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COMPACT ORBITS OF PARABOLIC SUBGROUPS

Part of: Complex spaces with a group of automorphisms Automorphic functions

Published online by Cambridge University Press:  14 December 2021

LEONARDO BILIOTTI Open the ORCID record for LEONARDO BILIOTTI [Opens in a new window]  and
OLUWAGBENGA JOSHUA WINDARE Open the ORCID record for OLUWAGBENGA JOSHUA WINDARE [Opens in a new window]
LEONARDO BILIOTTI
Affiliation:
Dipartimento di Scienze Matematiche, Fisiche e Informatiche Università di Parma, Parma, Italy [email protected]
OLUWAGBENGA JOSHUA WINDARE
Affiliation:
Dipartimento di Scienze Matematiche, Fisiche e Informatiche Università di Parma, Parma, Italy [email protected]
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Abstract

We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected Lie group U with Lie algebra $\mathfrak {u}$ extends holomorphically to an action of the complexified group $U^{\mathbb {C}}$ and that the U-action on Z is Hamiltonian. If $G\subset U^{\mathbb {C}}$ is compatible, there exists a gradient map $\mu _{\mathfrak p}:X \longrightarrow \mathfrak p$ where $\mathfrak g=\mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of $\mathfrak g$ . In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map $\mu _{\mathfrak p}$ .

MSC classification

Primary: 57S20: Noncompact Lie groups of transformations
Secondary: 32M05: Complex Lie groups, automorphism groups acting on complex spaces
Type
Article
Information
Nagoya Mathematical Journal , Volume 247 , September 2022 , pp. 615 - 623
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Copyright
© (2021) The Authors. Copyright in the Journal, as distinct from the individual articles, is owned by Foundation Nagoya Mathematical Journal

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Footnotes

Biliotti was partially supported by the Project PRIN 2015, Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis, by the Project PRIN 2017, Real and Complex Manifolds: Topology, Geometry and Holomorphic Dynamics, and by the GNSAGA INdAM.

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