114
F. Barbaresco
To write the Gibbs density with respect to its statistical moments, we
have to express the density with respect to Q = E[J (z)]. Then, we have
to invert the relation between Q and β, to replace this last variable β =
ir η
η
∗
−ir
∈ β by
( )
1 Q
β
−
Θ
∈
=
g where
( )
( )
*
Q
β
β
β
∂Φ
=
=Θ
∈
∂
g with (β) =
− log
D
e
−J (z)β dλ(z), deduce from Legendre tranform. The mean moment map is
given by:
Q = E[J (z)] = E
⎡
⎣ ρ
⎡
⎣
1+|w|
2
(1−|w|
2
)
−2w
∗
(1−|w|
2
)
2w
(1−|w|
2
)
−
1+|w|
2
(1−|w|
2
)
⎞
⎠
⎤
⎦ where w ∈ D
Moment map interpretation by stereographic projection is well explained in a
paper of Charles-Michel Marle [121].
The extension of Kostant-Sekiguchi-Vergne correspondence is illustrated by
Olivier Biquard also with the simplest case G = SU (1, 1). From its Lie algebra, we
have the Cartan decomposition:
*
*
/
,
0
0
with =
and
0
0
ir
r R
C
ir
ir
ir
η
η
η
η
η
⎧
⎫
⎛
⎞
⎪
⎪
=
∈
∈
= ⊕
⎨
⎬
⎜
⎟
−
⎪
⎪
⎝
⎠
⎩
⎭
⎛
⎞
⎛
⎞
= ⎜
⎟
⎜
⎟
−
⎝
⎠
⎝
⎠
g
hm
h
m
(5.49)
The complexified action of H = SO(2, R) on
2
R
=
m
is that of SO(2, C) on C
2 ,
so the nonzero H
C -orbits in
C
m are copies of C
∗ . The nonzero G-orbits in gare the
connected components of:
det
ir η
∗
η −ir
= r
2
− |η|
2
= λ
(5.50)
More especially for λ < 0, ⊂
g
t
mand the orbit is a hyperboloid, diffeomorphic to the semi-simple orbits in
C
m (so the complex structure is that of C
∗ ); the
corresponding orbits in
C
m depend on a parameter 3 i i
τ ∈ =
a t.