6 P. BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, AND Y. VOGLAIRE

The setting and (i-ii) may be found in

[BiMs],

while (iii) is a straightforward

adaptation to

R

0

of

[BBM].

2.2. Geometry underlying the product formula.

We start with prelimi-

nary material concerning the symmetric spaces.

2.2.1.

Symmetric spaces.

A

symplectic symmetric space [Bie95, BCG]

is a

triple

(M,

w,

s)

where

M

is a connected smooth manifold, w is a non-degenerate

two-form on

M

and s :

M x M

-t

M : (x, y)

f---+

sx(Y) is a smooth map such that

Vx

EM the map sx :

M

-t

M

is an involutive diffeomorphism of

M

preserving w

and admitting

x

as an isolated fixed point. Moreover, one requires that the identity

Sx

o

Sy

o

Sx

=

Ssx(Y) holds for all

x,

y EM

[Loo69].

In this situation, if

x

EM and

X, Y and Z are smooth tangent vector fields on M,

1

(2.3) Wx(\7

X

Y, Z)

:=

2

Xx.w(Y

+

Sx. Y, Z)

defines an affine connection \7 on

M,

the unique affine connection on

M

which is

invariant under the symmetries { sx}xEM. It is moreover torsion-free and such that

\i'w

=

0. In particular, the two-form w is necessarily symplectic.

It then follows that the group

g =

Q(M,s)

generated by the compositions

{ Sx o sy }x,yEM is a Lie group of transformations acting transitively on

M.

The

group

g

is called the

transvection

group of

M.

Given a base point o in

M,

the conjugation by the symmetry s

0

defines an involutive automorphism if of

g.

Its differential at the unit element

0'

:=

(j*e

induces a decomposition into ±1-

eigenspaces of the Lie algebra g of

9:

g

=

£ EB

p.

The subspace £ of fixed vectors

turns out to be a Lie subalgebra which acts faithfully on the subspace

p

of "anti-

fixed" vectors (O'x

=

-x for x E

p).

Moreover

[p,p]

=

£. A pair (g,O') as above

is called a transvection pair. The subalgebra £ corresponds to the Lie algebra of

the stabilizer of o in

g,

while the vector space

p

is naturally identified with the

tangent space

T0 (M)

toM at point

o.

In particular the symplectic form at

o,

w0

,

induces a £-invariant symplectic bilinear form on p. Extending the latter by 0 on

£ yields a Chevalley 2-cocycle

n

on g with respect to the trivial representation of

g on

R

A triple (g,

0',

0)

as above is called a symplectic transvection triple. It it

said to be exact when there exists an element

~

in

g*

such that

c5~

= n,

where

c5

denotes the Chevalley coboundary operator. Up to coverings, the correspondence

which associates a symplectic transvection triple to a symplectic symmetric space is

bijective. More precisely there is an equivalence of categories between the category

of connected simply connected symplectic symmetric spaces and that of symplectic

transvection triples - the notion of morphism being the natural one in both cases.

In the above setting, exactness corresponds to the fact that the transvection group

acts on (

M, w)

in a strongly Hamiltonian manner.

The above considerations can be adapted in a natural manner to the Riemann-

ian or pseudo-Riemannian setting4

,

essentially by replacing

mutatis-mutandis

the

symplectic structure by a metric tensor. The canonical connection

(cf.

Formula

(2.3) above) corresponds in this case to the Levi-Civita connection.

2.2.2.

Symmetric spaces of group type.

We observe that in coordinates

(a, x,

z)

the map

¢ : R

0

-t

R

0

:

(a, x, z)

f---+

(-a, -x, -z)

preserves the symplectic form

w (because the coordinates are Darboux coordinates), is involutive and admits

the unit element e

=

(0, 0, 0) as an isolated fixed point. It may be therefore called

4See [CP80]

for an excellent reference.