Tohoku Mathematical Journal

A characterization of symmetric Siegel domains by convexity of Cayley transform images

Chifune Kai

Full-text: Open access


We show that a homogeneous Siegel domain is symmetric if and only if its Cayley transform image is convex. Moreover, this convexity forces the parameter of the Cayley transform to be specific, so that the Cayley transform coincides with the inverse of the Cayley transform introduced by Korányi and Wolf.

Article information

Tohoku Math. J. (2), Volume 59, Number 1 (2007), 101-118.

First available in Project Euclid: 16 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15]
Secondary: 43A85: Analysis on homogeneous spaces 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions

Homogeneous Siegel domain symmetric Siegel domain Cayley transform normal $j$-algebra


Kai, Chifune. A characterization of symmetric Siegel domains by convexity of Cayley transform images. Tohoku Math. J. (2) 59 (2007), no. 1, 101--118. doi:10.2748/tmj/1176734750.

Export citation


  • H. Asano, On the irreducibility of homogeneous convex cones, J. Fac. Sci. Univ. Tokyo Sect. I 15 (1968), 201--208.
  • K. Azukawa, Curvature operator of the Bergman metric on a homogeneous bounded domain, Tohoku Math. J. (2) 37 (1985), 197--223.
  • J. E. D'Atri, J. Dorfmeister and Zhao Yan Da, The isotropy representation for homogeneous Siegel domains, Pacific J. Math. 120 (1985), 295--326.
  • J. E. D'Atri and I. Dotti Miatello, A characterization of bounded symmetric domains by curvature, Trans. Amer. Math. Soc. 276 (1983), 531--540.
  • J. Dorfmeister, Homogene Siegel-Gebiete, Habilitationsschrift, Münster, 1978.
  • J. Dorfmeister, Quasisymmetric Siegel domains and the automorphisms of homogeneous Siegel domains, Amer. J. Math. 102 (1980), 537--563.
  • C. Kai, Symmetry characterization of quasisymmetric Siegel domains by convexity of Cayley transform images, J. Lie Theory 16 (2006), 47--56.
  • C. Kai and T. Nomura, A characterization of symmetric cones through pseudoinverse maps, J. Math. Soc. Japan 57 (2005), 195--215.
  • C. Kai and T. Nomura, A characterization of symmetric tube domains by convexity of Cayley transform images, Differential Geom. Appl. 23 (2005), 38--54.
  • S. Kaneyuki, On the automorphism groups of homogeneous bounded domains, J. Fac. Sci. Univ. Tokyo Sect. I 14 (1967), 89--130.
  • A. Korányi and J. A. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Ann. of Math. (2) 81 (1965), 265--288.
  • O. Loos, Bounded symmetric domains and Jordan pairs, Lecture Notes, Univ. California at Irvine, Cal., 1977.
  • N. Mok and I.-H. Tsai, Rigidity of convex realizations of irreducible bounded symmetric domains of rank $\ge$ 2, J. Reine Angew. Math. 431 (1992), 91--122.
  • T. Nomura, On Penney's Cayley transform of a homogeneous Siegel domain, J. Lie Theory 11 (2001), 185--206.
  • T. Nomura, A characterization of symmetric Siegel domains through a Cayley transform, Transform. Groups 6 (2001), 227--260.
  • T. Nomura, Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains, Differential Geom. Appl. 15 (2001), 91--106.
  • T. Nomura, Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms, Differential Geom. Appl. 18 (2003), 55--78.
  • T. Nomura, Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains, J. Funct. Anal. 198 (2003), 229--267.
  • R. Penney, The Harish-Chandra realization for non-symmetric domains in $\boldsymbolC^n$, Topics in Geometry, 295--313, Progr. Nonlinear Differential Equations Appl. 20, Birkhäuser, Boston, Mass., 1996.
  • I. I. Pyatetskii-Shapiro, Automorphic functions and the geometry of classical domains, Gordon and Breach, New York, 1969.
  • H. Rossi, Lectures on representations of groups of holomorphic transformations of Siegel domains, Lecture Notes, Brandeis University, 1972.
  • H. Rossi and M. Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Funct. Anal. 13 (1973), 324--389.
  • I. Satake, Algebraic structures of symmetric domains, Iwanami Shoten, Tokyo; Princeton Univ. Press, Princeton, N.J., 1980.
  • E. B. Vinberg, Structure of the group of automorphisms of a homogeneous convex cone, Trudy Moskov Mat. Obshch. 13 (1965), 56--83; Trans. Moscow Math. Soc. 13 (1967), 63--93.