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**Compactification of parahermitian symmetric spaces and its applications. I: Tube-type realizations.**
*(English)*
Zbl 1073.32009

Doebner, H.-D. (ed.) et al., Lie theory and its applications in physics III. Proceedings of the 3rd international workshop, Technical University of Clausthal, Clausthal, Germany, July 11–14, 1999. Singapore: World Scientific (ISBN 981-02-4421-5/hbk). 63-74 (2000).

Let \(X_{0}=G_{0}/K_{0}\) be an irreducible noncompact Hermitian symmetric space, where \(G_{0}\) is a simple connected Lie group of automorphisms and \(K_{0}\) a maximal compact subgroup, stabilizer of a base point in \(X_{0}\). A complexification of \(X_{0}\) is given by \(X^{\mathbb{C}}=G^{\mathbb{C} }/K^{\mathbb{C}}\), where \(G^{\mathbb{C}}\) and \(K^{\mathbb{C}}\) are complexifications of \(G_{0}\) and \(K_{0}\) respectively.

The real forms of \(X^{\mathbb{C}}\) fall into two classes:

- elliptic real forms, including \(X_{0}\), its compact dual and some spaces called pseudo-Hermitian symmetric spaces;

- hyperbolic real forms, called para-Hermitian symmetric spaces.

The paper gives realizations of the para-Hermitian symmetric spaces as tubes over non-convex cones of a simple Jordan algebra, assuming that the domain \(X_0\) is of tube type. This generalizes a result of K. Koufany [C. R. Acad. Sci., Paris, Sér. I 318, No. 5, 425–428 (1994; Zbl 0839.53035)] for symmetric spaces of Cayley type, which are a special type of para-Hermitian symmetric domains. The proof of Koufany was Jordan theoretic, while the method in this paper relies on semisimple graded Lie algebras. However the result, which is entirely expressed in terms of simple Jordan algebras and their generic norm, strongly suggests there should be a purely Jordan theoretic proof of the main result of this paper.

For the entire collection see [Zbl 0994.00025].

The real forms of \(X^{\mathbb{C}}\) fall into two classes:

- elliptic real forms, including \(X_{0}\), its compact dual and some spaces called pseudo-Hermitian symmetric spaces;

- hyperbolic real forms, called para-Hermitian symmetric spaces.

The paper gives realizations of the para-Hermitian symmetric spaces as tubes over non-convex cones of a simple Jordan algebra, assuming that the domain \(X_0\) is of tube type. This generalizes a result of K. Koufany [C. R. Acad. Sci., Paris, Sér. I 318, No. 5, 425–428 (1994; Zbl 0839.53035)] for symmetric spaces of Cayley type, which are a special type of para-Hermitian symmetric domains. The proof of Koufany was Jordan theoretic, while the method in this paper relies on semisimple graded Lie algebras. However the result, which is entirely expressed in terms of simple Jordan algebras and their generic norm, strongly suggests there should be a purely Jordan theoretic proof of the main result of this paper.

For the entire collection see [Zbl 0994.00025].

Reviewer: Guy Roos (St. Peterburg)

### MSC:

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |

22E46 | Semisimple Lie groups and their representations |

53C35 | Differential geometry of symmetric spaces |

17C36 | Associated manifolds of Jordan algebras |