QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES
7
"centered symmetry" of the associative kernel (2.2). Since the kernel is left-invariant
as well, such a centered symmetry s9
:
R
0
---+
R
0
:
g'
~
s9 (g')
:=
L
9
o ¢ o Lg-1 (g')
is attached to every point gin Ro. It turns out that endowed with the above family
of symmetries, the manifold R
0
becomes a symplectic symmetric space. More
precisely:
PROPOSITION 2.2. Lets: R
0
X
Ro---+ R
0
be the map (g, g')
~
s9 (g'). Then the
triple (R0
,
w,
s) is a symplectic symmetric space whose transvection group is solv-
able. The underlying affine connection is the unique affine symplectic connection
which is invariant under the group of symmetries of the oscillatory kernel (2.2}.
PROOF. In coordinates (a, x, z), the symmetry map is expressed as
(2.4)
S(a,x,z)(a', x', z')
=
(2a- a', 2 cosh( a- a')x- x',
2 cosh(2(a- a'))z
+
Ov(x, x') sinh( a- a')- z').
One then verifies that it satisfies the defining identity: s9 o s9
,
o s9
=
s
89
(g')·
Concerning the solvability of the transvection group, four-dimensional symplec-
tic transvection triples have been classified in [Bie95, Bie98l. The one we are
concerned with here is given by Table (1) (c
=
1) in Proposition 2.3 of [Bie98l;
denoting here
t
=
Span{u1,u2,u3} and
p
=
Span{el.e2,fl,/2}, it writes:
[u2, u3l
=
u1; [u2, e2l
=
e1; [u2, Ill =-/2; [u3, hl
=
e1; [ul. Ill
=
2e1;
[e1, Ill
=
2u1; [e2, Ill
=
u3; [e2, hl
=
u1; [!1, hl
=
u2.
In these notations the Lie algebra to
=
aoEB V EBIRZ of Ro (with Darboux chart (2.1))
is generated by ao
=
R(-!1), V
=
Span{u2- /2,u3
+
e2}, Z
=
2(ul
+
e1). We
now verify (by a short computation) that for all a, z E lR and x E V, one has
iT(exp( -afl) exp(x
+
2z(ul
+
el))K
=
(exp(afl) exp( -x- 2z(ul
+
el))K,
where K denotes the analytic subgroup associated with
t.
Thus iT gives
Se
=
¢ on
R
0
.
The triple considered here is therefore precisely the triple that induces on R
0
the present symmetric space structure, hence the transvection group is solvable.
The higher dimensional case is similar to the 4-dimensional one.
Note that the above symmetric space structure on R
0
is canonically associated
with the data of the oscillatory kernel (2.2). Indeed, coming from the stationary
phase expansion of an oscillatory integral, the formal star product
*o
mentioned in
item
(iii)
of Theorem 2.1 is natural in the sense that for all positive integer
r
the
r-th cochain of the star product is a bidifferential operator of order r. To every such
natural star product is uniquely attached a symplectic connection [Lic82, GR03l.
In our case the latter, being invariant under the symmetries, must coincide with
the canonical connection associated with the symmetries { s9
}
gE'Ro. D
The above considerations lead to the following definitions.
DEFINITION 2.3. Let R be a Lie group.
A
symmetric structure on R is a
diffeomorphism ¢ : R
---+
R such that ¢
2
=
id'R; ¢(e)
=
e;
¢*e
=
-idre('R); and
setting, for all g E R, s9
:=
L
9
o¢oL9 -1 then, for all g', we define s9 os9
,
os9
=
S89
(g').
Or equivalently:
DEFINITION 2.4. A (symplectic) symmetric space, or more generally a homo-
geneous space, M of dimension m is locally of group type if there exists a
m-dimensional (symplectic) Lie subgroup R ofits automorphism group which acts
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