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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Theta lifting of nilpotent orbits for symmetric pairs

Authors: Kyo Nishiyama, Hiroyuki Ochiai and Chen-bo Zhu
Journal: Trans. Amer. Math. Soc. 358 (2006), 2713-2734
MSC (2000): Primary 22E46, 11F27
Published electronically: December 20, 2005
MathSciNet review: 2204053
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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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Kyo Nishiyama
Affiliation: Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo, Kyoto 606-8502, Japan

Hiroyuki Ochiai
Affiliation: Department of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

Chen-bo Zhu
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543

Keywords: Reductive dual pair, theta lifting, nilpotent orbits, harmonic polynomial, invariant theory
Received by editor(s): December 18, 2003
Received by editor(s) in revised form: August 14, 2004
Published electronically: December 20, 2005
Dedicated: Dedicated to Roger Howe on his sixtieth birthday
Article copyright: © Copyright 2005 American Mathematical Society

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