5 Invariant Koszul Form of Homogeneous Bounded Domains …
95
on M, such that p
∗
(( pX). f ) = X.( p
∗ f ). The projection p : E → M defines an
injective homomorphism p
∗ of the space of differential forms of M in the space
of the differential forms of E such that for any form α of degree n on M and any
sequence of n projectable vectors fields, we have p
∗
(α( pX 1 , pX 2 , . . . , pX n )) =
( p
∗
α)(X 1 , X 2 , . . . , X n ). Let I be the tensor of an almost complex structure on the
basis M, there exists on E a tensor J of type (1,1) and only one which possesses
the following properties p(J X) = I ( pX) and J
2 X = −X mod h, X ∈ g for any
vector field X on E. Let G be a connected Lie group and B a closed subgroup of G,
we note g the Lie algebra left invariant vector fields on G and b sub-algebra of g
corresponding to B. The canonical mapping of G on G/B is denoted p (defining E as
before). We assume that there exists on G/B an invariant volume by G, which consist
in assuming that, for all s ∈ B, the automorphism X → Xs of g defines by passing
to the quotient an automorsphism of determinant 1 in g/b . Let r be the dimension
of G/B and (X i ) 1≤i≤m a base of g such that X i ∈ b, for r ≤ i ≤ m. Let (ξ i ) 1≤i≤m
the base of the space of differential forms of degree 1 left invariant on G such that
ξ i
X j
= δ i j . If ω is an invariant volume on G/B, then = p
∗
ω is equal, up to a
constant factor, to ξ 1 ∧ ξ 2 ∧ . . . ∧ ξ r . We will assume the base
X j
chosen so that
this factor is equal to 1, let = ξ 1 ∧ ξ 2 ∧ . . . ∧ ξ r . For any vector field that can be
projected X on G, we have:
p
∗
(div( pX))Ω = p
∗
((div( pX))ω) = p
∗
(( pX)ω)
= X Ω =
r
j=1
ξ j ([X j , X ])Ω
(5.1)
p
∗
(div( pX)) =
r
j=1
ξ j
X j , X
(5.2)
These elements being defined, Koszul calculates the Hermitian canonical form of
G/B, denoted h, more particularly η = p
∗ h on G. Let X and Y both right invariant
vector fields on G. They are projectable and the fields pX and pY are conformal
vector fields on G/B such that div( pX) = div( pY ) = 0, because the volume and
the complex structure of G/B are invariant under G. As a result, if κ is the Kähler
form of h and if α = p
∗
κ, then:
4α(X, Y ) = 4 p
∗
(κ( pX, pY )) = p
∗ div(I [ pX, pY ])
(5.3)
and as p(J [X, Y ]) = I [ pX, pY ], we obtain:
4α(X, Y ) = p
∗ div(J [X, Y ]) =
2n
i=1
ξ i ([X i , J [X, Y ]])
(5.4)
95
on M, such that p
∗
(( pX). f ) = X.( p
∗ f ). The projection p : E → M defines an
injective homomorphism p
∗ of the space of differential forms of M in the space
of the differential forms of E such that for any form α of degree n on M and any
sequence of n projectable vectors fields, we have p
∗
(α( pX 1 , pX 2 , . . . , pX n )) =
( p
∗
α)(X 1 , X 2 , . . . , X n ). Let I be the tensor of an almost complex structure on the
basis M, there exists on E a tensor J of type (1,1) and only one which possesses
the following properties p(J X) = I ( pX) and J
2 X = −X mod h, X ∈ g for any
vector field X on E. Let G be a connected Lie group and B a closed subgroup of G,
we note g the Lie algebra left invariant vector fields on G and b sub-algebra of g
corresponding to B. The canonical mapping of G on G/B is denoted p (defining E as
before). We assume that there exists on G/B an invariant volume by G, which consist
in assuming that, for all s ∈ B, the automorphism X → Xs of g defines by passing
to the quotient an automorsphism of determinant 1 in g/b . Let r be the dimension
of G/B and (X i ) 1≤i≤m a base of g such that X i ∈ b, for r ≤ i ≤ m. Let (ξ i ) 1≤i≤m
the base of the space of differential forms of degree 1 left invariant on G such that
ξ i
X j
= δ i j . If ω is an invariant volume on G/B, then = p
∗
ω is equal, up to a
constant factor, to ξ 1 ∧ ξ 2 ∧ . . . ∧ ξ r . We will assume the base
X j
chosen so that
this factor is equal to 1, let = ξ 1 ∧ ξ 2 ∧ . . . ∧ ξ r . For any vector field that can be
projected X on G, we have:
p
∗
(div( pX))Ω = p
∗
((div( pX))ω) = p
∗
(( pX)ω)
= X Ω =
r
j=1
ξ j ([X j , X ])Ω
(5.1)
p
∗
(div( pX)) =
r
j=1
ξ j
X j , X
(5.2)
These elements being defined, Koszul calculates the Hermitian canonical form of
G/B, denoted h, more particularly η = p
∗ h on G. Let X and Y both right invariant
vector fields on G. They are projectable and the fields pX and pY are conformal
vector fields on G/B such that div( pX) = div( pY ) = 0, because the volume and
the complex structure of G/B are invariant under G. As a result, if κ is the Kähler
form of h and if α = p
∗
κ, then:
4α(X, Y ) = 4 p
∗
(κ( pX, pY )) = p
∗ div(I [ pX, pY ])
(5.3)
and as p(J [X, Y ]) = I [ pX, pY ], we obtain:
4α(X, Y ) = p
∗ div(J [X, Y ]) =
2n
i=1
ξ i ([X i , J [X, Y ]])
(5.4)