Theta lifts of strongly positive discrete series: the case of (˜ Sp(n),O(V ))

GL(n j ) × Sp(n − |s|, F ), where |s| = m = ∑ ji=1 n i. Then the standardparabolic subgroup ˜P s of ˜Sp(n) is the preimage of P s in ˜Sp(n). We have theanalogous notation for the Levi subgroups of the metaplectic groups, whichare described in more detail in Section 2.2 of [5]. The standard parabolicsubgroups (containing the upper triangular Borel subgroup) of O(V r ) havethe analogous description as the standard parabolic subgroups of Sp(n, F ). If˜P s is a standard parabolic subgroup of ˜Sp(n) described above, or P s a similarstandard parabolic subgroup of O(V r ), the normalized Jacquet module of asmooth representation σ of ˜Sp(n) (resp., O(V r )) with respect to ˜P s (resp., P s )is denoted by R fP s(σ) (resp., R Ps (σ)). From now on, R P1 (π)(χ) (or R fP 1(π)(χ))stands for the isotypic component of R P1 (π) along the generalized characterχ.Also, when dealing with Jacquet modules of ω n,r , we write shortly R P1 (ω n,r )(resp., R eP 1(ω n,r )) for R˜Sp(n)×P1(ω n,r ) (resp., R eP 1 ×O(V m) (ω n,r)), following thenotation from [6].For any irreducible representation π ∈ S ′ (n) there exist an ordered partitions = (n 1 , n 2 , . . . , n j ) of some m ≤ n, cuspidal representations ρ i ∈Irr(R ′ (n i )) and π cusp ∈ S ′ (n − |s|) such that π is an irreducible subquotientof the induced representation ρ 1 × ρ 2 × · · · × ρ j ⋊ π cusp . In this situation, wewrite [π] = [ρ 1 , ρ 2 , . . . , ρ j ; π cusp ], following the notation used in [8].Let σ ∈ S ′ (n) denote an irreducible representation. To simplify notation,set P ′ s = P s in orthogonal case and P ′ s = ˜P s in the metaplectic one. Weintroduce µ ∗ (σ) ∈ R ′ ⊗ S ′ byµ ∗ (σ) =n∑s.s.(P (k)(σ)),′k=0where s.s. denotes the semisimplification. We extend µ ∗ linearly to the wholeof S ′ .In the following lemma we recall useful formula for calculations withJacquet modules which is valid in both orthogonal and metaplectic case([20, 5]). Let α ′ = α in the metaplectic case, while in the orthogonal case α ′denotes a trivial character.Lemma 2.1. Let ρ ∈ R ′ be an irreducible cuspidal representation and a, b ∈R such that a + b ∈ Z ≥0 . Let σ ∈ S ′ be an admissible representation of finite6

length. Write µ ∗ (σ) = ∑ π,σ ′ π ⊗ σ ′ . Then the following holds:µ ∗ (δ([ν −a ρ, ν b ρ]) ⋊ σ) =We omit δ([ν x ρ, ν y ρ]) if x > y.b∑i=−a−1 j=ib∑ ∑π,σ ′ δ([ν −i α ′˜ρ, ν a α ′˜ρ]) × δ([ν j+1 ρ, ν b ρ]) × π⊗ δ([ν i+1 ρ, ν j ρ]) ⋊ σ ′ . (1)We take a moment to recall the formulation of the second Frobeniusisomorphism.Generally, for some reductive group G ′ , its parabolic subgroup P ′ with theLevi subgroup M ′ and opposite parabolic subgroup P ′ , the second Frobeniusisomorphism isHom G ′(Ind G′M ′(π), Π) ∼ = Hom M ′(π, R P ′(Π)),for some smooth representation π (resp., Π) of the group M ′ (resp., G ′ ). Wedenote the space of the representation π by V π .Above isomorphism can be explicitly described in the following way:Let Ψ denote the embeddingΨ : V π ↩→ R P ′(Ind G′M ′(V π)),which corresponds to the open cell P ′ P ′ in G ′ ([3]). Now, for some T ∈Hom G ′(Ind G′M ′(π), Π), compose Ψ with the corresponding mappingT P ′ : R P ′(Ind G′M ′(π)) → R P ′(Π).3 Embeddings of discrete seriesIn this section we recall the classification of strongly positive discrete seriesand obtain further embeddings of general discrete series which will be usedafterwards in the paper.In the following theorem we gather the results obtained in the Section 5 ofthe paper [10]. The arguments used there rely on Jacquet module methods,and build up in an essentially combinatorial way from the cuspidal reducibilityvalues. Moreover, the underlying combinatorics are essentially the samefor classical groups. Thus, our classification is valid for both metaplectic andorthogonal groups.7

Theorem 3.1. We define a collection of pairs (Jord, σ ′ ), where σ ′ is anirreducible cuspidal representation of some S ′ (n σ ′) and Jord has the followingform: Jord = ⋃ k ⋃ kii=1 j=1 {(ρ i, b (i)j )}, where• {ρ 1 , ρ 2 , . . . , ρ k } is a (possibly empty) set of mutually nonisomorphic irreducibleself-dual cuspidal representations of some R ′ (m 1 ),, R ′ (m 2 ), . . . ,R ′ (m k ) such that ν aρ iρ i ⋊ σ ′ reduces for a ρi > 0 (this defines a ρi ).• k i = ⌈a ρi ⌉, the smallest integer which is not smaller that a ρi .• For each i = 1, . . . , k, b (i)1 , . . . , b (i)k ithat a ρi − b (i)j· · · < b (i)k i.is a sequence of real numbers suchis an integer, for j = 1, 2, . . . , k i and −1 < b (i)1 < b (i)2 0, i = 1, 2, . . . , k, denote the unique positive s ∈ R such that therepresentation ν s ρ i ⋊ σ cusp reduces. Set k i = ⌈a ρi ⌉. For each i = 1, 2, . . . , kthere exists a unique increasing sequence of real numbers b (i)1 , b (i)2 , . . . , b (i)where a ρi − b (i)j is an integer, for j = 1, 2, . . . , k i and b (i)1 > −1, such that σis the unique irreducible subrepresentation of the induced representation(k∏ ∏k iδ([ν aρ i −ki+j ρ i , ν b(i) jρ i ])) ⋊ σ cusp .i=1 j=1Now, Jord(σ) = ⋃ k ⋃ kii=1 j=1 {(ρ i, b (i)j )} and σ′ (σ) = σ cusp .We note that results of [1] should imply that every a ρi in the previoustheorem is half integral.This classification implies some interesting properties of strongly positivediscrete series, which are listed in the next two lemmas. We note that firstof them is Lemma 3.5 in [11].8k i,

