A family of adapted complexifications for SL 2 ()
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 17-49.

Let G be a non-compact, real semisimple Lie group. We consider maximal complexifications of G which are adapted to a distinguished one-parameter family of naturally reductive, left-invariant metrics. In the case of G=SL 2 () their realization as equivariant Riemann domains over G =SL 2 () is carried out and their complex-geometric properties are investigated. One obtains new examples of non-univalent, non-Stein, maximal adapted complexifications.

Classification : 53C30, 53C22, 32C09, 32Q99, 32M05
Halverscheid, Stefan 1 ; Iannuzzi, Andrea 2

1 Universität Bremen, Bibliothekstr. 1, D-28359 Bremen, Germany
2 Dipartimento di Matematica, II Università di Roma “Tor Vergata", Via della Ricerca Scientifica, I-00133 Roma, Italia
@article{ASNSP_2009_5_8_1_17_0,
     author = {Halverscheid, Stefan and Iannuzzi, Andrea},
     title = {A family of adapted complexifications for $SL_2(\mathbb{R})$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {17--49},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {1},
     year = {2009},
     mrnumber = {2512199},
     zbl = {1180.53053},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2009_5_8_1_17_0/}
}
TY  - JOUR
AU  - Halverscheid, Stefan
AU  - Iannuzzi, Andrea
TI  - A family of adapted complexifications for $SL_2(\mathbb{R})$
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2009
SP  - 17
EP  - 49
VL  - 8
IS  - 1
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2009_5_8_1_17_0/
LA  - en
ID  - ASNSP_2009_5_8_1_17_0
ER  - 
%0 Journal Article
%A Halverscheid, Stefan
%A Iannuzzi, Andrea
%T A family of adapted complexifications for $SL_2(\mathbb{R})$
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2009
%P 17-49
%V 8
%N 1
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2009_5_8_1_17_0/
%G en
%F ASNSP_2009_5_8_1_17_0
Halverscheid, Stefan; Iannuzzi, Andrea. A family of adapted complexifications for $SL_2(\mathbb{R})$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 17-49. http://www.numdam.org/item/ASNSP_2009_5_8_1_17_0/

[1] R. Aguilar, Symplectic reduction and the homogeneous complex Monge-Ampère equation, Ann. Global Anal. Geom. 19 (2001), 327–353. | MR | Zbl

[2] D. N. Akhiezer and S. G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 1–12. | EuDML | MR | Zbl

[3] R. Bielawski, Complexification and hypercomplexification of manifolds with a linear connection, Internat. J. Math. 14 (2003), 813–824. | MR | Zbl

[4] R. Bremigan, Pseudokähler forms on complex Lie groups, Doc. Math. 5 (2000), 595–611. | EuDML | MR | Zbl

[5] D. Burns, On the uniqueness and characterization of Grauert tubes, In: “Complex Analysis and Geometry”, V. Ancona and A. Silva (eds.), Lecture Notes in Pure and Applied Math., Vol. 173, 1995, 119–133. | Zbl

[6] D. Burns, S. Halverscheid and R. Hind, The geometry of grauert tubes and complexification of symmetric spaces, Duke Math. J. 118 (2003), 465–491. | MR | Zbl

[7] J. Berndt, F. Tricerri and L. Vanhecke, “Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces”, Lecture Notes Math., Vol. 1598, Springer-Verlag, Berlin 1995. | MR | Zbl

[8] J. E. D’Atri and W. Ziller, “Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups”, Mem. Amer. Math. Soc., Vol. 215, 1979. | MR | Zbl

[9] O. Fels, Pseudo-Kählerian structure on domains over a complex semisimple Lie group, Math. Ann. 232 (2002), 1–29. | MR | Zbl

[10] G. Fels, A. T. Huckleberry and J. A. Wolf, “Cycle Spaces of Flag Domains: A Complex Geometric Viewpoint”, Progress in Mathematics, Vol. 245, Birkhäuser, Boston, 2005. | MR | Zbl

[11] L. Geatti and A. Iannuzzi, On univalence of equivariant Riemann domains over the complexification of a non-compact, Riemannian symmetric space, Pacific J. Math. 238 (2008), 275–330. | MR | Zbl

[12] C. S. Gordon, Naturally reductive homogeneous Riemannian manifolds, Canad. J. Math. 37 (1985), 467–487. | MR | Zbl

[13] V. Guillemin and M. Stenzel, Grauert tubes and the homogeneous Monge-Ampère equation, J. Differential Geom. 34 (1991), 561–570 (first part); J. Differential Geom. 35 (1992), 627–641 (second part). | MR | Zbl

[14] S. Halverscheid and A. Iannuzzi, Maximal complexifications of certain Riemannian homogeneous spaces, Trans. Amer. Math. Soc. 355 (2003), 4581–4594. | MR | Zbl

[15] S. Halverscheid and A. Iannuzzi, On naturally reductive left-invariant metrics of SL 2 (). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 5 (2006), 171–187. | EuDML | Numdam | MR | Zbl

[16] P. Heinzner, Geometric invariant theory on Stein spaces. Math. Ann. 289 (1991), 631–662. | EuDML | MR | Zbl

[17] P. Heinzner and A. Iannuzzi, Integration of local actions on holomorphic fiber spaces, Nagoya Math. J. 146 (1997), 31–53 | MR | Zbl

[18] S. Helgason, “Differential Geometry, Lie Groups and Symmetric Spaces”, Graduate Studies in Mathematics, Vol. 34, Amer. Math. Soc., Providence R.I., 2001. | Zbl

[19] L. Lempert and R. Szőke, Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundles of Riemannian manifolds, Math. Ann. 290 (1991), 689–712. | EuDML | MR | Zbl

[20] B. O’Neill, “Semi-Riemannian Geometry”, Academic Press, New York, 1983. | MR

[21] G. Patrizio and P.-M. Wong, Stein manifolds with compact symmetric center, Math. Ann. 289 (1991), 355–382. | EuDML | MR | Zbl

[22] H. Rossi, On envelopes of holomorphy, Commun. Pure Appl. Math. 16 (1963), 9–17. | MR | Zbl

[23] W. Stoll, The characterization of strictly parabolic manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 87–154. | EuDML | Numdam | MR | Zbl

[24] R. Szőke, Complex structures on tangent bundles of Riemannian manifolds, Math. Ann. 291 (1991), 409–428. | EuDML | MR | Zbl

[25] R. Szőke, Adapted complex structures and Riemannian homogeneous spaces, In: “Complex Analysis and Applications” (Warsaw, 1997) Ann. Polon. Math. 70 (1998), 215–220. | EuDML | MR | Zbl

[26] R. Szőke, Canonical complex structures associated to connections, Math. Ann. 329 (2004), 553-591. | MR | Zbl

[27] V. S. Varadarajan, “Lie Groups, Lie Algebras, and their Representations”, Springer-Verlag, New York, 1984. | MR | Zbl

[28] P. Zhao, “Invariant Stein Domains: A Contribution to the Program of Gelfand and Gindikin”, PhD Thesis, Berichte aus der Mathematik, Shaker Verlag, Aachen 1996. | Zbl