8 P.
BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE
freely on one of its orbits in
M.
One says that it is globally of group type if it is
locally and if R has only one orbit5
.
Lie groups are themselves examples of symmetric spaces (globally) of group
type. In the symplectic situation, however, a symplectic symmetric Lie group must
be Abelian [Bie95]. We will see in what follows other non-Abelian examples.
2.2.3. Strategy. Our strategy for constructing UDF's for certain Lie groups
may now be easily described: starting with a symplectic symmetric space of group
type admitting an invariant deformation quantization
~
obtained by geometric
considerations at the level of the symmetric space structure
~
one deduces a UDF
for every Lie subgroup
R
as above by either identifying
M
toR, or, in the formal
case, by restricting the deformation quantization to an open R-orbit. This type of
strategy, in contexts other than symmetric spaces, has already been proposed within
a formal framework, see for example [Xu93, CP95, Huy82, CM04, GZ98].
3. Construction of UDF's for symmetric spaces of group type
The first example presented in the preceding section as well as the elementary
solvable exact triples (briefly "ESET") of [Bie02] are special cases of the following
situation.
3.1. Weakly nilpotent solvable symmetric spaces. In this section (g,
0')
denotes a complex solvable involutive Lie algebra (in short, iLa) such that if g
=
tEBp
is the decomposition into eigenspaces of
O",
the action of
e
on
p
is nilpotent6
.
We
denote by prp : g
~
p
the projection onto
p
parallel to
e.
DEFINITION 3.1. A good Abelian subalgebra (in short, gAs) of g is an
Abelian subalgebra
a
of g, contained in
p,
supplementary to a 0'-stable ideal b in g
and such that the homomorphism
p :
a~
Der(b) associated with the split extension
0
~
b
~
g
~
a
~
0 is injective.
LEMMA 3.2. If (g,
0')
is not flat (i.e.
[p, p]
acts nontrivially on
p)
then a gAs
always exists.
PROOF. Since
e
is nilpotent,
[t,p]
i=-
p.
Moreover by non-flatness there exists
X
E
P\[t,
p]
not central in g. Hence for every choice of a subspace
V
supplementary
to
a
:=
JRX in
p
and containing
[t,
p],
one has that b
:=
e
EB
V is an ideal of g
supplementary to
a
and on which
X
acts nontrivially. D
Note that the centralizer
3o(a)
of
a
in
b
is stable by the involution
O'i
indeed,
for all
a
E
a
and X E
3o(a),
one has [a, O"X]
=
-O"[a,
X]
=
0. Moreover, the map
p : a
~
End(b) being injective, we may identify
a
with its image:
a
=
p(a).
Let
I:: End(b)
~
End(b) be the conjugation with respect to the involution
O'lo
E
GL(b),
i.e. I:
=
Ad(O"I
0
).
The automorphism I: is involutive and preserves the canonical
Levi decomposition End(b)
=
Z
EB
st(b), where
Z
denotes the center of End(b).
Writing the element a
=
p(a) E a as a
=
az
+
a0 within this decomposition,
one has: I:(a)
=
az
+
I:(a0
)
=
-a
=
-az - a0
,
because the endomorphisms a
and
O'lb
anticommute. Hence I:(ao)
=
-2az - a0 and therefore az
=
0. So, a
5In
this case, for every choice of a base point
o
in
M,
the map
R ---. M :
g - g.o
is a
diffeomorphism.
6This
condition is automatic for a transvection algebra, but not in general. Indeed, consider
the 2-dimensional non-Abelian (solvable) algebra g = Span{k,p} with table [k,p] =
p.
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