Invariants and $K$-spectrums of local theta lifts. (17th September 2014) - Loke, Hung Yean; Ma, Jiajun
Abstract: Let $(G, G^{′ })$ be a type I irreducible reductive dual pair in Sp $(W_{\mathbb{R}})$ . 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 $\mathfrak{g}=\mathfrak{k}\bigoplus \mathfrak{p}$ and $\mathfrak{g}^{′ }=\mathfrak{k}^{′ }\bigoplus \mathfrak{p}^{′ }$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G^{′ }$ respectively. Let $\widetilde{K}$ and $\widetilde{K}^{′ }$ be the inverse images of $K$ and $K^{′ }$ in the metaplectic double cover $\widetilde{\text{Sp}}(W_{\mathbb{R}})$ of Sp $(W_{\mathbb{R}})$ . Let ${\itρ}$ be a genuine irreducible $(\mathfrak{g}, \widetilde{K})$ -module. Our first main result is that if ${\itρ}$ is unitarizable, then except for one special case, the full local theta lift ${\itρ}^{′ }={\rmΘ}({\itρ})$ is equal to the local theta lift ${\itθ}({\itρ})$ . Thus excluding the special case, the full theta lift ${\itρ}^{′ }$ is an irreducible and unitarizable $(\mathfrak{g}^{′ }, \widetilde{K}^{′ })$ -module. Our second main result is that the associated variety and the associated cycle of ${\itρ}^{′ }$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation ${\itρ}^{* }$ respectively. Finally we obtain some interesting $(\mathfrak{g}, \widetilde{K})$ -modules whose $\widetilde{K}$ -spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent $K_{\mathbb{C}}$ -orbits in $\mathfrak{p}^{* }$ .