case t “ π is acommutative product, we order these π i in the following way: We first group these π i bycuspidal supports. Namely, representations with cuspidal supports contained in the unionof one cuspidal line and its contragredient are put in the same group. The ordering ofthe groups can be arbitrary. For representations within the same group, if their cuspidalsupports are contained in one cuspidal line, we arrange the ordering such that when i ă j ,we have either b p π i q ă b p π j q , or b p π i q “ b p π j q and ht p π i q ď ht p π j q ; if their cuspidalsupports are contained in two di ff erent cuspidal lines, we arrange the ordering such thatwhen i ă j , we have ht p π i q ď ht p π j q .By our assumption, π is H m , m -distinguished. We apply Propositon 6.6 and discuss caseby case.Case A. There exists a division of π t , π t “ π t \ π t , where π t is neither π t nor the trivialrepresentation of G , such that, among other things, π t is p H r , s , µ p r ´ s q{ q -distinguished fortwo nonnegative integers r and s . We have two subcases.(1) The representation π t is not one dimensional . By Theorem 6.18, we know π t is anessentially Speh representation. So, by Theorem 6.11, we have r “ s . That is, π t is H r , r -distinguished. In particular, π t is self-dual, and hence π t is self-dual. Thisfurther shows that π t is a Speh representation. By Proposition 6.6, there exists i ,1 ď i ď t ´
1, and a division of π i , π i “ π i \ π i such that p π t q _ – π i and that π ˆ ¨ ¨ ¨ ˆ π i ´ ˆ π i ˆ π i ` ˆ ¨ ¨ ¨ ˆ π t ´ (7.6) is H m , m -distinguished for some positive integer m . Thus we have b p π t q “ b p π i q .By our assumption on the ordering of representations, we have ht p π i q ď ht p π t q .As π t is a self-dual Speh representation that does not equal to π t , we have ht p π t q “ ht p π t q . As ht p π i q ď ht p π i q , we have ht p π t q “ ht p π i q due to the fact that p π t q _ – π i . Thus we have π i – π t , and π i – π t , which is a H r , r -distinguished Spehrepresentation. So, by induction hypothesis, the representation (7.6) is of theform (7.5). After removing π i in the product, we still get a representation of theform (7.5). Therefore, by adding π t ˆ π i , we get that π is of the form (7.5). INEAR PERIODS FOR UNITARY REPRESENTATIONS 31 (2) The representation π t is one dimensional . If r ą s ą
0, then π t is thetrivial representation of G r by Lemma 3.2. Note that, in this case, π t is not aone dimensional representaion. Then by the same arguments as in Case A (1),we are done in this case. If one of r , s is 0, then π t is the character ν h { of G h , h “ max p r , s q . Thus we have b p π t q “ b p π t q “ ν h ´ { . In particular, π t is self-dual. By Proposition 6.6, there exists i , 1 ď i ď t ´
1, and a division of π i , π i “ π i \ π i such that p π t q _ – π i and that π ˆ ¨ ¨ ¨ ˆ π i ´ ˆ π i ˆ π i ` ˆ ¨ ¨ ¨ ˆ π t ´ (7.7) is p H m ´ n t , m ´ n t ` h , µ h { q -distinguished with n t the degree of π t . Thus we have b p π i q “ b p π t q “ ν h ´ { , and π i is also self-dual. By our assumption on the or-dering of representations, we have ht p π i q “ ht p π t q , and hence π i – π t . Thus, therepresentation π i is the character ν ´ h { of G h . By Lemma 6.7, the representation(7.7) is isomorphic to the representation π ˆ ¨ ¨ ¨ ˆ π i ´ ˆ π i ` ˆ ¨ ¨ ¨ ˆ π t ´ ˆ ν ´ h { . By Proposition 7.2, the representation π ˆ ¨ ¨ ¨ ˆ π i ´ ˆ π i ` ˆ ¨ ¨ ¨ ˆ π t ´ is H m ´ n t , m ´ n t -distinguished, and hence is of the form (7.5) by induction hypothesis.Therefore, by adding π i ˆ π t , we get that π is of the form (7.5).Case B. In this case the representation π t is p H r , s , µ p r ´ s q{ q -distinguished for two non-negative integers r and s , and the representation π ˆ ¨ ¨ ¨ ˆ π t ´ is p H m ´ r , m ´ s , µ p s ´ r q{ q -distinguished. As π t is a Speh representation, by Theorem 6.11 andLemma 6.15, we have r “ s . Therefore, by induction hypothesis we are done.Case C. There exists i , 1 ď i ď t ´
