QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES
15
particular, every open orbit Mo of R in M containing Mphys is itself endowed
with a black hole structure.
Recall that if J denotes an element of Z(K) whose associated conjugation coin-
cides with the Cartan involution () then the R-orbit
Mo
in
9
/H.
of an element
uH.
with u
2
=
J is open and contains
Mphys,
see
[CD07].
Remark that the extension
lemma 3.8 yields an oscillatory integral UDF for proper actions of
R.
But here
the situation is simplified by the following observation - for convenience of the
presentation we write it below for
n
=
4 but the results are valid for any
n
2
3.
PROPOSITION 4.2.
The R-homogeneous space Mo admits a unique structure
of globally group type symplectic symmetric space. The latter is isomorphic to
(R0
,
w,
s) described in section 2.
For the proof, recall first (see
[CD07])
that the solvable part of the Iwasawa
decomposition of
.so(2, 3)
may be realized with as nilpotent part
n
=
{W, V, M,
L}
and Abelian
a=
{J1
,
h}
with the commutator table
[V, W] = M, [V, L] = 2W,
[J1, W] =
W,
[J2, V] =
V,
[J1,L] =
L,
[J2,L] =
-L,
[J1,M] =
M,
[J2,M] =
M.
Notice that W, J1
E
~'
and
h
E
q.
This decomposition is related to the one given
in (3.4) by
No =L
N1
=V
H1
=
J1
+
J2
H2
=
J1- J2.
We choose to study the orbit of the element {)
=
uH.
with
We denote by
Re
its stabilizer group in
R,
byte the Lie algebra of
Re;
t' is the
subalgebra of t generated by elements of t minus the generator of te and
R'
is the
analytic subgroup of
n
whose algebra is t'.
LEMMA 4.3.
The action ofR' on
U
is simply transitive,
i.e.
R'uH.
=
RuH..
PROOF. The first step is to prove that
Re
is connected and
U
simply connected
in order to prevent any double covering problem. The stabilizer of
uH.
is
(4.1) Re = {r En IT. uH. = uH.} ={TEn I Cu-l(T) E H.}.
Since
n
is an exponential group, we have te
=
{X
E
n
I
Ad(u- 1
)X E
~}
with
Re
=
exp te. The set te being connected,
Re
is connected too. A long exact
sequence argument using the fibration
Re---t
n
---t
u
shows that
H
0
(Re)
~
H
1
(U),
which proves that U is simply connected.
As an algebra, tis a split extension t
=
te
EBact
t'. Hence, as group,
R
=
ReR',
or equivalently
R
=
R'Re.
This proves that the action is transitive. The action is
even simply transitive because U is simply connected. D
Let us now find the algebra t
6
.
The Cartan involution
X
f---+
-Xt
is imple-
mented as
C
J
with
J
= (
-I2x2 I3x3) .
Using the relations u
2
=
J and
a(u)
=
u-I,
one sees that
Cu-l(T)
E
H.
if and only
if
a( Cu-lT) =Cu-lT.
This condition is equivalent to
()a(T)
=
T.
The involution
a
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