Lemma 3.2. Let σ ∈ S ′ be a strongly positive discrete series. The σ isuniquely determined by [σ].Next result is rather straightforward from the mentioned classification:Lemma 3.3. Let σ ∈ S ′ denote a strongly positive discrete series and supposethat ν x ρ appears in [σ], where ρ ∈ R ′ is an irreducible unitarizable cuspidalrepresentation and |x| ≤ 1. Then the representation ν x ρ appears in [σ] withmultiplicity one. Also, if ν y ρ appears in [σ] for some y ≠ x, then |y| > 1.Proof. It is enough to prove the lemma for x ≥ 0, since otherwise the sameconclusion can be drawn for |x|.We write σ as the unique irreducible subrepresentation of the induced representationof the form ( ∏ k ∏ kii=1 j=1 δ([νaρ i −ki+j ρ i , ν b(i) jρ i ]))⋊σ cusp . Obviously,ρ is isomorphic to ρ l for some l ∈ {1, 2, . . . , k}.By the assumption of lemma, there is some j ∈ {1, 2, . . . , k l } such thata ρl −k l +j ≤ x ≤ b (l)j . Strong positivity of σ implies x > 0. Since a ρ l−k l +j >1 for j ≥ 2, it follows that ν x ρ appears in the segment [ν aρ l −kl+1 ρ l , ν b(l) 1 ρl ]and ν x ρ does not appear in [ν aρ l −kl+j ρ l , ν b(l) j ρl ], for j ≥ 2. Further, usingx − 1 ≤ 0 we obtain x = a ρl − k l + 1.Consequently, ν x ρ appears in [σ] with multiplicity one.The inequality |y| > 1 for y ≠ x such that ν y ρ appears in [σ] is a consequenceof the fact that |y| − x is a positive integer and x > 0.The principal significance of the following lemma is that it allows us toobtain certain embeddings of general discrete series.Lemma 3.4. Suppose that π ∈ S ′ (n) is an irreducible representation, whichis not in the discrete series. Then there exists an embedding of the formπ ↩→ δ([ν a ρ, ν b ρ]) ⋊ π ′ ,where a + b ≤ 0, ρ ∈ R ′ and π ′ ∈ S ′ are irreducible representations.Proof. We adopt the approach from the Section 3 of [10], which was motivatedby [16]. Suppose thatπ ↩→ ρ 1 × ρ 2 × · · · × ρ k ⋊ π cuspis an embedding of the representation π contradicting Casselman’s squareintegrabilitycriterium (whose metaplectic version is written in non-published9

where each ρ i ∈ R ′ is an irreducible cuspidal representation and σ cusp ∈ S ′ (n ′ )is a partial cuspidal support of σ, and consider all possible embeddings ofthe formσ ↩→ δ(∆ 1 ) × δ(∆ 2 ) × · · · × δ(∆ m ) ⋊ σ cusp ,where ∆ 1 + ∆ 2 + · · · + ∆ m = {ρ 1 , ρ 2 , . . . , ρ l }, viewed as the equality of multisets.In the same way as in the proof of the previous lemma, to every suchembedding we attach an element of R n−n′ and denote byσ ↩→ δ(∆ ′ 1) × δ(∆ ′ 2) × · · · × δ(∆ ′ m ′) ⋊ σ cusp (2)minimal such embedding with respect to the lexicographic ordering on R n−n′ .Analysis similar to that in the proof of Theorem 3.3 from [10] shows e(∆ ′ 1) ≤e(∆ ′ 2) ≤ · · · ≤ e(∆ ′ m ′).Write each element of the multiset {ρ 1 , ρ 2 , . . . , ρ l } in a form ρ i = ν a iρ i,u ,where ρ i,u is an irreducible unitary cuspidal representation. Define a =min{a i : 1 ≤ i ≤ l}. The assumption that σ is not strongly positive yieldsa ≤ 0. Suppose that ν a ρ appears in the segment ∆ ′ i, with i minimal (forappropriate ρ). Then ∆ ′ i = [ν a ρ, ν b ρ], for some b.If the segment ∆ ′ i is not connected in the sense of Zelevinsky with any ofthe segments ∆ ′ 1, . . . , ∆ ′ i−1, we obtain the embeddingσ ↩→ δ(∆ ′ i) × δ(∆ ′ 1) × · · · × δ(∆ ′ m ′) ⋊ σ cusp.Suppose that there is some segment ∆ ′ j, 1 ≤ j ≤ i−1, such that the segments∆ ′ i and ∆ ′ j are connected in the sense of Zelevinsky. We choose largest such jand denote it with j again. Also, we write ∆ ′ j = [ν a′ ρ, ν b′ ρ]. The intertwiningoperator δ(∆ ′ j) × δ(∆ ′ i) → δ(∆ ′ i) × δ(∆ ′ j) gives the following mapsσ ↩→ δ(∆ ′ 1) × · · · × δ(∆ ′ j) × δ(∆ ′ i) × · · · × δ(∆ ′ m ′) ⋊ σ cusp→ δ(∆ ′ 1) × · · · × δ(∆ ′ i) × δ(∆ ′ j) × · · · × δ(∆ ′ m ′) ⋊ σ cusp.Observe that the kernel of previous intertwining operator equals δ(∆ ′ 1) ×· · ·×δ([ν a ρ, ν b′ ρ])×δ([ν a′ ρ, ν b ρ])×· · ·×δ(∆ ′ m ′)⋊σ cusp. Since e(∆ ′ j) ≤ e(∆ ′ i),the inequality a < a ′ implies e([ν a ρ, ν b′ ρ]) < e(∆ ′ j). Thus, minimality ofthe embedding (2) shows that σ is not contained in the kernel of observedintertwining operator, which givesσ ↩→ δ(∆ ′ 1) × · · · × δ(∆ ′ i) × δ(∆ ′ j) × · · · × δ(∆ ′ m ′) ⋊ σ cusp.11