1, and a division of π i , π i “ π i \ π i , such that p π t q _ – π i and that the representation π ˆ ¨ ¨ ¨ ˆ π i ´ ˆ π i ˆ π i ` ˆ ¨ ¨ ¨ ˆ π t ´ is H m ´ n t , m ´ n t -distinguished. By our assumption on the ordering of representations, wehave π i – p π t q _ . Thus π i is the trivial representation of G . By induction hypothesis, therepresentation π ˆ ¨ ¨ ¨ˆ π i ´ ˆ π i ` ˆ ¨ ¨ ¨ ˆ π t ´ is of the form (7.5). Therefore, by adding π t ˆ p π t q _ , the representation π is of the form (7.5). Q.E.D.To classify distinguished representations in the entire unitary dual, it remains to con-sider distinction of complementary series representations. Recall that a complementary se-ries representation is an irreducible unitary representation of the form ν α Sp p δ, k qˆ ν ´ α Sp p δ, k q with 0 ă α ă {
2, and is denoted by Sp p δ, k qr α, ´ α s . By the work of Blanc and Delorme[ BD ], one sees that Sp p δ, k qr α, ´ α s is H m , m -distinguished if and only if it is self-dual,where m is the degree of Sp p δ, k q . To apply the geometric lemma, we first note the follow-ing lemma. L emma Let ρ be a unitary supercuspidal representation of G d and c a fixed integer.Let π be a ladder representation of G n with cuspidal supports contained in the cuspidal line Z ν α ` c { ρ or Z ν ´ α ` c { ρ with ă α ă { , then π cannot be self-dual. If, moreover, π is leftaligned, then π cannot be p H p , q , µ p p ´ q q{ q -distinguished for certain nonnegative integers p,q with p ` q “ n. P roof . As 0 ă α ă {
2, the cuspidal line Z ν α ` c { ρ (or Z ν ´ α ` c { ρ ) is not self-dual.Thus π cannot be self-dual by Lemma 6.1. For the second statement, if π is one dimen-sional, then by Lemma 6.15, the cuspidal supports of π is contained in Z ν or Z ν ´ { .This contradicts with our assumption; if π is not one dimensional, then by Theorem 6.18and Theorem 6.11, one sees π is self-dual. This is absurd as shown by the first state-ment. Q.E.D.T heorem An irreducible unitary representation π of G n is H n , n -distinguished ifand only if it is self-dual and its Arthur part π Ar is of the form (7.5) . P roof . To simplify notation, we will say a representaion H -distinguished for H m , m -distinguished when there is no need to address m . Write π “ π Ar ˆ π c . If π is self-dual, byuniqueness of Tadi´c decomposition, we have π c is also self-dual. As π c is a commutativeproduct of complementary series representations, we have π c is H -distinguished. The ‘if’part then follows from Lemma 5.7. For the ‘only if’ part, write π as a product of essentiallySpeh representations π ˆ ¨ ¨ ¨ ˆ π t ˆ ν α Sp p δ , k q ˆ ν ´ α Sp p δ , k q ˆ ¨ ¨ ¨ ˆ ν α r Sp p δ r , k r q ˆ ν ´ α r Sp p δ r , k r q (7.8)such that k ď k ď ¨ ¨ ¨ ď k r , and that π i is a Speh representation for i “ , ¨ ¨ ¨ , t . Nowwe appeal to Proposition 6.6. By Lemma 7.4, only Case C can happen. Note that we have k ď ¨ ¨ ¨ ď k r and 0 ă α i ă { i “ , ¨ ¨ ¨ , r . By simple arguments we can show thateach time after applying Proposition 6.6, we can delete two non-unitary essentially Spehrepresentations in the product (7.8), and the new representation is H -distinguished. Thusby a repeated use of Proposition 6.6, we get π Ar “ π ˆ ¨ ¨ ¨ ˆ π t is H -distinguished. The‘only if’ part then follows from Theorem 7.3. Q.E.D. Acknowledgement
The author thanks Wee Teck Gan for his kindness of sharing his paper [
Gan ] withus. He would also like to thank Dipendra Prasad for helpful comments and suggestions toimprove the paper.
References [BD] P. Blanc and P. Delorme. Vecteurs distributions H -invariants de repr´esentations induites, pour un es-pace sym´etrique r´eductif p -adique G { H . In Annales de l’institut Fourier , volume 58, pages 213–261,2008.[BZ] I. N. Bernstein and A. V. Zelevinsky. Induced representations of reductive p -adic groups. I. In Annalesscientifiques de l’ ´Ecole normale sup´erieure , volume 10, pages 441–472, 1977.
INEAR PERIODS FOR UNITARY REPRESENTATIONS 33 [CPS] J. W. Cogdell and I. I. Piatetski-Shapiro. Derivatives and L -functions for GL p n q . In Representationtheory, number theory, and invariant theory , pages 115–173. Springer, 2017.[CS] F. Chen and B. Sun. Uniqueness of twisted linear periods and twisted Shalika periods.
Science ChinaMathematics , 63(1):1–22, 2020.[FJ] S. Friedberg and H. Jacquet. Linear periods.
