Invariant Koszul Form of Homogeneous Bounded Domains and Information Geometry Structures

In 1955, Jean-Louis Koszul has written a seminal paper entitled “Sur la forme hermitienne canonique des espaces homogenes complexes“. Let G be a Lie group and let B be a closed subgroup of G, Koszul introduced a hermitian form that he called canonical hermitian form of the complex homogeneous space G/B with an invariant volume form. In this seminal paper, Koszul has investigated homogeneous spaces and necessary conditions to carry a nondegenerate 2-form derived from the invariant volume element. The advantage of this form for the determination of homogeneous bounded domains was underlined by Elie Cartan in last sentence of his 1932 paper “Sur les domaines bornes de l’espace de n variables complexes”, observing that a necessary condition for G/B to be a bounded domain is that this form is positive definite. Hirohiko Shima has first observed that this geometry of Koszul Hessian structures is linked with Information Geometry. Jean-Louis Koszul also developed these structures in a Lecture “Exposes sur les Espaces Homogenes Symetriques” given at Sao-Paulo in 1958. We will synthetize these Jean-Louis Koszul works of 1955 and 1958 on this invariant form. We will put these elements in the context of more recent researches as a recent work of Della Vedova on covering spaces of symplectic manifolds (co-adjoint orbits, endowed with their canonical Kirillov-Kostant-Souriau symplectic structure) admitting homogeneous non Chern-Ricci flat special compatible almost complex structures, or Biquard extension of Kostant-Sekiguchi-Vergne correspondance. In last part, we develop the role of homogeneous bounded domains and invariant Koszul form in the framework of Information Geometry with use-case of Gaussian laws.