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.

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.