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.
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