5 Invariant Koszul Form of Homogeneous Bounded Domains …
97
(Xs) = (X ) + T r g/b
J − r (s)
−1 Jr(s)
ad(X )
, ∀X ∈ g, s ∈ B (5.11)
As
J − r (s)
−1 Jr(s)
maps g in b, we get (Xs) = (X ). The form is not
zero on b. This is not the image by p
∗ of a differential form of G/B. However, the
right invariance of on B is translated, infinitesimally by the relation:
([b, g]) = (0)
(5.12)
Koszul proved that the canonical hermitian form h of a homogeneous Kähler
manifold G/B has the following expression:
η(X, Y ) =
1
2
([J X, Y ])
with
([X, Y ]) = ([J X, J Y ])
η([J X, J Y ]) = η(X, Y )
∀X, Y ∈ g
(5.13)
To do, the link with the first chapters, I can summarize the main result of Koszul
that there is an integrable structure almost complex J on g, and for l ∈ g
∗ defined by
a positive J -invariant inner product on g:
X, Y i = [J X, Y ], l
(5.14)
Koszul has proposed as admissible form, l ∈ g
∗ , the form ξ :
(X ) = =X, ξ = Tr[ad(J X) − J ad(X )] ∀X ∈ g
(5.15)
Koszul proved that X, Y ξ coincides, up to a positive multiplicative constant;
with the real part of the Hermitian inner product obtained by the Bergman metric of
symmetric homogeneous bounded domains by identifying g with the tangent space.
(X ) is the restriction to g of a differential form of degree 1, with left invariance
on G. This form is fully defined by the invariant complex structure of G/B. This
form is invariant to the choice of J. This form is invariant on the right by B. We
have ([X, Y ]) = 0 with X ∈ g, Y ∈ b. The exterior differential d of is the
inverse image by the projection G → G/B of degree 2 form . This form is, up
to a constant, the Kähler form h, defined by the canonical Hermitian form of G/B:
h(π.X, π.Y ) =
1
2 (d)(X, J.Y ), ∀X, Y ∈ G as it is proved in Bourbaki seminar
by Koszul in [129].
The 1st Koszul form is then given by:
α = −
1
4
d(X )
(5.16)
Koszul has illustrated this structure for the simplest example of Siegel Domains.
First, the Poincaré upper half-plane V = {z = x + iy/y > 0} which is isomorphic
to the open zz
∗
< 1, which is a bounded domain. The group G of transformations
97
(Xs) = (X ) + T r g/b
J − r (s)
−1 Jr(s)
ad(X )
, ∀X ∈ g, s ∈ B (5.11)
As
J − r (s)
−1 Jr(s)
maps g in b, we get (Xs) = (X ). The form is not
zero on b. This is not the image by p
∗ of a differential form of G/B. However, the
right invariance of on B is translated, infinitesimally by the relation:
([b, g]) = (0)
(5.12)
Koszul proved that the canonical hermitian form h of a homogeneous Kähler
manifold G/B has the following expression:
η(X, Y ) =
1
2
([J X, Y ])
with
([X, Y ]) = ([J X, J Y ])
η([J X, J Y ]) = η(X, Y )
∀X, Y ∈ g
(5.13)
To do, the link with the first chapters, I can summarize the main result of Koszul
that there is an integrable structure almost complex J on g, and for l ∈ g
∗ defined by
a positive J -invariant inner product on g:
X, Y i = [J X, Y ], l
(5.14)
Koszul has proposed as admissible form, l ∈ g
∗ , the form ξ :
(X ) = =X, ξ = Tr[ad(J X) − J ad(X )] ∀X ∈ g
(5.15)
Koszul proved that X, Y ξ coincides, up to a positive multiplicative constant;
with the real part of the Hermitian inner product obtained by the Bergman metric of
symmetric homogeneous bounded domains by identifying g with the tangent space.
(X ) is the restriction to g of a differential form of degree 1, with left invariance
on G. This form is fully defined by the invariant complex structure of G/B. This
form is invariant to the choice of J. This form is invariant on the right by B. We
have ([X, Y ]) = 0 with X ∈ g, Y ∈ b. The exterior differential d of is the
inverse image by the projection G → G/B of degree 2 form . This form is, up
to a constant, the Kähler form h, defined by the canonical Hermitian form of G/B:
h(π.X, π.Y ) =
1
2 (d)(X, J.Y ), ∀X, Y ∈ G as it is proved in Bourbaki seminar
by Koszul in [129].
The 1st Koszul form is then given by:
α = −
1
4
d(X )
(5.16)
Koszul has illustrated this structure for the simplest example of Siegel Domains.
First, the Poincaré upper half-plane V = {z = x + iy/y > 0} which is isomorphic
to the open zz
∗
< 1, which is a bounded domain. The group G of transformations