5 Invariant Koszul Form of Homogeneous Bounded Domains …
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of G, the sub-algebra corresponding to B and an endomorphism algebra defining
the invariant complex structure of G/B. The results obtained by Koszul proved that
the homogeneous bounded domains whose group of automorphisms is semi-simple
are bounded symmetric domains in the sense of Elie Cartan. In this seminal paper,
Koszul introduced a left invariant form of degree 1 on G that is given by the following:
(X ) = T r g/b [ad(J X) − J.ad(X )] ∀X ∈ g with J an endomorphism of the Lie
algebra space and the trace T r g/b [.] corresponding to that of the endomorphism g/b.
The Kähler form of the canonical Hermitian form is given by the differential of
−(X ) of this form of degree 1.
These results were deepened by Koszul in a Lecture «Exposés sur les espaces
homogènes symétriques» [78] published in 1959 after a seminar held in September
and October 1958 at the University of Sao Paulo, which details the determination of
homogeneous bounded domains. He returned to [77] and showed that any symmetric
bounded domain is a direct product of irreducible symmetric bounded domains, determined by Elie Cartan (4 classes corresponding to classical groups and 2 exceptional
domains). For the study of irreducible symmetric bounded domains, Koszul refered
to Elie Cartan, Carl-Ludwig Siegel and Loo-Keng Hua. Koszul illustrated the subject
with two particular cases, the upper half-plane of Poincaré and the upper half-space
of Siegel, and showed that with its trace formula of endomorphism g/h, he found
that the canonical Kähler hermitian form and the associated metrics allow to recover
those introduced by Henri Poincaré and Carl-Ludwig Siegel [6] (who introduced
them as invariant metric under action of the automorphisms of these spaces). Koszul
has extended these results in four other papers published until 1970.
In the paper «Domaines bornées homogènes et orbites de groupes de transformations affines» [79] of 1961 written by Koszul at the Institute for Advanced Study at
Princeton during a stay funded by the National Science Foundation, Koszul demonstrated the reciprocal of its 1955 result for a class of complex homogeneous spaces.
This class consists of some open orbits of complex affine transformation groups and
contains all homogeneous bounded domains. Koszul addressed again the problem
of knowing if a complex homogeneous space, whose canonical Hermitian form is
positive definite is isomorphic to a bounded domain, but via the study of the invariant
bilinear form defined on a real homogeneous space by an invariant volume and an
invariant flat connection. Koszul demonstrated that if this bilinear form is positive
definite then the homogeneous space with its flat connection is isomorphic to a convex
open domain containing no straight line in a real vector space and extended it to the
initial problem for the complex homogeneous spaces obtained in defining a complex
structure in the variety of vectors of a real homogeneous space provided with an
invariant flat connection. It is in this article that Koszul used the affine representation
of Lie groups and algebras.
In the paper «Ouverts convexes homogènes des espaces affines» [80] of 1962,
Koszul was interested by the structure of the convex open non-degenerate (with
no straight line) and homogeneous (the group of affine transformations of E leaving
stable operates transitively in ) in a real affine space of finite dimension. Koszul
demonstrated that they can be all deduced from non-degenerate and homogeneous
convex open cones built in [79]. He used for this the properties of the group of
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of G, the sub-algebra corresponding to B and an endomorphism algebra defining
the invariant complex structure of G/B. The results obtained by Koszul proved that
the homogeneous bounded domains whose group of automorphisms is semi-simple
are bounded symmetric domains in the sense of Elie Cartan. In this seminal paper,
Koszul introduced a left invariant form of degree 1 on G that is given by the following:
(X ) = T r g/b [ad(J X) − J.ad(X )] ∀X ∈ g with J an endomorphism of the Lie
algebra space and the trace T r g/b [.] corresponding to that of the endomorphism g/b.
The Kähler form of the canonical Hermitian form is given by the differential of
−(X ) of this form of degree 1.
These results were deepened by Koszul in a Lecture «Exposés sur les espaces
homogènes symétriques» [78] published in 1959 after a seminar held in September
and October 1958 at the University of Sao Paulo, which details the determination of
homogeneous bounded domains. He returned to [77] and showed that any symmetric
bounded domain is a direct product of irreducible symmetric bounded domains, determined by Elie Cartan (4 classes corresponding to classical groups and 2 exceptional
domains). For the study of irreducible symmetric bounded domains, Koszul refered
to Elie Cartan, Carl-Ludwig Siegel and Loo-Keng Hua. Koszul illustrated the subject
with two particular cases, the upper half-plane of Poincaré and the upper half-space
of Siegel, and showed that with its trace formula of endomorphism g/h, he found
that the canonical Kähler hermitian form and the associated metrics allow to recover
those introduced by Henri Poincaré and Carl-Ludwig Siegel [6] (who introduced
them as invariant metric under action of the automorphisms of these spaces). Koszul
has extended these results in four other papers published until 1970.
In the paper «Domaines bornées homogènes et orbites de groupes de transformations affines» [79] of 1961 written by Koszul at the Institute for Advanced Study at
Princeton during a stay funded by the National Science Foundation, Koszul demonstrated the reciprocal of its 1955 result for a class of complex homogeneous spaces.
This class consists of some open orbits of complex affine transformation groups and
contains all homogeneous bounded domains. Koszul addressed again the problem
of knowing if a complex homogeneous space, whose canonical Hermitian form is
positive definite is isomorphic to a bounded domain, but via the study of the invariant
bilinear form defined on a real homogeneous space by an invariant volume and an
invariant flat connection. Koszul demonstrated that if this bilinear form is positive
definite then the homogeneous space with its flat connection is isomorphic to a convex
open domain containing no straight line in a real vector space and extended it to the
initial problem for the complex homogeneous spaces obtained in defining a complex
structure in the variety of vectors of a real homogeneous space provided with an
invariant flat connection. It is in this article that Koszul used the affine representation
of Lie groups and algebras.
In the paper «Ouverts convexes homogènes des espaces affines» [80] of 1962,
Koszul was interested by the structure of the convex open non-degenerate (with
no straight line) and homogeneous (the group of affine transformations of E leaving
stable operates transitively in ) in a real affine space of finite dimension. Koszul
demonstrated that they can be all deduced from non-degenerate and homogeneous
convex open cones built in [79]. He used for this the properties of the group of