Repeated application of the above procedure enables us to obtain theembeddingσ ↩→ δ(∆ ′ i) × δ(∆ ′ 1) × · · · × δ(∆ ′ m ′) ⋊ σ cusp.Lemma 3.2 from [13] implies that there is some irreducible representationσ 1 such that σ ↩→ δ([ν a ρ, ν b ρ])⋊σ 1 . Square-integrability of σ shows a+b > 0.We claim that σ 1 is a discrete series representation.Suppose on the contrary that σ 1 is not in the discrete series. Then previouslemma shows that it can be written as a subrepresentation of the inducedrepresentation of the form δ([ν x ρ ′ , ν y ρ ′ ]) ⋊ σ ′ 1, where x + y ≤ 0. Thus,σ ↩→ δ([ν a ρ, ν b ρ])×δ([ν x ρ ′ , ν y ρ ′ ])⋊σ ′ 1. Square-integrability of the representationsσ shows that the segments [ν a ρ, ν b ρ] and [ν x ρ ′ , ν y ρ ′ ] are connected in thesense of Zelevinsky, and consequently σ ↩→ δ([ν a ρ, ν y ρ]) × δ([ν x ρ ′ , ν b ρ ′ ]) ⋊ σ ′ 1.The choice of a shows that a ≤ x, which leads to a + y ≤ x + y ≤ 0,i.e., e([ν a ρ, ν y ρ]) ≤ 0, contradicting square-integrability of σ. In this way wehave proved that σ 1 is also a discrete series representation.We continue in this fashion to obtain that either σ 1 is strongly positiveor it can be written as a subrepresentation of the induced representation ofthe form δ(ν a′ ρ ′ , ν b′ ρ ′ ]) ⋊ σ 2 , where a ′ ≤ 0 and σ 2 ∈ S ′ is a discrete seriesrepresentations. Repeating this procedure, after a finite number of steps weobtain the claim of the theorem.4 Howe’s correspondence and results of Gan& Savin and of KudlaIn this section we review some results about Howe correspondence.For an irreducible genuine smooth representation σ ∈ S 2 (n), let Θ(σ, r)be a smooth representation of O(V r ), given as the full lift of σ to the r-levelof the orthogonal tower, i.e., the biggest quotient of ω n,ron which ˜Sp(n)acts as a multiple of σ. As a representation of ˜Sp(n) × O(V r ) it has a formσ ⊗ Θ(σ, r). We write Θ + (σ, r) (resp., Θ − (σ, r)) for the lift on the +–tower(resp., −–tower), when emphasizing the tower.Similarly, if τ is an irreducible representation of O(V r ), then one has itsfull lift Θ(τ, n), which is a smooth representation of ˜Sp(n).12

the representation ν s 2χ V,ψ ⋊ σ cusp reduces for s 2 = |m r ′ − n ′ − 1|, wherem r ′ = 1 2 dimV r ′.We take a moment to state the results from the Section 2 of the paper[7], which happen to be crucial for our investigation.Theorem 4.3. Let τ ∈ S 1 (r) denote an irreducible representation and suppose[τ] = [ρ 1 , ρ 2 , . . . , ρ k ; τ cusp ], with τ cusp ∈ S 1 (r ′ ) being an irreducible cuspidalrepresentation. Let σ cusp = τ(n ′ ) be the first non-zero lift of the representationτ cusp , observe that σ cusp ∈ S 2 (n ′ ) is an irreducible cuspidal representation.Let σ denote an irreducible quotient of Θ(τ, n). We have the followingpossibilities:• If n ≥ n ′ +r−r ′ , then [σ] = [χ V,ψ ν mr−n , χ V,ψ ν mr−n+1 , . . . , χ V,ψ ν m r ′−n′ −1 ,χ V,ψ ρ 1 , χ V,ψ ρ 2 , . . . , χ V,ψ ρ k ; σ cusp ],• If n < n ′ + r − r ′ , set t = r − r ′ − n + n ′ . Then there exist i 1 , i 2 , . . . , i t ∈{1, 2, . . . , k} such that ρ ij = ν mr−n−j for j = 1, 2, . . . , t and [σ] =[χ V,ψ ρ 1 , . . . , ̂χ V,ψ ρ i1 , . . . , ̂χ V,ψ ρ it , . . . , χ V,ψ ρ k ; σ cusp ], where ̂χ V,ψ ρ i meansthat we omit χ V,ψ ρ i .Similarly, let σ ∈ S 2 (n) denote an irreducible representation and suppose[σ] = [χ V,ψ ρ 1 , χ V,ψ ρ 2 , . . . , χ V,ψ ρ k ; σ cusp ], with σ cusp ∈ S 2 (n ′ ) being an irreduciblecuspidal representation. Let τ cusp = σ(r ′ ) be the first non-zero lift ofthe representation σ cusp , observe that τ cusp ∈ S 1 (r ′ ) is an irreducible cuspidalrepresentation. Let τ denote an irreducible quotient of Θ(σ, r). We have thefollowing possibilities:• If r ≥ r ′ +n−n ′ , then [τ] = [ν mr−n−1 , ν mr−n−2 , . . . , ν m r ′−n′ , ρ 1 , ρ 2 , . . . , ρ k ;τ cusp ],• If r < r ′ + n − n ′ , set t = r ′ − n ′ + n − r. Then there exist i 1 , i 2 , . . . , i t ∈{1, 2, . . . , k} such that ρ ij = ν mr−n+j−1 for j = 1, 2, . . . , t and [τ] =[ρ 1 , . . . , ̂ρ i1 , . . . , ̂ρ it , . . . , ρ k ; τ cusp ], where ̂ρ i means that we omit ρ i .The next theorem that we need is Kudla’s filtration of Jacquet modulesof the oscillatory representation ([7, Theorem 2.8]):Theorem 4.4. Let ω n,r denote the oscillatory representation of the group˜Sp(n) × O(V r ) corresponding to the non-trivial additive character ψ. Thefollowing holds:14

• Let P j denote the standard maximal parabolic subgroup of O(V r ). ThenJacquet module R Pj (ω n,r ) has ˜Sp(n) × M j -invariant filtration given byI jk , 0 ≤ k ≤ j, whereI jk ≃ Ind˜Sp(n)×M jP jk × fP k ×O(V r−j ) (γ jk ⊗ Σ ′ k ⊗ ω n−k,r−j ).Here, P jk is a standard parabolic subgroup of GL(j, F ) which correspondsto the partition (j − k, k), γ jk is a character of GL(j − k, F ) ×GL(k, ˜F ) given byj−k+1−(mr−n−γ jk (g 1 , g 2 ) = ν 2 ) (g 1 )χ V,ψ (g 2 ),and Σ ′ k is a twist of the standard representation of GL(k, F )×GL(k, F )on the space of smooth locally constant compactly supported complexvalued functions Cc ∞ (GL(k, F )):Σ ′ k(g 1 , g 2 )f(g) = ν −(m r−j+ k−12 ) ν m r−j+ k−12 f(g−11 gg 2 ).j+1−(mr−n−Especially, a quotient I j0 equals ν 2 ) ⊗ ω n,r−j and a subrepresentationI jj equals Ind˜Sp(n)×M jGL(j,F )× fP j ×O(V r−j ) (χ V,ψ ⊗ Σ ′ j ⊗ ω n−j,r−j ).• Let ˜P j denote the standard maximal parabolic subgroup of ˜Sp(n). ThenJacquet module R fP j(ω n,r ) has ˜M j × O(V r )-invariant filtration given byJ jk , 0 ≤ k ≤ j, wherefMJ jk ≃ Indj ×O(V r)(β gP jk ×P k × Sp(n−j) ˜ jk ⊗ Σ ′ k ⊗ ω n−j,r−k ).Here, ˜Pjk is a standard parabolic subgroup of GL(j, ˜ F ) which correspondsto the partition (j − k, k), β jk is a character of GL(j ˜ − k, F ) ×GL(k, ˜F ) given byj−k−1mr−n−β jk (g 1 , g 2 ) = (χ V,ψ ν 2 )(g 1 )χ V,ψ (g 2 ),and Σ ′ k is a twist of the standard representation of GL(k, F )×GL(k, F )on the space of smooth locally constant compactly supported complexvalued functions Cc ∞ (GL(k, F )):Σ ′ k+1mr+k(g 1 , g 2 )f(g) = ν 2 ν−(m r+ k+12 ) f(g1 −1 gg 2 ).15

