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Pseudo-Hermitian symmetric spaces of tube type. (English) Zbl 0933.32034

Gindikin, Simon (ed.), Topics in geometry. In memory of Joseph D’Atri. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 20, 123-154 (1996).
This paper presents a geometric study of pseudo-Hermitian symmetric spaces “of tube type”, as a geometric preliminary for a study of Hardy spaces of \(\overline\partial\)-cohomology on these domains.
The authors first give a survey of the theory of (non necessarily convex) symmetric cones, their relation with simple Jordan algebras, and the characterization of maximal convex cones contained in a given symmetric cone. They then construct and prove a realization of the conformal compactification of a simple complex Jordan algebra \(V\), as an algebraic variety in the projective space of \[ W=\mathbb{C} \oplus {\mathcal P}_{(1)}\oplus \cdots\oplus {\mathcal P}_{(r)}, \] where \(r\) is the rank of \(V\) and \({\mathcal P}_{(1)},\dots,{\mathcal P}_{(r)}\) are spaces of polynomials on \(V\), related to the Koecher norm function of the Jordan algebra \(V\). If \(\Omega\) is a symmetric cone, realized as the connected component of the unit element in the set of invertible elements of some Jordan algebra \(V\), the associated tube domain in the complexification \(V_\mathbb{C}\) is \(T_\Omega= V+i\Omega\). This tube domain is shown to be a globally pseudo-Hermitian symmetric space iff \(V\) is a Euclidean Jordan algebra. If \(V\) is non-Euclidean, then \(T_\Omega\) is an open dense subset in a pseudo-Hermitian globally symmetric space \(X\), which will be called a pseudo-Hermitian symmetric space “of tube type”.
For the entire collection see [Zbl 0842.00040].

MSC:

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
22E10 General properties and structure of complex Lie groups
17C50 Jordan structures associated with other structures
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