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Invariants and $K$ -spectrums of local theta lifts

  • Hung Yean Loke (a1) and Jiajun Ma (a2)

Abstract

Let $(G,G^{\prime })$ be a type I irreducible reductive dual pair in Sp $(W_{\mathbb{R}})$ . We assume that $(G,G^{\prime })$ is in the stable range where $G$ is the smaller member. Let $K$ and $K^{\prime }$ be maximal compact subgroups of $G$ and $G^{\prime }$ respectively. Let $\mathfrak{g}=\mathfrak{k}\bigoplus \mathfrak{p}$ and $\mathfrak{g}^{\prime }=\mathfrak{k}^{\prime }\bigoplus \mathfrak{p}^{\prime }$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G^{\prime }$ respectively. Let $\widetilde{K}$ and $\widetilde{K}^{\prime }$ be the inverse images of $K$ and $K^{\prime }$ in the metaplectic double cover $\widetilde{\text{Sp}}(W_{\mathbb{R}})$ of Sp $(W_{\mathbb{R}})$ . Let ${\it\rho}$ be a genuine irreducible $(\mathfrak{g},\widetilde{K})$ -module. Our first main result is that if  ${\it\rho}$ is unitarizable, then except for one special case, the full local theta lift ${\it\rho}^{\prime }={\rm\Theta}({\it\rho})$ is equal to the local theta lift ${\it\theta}({\it\rho})$ . Thus excluding the special case, the full theta lift  ${\it\rho}^{\prime }$ is an irreducible and unitarizable $(\mathfrak{g}^{\prime },\widetilde{K}^{\prime })$ -module. Our second main result is that the associated variety and the associated cycle of  ${\it\rho}^{\prime }$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation ${\it\rho}^{\ast }$ 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  $\mathfrak{p}^{\ast }$ .

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