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

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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
Page 6 and 7: GL(n j ) × Sp(n − |s|, F ), wher
Page 8 and 9: Theorem 3.1. We define a collection
Page 11 and 12: where each ρ i ∈ R ′ is an irr
Page 14 and 15: the representation ν s 2χ V,ψ
Page 18: contradicting square-integrability
Page 21 and 22: In this case, the representation χ
Page 23 and 24: From Proposition 5.2 we conclude th
Page 25 and 26: Since a ′ 1 ≤ 0, using Proposit
Page 27 and 28: Theorem 5.3 from [11] implies R P1
Page 29 and 30: The induced representation ν s ⋊

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

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