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Examples of unbounded homogeneous domains in complex space. (English) Zbl 1128.32013

From the text: The study of bounded holomorphically homogeneous domains in complex space goes back to É. Cartan [Abh. Math. Semin. Univ. Hamb. 11, 116–162 (1935; Zbl 0011.12302)] who determined all bounded symmetric domains in \(\mathbb C^n\) as well as all bounded homogeneous domains in \(\mathbb C^2\) and \(\mathbb C^3\) . A fundamental theorem due to Vinberg, Gindikin, and Pyatetskii-Shapiro states that every bounded homogeneous domain is biholomorphically equivalent to a Siegel domain of the second kind [see I. Pyatetskii-Shapiro, Automorphic functions and the geometry of classical domains (translated from the Russian), New York-London-Paris: Gordon and Breach (1969; Zbl 0196.09901)]. Although this result does not immediately imply a complete classification of bounded homogeneous domains, it reduces the classification problem to that for domains of a very special form. A generalisation of the above theorem to the case of unbounded domains for the class of rational homogeneous domains was obtained by R. Penney in his remarkable paper [Ann. Math. (2) 126, No. 2, 389–414 (1987; Zbl 0655.32027)], where the role of models is played by so-called Siegel domains of type \(N\)-\(P\). Nevertheless, the classification problem for the unbounded case is far from fully understood, and any examples of homogeneous domains that possess no bounded realisations are of interest. In this paper we discuss several nontrivial examples of such domains: The domains are equivalent to tubes over affinely homogeneous domains in real space.

MSC:

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
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