Bounded symmetric domains and derived geometric structures

Wilhelm Kaup

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2002)

  • Volume: 13, Issue: 3-4, page 243-257
  • ISSN: 1120-6330

Abstract

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Every homogeneous circular convex domain D ⊂ C n (a bounded symmetric domain) gives rise to two interesting Lie groups: The semi-simple group G = A ⁢ u ⁢ t ⁢ D of all biholomorphic automorphisms of D and its isotropy subgroup K ⊂ G ⁢ L ⁢ n , C at the origin (a maximal compact subgroup of G ). The group G acts in a natural way on the compact dual X of D (a certain compactification of C n that generalizes the Riemann sphere in case D is the unit disk in C ). Various authors have studied the orbit structure of the G -space X , here we are interested in the Cauchy-Riemann structure of the G -orbits in X (which in general are only real-analytic submanifolds of X ). Also, we discuss certain K -orbits in the Grassmannian of all linear subspaces of C n that are closely related to the geometry of the bounded symmetric domain D .

How to cite

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Kaup, Wilhelm. "Bounded symmetric domains and derived geometric structures." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13.3-4 (2002): 243-257. <http://eudml.org/doc/252341>.

@article{Kaup2002,
abstract = {Every homogeneous circular convex domain $D \subset \mathbb\{C\}^\{n\}$ (a bounded symmetric domain) gives rise to two interesting Lie groups: The semi-simple group $G = Aut(D)$ of all biholomorphic automorphisms of $D$ and its isotropy subgroup $K \subset GL(n,\mathbb\{C\})$ at the origin (a maximal compact subgroup of $G$). The group $G$ acts in a natural way on the compact dual $X$ of $D$ (a certain compactification of $\mathbb\{C\}^\{n\}$ that generalizes the Riemann sphere in case $D$ is the unit disk in $\mathbb\{C\}$). Various authors have studied the orbit structure of the $G$-space $X$, here we are interested in the Cauchy-Riemann structure of the $G$-orbits in $X$ (which in general are only real-analytic submanifolds of $X$). Also, we discuss certain $K$-orbits in the Grassmannian of all linear subspaces of $\mathbb\{C\}^\{n\}$ that are closely related to the geometry of the bounded symmetric domain $D$.},
author = {Kaup, Wilhelm},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Bounded symmetric domains; Lie groups; Jordan triple systems; CR-structures; Orbit structures; Grassmannians; bounded domains; orbit structures},
language = {eng},
month = {12},
number = {3-4},
pages = {243-257},
publisher = {Accademia Nazionale dei Lincei},
title = {Bounded symmetric domains and derived geometric structures},
url = {http://eudml.org/doc/252341},
volume = {13},
year = {2002},
}

TY - JOUR
AU - Kaup, Wilhelm
TI - Bounded symmetric domains and derived geometric structures
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2002/12//
PB - Accademia Nazionale dei Lincei
VL - 13
IS - 3-4
SP - 243
EP - 257
AB - Every homogeneous circular convex domain $D \subset \mathbb{C}^{n}$ (a bounded symmetric domain) gives rise to two interesting Lie groups: The semi-simple group $G = Aut(D)$ of all biholomorphic automorphisms of $D$ and its isotropy subgroup $K \subset GL(n,\mathbb{C})$ at the origin (a maximal compact subgroup of $G$). The group $G$ acts in a natural way on the compact dual $X$ of $D$ (a certain compactification of $\mathbb{C}^{n}$ that generalizes the Riemann sphere in case $D$ is the unit disk in $\mathbb{C}$). Various authors have studied the orbit structure of the $G$-space $X$, here we are interested in the Cauchy-Riemann structure of the $G$-orbits in $X$ (which in general are only real-analytic submanifolds of $X$). Also, we discuss certain $K$-orbits in the Grassmannian of all linear subspaces of $\mathbb{C}^{n}$ that are closely related to the geometry of the bounded symmetric domain $D$.
LA - eng
KW - Bounded symmetric domains; Lie groups; Jordan triple systems; CR-structures; Orbit structures; Grassmannians; bounded domains; orbit structures
UR - http://eudml.org/doc/252341
ER -

References

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