j−1mr−n+Especially, a quotient J j0 equals χ V,ψ ν 2 ⊗ ω n−j,r and a subrepresentationJ jj equals IndfM j ×O(V r)GL(j,F ˜ )×P j × ˜5 Some technical results on liftsSp(n−j) (χ V,ψ ⊗ Σ ′ j ⊗ ω n−j,r−j ).The purpose of this section is to state and prove many technical results whichwill be of particular importance in the following sections.An elementary but useful criterion for pushing down the lifts of irreduciblerepresentations is established by the following two propositions.Proposition 5.1. Let τ ∈ S 1 (r) be an irreducible representation. Then thefollowing hold:1. Suppose that Θ(τ, n) ≠ 0. Then R fP 1(Θ(τ, n + 1))(χ V,ψ ν mr−(n+1) ) ≠ 0.2. Suppose that R P1 (τ)(ν mr−(n+1) ) = 0. Then Θ(τ, n) ≠ 0 if and only ifR fP 1(Θ(τ, n + 1))(χ V,ψ ν mr−(n+1) ) ≠ 0.Proof. The proof follows the same lines as that of Theorem 4.5 of [6].Assume that Θ(τ, n) ≠ 0. Then there exists an epimorphism ω n,r →τ ⊗ Θ(τ, n). Kudla’s filtration gives the epimorphismsR fP 1(ω n+1,r ) → χ V,ψ ν mr−(n+1) ⊗ ω n,r → χ V,ψ ν mr−(n+1) ⊗ τ ⊗ Θ(τ, n).Using Frobenius reciprocity we get a non-trivial intertwining Θ(τ, n + 1) →χ V,ψ ν mr−(n+1) ⋊ Θ(τ, n). This obviously proves the first statement of theproposition.It remains to prove sufficiency in the second statement. The conditionR fP 1(Θ(τ, n + 1))(χ V,ψ ν mr−(n+1) ) ≠ 0 gives Θ(τ, n + 1) ≠ 0, that gives anepimorphism ω n+1,r → τ ⊗ Θ(τ, n + 1). Applying Jacquet modules, we getan epimorphism R fP 1(ω n+1,r ) → τ ⊗ χ V,ψ ν mr−(n+1) ⊗ σ ′ for some irreduciblerepresentation σ ′ ∈ S 1 (n). If we suppose that the restriction of this epimorphismto a subrepresentation J 11 is non-zero, second Frobenius reciprocitygives a non-zero intertwining mapχ V,ψ ⊗ Σ ′ 1 ⊗ ω n,r−1 → ˜R P1 (˜τ) ⊗ χ V,ψ ν mr−(n+1) ⊗ σ ′ .From this intertwining we deduce τ ↩→ ν mr−(n+1) ⋊ τ ′ , for some irreduciblerepresentation τ ′ ∈ S 2 (r − 1), contradicting the assumption of proposition.Consequently, there exists a non-zero intertwining J 10 → τ ⊗χ V,ψ ν mr−(n+1) ⊗σ ′ , which gives Θ(τ, n) ≠ 0.16

We omit the proof of the next proposition since it is completely analogousto the proof of the previous one.Proposition 5.2. Let σ ∈ S 2 (n) be an irreducible representation. Then thefollowing hold:1. Suppose that Θ(σ, r) ≠ 0. Then R P1 (Θ(σ, r + 1))(ν −(m r+1−n−1) ) ≠ 0.2. Suppose that R fP 1(σ)(χ V,ψ ν −(m r+1−n−1) ) = 0. Then Θ(σ, r) ≠ 0 if andonly if R P1 (Θ(σ, r + 1))(ν −(m r+1−n−1) ) ≠ 0.Now we prove an important result regarding the square-integrability ofthe lifts of strongly positive discrete series. In particular, this result gives analternative and essentially combinatorial proof of a special case of the resultsof [17].Proposition 5.3. Let σ ∈ S 2 (n) denote a strongly positive discrete series.Suppose that Θ(σ, r) ≠ 0, for some r such that m r ≤ n + 1 2 , andR fP 1(σ)(χ V,ψ ν −(m k−n−1) ) = 0 for k ≥ r + 1. Then σ(r) is a discrete seriesrepresentation.Proof. We prove this proposition by downwards induction on r, starting withan r such that m r = n + 1. If m 2 r = n + 1 , Theorem 4.2 shows our claim.2Thus, suppose that the claim holds for some r + 1 such that m r+1 ≤ n + 1.2We prove it for r.It may be easily concluded from the proof of Lemma 5.1 (in the same wayas in the proof of Lemma 5.1 from [15]) that there is a non-zero intertwiningσ(r) ↩→ ν −(mr−n−1) ⋊ σ(r − 1).Note that in our case m r < n + 1, which implies −(m 2 r − n − 1) ≥ 3.2Now, suppose that σ(r − 1) is not a discrete series representation. Accordingto Lemma 3.4, there is an embedding σ(r − 1) ↩→ δ([ν a ρ, ν b ρ]) ⋊ σ ′ , wherea + b ≤ 0. Obviously, a ≤ 0.Since m r − n − 1 ≤ − 3 , strong positivity of the representation σ and2Lemma 3.3 together with Theorem 4.3 imply there is at most one x ∈ R,0 < |x| ≤ 1 such that ν x ρ appears in [σ(r − 1)]. Therefore, b ≤ 0 andthe representation ν −(mr−n−1) × ν b ρ is irreducible and isomorphic to ν b ρ ×ν −(mr−n−1) .We thus get the following embeddings and isomorphisms:σ(r) ↩→ ν −(mr−n−1) ⋊ σ(r − 1) ↩→ ν −(mr−n−1) × δ([ν a ρ, ν b ρ]) ⋊ σ ′↩→ ν −(mr−n−1) × ν b ρ × δ([ν a ρ, ν b−1 ρ]) ⋊ σ ′≃ ν b ρ × ν −(mr−n−1) × δ([ν a ρ, ν b−1 ρ]) ⋊ σ ′ ,17

