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

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