100
F. Barbaresco
of G, and [v, X ] = 0. v has compact isotropy subgroup V ⊂ G. The canonical
symplectic form ρ = λω on the orbit is determined by the exact two-cocycle σ .
The canonical almost complex structure J on G/V is compatibile with the KirillovKostant-Souriau symplectic form ω. The author in [88] proves also the converse of
previous theorem that a coadjoint orbit having the first Chern class equal to a nonzero multiple of the class of the canonical symplectic form admits a homogeneous
special compatible almost complex structure, under the additional assumptions that
G is semi-simple and the coadjoint orbit has compact isotropy.
In a preprint [91], O. Biquard has extended the Kostant-Sekiguchi-Vergne [92]
correspondence. This classical Kostant-Sekiguchi-Vergne correspondence concerns
G/H a symmetric space of noncompact type, with H a maximal compact subgroup
of G, associated to a Cartan decomposition of the Lie algebra
⊕
g = h m, and gives
a diffeomorphism between the nilpotent G-orbits in gand the nilpotent H
C -orbits in
C
m . Olivier Biquard has extend this correspondence to all G-orbits in g, turns out
that each G-orbit in gis diffeomorphic to each orbit in a family of H
C -orbits in
C
m ,
providing a set of H-invariant Kähler metrics on any G-orbit in g, such that the Kähler
form equals the Kirillov-Kostant-Souriau symplectic form of the orbit. = ⊕
g
t t a,
and any semi-simple element in gis G-conjugate to τ = τ 1 + iτ 2 , where 1
τ ∈ tand
2
i
τ ∈ a, and = ⊕
g
t t ais a Cartan subalgebra of g. Any element of gis conjugate to
τ = τ 1 +iτ 2 +σ 1 +iσ 2 , and the representation σ = (σ 1 , σ 2 , σ 3 ) satisfying 1
σ ∈ hand
2
3
,
i
σ σ ∈ m, and commutes with the τ i , such that
σ i , τ j
= 0.The Cartan subalgebra
of gis one dimensional, so either it is compact ( =
g
t h), either it is noncompact (
= ⊂
g
t a m). For all cases, the diffeomorphism with a complex H
C -space gives a
H-invariant Kähler structure on the orbit.
Theorem [91]
(1) For any 3 i
τ ∈ asuch that [τ 3 , σ i ] = 0 and the regularity assumptions
1 2
2 3
1 2 3
( , )
( , )
( , , )
C
C
C
τ τ
τ τ
τ τ τ
=
=
g
g
g
are satisfied, there exist an H-invariant
diffeomorphism from the G-orbit of τ 1 + iτ 2 + σ 1 + iσ 2 in gto the H
C -orbit
of τ 2 + iτ 3 + σ 2 + iσ 3 in
C
m . This diffeomorphism gives a H-invariant kähler
metric on the G-orbit of τ 1 + iτ 2 + σ 1 + iσ 2 in g, whose kähler form is the
Krillov-Kostant-Souriau Symplectic form.
(2) If 3 i
τ ∈ asatisfies [τ 3 , σ i ] = 0 but we have only the regularity
1 2
1 2 3
( , )
( , , )
C
C
τ τ
τ τ τ
=
g
g
, then there still exists a H-invariant diffeomorphism
from the G-orbit of τ 1 + iτ 2 + σ 1 + iσ 2 in gto a H
C -space, which gives a
H-invariant Kähler structure on the orbit.
Other results on this extension have been proven by Roger Bielawski [93], with
a natural interpretation in terms of Nahm’s equations with reference to Vergne [92].
Other elements on links between Kähler form and Symplectic KKS 2-form are
considered in [94–96]. In [94], author has considered the regular coadjoint orbits of
G, a noncompact real semi-simple Lie group, on which, the Iwasawa decomposition
F. Barbaresco
of G, and [v, X ] = 0. v has compact isotropy subgroup V ⊂ G. The canonical
symplectic form ρ = λω on the orbit is determined by the exact two-cocycle σ .
The canonical almost complex structure J on G/V is compatibile with the KirillovKostant-Souriau symplectic form ω. The author in [88] proves also the converse of
previous theorem that a coadjoint orbit having the first Chern class equal to a nonzero multiple of the class of the canonical symplectic form admits a homogeneous
special compatible almost complex structure, under the additional assumptions that
G is semi-simple and the coadjoint orbit has compact isotropy.
In a preprint [91], O. Biquard has extended the Kostant-Sekiguchi-Vergne [92]
correspondence. This classical Kostant-Sekiguchi-Vergne correspondence concerns
G/H a symmetric space of noncompact type, with H a maximal compact subgroup
of G, associated to a Cartan decomposition of the Lie algebra
⊕
g = h m, and gives
a diffeomorphism between the nilpotent G-orbits in gand the nilpotent H
C -orbits in
C
m . Olivier Biquard has extend this correspondence to all G-orbits in g, turns out
that each G-orbit in gis diffeomorphic to each orbit in a family of H
C -orbits in
C
m ,
providing a set of H-invariant Kähler metrics on any G-orbit in g, such that the Kähler
form equals the Kirillov-Kostant-Souriau symplectic form of the orbit. = ⊕
g
t t a,
and any semi-simple element in gis G-conjugate to τ = τ 1 + iτ 2 , where 1
τ ∈ tand
2
i
τ ∈ a, and = ⊕
g
t t ais a Cartan subalgebra of g. Any element of gis conjugate to
τ = τ 1 +iτ 2 +σ 1 +iσ 2 , and the representation σ = (σ 1 , σ 2 , σ 3 ) satisfying 1
σ ∈ hand
2
3
,
i
σ σ ∈ m, and commutes with the τ i , such that
σ i , τ j
= 0.The Cartan subalgebra
of gis one dimensional, so either it is compact ( =
g
t h), either it is noncompact (
= ⊂
g
t a m). For all cases, the diffeomorphism with a complex H
C -space gives a
H-invariant Kähler structure on the orbit.
Theorem [91]
(1) For any 3 i
τ ∈ asuch that [τ 3 , σ i ] = 0 and the regularity assumptions
1 2
2 3
1 2 3
( , )
( , )
( , , )
C
C
C
τ τ
τ τ
τ τ τ
=
=
g
g
g
are satisfied, there exist an H-invariant
diffeomorphism from the G-orbit of τ 1 + iτ 2 + σ 1 + iσ 2 in gto the H
C -orbit
of τ 2 + iτ 3 + σ 2 + iσ 3 in
C
m . This diffeomorphism gives a H-invariant kähler
metric on the G-orbit of τ 1 + iτ 2 + σ 1 + iσ 2 in g, whose kähler form is the
Krillov-Kostant-Souriau Symplectic form.
(2) If 3 i
τ ∈ asatisfies [τ 3 , σ i ] = 0 but we have only the regularity
1 2
1 2 3
( , )
( , , )
C
C
τ τ
τ τ τ
=
g
g
, then there still exists a H-invariant diffeomorphism
from the G-orbit of τ 1 + iτ 2 + σ 1 + iσ 2 in gto a H
C -space, which gives a
H-invariant Kähler structure on the orbit.
Other results on this extension have been proven by Roger Bielawski [93], with
a natural interpretation in terms of Nahm’s equations with reference to Vergne [92].
Other elements on links between Kähler form and Symplectic KKS 2-form are
considered in [94–96]. In [94], author has considered the regular coadjoint orbits of
G, a noncompact real semi-simple Lie group, on which, the Iwasawa decomposition