contradicting square-integrability of σ(r). This proves the proposition.In pretty much the same way one can also proveCorollary 5.4. Let τ ∈ S 1 (r) denote a strongly positive discrete series.Suppose that Θ(τ, n) ≠ 0, for some n such that m r ≥ n + 1 . Then τ(n) is a2discrete series representation.The last two propositions of this section contain rather important resulton the transfer of certain embeddings by the theta lifts. We omit the proofs,since this results can be obtained in completely analogous way as in [15,Remark 5.2], i.e., by precise examination of the filtration of Jacquet modulesquoted in the Theorem 4.4.Proposition 5.5. Suppose that the representation σ ∈ Irr(˜Sp(n)) may bewritten as an irreducible subrepresentation of the induced representation ofthe form δ([ν a ρ, ν b ρ]) ⋊ σ ′ , where ρ is an irreducible cuspidal genuine representation,σ ′ ∈ Irr( ˜Sp(n ′ )) and b − a ≥ 0. Let Θ(σ, r) ≠ 0. Then one of thefollowing hold:• There is an irreducible representation τ of some O(V r ′) such that σ(r)is a subrepresentation of δ([ν a χ −1V,ψ ρ, νb χ −1V,ψρ]) ⋊ τ.• There is an irreducible representation τ of some O(V r ′) such that σ(r)is a subrepresentation of δ([ν a+1 χ −1V,ψ ρ, νb χ −1V,ψρ]) ⋊ τ.The latter situation is impossible unless (a, ρ) = (m r − n, χ V,ψ ).Proposition 5.6. Suppose that the representation τ ∈ Irr(O(V r )) may bewritten as an irreducible subrepresentation of the induced representation ofthe form δ([ν a ρ, ν b ρ]) ⋊ τ ′ , where ρ is an irreducible cuspidal representation,τ ′ ∈ Irr(O(V r ′)) and b − a ≥ 0. Let Θ(τ, n) ≠ 0. Then one of the followingholds:• There is an irreducible representation σ of some ˜Sp(n ′ ) such that τ(n)is a subrepresentation of δ([ν a χ V,ψ ρ, ν b χ V,ψ ρ]) ⋊ σ.• There is an irreducible representation σ of some ˜Sp(n ′ ) such that τ(n)is a subrepresentation of δ([ν a+1 χ V,ψ ρ, ν b χ V,ψ ρ]) ⋊ σ.The latter situation is impossible unless (a, ρ) = (n − m r + 1, 1 F ×).18

Observe that Proposition 5.2 implies that the representation σ(l) is nota discrete series representation for l > r.There are two main cases which we consider.Suppose that the representation χ V,ψ ν 1 2 does not appear in [σ]. Sincem r − n = 1, Theorem 4.3 yields 2 n′ ≥ r ′ + 1(dim(V ɛ2 0 ) − 1). We have twopossibilities:• n ′ = r ′ + 1 2 (dim(V ɛ0 ) − 1)In this case both representation χ V,ψ ν s ⋊σ cusp and ν s ⋊τ cusp reduce for s =1. Therefore, by Theorem 3.1, there is no representations of the form χ 2 ,ψν sappearing in [σ]. Further, Theorem 3.5 of [6] implies that the representationχ V,ψ ν s ρ i ⋊ σ cusp reduces if and only if the representation ν s ρ i ⋊ τ cusp reduces.One of the main results of the paper [4] states that σ(r) is a discrete seriesrepresentation. Applying Theorem 2 we obtain the embeddingσ(r) ↩→ δ([ν a 1ρ ′ 1, ν b 1ρ ′ 1]) × δ([ν a 2ρ ′ 2, ν b 2ρ ′ 2]) × · · · × δ([ν a lρ ′ l, ν b lρ ′ l]) ⋊ τ sp ,where a i ≤ 0 and ρ ′ i ∈ {ρ 1 , ρ 2 , . . . , ρ k } for i = 1, 2, . . . , l, and τ sp ∈ Irr(O(V ɛr ′))a strongly positive discrete series, for some r ′ .Since the representation σ is strongly positive, Theorem 4.3 and Lemma3.3 show that for every i ∈ {1, 2, . . . , k} there is at most one representationof the form ν x ρ i that appears in [σ(r)] with 0 ≤ |x| < 1. In the same way asin the proof of Proposition 5.3 we deduce σ(r) ≃ τ sp , i.e., σ(r) is a stronglypositive representation.It is now easy to see, using Lemma 3.2, that σ(r) is unique irreduciblesubrepresentation of the induced representation(k∏ ∏k iδ([ν aρ i −ki+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp .i=1 j=1Suppose that Θ(σ, r−1) ≠ 0. Then Proposition 5.2 implies R P1 (Θ(σ, r))(ν 1 2 ) ≠0, which is impossible. Thus, r is the first occurrence index of σ.• n ′ > r ′ + 1 2 (dim(V ɛ0 ) − 1)20

In this case, the representation χ V,ψ ν s ⋊σ cusp reduces for s = n ′ −m r ′ +1,and the representation ν s ⋊ τ cusp reduces for s = n ′ − m r ′.Observe that [σ(r)] is obtained from [σ] by multiplying with χ −1V,ψall representationsof the form χ V,ψ ν x ρ i appearing in [σ], adding the representationsν − 1 2 , ν − 3 2 , . . . , ν m r ′−n′ and replacing σ cusp with τ cusp .There are two possible cases which we consider:1. Some representation of the form χ V,ψ ν s , s ∈ R, appears in [σ]: We maysuppose that ρ 1 is a trivial representation. Note that a ρ1 − k 1 + 1 isstrictly greater than 1 2 and a ρ 1equals n ′ − m r ′ + 1.For simplicity of notation, let a j stand for a ρ1 −k 1 +j, for j = 1, 2, . . . , k 1 .Again, we know that σ(r) is a discrete series representation. Inspectingits cuspidal support more precisely, it is not hard to see that it has to bestrongly positive. Using Lemma 3.2 we get that σ(r) can be obtainedas the unique irreducible subrepresentation ofν 1 2 × ν32 × · · · × νa 1 −2 ×(i=2 j=1∏k 1j=1δ([ν a j−1 , ν b(1) j]))×k∏ ∏k iδ([ν aρ i −ki+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp .If a 1 ≥ 5, Theorem 5.3 from [11] implies R 2 P 1(σ(r))(ν 1 2 ) ≠ 0. If a 1 = 3,2the same result shows that R P1 (σ(r))(ν 1 2 ) = 0 (since b (1)1 ≥ a 1 > 1).2Using Proposition 5.2 we conclude that Θ ɛ (σ, r − 1) ≠ 0 if a 1 ≥ 5 and 2Θ ɛ (σ, r − 1) = 0 otherwise.If a 1 ≥ 5 , combining the square-integrability of σ(r−1) (by Proposition25.3) with the fact that [σ(r − 1)] is obtained from [σ(r)] by subtractingν 1 2 , we get that σ(r − 1) is strongly positive discrete series which canbe realized as a unique irreducible subrepresentation ofν 3 2 × ν52 × · · · × νa 1 −2 ×(i=2 j=1∏k 1j=1δ([ν a j−1 , ν b(1) j]))×k∏ ∏k iδ([ν aρ i −ki+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp .21

