On classification of quasi-symmetric domains

Abstract

The notion of "Siegel domains" was introduced by . It was then shown that every homogeneous bounded domain is holomorphically equivalent to a Siegel domain (of the second kind) determined uniquely up to an affine isomorphism ([15] . The purpose of the present note is to show that this classification can also be obtained immediately from my previous result on linear imbeddings of self-dual cones ([10a]).Our method is based on the following two observations: (I) There are natural equivalences between the three categories of (punctured) self-dual cones, the corresponding reductive Lie algebras (with fixed Cartan involutions), and formally real Jordan algebras ( § 1).(II) There is a natural bisection between the set of isomorphism classes of quasi-symmetric Siegel domains and that of equivalence classes of the pairs formed of a self-dual cone and a (linear) "representation" of it ( §3, Proposition 2).

Homogeneous Siegel domains

“…Using the theory of representations of Jordan algebras [24] and the theory of Clifford algebras [2], [8] it is possible to find an explicite description of all quasisymmetric Siegel domains. This in particular reproduces results of I. Satake [46], [47], [48], and M. Takeuchi [53]. Moreover, a Cayley transformation for quasisymmetric Siegel domains and the symmetries of a symmetric Siegel domain can be stated explicitly.…”

Flat totally geodesic submanifolds of quasisymmetric Siegel domains

“…REMARK. A classification of the symmetric Siegel domains has been obtained by different methods in [13], [14], [15].…”

“…The notion of a quasisymmetric domain was originally defined by Satake [13]; Dorfmeister [9] showed that this was equivalent to the condition that be a Jordan algebra. D'Atri and Miatello [6] proved that an irreducible domain is quasisymmetric if and only if there are constants a and b so that dim n(l/2)(ek+em) : b, 1 ~< k < m ~< r, and dim n(1/2)t k ----a, 1 ~< k ~< r.…”

“…We shall therefore assume Rank A < Rank £f. Splitting q/= °/11 + q/o relative to the eigenvalues of ~o(A) we obtain by a straightforward (though lengthy) computation It is easy to see ( [13], [9]) that a quasisymmetric Siegel domain splits into a product of irreducible quasisymmetric Siegel (IQS for short) domains, where the irreducible factors correspond to the simple summands of ft. [13], [151, [9]. From the above remarks it is clear that such a classification will follow from the classification of all formally real Jordan algebras and their complex representations.…”

Holomorphic Sectional Curvature of Quasi-Symmetric Domains

“…Here we refine this result and prove that the holomorphic sectional curvature is bounded above by a negative constant. (see [4]). (see [4]).…”