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

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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· · · of real numbers suchis an integer, for j = 1, 2, . . . , k i and −1 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
Page 6 and 7: GL(n j ) × Sp(n − |s|, F ), wher
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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