Geometry of invariant domains in complex semi-simple Lie groups
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 3, pp. 509-541.

We investigate the joint action of two real forms of a semi-simple complex Lie group U by left and right multiplication. After analyzing the orbit structure, we study the CR structure of closed orbits. The main results are an explicit formula of the Levi form of closed orbits and the determination of the Levi cone of generic orbits. Finally, we apply these results to prove q-completeness of certain invariant domains in U .

Classification : 22E46, 32V40
Miebach, Christian 1

1 Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D - 44780 Bochum, Germany
@article{ASNSP_2009_5_8_3_509_0,
     author = {Miebach, Christian},
     title = {Geometry of invariant domains in complex semi-simple {Lie} groups},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {509--541},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {3},
     year = {2009},
     mrnumber = {2581425},
     zbl = {1184.22006},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2009_5_8_3_509_0/}
}
TY  - JOUR
AU  - Miebach, Christian
TI  - Geometry of invariant domains in complex semi-simple Lie groups
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2009
SP  - 509
EP  - 541
VL  - 8
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2009_5_8_3_509_0/
LA  - en
ID  - ASNSP_2009_5_8_3_509_0
ER  - 
%0 Journal Article
%A Miebach, Christian
%T Geometry of invariant domains in complex semi-simple Lie groups
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2009
%P 509-541
%V 8
%N 3
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2009_5_8_3_509_0/
%G en
%F ASNSP_2009_5_8_3_509_0
Miebach, Christian. Geometry of invariant domains in complex semi-simple Lie groups. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 3, pp. 509-541. http://www.numdam.org/item/ASNSP_2009_5_8_3_509_0/

[1] A. Andreotti and H. Grauert, Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193–259. | EuDML | Numdam | MR | Zbl

[2] H. Azad and J.-J. Loeb, Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. (N.S.) 3 (1992), 365–375. | MR | Zbl

[3] M. Salah Baouendi, P. Ebenfelt and L. Preiss Rothschild, “Real Submanifolds in Complex Space and their Mappings”, Princeton Mathematical Series, Vol. 47, Princeton University Press, Princeton, NJ, 1999. | MR | Zbl

[4] A. Boggess, “CR Manifolds and the Tangential Cauchy-Riemann complex”, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. | MR | Zbl

[5] R. J. Bremigan, Invariant analytic domains in complex semisimple groups, Transform. Groups 1 (1996), 279–305. | MR | Zbl

[6] C. Chevalley, “Theory of Lie Groups. I”, Princeton University Press, Princeton, N. J., 1946 [Eighth Printing, 1970]. | MR | Zbl

[7] J.-P. Demailly, “Complex Analytic and Algebraic Geometry”, available at http://www-fourier.ujf-grenoble.fr/ demailly/books.html.

[8] F. Docquier and H. Grauert, Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140 (1960), 94–123. | EuDML | MR | Zbl

[9] M. G. Eastwood and G. Vigna Suria, Cohomologically complete and pseudoconvex domains, Comment. Math. Helv. 55 (1980), 413–426. | EuDML | MR | Zbl

[10] G. Fels and L. Geatti, Geometry of biinvariant subsets of complex semisimple Lie groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), 329–356. | EuDML | Numdam | MR | Zbl

[11] M. Flensted-Jensen, Spherical functions of a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), 106–146. | MR | Zbl

[12] J. Hilgert and K.-H. Neeb, “Lie Semigroups and their Applications”, Lecture Notes in Mathematics, Vol. 1552, Springer-Verlag, Berlin, 1993. | MR | Zbl

[13] P. Heinzner and G. W. Schwarz, Cartan decomposition of the moment map, Math. Ann. 337 (2007), 197–232. | MR | Zbl

[14] J. E. Humphreys, “Conjugacy Classes in Semisimple Algebraic Groups”, Mathematical Surveys and Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1995. | MR | Zbl

[15] L. Kaup, “Vorlesungen über Torische Varietäten”, Konstanzer Schriften in Mathematik und Informatik, Nr. 130, Fassung vom Herbst 2001.

[16] S. Kobayashi and K. Nomizu, “Foundations of Differential Geometry. Vol I”, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. | MR | Zbl

[17] A. W. Knapp, “Lie Groups Beyond an Introduction”, second ed., Progress in Mathematics, Vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002. | MR | Zbl

[18] M. Lassalle, Sur la transformation de Fourier-Laurent dans un groupe analytique complexe réductif, Ann. Inst. Fourier (Grenoble) 28 (1978), 115–138. | EuDML | Numdam | MR | Zbl

[19] T. Matsuki, Double coset decompositions of reductive Lie groups arising from two involutions, J. Algebra 197 (1997), 49–91. | MR | Zbl

[20] T. Matsuki, Classification of two involutions on compact semisimple Lie groups and root systems, J. Lie Theory 12 (2002), 41–68. | EuDML | MR | Zbl

[21] C. Miebach, Geometry of invariant domains in complex semi-simple Lie groups, Dissertation, Bochum, 2007.

[22] K.-H. Neeb, Invariant convex sets and functions in Lie algebras, Semigroup Forum 53 (1996), 230–261. | EuDML | MR | Zbl

[23] K.-H. Neeb, On the complex and convex geometry of Ol’shanskiĭsemigroups, Ann. Inst. Fourier (Grenoble) 48 (1998), 149–203. | EuDML | Numdam | MR | Zbl

[24] K.-H. Neeb, “Holomorphy and Convexity in Lie Theory”, de Gruyter Expositions in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2000. | MR | Zbl

[25] R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. | MR | Zbl

[26] R. Steinberg, “Endomorphisms of Linear Algebraic Groups”, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. | MR | Zbl

[27] È. B. Vinberg (ed.), “Lie Groups and Lie Algebras, III”, Encyclopaedia of Mathematical Sciences, Vol. 41, Springer-Verlag, Berlin, 1994. | Zbl