Some remarks and problems on complex homogeneous domains (Q867816)

The author discusses realizations of homogeneous domains in \({\mathbb C}^n\) which satisfy some natural geometric conditions. He formulates 5 problems and illustrates them by several examples. Problem 1. (1) Give an affine classification of convex homogeneous domains in \({\mathbb C}^n\) which do not contain complex lines. Is it true that any such domain is affine equivalent to a Siegel domain of the third kind? (2) Is it true that any convex bounded homogeneous domain in \({\mathbb C}^n\) is symmetric? Problems~2 and~3 concern the universality of the Cauchy-Szegő kernels. In \({\mathbb C}\), the Cauchy kernel \({{1}\over{2\pi i}}{{1}\over{z-u}}\) is independent of the domain. Let \(D\) be a symmetric tube domain. Is it true that the Cauchy-Szegő kernel for \(D\) of the type \(K(z-u)\) is universal in the class of convex domains which are biholomorphically equivalent to \(D\)? Conversely, if \(D'\) has the same kernel, is it equivalent to \(D\) (under some additional restrictions)? A cone \(V\subseteq {\mathbb C}^n\) is called \(k\)-linear concave if it is the union of \(k\)-planes which are contained in \(V\). A strictly \(k\)-concave cone is a \(k\)-linear concave cone such that any maximal convex subcone is a \(k\)-wedge (i.e., it is a direct product of a linear \(k\)-subspace and an \((n-k)\)-dimensional convex cone which does not contain lines). Problem 4. (1) Is it true that strictly \(k\)-linear concave self-dual homogeneous cones are exactly cones corresponding to pseudo-Hermitian symmetric spaces of tube type? In other words, do they correspond to Jordan algebras? (2) Is it true that corresponding tubes in the non-Hermitian case are rigid such that any biholomorphically equivalent to such a tube \(k\)-linear concave domain is affine equivalent? The definition of the self-duality is natural but too complicated to be formulated here. Problem 5. (1) Is it possible to describe the structure of homogeneous strictly \(k\)-concave cones in \({\mathbb R}^n\)? (2) Is it possible to describe the structure of homogeneous strictly \(k\)-concave domains in \({\mathbb C}^n\)?