Theta lifting of nilpotent orbits for symmetric pairs

Nishiyama, Kyo; Ochiai, Hiroyuki; Zhu, Chen-bo

doi:10.1090/S0002-9947-05-03826-2

Theta lifting of nilpotent orbits for symmetric pairs
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by Kyo Nishiyama, Hiroyuki Ochiai and Chen-bo Zhu PDF

Trans. Amer. Math. Soc. 358 (2006), 2713-2734 Request permission

Abstract:

We consider a reductive dual pair $(G, G’)$ in the stable range with $G’$ the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent $K’_{\mathbb {C}}$-orbits, where $K’$ is a maximal compact subgroup of $G’$ and we describe the precise $K_{\mathbb {C}}$-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair $( G, K)$. As an application, we prove sphericality and normality of the closure of certain nilpotent $K_{\mathbb {C}}$-orbits obtained in this way. We also give integral formulas for their degrees.

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