Invariants and K-spectrums of local theta lifts

Abstract

Let $(G,G^{\prime })$ be a type I irreducible reductive dual pair in $\mathrm{Sp}(W_{\mathbb{R}})$. 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 $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ and ${\mathfrak{g}}^{\prime }={\mathfrak{k}}^{\prime }\oplus {\mathfrak{p}}^{\prime }$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G^{\prime }$ respectively. Let $\tilde{K}$ and ${\tilde{K}}^{\prime }$ be the inverse images of $K$ and $K^{\prime }$ in the metaplectic double cover $\widetilde{\mathrm{Sp}}(W_{\mathbb{R}})$ of $\mathrm{Sp}(W_{\mathbb{R}})$. Let $\rho$ be a genuine irreducible $(\mathfrak{g},\tilde{K})$-module. Our first main result is that if $\rho$ is unitarizable, then except for one special case, the full local theta lift ${\rho }^{\prime }=\Theta (\rho )$ is equal to the local theta lift $\theta (\rho )$. Thus excluding the special case, the full theta lift ${\rho }^{\prime }$ is an irreducible and unitarizable $({\mathfrak{g}}^{\prime },{\tilde{K}}^{\prime })$-module. Our second main result is that the associated variety and the associated cycle of ${\rho }^{\prime }$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation ${\rho }^{\ast }$ respectively. Finally we obtain some interesting $(\mathfrak{g},\tilde{K})$-modules whose $\tilde{K}$-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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