Invariants and $K$-spectrums of local theta lifts. (17th September 2014)

• Record Type: Journal Article
• Authors: Loke, Hung Yean; Ma, Jiajun
• Is Part Of: Compositio mathematica. Volume 151:Number 1(2015:Jan.)
• Page Start: 179
• Page End: 206
• Publication Date: 2014-09-17
• Subjects: 22E46, -- 22E47 (primary); local theta lifts, -- nilpotent orbits, -- associated varieties, -- associated cycles, -- moment maps; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X14007520
• Languages: English
• ISSNs: 0010-437X

Abstract

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 $\mathfrak{p}^{\ast }$ .