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Invariants and $K$-spectrums of local theta lifts

Part of: Lie groups

Published online by Cambridge University Press:  17 September 2014

Hung Yean Loke  and
Jiajun Ma
Hung Yean Loke
Affiliation:
Department of Mathematics, National University of Singapore, Block S17, 10, Lower Kent Ridge Road, Singapore 119076, Singapore email [email protected]
Jiajun Ma
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er Sheva 84105, Israel email [email protected]
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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 $\mathfrak{p}^{\ast }$.

Keywords

local theta liftsnilpotent orbitsassociated varietiesassociated cyclesmoment maps

MSC classification

Primary: 22E46: Semisimple Lie groups and their representations 22E47: Representations of Lie and real algebraic groups
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 1 , January 2015 , pp. 179 - 206
Copyright
© The Author(s) 2014 

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