Proceeding with the same analysis as above, we obtain that Θ ɛ (σ, r −l) ≠ 0 for l = 1, 2, . . . , r − a 1 + 3 , and σ(r − l) is a strongly positive2discrete series which can be realized as a unique irreducible subrepresentationof∏k 1ν l+ 1 2 × νl+ 3 2 × · · · × νa 1 −2 × δ([ν aj−1 , ν b(1) j])×(i=2 j=1j=1k∏ ∏k iδ([ν aρ i −ki+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp .Further, it is easy to check that the first occurrence index of σ equalsr − a 1 + 3 2 .2. There is no representation of the form χ V,ψ ν s , s ∈ R, appearing in[σ]: As in the previous case we conclude that σ(r) is strongly positivediscrete series. An easy computation shows that σ(r) is a uniqueirreducible subrepresentation of the induced representationν 1 2 × ν32 × · · · × νn ′ −m r′× (k∏ ∏k ii=1 j=1δ([ν aρ i −k i+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp .Now Theorem 5.3 from [11] shows that R P1 (σ(r))(ν 1 2 ) ≠ 0. SinceR fP 1(σ)(χ V,ψ ν 1 2 ) = 0, part 2 of Proposition 5.2 implies Θ ɛ (σ, r − 1) ≠ 0.Note that [σ(r − 1)] and [σ(r)] differ by ν 1 2 . Proposition 5.3 now showsthat σ(r − 1) is a discrete series representation, and we again concludethat it must be strongly positive. Thus, σ(r −1) is a unique irreduciblesubrepresentation of the induced representationν 3 2 × ν52 × · · · × νn ′ −m r′× (k∏ ∏k ii=1 j=1δ([ν aρ i −k i+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp .If n ′ −m r ′ > 1, in the same way as above we deduce 2 Θɛ (σ, r−2) ≠ 0. Wecontinue in this fashion obtaining Θ ɛ (σ, r − j) ≠ 0 for j = 1, 2, . . . , n ′ −m r ′ + 1 , while σ(r − j) is a strongly positive discrete series which can2be characterized as the unique irreducible subrepresentation ofν j+ 1 2 × νj+ 3 2 × · · · × νn ′ −m r′× (k∏ ∏k iδ([ν aρ i −ki+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp .i=1 j=122

From Proposition 5.2 we conclude that the first occurrence index of σequals r − n ′ + m r ′ − 1 2 = r − (n′ − 1 2 dimV ɛr ′ + 2) + 3 2 .Second, suppose that the representation χ V,ψ ν 1 2 appears in [σ]. There isno loss of generality in assuming that ρ 1 is a trivial representation. We haveto examine the following three possibilities:• n ′ = r ′ + 1 2 (dim(V ɛ0 ) − 1)Observe that in this case both representation χ V,ψ ν s ⋊σ cusp and ν s ⋊τ cuspreduce for s = 1. Obviously, Theorem 3.1 implies k 2 1 = 1.Observe that [σ(r)] is obtained from [σ] simply by replacing σ cusp withτ cusp and multiplying all ˜GL-members of [σ] with χ−1V,ψ, discrete series σ(r)may be realized as the unique irreducible subrepresentation ofk∏ ∏k iδ([ν 1 2 , νb (1)1 ]) × ( δ([ν aρ i −ki+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp .i=2 j=1We just note that for each i ∈ {1, 2, . . . , k} there is at most one x ∈ R,0 ≤ |x| ≤ 1, such that ν x ρ i appears in [σ(r)], thus τ has to be stronglypositive.Obviously, R P1 (σ(r))(ν 1 2 ) ≠ 0 if and only if b (1)1 = 1 2 .If b (1)1 > 1 2 , using Proposition 5.2 we directly conclude Θɛ (σ, r − 1) = 0.Suppose that b (1)1 = 1 2 . If Θɛ (σ, r − 1) ≠ 0, we get that ν 1 2 does not appearin [σ(r − 1)], contradicting Proposition 5.5 (we are in the first case there).Thus, r is the first occurrence index of σ.• n ′ < r ′ + 1 2 (dim(V ɛ0 ) − 1)In this case the representation χ V,ψ ν s ⋊ σ cusp reduces for s = m r ′ − n ′ − 1and the representation ν s ⋊ τ cusp reduces for s = m r ′ − n ′ .According to Theorem 4.3, [σ(r)] is obtained from [σ] by multiplyingwith χ −1V,ψ all ˜GL-members of [σ], subtracting the representations ν 1 2 , ν 3 2 , . . . ,ν m r ′−n′ −1 and replacing σ cusp with τ cusp . In the same way as before, weconclude that σ(r) is strongly positive discrete series, which is characterizedas a unique irreducible subrepresentation ofδ([ν 3 2 , νb (1)1 ]) × δ([ν52 , νb (1)2 ]) × · · · × δ([νm r ′−n ′ , ν b(1) k 1 ])×23

k∏ ∏k i( δ([ν aρ i −ki+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp .i=2 j=1Since ν 1 2 does not appear in [σ(r)], it follows that r is the first occurrenceindex of σ.• n ′ > r ′ + 1 2 (dim(V ɛ0 ) − 1)Now the representation χ V,ψ ν s ⋊ σ cusp reduces for s = n ′ − m r ′ + 1, andthe representation ν s ⋊ τ cusp reduces for s = n ′ − m r ′.Theorem 4.3 now shows that [σ(r)] is obtained from [σ] by multiplyingwith χ −1V,ψ all ˜GL-members of [σ], adding the representations ν− 1 2 , ν − 3 2 , . . . ,ν m r ′−n′ and replacing σ cusp with τ cusp .From Theorem 4.2 we know that the representation σ(r) is in the discreteseries. But, ν 1 2 appears in [σ(r)] with the multiplicity two and consequentlyσ(r) can’t be a strongly positive representation (by Lemma 3.3).In the sequel, we use Theorem 3.5 to describe discrete series σ(r) asprecise as we can. So, we write σ(r) as a subrepresentation of the inducedrepresentation of the formδ([ν a′ 1 ρ′1 , ν b′ 1 ρ′1 ]) × δ([ν a′ 2 ρ′2 , ν b′ 2 ρ′2 ]) × · · · × δ([ν a′ l ρ′l , ν b′ l ρ′l ]) ⋊ τ sp ,where ρ ′ i ∈ {ρ 1 , ρ 2 , . . . , ρ k }, a ′ i ≤ 0 and a ′ i + b ′ i > 0 for i = 1, 2, . . . , l. Further,τ sp is an irreducible strongly positive representation such that [τ sp ] iscontained in [σ(r)]. Hence, at least one of the representations ν 1 2 and ν − 1 2has to appear in some segment [ν a′ i ρ′i , ν b′ i ρ′i ], i ∈ {1, 2, . . . , l}. Since a ′ i ≤ 0and b ′ i > 0, both these representations appear in this segment.Our next claim is that l = 1. Suppose, on the contrary, that l > 1.We conclude that there is some j ∈ {1, 2, . . . , l}, j ≠ i, such thatν 1 2 /∈ [ν a′ j ρ′j , ν b′ j ρ′j ]. But, the union of the segments [ν a′ i , νb ′ i ] and [νa ′ j ρ′j , ν b′ j ρ′j ]is contained in [σ(r)], so there is at most one x, 0 ≤ |x| ≤ 1, such thatν x ρ ′ j appears in [ν a′ j ρ′j , ν b′ j ρ′j ]. This contradicts the fact that the ends of thesegment [ν a′ j ρ′j , ν b′ j ρ′j ] satisfy a ′ j ≤ 0 and b ′ j > 0. Thus, l = 1 and ρ ′ 1 ∼ = 1 F ×.In this way we obtained the following embedding:σ(r) ↩→ δ([ν a′ 1 , νb ′ 1 ]) ⋊ τsp .24

