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

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