J. reine angew. Math , 443(91):139, 1993.[FLO] B. Feigon, E. Lapid, and O. O ff en. On representations distinguished by unitary groups. Publicationsmath´ematiques de l’IH ´ES , 115(1):185–323, 2012.[FSX] Y. Fang, B. Sun, and H. Xue. Godement-Jacquet L -functions and full theta lifts. MathematischeZeitschrift , 289(1-2):593–604, 2018.[Gan] W. T. Gan. Periods and theta correspondence. In
Proceedings of Symposia in Pure Mathematics , vol-ume 101, pages 113–132, 2019.[GGP] W. T. Gan, B. H. Gross, and D. Prasad. Branching laws for classical groups: the non-tempered case. arXiv:1911.02783 , 2019.[Gur] M. Gurevich. On a local conjecture of Jacquet, ladder representations and standard modules.
Mathe-matische Zeitschrift , 281(3-4):1111–1127, 2015.[JNQ] D. Jiang, C. Nien, and Y. Qin. Local shalika models and functoriality. manuscripta mathematica ,127(2):187–217, 2008.[JPSS] H. Jacquet, I. I Piatetskii-Shapiro, and J. Shalika. Rankin-Selberg convolutions.
American journal ofmathematics , 105(2):367–464, 1983.[JR] H. Jacquet and S. Rallis. Uniqueness of linear periods.
Compositio Mathematica , 102(1):65–123,1996.[JS] D. Jiang and D. Soudry. Generic representations and local langlands reciprocity law for p -adic SO n ` . Contributions to automorphic forms, geometry, and number theory , pages 457–519, 2004.[Kab] A. C. Kable. Asai L -functions and Jacquet’s conjecture. American journal of mathematics ,126(4):789–820, 2004.[KL] A. Kret and E. Lapid. Jacquet modules of ladder representations.
Comptes Rendus Mathematique ,350(21-22):937–940, 2012.[LM1] E. Lapid and A. M´ınguez. On a determinantal formula of Tadi´c.
American Journal of Mathematics ,136(1):111–142, 2014.[LM2] E. Lapid and A. M´ınguez. On parabolic induction on inner forms of the general linear group over anon-archimedean local field.
Selecta Mathematica , 22(4):2347–2400, 2016.[Mat1] N. Matringe. Distinguished generic representations of GL p n q over p -adic fields. International mathe-matics research notices , 2011(1):74–95, 2011.[Mat2] N. Matringe. Linear and shalika local periods for the mirabolic group, and some consequences.
Journalof Number Theory , 138:1–19, 2014.[Mat3] N. Matringe. Unitary representations of GL p n , K q distinguished by a Galois involution for a p -adicfield K . Pacific Journal of Mathematics , 271(2):445–460, 2014.[Mat4] N. Matringe. On the local Bump–Friedberg L -function. Journal f¨ur die reine und angewandte Mathe-matik (Crelles Journal) , 2015(709):119–170, 2015.[MeJLW] C. Moeglin and J.-L. Waldspurger. Sur l’involution de Zelevinski.
Journal f¨ur die reine und ange-wandte Mathematik (Crelles Journal) , 1986(372):136–177, 1986.[M´ın] A. M´ınguez. Correspondance de Howe explicite: paires duales de type II. In
Annales scientifiques del’ ´Ecole Normale Sup´erieure , volume 41, pages 717–741, 2008.[MOS] A. Mitra, O. O ff en, and E. Sayag. Klyachko models for ladder representations. Documenta Mathemat-ica , 22:611–657, 2017.[O ff ] O. O ff en. On parabolic induction associated with a p -adic symmetric space. Journal of Number The-ory , 170:211–227, 2017. [OS1] O. O ff en and E. Sayag. On unitary representations of GL n distinguished by the symplectic group. Journal of Number Theory , 125(2):344–355, 2007.[OS2] O. O ff en and E. Sayag. Uniqueness and disjointness of Klyachko models. Journal of Functional Anal-ysis , 254(11):2846–2865, 2008.[S´ec] V. S´echerre. Repr´esentations cuspidales de GL p r , D q distingu´ees par une involution int´erieure. 2020.[Tad1] M. Tadi´c. Classification of unitary representations in irreducible representations of general linear group(non-Archimedean case). In Annales scientifiques de l’Ecole normale sup´erieure , volume 19, pages335–382, 1986.[Tad2] M. Tadi´c. Induced representations of GL p n , A q for p-adic division algebras A . J. reine angew. Math ,405:48–77, 1990.[Tad3] M. Tadi´c. On characters of irreducible unitary representations of general linear groups. In
Abhandlun-gen aus dem Mathematischen Seminar der Universit¨at Hamburg , volume 65, pages 341–363. Springer,1995.[Zel] A. V. Zelevinsky. Induced representations of reductive p-adic groups. II. on irreducible representationsof GL p n q . Annales Scientifiques de l’ ´Ecole Normale Sup´erieure , 13(2):165–210, 1980.K ey L aboratory of H igh P erformance C omputing and S tochastic I nformation P rocessing (HPCSIP), C ol - lege of M athematics and S tatistics , H unan N ormal U niversity , C hangsha , 410081, C hina E-mail address ::