Since a ′ 1 ≤ 0, using Proposition 5.6 we obtain a contradiction with thestrong positivity of σ. Therefore, this case is impossible and Theorem 6.1 isproved.We point out that the obtained results closely parallel those contained inthe Theorem 4.2 of the manuscript [15] for the dual pair (Sp(n), O(V )).7 Lifts of strongly positive discrete series ofthe orthogonal groupsThe purpose of this section is to determine the first occurrence indices ofstrongly positive discrete series of the odd orthogonal groups which appearin the correspondence given by Theorem 4.2 and to provide a description ofthe lifts of such representations in the metaplectic tower.Thus, we let τ ∈ Irr(O(V r )) denote a strongly positive discrete series suchthat Θ(τ, m r − 1 ) ≠ 0 and realize it as a unique irreducible subrepresentation2of the induced representation of the form(k∏ ∏k iδ([ν aρ i −ki+j ρ i , ν b(i) jρ i ])) ⋊ τ cusp ,i=1 j=1with k minimal and k i minimal for i = 1, 2, . . . , k, where τ cusp ∈ Irr(O(V r ′))is a cuspidal representation and ρ i an irreducible cuspidal representation ofGL(n ρi , F ) (this defines n ρi ) for i = 1, 2, . . . , k.Note that Proposition 5.1 yields that the representation τ(l) is not adiscrete series representation for l > m r − 1 2 .In the following theorem we describe the first occurrence indices of certainstrongly positive discrete series of the odd orthogonal groups.Theorem 7.1. Let τ ∈ Irr(O(V r )) be a strongly positive discrete series witha non-zero lift on the (m r − 1 )–th level of the metaplectic tower. Suppose that2τ cusp ∈ Irr(O(V r ′)) is a partial cuspidal support of τ and σ cusp ∈ Irr( ˜Sp(n ′ ))the first non-zero lift of τ cusp . Let n = m r − 1 . Further, define M = {|x| : νx2appears in [τ]} and denote by a min the minimal element of M. If M = ∅, leta min = m r ′ − n ′ + 1.If a min = 1 or 2 r′ = n ′ − 1(dim(V 2 0) − 1), then the first occurrence index ofτ is n. Otherwise, the first occurrence index of τ is n − a min + 3.225

The remaining part of this section is devoted to the proof this theorem.Again, we have two main cases to discuss.First, assume that ν 1 2 does not appear in [τ]. This implies r ′ ≥ n ′ −1(dim(V 2 0) − 1). This leaves us two possibilities:• r ′ = n ′ − 1 2 (dim(V 0) − 1):In this case both representation χ V,ψ ν s ⋊σ cusp and ν s ⋊τ cusp reduce for s =1. From the classification of strongly positive discrete series, elaborated in2section 2, we deduce that there is no representations of the form ν s appearingin [τ].Applying Theorem 4.2 we obtain that τ(n) is a discrete series representationand in the same way as before we may conclude that it is stronglypositive. This yields the following embedding:τ(n) ↩→ (k∏ ∏k iδ([χ V,ψ ν aρ i −ki+j ρ i , χ V,ψ ν b(i) jρ i ])) ⋊ σ cusp .i=1 j=1Proposition 5.1 implies Θ(τ, n−1) = 0. So, n is the first occurrence indexof τ.• r ′ > n ′ − 1 2 (dim(V 0) − 1):In this case, the representation ν s ⋊ τ cusp reduces for s = m r ′ − n ′ and therepresentation χ V,ψ ν s ⋊ σ cusp reduces for s = m r ′ − n ′ − 1.Theorem 4.3 shows that [τ(n)] is obtained from [τ] by multiplying withχ V,ψ all the elements of R appearing in [τ], adding the representations χ V,ψ ν 1 2 ,χ V,ψ ν 3 2 , . . . , χ V,ψ ν m r ′−n′ −1 and replacing τ cusp with σ cusp .There are two main cases to consider:1. There is no representation of the form ν s appearing in [τ], for s ∈ R:As before, we conclude that τ(n) is a strongly positive discrete serieswhich is a unique irreducible subrepresentation of(χ V,ψ ν 1 2 × χV,ψ ν 3 2 × · · · × χV,ψ ν m r ′−n′ −1 ×k∏ ∏k iδ([χ V,ψ ν aρ i −ki+j ρ i , χ V,ψ ν b(i) jρ i ])) ⋊ σ cusp .i=1 j=126

