Abstract
In this paper we give a characterization of symmetric Siegel domains in terms of a certain norm equality which involves a Cayley transform.
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H. Asano,On the irreducibility of homogeneous convex cones, J. Fac. Sci. Univ. Tokyo15 (1968), 201–208.
H. Braun, M. Koecher,Jordan-Algebren, Springer, Berlin, 1966.
J. E. D'Atri,Holomorphic sectional curvatures of bounded homogeneous domains and related questions, Trans. Amer. Math. Soc.256 (1979), 405–413.
J. E. D'Atri, J. Dorfmeister,Flat totally geodesic submanifolds of quasisymmetric Siegel domains, Geom. Dedicata28 (1988), 321–336.
J. E. D'Atri, J. Dorfmeister, Zhao Yan Da,The isotropy representation for homogeneous Siegel domains, Pacific J. Math.,120 (1985), 295–326.
J. E. D'Atri, I. D. Miatello,A characterization of bounded symmetric domains, Trans. Amer. Math. Soc.276 (1983), 531–540.
J. Dorfmeister,Quasisymmetric Siegel domains and the automorphisms of homogeneous Siegel domains, Amer. J. Math.102 (1980), 537–563.
J. Dorfmeister,Homogeneous Siegel domains, Nagoya Math. J.86 (1982), 39–83.
J. Faraut, A. Korányi,Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994.
С. Г. Гиндикин,Анализ в однородных одласмях, YMHXIX (1964), в. 4, 3–92. English transl.: S. G. Gindikin,Analysis in homogeneous domains, Russian Math. Surveys19–4 (1964), 1–89.
H. Ishi,Representations of the affine transformation groups acting simply transitively on Siegel domains, J. Funct. Anal.167 (1999), 425–462.
S. Kaneyuki,On the automorphism groups of homogeneous bounded domains, J. Fac. Sci. Univ. Tokyo14 (1967), 89–130.
S. Kaneyuki,Geometry of Complex Bounded Domains, Lecture Notes3, Sophia University, 1978 (in Japanese).
M. Koecher,The Minnesota Notes on Jordan Algebras and Their Applications, Lecture Notes in Math.1710, 1999, Springer, Berlin.
J. L. Koszul,Sur la forme hermitienne canonique des espaces homogènes complexes, Canad. J. Math.7 (1955), 562–576.
O. Loos,Bounded Symmetric Domains and Jordan Pairs, Lecture Notes, Univ. California at Irvine, 1977.
T. Nomura,On Penney's Cayley transform of a homogeneous Siegel domain, J. Lie Theory11 (2001), 185–206.
T. Nomura,Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains, Diff. Geom. Appl.15 (2001), 91–106.
R. Penney, The Harish-Chandra realization for non-symmetric domains in ℂn, in:Topics in Geometry in Memory of Joseph D'Atri (S. Gindikin, ed.), Birkhäuser, Boston, 1996, pp. 295–313.
И. И. Пятецкий-Цапиро,Геомемрия классических обласмей и меория авмоморфных функций, Физматгиз, М., 1961. English transl.: 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 Univ., 1972.
H. Rossi, 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 and Princeton Univ. Press, Tokyo, Princeton, 1980.
T. Tsuji,A characterization of homogeneous self-dual cones, Tokyo J. Math.5 (1982), 1–12.
A. Unterberger, H. Upmeier,The Berezin transform and invariant differential operators, Comm. Math. Phys.164 (1994), 563–597.
H. Upmeier,Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics, CBMS Regional Conf. Ser.67, Amer. Math. Soc., Providence, Rhode Island, 1987.
З. Б. Винберг,Теория однородных выпуклых конусов, Труды MMO12 (1963), 303–358. English transl.: È. B. Vinberg,The theory of convex homogeneous cones, Trans. Moscow Math. Soc.12 (1963), 340–403.
З. Б. Винберг,Смрукмура группы авмоморфизмов однородного выпуклого конуса, Труды MMO13 (1965), 56–83. English transl.: È. B. Vinberg,The structure of the group of automorphisms of a homogeneous convex cone, Trans. Moscow Math. Soc.13 (1965), 63–93.
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Nomura, T. A characterization of symmetric Siegel domains through a Cayley transform. Transformation Groups 6, 227–260 (2001). https://doi.org/10.1007/BF01263091
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DOI: https://doi.org/10.1007/BF01263091