Abstract
We consider a real reductive dual pair (G′, G) of type I, with rank \({({\rm G}^{\prime}) \leq {\rm rank(G)}}\). Given a nilpotent coadjoint orbit \({\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}\), let \({\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}\) denote the complex orbit containing \({\mathcal{O}^{\prime}}\). Under some condition on the partition λ′ parametrizing \({\mathcal{O}^{\prime}}\), we prove that, if λ is the partition obtained from λ by adding a column on the very left, and \({\mathcal{O}}\) is the nilpotent coadjoint orbit parametrized by λ, then \({\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}\), where \({\tau, \tau^{\prime}}\) are the moment maps. Moreover, if \({chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}\), where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to \({\mu_{\mathcal{O}^{\prime}}}\) with corresponds to \({\mathcal{O}_\mathbb{C}}\) via the Springer correspondence.
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Aubert, AM., Kraśkiewicz, W. & Przebinda, T. Howe correspondence and Springer correspondence for real reductive dual pairs. manuscripta math. 143, 81–130 (2014). https://doi.org/10.1007/s00229-013-0617-y
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DOI: https://doi.org/10.1007/s00229-013-0617-y