Siegel domain realization of pseudo-Hermitian symmetric manifolds
type: Item
instance of: scholarly article
title: Siegel domain realization of pseudo-Hermitian symmetric manifolds
zbMATH Open document ID: 0795.32012
DOI: 10.1007/BF01264096
author: J. E. D'Atri
author: S. G. Gindikin
published in: Geometriae Dedicata
publication date: 1994-08-29T00:00:00Z
review text: A description of some pseudo Hermitian symmetric spaces is given in the following way: Take \(Y\) a complex symmetric space (In general the complex Grassmann manifold \(\text{Gr}^ \mathbb{C}(2n,n)\) of complex linear \(n\) planes in \(\mathbb{C}^{2n}\), or a submanifold). The space \(X\) is an orbit of a noncompact real form acting on \(Y\). Some Zariski open set \(X_ 0\) of \(X\) (obtained by using inhomogeneous coordinates) is a tube domain over a nonconvex cone \(V\) which is homogeneous. Then, in the situation considered by the author, there exists some natural convex subcone \(V_ 1\) of \(V\) such that the associated tube domain \(X^ 1\) is a Hermitian symmetric space. Finally, the study of compact cycles is done. For instance, if \(Y=\text{Gr}^ \mathbb{C} (2n,n)\), one takes \(G=S1 (n,\mathbb{R})\) and the \(G\)-orbit associated is the orbit of \(Z_ 0=(I_ n, i I_ n)\). Then the isotropy group is isomorphic to \(Sl(n,\mathbb{C})\times t\) where \(T\) is the circle. The tube \(X_ 0\) associated to \(X\) has as basis the cone \(V\) of all real matrices of order \(n\) of positive determinant. The cone \(V_ 1\) is the cone of all positive definite matrices of order \(n\). A collection of compact complex submanifolds is also given.
Mathematics Subject Classification ID: 32M15
zbMATH DE Number: 205525
zbMATH Keywords: Siegel domain
zbMATH Keywords: Hermitian symmetric spaces
reviewed by: Jean-Jacques Loeb
MaRDI profile type: Publication
cites work: Les espaces symétriques noncompacts
cites work: Pseudo-Hermitian symmetric spaces and Siegel domains over nondegenerate cones
cites work: Poincaré series and automorphic cohomology on flag domains
cites work: The action of a real semisimple group on a complex flag manifold. I: Orbit structure and holomorphic arc components
cites work: Q3846750
cites work: Q4033552
full work available at URL: https://doi.org/10.1007/bf01264096
OpenAlex ID: W2094799737
Recommended article: Q3349171
Recommended article: Symmetries on compact pseudo-Hermitian manifolds
Recommended article: On hypercomplex pseudo-Hermitian manifolds
Recommended article: Pseudo-harmonic Hermitian structures on Weyl manifolds
Recommended article: Q5025637
Recommended article: Standard pseudo-Hermitian structure on manifolds and Seifert fibration
Recommended article: Q3626639
Recommended article: Q5217325
Recommended article: Higher Laplacians on pseudo-Hermitian symmetric spaces
description: scientific article; zbMATH DE number 205525