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

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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
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 16 and 17: j−1mr−n+Especially, a quotient
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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