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