90
F. Barbaresco
5.1 Preamble
«C’est au problème de la détermination des domaines bornés homogènes posé par E. Cartan
vers 1935 que se rattache [mes travaux]…Ce sont les travaux de Piatetskii Shapiro sur les
domaines de Siegel, puis ceux de E.B. Vinberg sur les cônes homogènes qui m’ont amené à
l’étude des groupes de transformation affines des variétés localement plates et en particulier
aux critères de convexité liés aux formes invariantes.»- Jean-Louis Koszul, 1995
Jean-Louis Koszul was influenced by Elie Cartan who determined all symmetric
bounded domains in C
n
, ∀n, and all homogeneous bounded domains in C
n
, n = 2, 3.
As soon as 1955, Koszul studied the class of homogeneous complex manifolds
G/B (not only for Kähler manifold) with an invariant volume form such that the Ricci
tensor is non-degenerate and proved that any bounded domain which is homogeneous
with respect to a semi-simple group of automorphisms is symmetric, followed by
Hano who proved the same for a unimodular group of matrices (if G/B is a homogeneous Kähler manifold with G unimodular and non-degenerate Ricci tensor, then G is
semi-simple, and then every homogeneous bounded domain G/B with G unimodular
is symmetric). In the following, we will synthetize Koszul approach for this result
in his 1955 paper “Sur la forme hermitienne canonique des espaces homogènes
complexes” but also detailed in his Lecture given at Sao Paulo “Exposés sur les
espaces homogènes symétriques” in September and October 1958. More especially,
we will underline importance of the left invariant form of degree 1 introduced by
Koszul in these documents to deduce invariant metric in homogeneous bounded
domains, that plays a fundamental role in the framework of Information Geometry. In Sao Paulo Lecture, Koszul illustrated these results with use-cases for the
Poincaré upper-half plane and for the half-space of Siegel, and showed that with its
trace formula of endomorphism g/b, he verify that with the canonical Kähler hermitian form and the associated metrics, we can recover metrics introduced by Henri
Poincaré and Carl-Ludwig Siegel in these bounded domains. Based on seminal work
of Elie Cartan [1], Jean-Louis Koszul has developed his model [2–4] of homogeneous
bounded domains. This topic has been studied by authors [5–12]. On information
Geometry, based on first papers [13–15], other works have been developed [16–30],
with extension of Fisher matrix by Souriau on Symplectic Manifolds [31, 32], with
links with other hessian structures as Kähler ones [33–35]. Biography and testimony
on Jean-Louis Loszul are available in [36–38]. For more classical introduction to
information geometry, we make reference to S.I. Amari papers [39, 40]. Koszul
seminal papers [41, 42] and their developments are given in [43–50]. For references
on moment map on homogeneous spaces, we make reference to [51–53] and other
extension [54–56].
Henri Cartan was Jean-Louis Koszul PhD supervisor and in a letter from André
Weil to Henri Cartan, cited in the proceedings of the conference “Elie Cartan and
today’s mathematics” in 1984, we can read “As to the symmetrical spaces, and more
particularly to the symmetric bounded domains at the birth of which you contributed,
I have kept alive the memory of the satisfaction I felt in finding some incarnations
in Siegel from his first works on quadratic forms, and later to convince Siegel of the
F. Barbaresco
5.1 Preamble
«C’est au problème de la détermination des domaines bornés homogènes posé par E. Cartan
vers 1935 que se rattache [mes travaux]…Ce sont les travaux de Piatetskii Shapiro sur les
domaines de Siegel, puis ceux de E.B. Vinberg sur les cônes homogènes qui m’ont amené à
l’étude des groupes de transformation affines des variétés localement plates et en particulier
aux critères de convexité liés aux formes invariantes.»- Jean-Louis Koszul, 1995
Jean-Louis Koszul was influenced by Elie Cartan who determined all symmetric
bounded domains in C
n
, ∀n, and all homogeneous bounded domains in C
n
, n = 2, 3.
As soon as 1955, Koszul studied the class of homogeneous complex manifolds
G/B (not only for Kähler manifold) with an invariant volume form such that the Ricci
tensor is non-degenerate and proved that any bounded domain which is homogeneous
with respect to a semi-simple group of automorphisms is symmetric, followed by
Hano who proved the same for a unimodular group of matrices (if G/B is a homogeneous Kähler manifold with G unimodular and non-degenerate Ricci tensor, then G is
semi-simple, and then every homogeneous bounded domain G/B with G unimodular
is symmetric). In the following, we will synthetize Koszul approach for this result
in his 1955 paper “Sur la forme hermitienne canonique des espaces homogènes
complexes” but also detailed in his Lecture given at Sao Paulo “Exposés sur les
espaces homogènes symétriques” in September and October 1958. More especially,
we will underline importance of the left invariant form of degree 1 introduced by
Koszul in these documents to deduce invariant metric in homogeneous bounded
domains, that plays a fundamental role in the framework of Information Geometry. In Sao Paulo Lecture, Koszul illustrated these results with use-cases for the
Poincaré upper-half plane and for the half-space of Siegel, and showed that with its
trace formula of endomorphism g/b, he verify that with the canonical Kähler hermitian form and the associated metrics, we can recover metrics introduced by Henri
Poincaré and Carl-Ludwig Siegel in these bounded domains. Based on seminal work
of Elie Cartan [1], Jean-Louis Koszul has developed his model [2–4] of homogeneous
bounded domains. This topic has been studied by authors [5–12]. On information
Geometry, based on first papers [13–15], other works have been developed [16–30],
with extension of Fisher matrix by Souriau on Symplectic Manifolds [31, 32], with
links with other hessian structures as Kähler ones [33–35]. Biography and testimony
on Jean-Louis Loszul are available in [36–38]. For more classical introduction to
information geometry, we make reference to S.I. Amari papers [39, 40]. Koszul
seminal papers [41, 42] and their developments are given in [43–50]. For references
on moment map on homogeneous spaces, we make reference to [51–53] and other
extension [54–56].
Henri Cartan was Jean-Louis Koszul PhD supervisor and in a letter from André
Weil to Henri Cartan, cited in the proceedings of the conference “Elie Cartan and
today’s mathematics” in 1984, we can read “As to the symmetrical spaces, and more
particularly to the symmetric bounded domains at the birth of which you contributed,
I have kept alive the memory of the satisfaction I felt in finding some incarnations
in Siegel from his first works on quadratic forms, and later to convince Siegel of the