Theorem 5.3 from [11] implies R P1 (τ(n))(χ V,ψ ν 1 2 ) ≠ 0. Since R P1 (τ)(ν 1 2 ) =0, part 2 of Proposition 5.1 shows Θ(σ, n − 1) ≠ 0.From Corollary 5.4 we obtain that τ(n − l) is a discrete series representationfor each l > 0 such that Θ(τ, n − l) ≠ 0. In the same way asabove we see that it must be strongly positive.Since [τ(n − l)] is obtained from [τ(n)] by subtraction of the representationsχ V,ψ ν 1 2 , χ V,ψ ν 3 2 , . . . , χ V,ψ ν 2l−12 , for l ∈ {1, 2, . . . , m r ′ − n ′ − 1},2it is not hard to see, using Proposition 5.1, that Θ(τ, n − l) ≠ 0 forl ∈ {1, 2, . . . , m r ′ − n ′ − 1 }. Furthermore, τ(n − l) is a unique irreduciblesubrepresentation of the induced2representation(χ V,ψ ν 2l+12 × χ V,ψ ν 2l+32 × · · · × χ V,ψ ν m r ′−n′ −1 ×k∏ ∏k iδ([χ V,ψ ν aρ i −ki+j ρ i , χ V,ψ ν b(i) jρ i ])) ⋊ σ cusp ,i=1 j=1for l ∈ {1, 2, . . . , m r ′ − n ′ − 1 2 }.Since there is no representation of the form χ V,ψ ν s appearing in [τ(n −m r ′ + n ′ + 1 )], Proposition 5.1 shows that the first occurrence index of2τ equals n − m r ′ + n ′ + 1.22. There is some representation of the form ν s appearing in [τ]: We maysuppose that ρ 1 is a trivial representation. Obviously, a ρ1 − k 1 + 1 isstrictly greater than 1 2 and a ρ 1equals m r ′ − n ′ .For abbreviation, let a j stand for a ρ1 −k 1 +j, for j = 1, 2, . . . , k 1 . Sinceχ V,ψ ν 1 2 appears in [τ(n)] with multiplicity one, it follows that τ(n 1 ) isstrongly positive representation for each n 1 ≤ n such that Θ(τ, n 1 ) ≠ 0.Also, τ(n) is the unique irreducible subrepresentation of∏k 1χ V,ψ ν 1 2 × χV,ψ ν 3 2 × · · · × χV,ψ ν a1−2 × δ([χ V,ψ ν aj−1 , χ V,ψ ν b(1) j]))×(i=2 j=1j=1k∏ ∏k iδ([χ V,ψ ν aρ i −ki+j ρ i , χ V,ψ ν b(i) jρ i ])) ⋊ σ cusp .Arguing in the same way as in the analogous situation in the metaplecticcase, we deduce that Θ(τ, n − l) ≠ 0 for l ∈ {1, 2, . . . , a 1 − 3 2 }27

and n − a 1 + 3 is the first occurrence index of τ. Further, τ(n − l) is a2unique irreducible representation of the induced representation∏k 1χ V,ψ ν l+ 1 2 × χV,ψ ν l+ 3 2 × · · · × χV,ψ ν a1−2 × δ([χ V,ψ ν aj−1 , χ V,ψ ν b(1) j]))×(i=2 j=1for l ∈ {1, 2, . . . , a 1 − 3 2 }.j=1k∏ ∏k iδ([χ V,ψ ν aρ i −ki+j ρ i , χ V,ψ ν b(i) jρ i ])) ⋊ σ cusp ,It remains to consider the case when the representation ν 1 2 appears in[τ]. Without loss of generality we may suppose that ρ 1 is a trivial character.Similarly as in the previous section, we have to examine three possibilities.• r ′ = n ′ − 1 2 (dim(V 0) − 1):The specificity of this case is that both induced representations ν s ⋊ τ cuspand χ V,ψ ν s ⋊ σ cusp reduce for s = 1 . On account of Theorem 3.1, we have2k 1 = 1 and a ρ1 = 1.2Furthermore, [τ(n)] is obtained from [τ] by replacing τ cusp with σ cusp andmultiplying all other members of [τ] by χ V,ψ .From the equality of cuspidal reducibilities for τ cusp and σ cusp , it may beconcluded that τ(n) is the strongly positive discrete series which is a uniqueirreducible subrepresentation ofk∏ ∏k iδ([χ V,ψ ν 1 2 , χV,ψ ν b(1) 1 ]) × ( δ([χ V,ψ ν aρ i −ki+j ρ i , χ V,ψ ν b(i) jρ i ])) ⋊ σ cusp .i=2 j=1Suppose that the lift Θ(τ, n−1) is non-zero. Then Proposition 5.1, enhancedby Theorem 5.3 of [11] , implies b (1)1 = 1 . From Theorem 4.3 it follows2that there is no representation χ V,ψ ν 1 2 appearing in [τ(n − 1)], contrary toProposition 5.6.It follows that n is the first occurrence index of τ.• r ′ < n ′ − 1 2 (dim(V 0) − 1):28

The induced representation ν s ⋊ τ cusp reduces for s = n ′ − m r ′ and the inducedrepresentation χ V,ψ ν s ⋊ σ cusp reduces for s = n ′ − m r ′ + 1. Accordingto Theorem 4.3, [τ(n)] is obtained from [τ] by replacing τ cusp with σ cusp , multiplyingGL-members of [τ] by χ V,ψ and then subtracting the representationsχ V,ψ ν 1 2 , χ V,ψ ν 3 2 , . . . , χ V,ψ ν n′ −m r′ .The strong positivity of the representation τ and the above discussionshow that for each i ∈ {1, 2, . . . , k} there is at most one x, |x| ≤ 1 such thatχ V,ψ ν x appears in [τ(n)]. Since τ(n) is in the discrete series, from Theorem3.5 we see that it is strongly positive.An easy computation shows that τ(n) is a unique irreducible subrepresentationof the induced representationδ([χ V,ψ ν 3 2 , χV,ψ ν b(1) 1 ])×δ([χV,ψ ν 5 2 , χV,ψ ν b(1) 2 ])×· · ·×δ([χV,ψ ν n′ −m r ′+1 , χ V,ψ ν b(1) k 1 ])×(k∏ ∏k iδ([χ V,ψ ν aρ i −ki+j ρ i , χ V,ψ ν b(i) jχ V,ψ ρ i ])) ⋊ σ cusp .i=2 j=1That n is the first occurrence index of τ follows directly from Proposition5.1.• r ′ > n ′ − 1 2 (dim(V 0) − 1):The induced representation ν s ⋊ τ cusp reduces for s = m r ′ − n ′ , and therepresentation χ V,ψ ν s ⋊σ cusp reduces for s = m r ′ −n ′ −1. The representationχ V,ψ ν 1 2 appears in [τ(n)] with multiplicity two, since [τ(n)] is obtained from[τ] by replacing τ cusp with σ cusp , multiplying other members of [τ] by χ V,ψand adding χ V,ψ ν 1 2 , χ V,ψ ν 3 2 , . . . , χ V,ψ ν m r ′−n′ −1 .According to Lemma 3.3, τ(n) is not a strongly positive discrete series,but the results of the paper [4] show that it is a discrete series representation.Applying Theorem 3.5 and analysis similar to that in the last case consideredin the previous section, we write τ(n) as an irreducible subrepresentationof the induced representation of the formδ([χ V,ψ ν a , χ V,ψ ν b ]) ⋊ σ sp ,where a ≤ 0, a + b > 0 and σ sp ∈ S 2 a strongly positive discrete series.Using Proposition 5.5 we obtain an embedding which contradicts thestrong positivity of τ. Consequently, this case is not possible.This completes the proof of Theorem 7.1.29

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