Invariants and K-spectrums of local theta lifts

Let (G,G′) be a type I irreducible reductive dual pair in Sp (W_ℝ). We assume that (G,G′) is in the stable range where G is the smaller member. Let K and K′ be maximal compact subgroups of G and G′ respectively. Let g{fraktur} = k{fraktur} ⊕ p{fraktur} and g{fraktur}′ = k{fraktur}′ ⊕ p{fraktur}′ be the complexified Cartan decompositions of the Lie algebras of G and G′ respectively. Let K and K′ be the inverse images of K and K′ in the metaplectic double cover Sp(W_ℝ) of Sp (W_ℝ). Let ρ be a genuine irreducible (g{fraktur}, K)-module. Our first main result is that if ρ is unitarizable, then except for one special case, the full local theta lift ρ′ = Θ(ρ) is equal to the local theta lift θ(ρ). Thus excluding the special case, the full theta lift ρ′ is an irreducible and unitarizable (g{fraktur}′, K′)-module. Our second main result is that the associated variety and the associated cycle of ρ′ are the theta lifts of the associated variety and the associated cycle of the contragredient representation ρ∗ respectively. Finally we obtain some interesting (g{fraktur}, K)-modules whose K-spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent K_ℂ-orbits in p{fraktur}∗.