Duke Mathematical Journal

The geometry of Grauert tubes and complexification of symmetric spaces

D. Burns, S. Halverscheid, and R. Hind

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Abstract

We consider complexifications of Riemannian symmetric spaces $X$ of nonpositive curvature. We show that the maximal Grauert domain of $X$ is biholomorphic to a maximal connected extension $\Omega\sb {{\rm AG}}$ of $X=G/K\subset G\sb {\mathbb {C}}/K\sb {\mathbb {C}}$ on which $G$ acts properly, a domain first studied by D. Akhiezer and S. Gindikin [1]. We determine when such domains are rigid, that is, when ${\rm Aut}\sb {\mathbb {C}}(\Omega\sb {{\rm AG}}=G$ and when it is not (when \Omega\sb {{\rm AG}}$ has "hidden symmetries"). We further compute the $G$-invariant plurisubharmonic functions on $\Omega\sb {{\rm AG}}$ and related domains in terms of Weyl group invariant strictly convex functions on a $W$-invariant convex neighborhood of $0\in \mathfrak {a}$. This generalizes previous results of M. Lassalle [25] and others. Similar results have also been proven recently by Gindikin and B. Krötz [8] and by Krötz and R. Stanton [24].

Article information

Source
Duke Math. J., Volume 118, Number 3 (2003), 465-491.

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1082744676

Digital Object Identifier
doi:10.1215/S0012-7094-03-11833-5

Mathematical Reviews number (MathSciNet)
MR1983038

Zentralblatt MATH identifier
1044.53039

Subjects
Primary: 32Q28: Stein manifolds
Secondary: 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15] 53C35: Symmetric spaces [See also 32M15, 57T15]

Citation

Burns, D.; Halverscheid, S.; Hind, R. The geometry of Grauert tubes and complexification of symmetric spaces. Duke Math. J. 118 (2003), no. 3, 465--491. doi:10.1215/S0012-7094-03-11833-5. https://projecteuclid.org/euclid.dmj/1082744676


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References

  • D. N. Akhiezer and S. G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 1--12.
  • A. Ash, D. Mumford, M. Rapoport, and Y. Tai, ``Smooth compactification of locally symmetric varieties'' in Lie Groups: History, Frontiers and Applications, Vol. IV, Math. Sci., Brookline, Mass., 1975.
  • H. Azad and J.-J. Loeb, Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. (N.S.) 3 (1992), 365--375.
  • L. Barchini, Stein extensions of real symmetric spaces and the geometry of the flag manifold, preprint, 2001.
  • A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111--122.
  • D. Burns, ``On the uniqueness and characterization of Grauert tubes'' in Complex Analysis and Geometry (Trento, Italy, 1993), Lecture Notes in Pure and Appl. Math. 173, Dekker, New York, 1996, 119--133.
  • D. Burns and R. Hind, Symplectic geometry and the uniqueness of Grauert tubes, Geom. Funct. Anal. 11 (2001), 1--10.
  • S. Gindikin and B. Krötz, Invariant Stein domains in Stein symmetric spaces and a nonlinear complex convexity theorem, Internat. Math. Res. Notices 2002, no. 18, 959--971.
  • S. Gindikin and T. Matsuki, Stein extensions of Riemann symmetric spaces and dualities of orbits on flag manifolds, Mathematical Sciences Research Institute preprint number 2001-028, http://msri.org/publications/preprints/2001.html
  • V. Guillemin and M. Stenzel, Grauert tubes and the homogeneous Monge-Ampère equation, J. Differential Geom. 34 (1991), 561--570.
  • S. Halverscheid, Maximal domains of definition of adapted complex structures for symmetric spaces of non-compact type, thesis, Ruhr-Universität Bochum, Bochum, Germany, 2001.
  • Harish-Chandra, Spherical functions on a semisimple Lie group, I, Amer. J. Math. 80 (1958), 241--310.
  • S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, New York, 1978.
  • --------, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Pure Appl. Math. 113, Academic Press, Orlando, Fla., 1984.
  • A. Huckleberry, On certain domains in cycle spaces of flag manifolds, Math. Ann. 323 (2002), 797--810.
  • A. Huckleberry and J. A. Wolf, Schubert varieties and cycle spaces, preprint.
  • H. A. Jaffee, Real forms of hermitian symmetric spaces, Bull. Amer. Math. Soc. 81 (1975), 456--458.
  • --. --. --. --., Anti-holomorphic automorphisms of the exceptional symmetric domains, J. Differential Geom. 13 (1978), 79--86.
  • S.-J. Kan, On the characterization of Grauert tubes covered by the ball, Math. Ann. 309 (1997), 71--80.
  • --. --. --. --., On the rigidity of non-positively curved Grauert tubes, Math. Z. 229 (1998), 349--363.
  • S.-J. Kan and D. Ma, On the rigidity of Grauert tubes over locally symmetric spaces, J. Reine Angew. Math. 524 (2000), 205--225.
  • --. --. --. --., On the rigidity of Grauert tubes over Riemannian manifolds of constant curvature, Math. Z. 239 (2002), 353--363. \CMP1 888 229
  • S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Pure Appl. Math. 2, Dekker, New York, 1970.
  • B. Krötz and R. Stanton, Holomorphic extensions of representations, II: Geometry and harmonic analysis, preprint, 2001, http://math.ohio-state.edu/~kroetz
  • M. Lassalle, Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact, Ann. Sci. École Norm. Sup. (4) 11 (1978), 167--210.
  • L. Lempert and R. Szőke, Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds, Math. Ann. 290 (1991), 689--712.
  • K.-H. Neeb, On the complex geometry of invariant domains in complexified symmetric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), 177--225.
  • G. Patrizio and P.-M. Wong, Stein manifolds with compact symmetric center, Math. Ann. 289 (1991), 355--382.
  • H. L. Royden, ``Remarks on the Kobayashi metric'' in Several Complex Variables, II (College Park, Md., 1970), Lecture Notes in Math. 185, Springer, Berlin, 1971..
  • G. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63--68.
  • N. Sibony, ``A class of hyperbolic manifolds'' in Recent Developments in Several Complex Variables (Princeton, 1979), Ann. of Math. Stud. 100, Princeton Univ. Press, Princeton, 1981, 357--372.
  • K. Stein, Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math. 7 (1956), 354--361.
  • R. Szőke, Complex structures on tangent bundles of Riemannian manifolds, Math. Ann. 291 (1991), 409--428.
  • --. --. --. --., Automorphisms of certain Stein manifolds, Math. Z. 219 (1995), 357--385.
  • J. A. Wolf, ``Fine structure of Hermitian symmetric spaces'' in Symmetric Spaces (St. Louis, 1969--1970.), Pure Appl. Math. 8, Dekker, New York, 1972, 271--357.
  • --. --. --. --., Exhaustion functions and cohomology vanishing theorems for open orbits on complex flag manifolds, Math. Res. Lett. 2 (1995), 179--191.
  • --. --. --. --., ``Complex geometry and representations of Lie groups'' in Global Differential Geometry: The Mathematical Legacy of Alfred Gray (Bilbao, Spain, 2000), Contemp. Math. 288, Amer. Math. Soc., Providence, 2001, 202--238.
  • J. A. Wolf and R. Zierau, Linear cycle spaces in flag domains, Math. Ann. 316 (2000), 529--545.