5 Invariant Koszul Form of Homogeneous Bounded Domains …
115
5.6 Extension to Siegel Disk and Moment Map of SU(P,Q)
Lie Group
To address computation of covariant Gibbs density for Siegel Unit Disk, we will
consider in this section SU ( p, q) Unitary Group:
G = SU ( p, q) and
K = S(U ( p) × U (q)) =
A 0
0 D
/A ∈ U ( p),
D ∈ U (q), det(A) det(D) = 1}
(5.51)
We can use the following decomposition for g ∈ G
C :
g =
A B
C D
∈ G
C
, g =
I p B D
−1
0 I q
A − B D
−1 C 0
0
D
I p 0
D
−1 C I q
(5.52)
and consider the action of g ∈ G
C on Siegel Unit Disk S D =
Z ∈ M pq (C)/I p − Z Z
+
> 0
given by:
g =
A B
C D
∈ G
C
, g =
I p B D
−1
0 I q
A − B D
−1 C 0
0
D
I p 0
D
−1 C I q
(5.53)
Benjamin Cahen has study this case and introduced the moment map by identifing
G-equivariantly
*
g with gby means of the Killing form β on
C
g :
(
)
( )
G-equivariant with by Killing form
,
2(
)
X Y
p q Tr XY
β
=
+
*
g
g
The set of all elements of gfixed by K is h:
{
} 0
0
0
element of fixed by
,
,
0
p
q
qI
G
K
i
pI
ξ
ξ
λ
−
⎛
⎞
∈
= ⎜
⎟
⎝
⎠
h =
h
(5.54)
Then, we the equivatiant moment map is given by:
∀X ∈ g
C
, Z ∈ D, ψ(Z ) = Ad
∗
exp
−Z
+
ζ
exp Z
+ exp Z
ξ 0
∀g ∈ G, Z ∈ D then ψ(g.Z ) = Ad
∗
g ψ(Z )
ψ is a diffeomorphism from SD onto orbit O(ξ 0 )
(5.55)
with: