14 P. BIELIAVSKY, L. CLAESSENS, D. STERNHEIMER, ANDY. VOGLAIRE

With this choice of generators, the minimal parabolic subalgebra .s :=

a

EB n with

n :=

Ef)~=O 9ai

has the following multiplication table:

(3.4a) [H1, N1] = N1, [H1, N2] = N2,

(3.4b) [H2, N1] = -N1, [H2, N2] = N2,

(3.4c)

(3.4d)

[H1, N3] = 2N3,

[No,N1] = -N2,

the other brackets being zero. Setting

[H2, No] = 2No,

[N1, N2] = 2N3;

(3.5) .s1 := Span{H2, No} and .s2 := Span{H1, N1, N2, N3},

one observes that .s is a split extension of .s2 by .s1:

(3.6)

Note that .s1 is a minimal parabolic subalgebra of .su(l, 1) while .s2 is a minimal

parabolic subalgebra of .su(l, n). In particular, the Lie algebra .sis exact symplectic

w.r.t. the

element~:=

6

EB

6

of .s* with

~i

E

.st

(i

= 1, 2) defined as

6

:=

N0

and

6

:=

N;,.

One therefore obtains a UDF for proper actions of

AN

by direct application

of the above extension lemma 3.8.

4. Isospectral deformations of anti de Sitter black holes

4.1.

Anti de Sitter black holes.

Anti de Sitter (AdS) black holes have

been introduced by Baiiados, Teitelbaum, Zannelli and Henneaux

[BTZ, BHTZ]

as connected locally AdS space-times

M

(possibly with boundary and corners)

admitting a

singular

causal structure in the following sense:

CONDITION BH. There exists a closed subset

S

in

M

called the

singularity

such that the subset Mbh constituted by all the points x such that every light like

geodesic issued from

x

ends in

S

within a finite time is a proper open subset of

Mphys :=

M\S.

Originally such solutions were constructed in space-time dimension 3, but they

exist in arbitrary dimension

n ;::::

3 (see

[BDSR, CD07]).

More precisely the

structure may be described as follows. Take

9

:= 80(2,

n

-1) (the AdS group), fix

a Cartan involution

e

and a B-commuting involutive automorphism

a

of

9

such that

the subgroup

1i

of

9

of the elements fixed by a is locally isomorphic to 80(1, n-1).

The quotient space

M

:=

9

/1i

is an n-dimensional Lorentzian symmetric space, the

anti

de

Sitter space-time.

It is a solution of the Einstein equations without source.

Let g denote the Lie algebra of

9

and denote by g =

~

EB q the ±!-eigenspace

decomposition with respect to the differential at e of a that we denote again by a.

Denote by g = t EB

j)

the Cartan decomposition induced by

e,

consider a a-stable

maximally Abelian subalgebra

a

in

j)

and choose accordingly a positive system of

roots. Denote by n the corresponding nilpotent subalgebra. Set

n

:= B(n),

t

:= nEBn

and

t

:=

a

EB

n.

Finally denote by R :=

AN

and R := AN the corresponding

analytic subgroups of

Q.

One then has

PROPOSITION 4.1.

[BDSR, CD07J The groups R and R admit open orbits

and finitely many closed orbits in the AdS space M. Prescribing as singular the

union of all closed orbits (of R and R) defines a structure of causal black hole

on an open subset Mphys in M (in the sense of the above condition (BH)). In

With this choice of generators, the minimal parabolic subalgebra .s :=

a

EB n with

n :=

Ef)~=O 9ai

has the following multiplication table:

(3.4a) [H1, N1] = N1, [H1, N2] = N2,

(3.4b) [H2, N1] = -N1, [H2, N2] = N2,

(3.4c)

(3.4d)

[H1, N3] = 2N3,

[No,N1] = -N2,

the other brackets being zero. Setting

[H2, No] = 2No,

[N1, N2] = 2N3;

(3.5) .s1 := Span{H2, No} and .s2 := Span{H1, N1, N2, N3},

one observes that .s is a split extension of .s2 by .s1:

(3.6)

Note that .s1 is a minimal parabolic subalgebra of .su(l, 1) while .s2 is a minimal

parabolic subalgebra of .su(l, n). In particular, the Lie algebra .sis exact symplectic

w.r.t. the

element~:=

6

EB

6

of .s* with

~i

E

.st

(i

= 1, 2) defined as

6

:=

N0

and

6

:=

N;,.

One therefore obtains a UDF for proper actions of

AN

by direct application

of the above extension lemma 3.8.

4. Isospectral deformations of anti de Sitter black holes

4.1.

Anti de Sitter black holes.

Anti de Sitter (AdS) black holes have

been introduced by Baiiados, Teitelbaum, Zannelli and Henneaux

[BTZ, BHTZ]

as connected locally AdS space-times

M

(possibly with boundary and corners)

admitting a

singular

causal structure in the following sense:

CONDITION BH. There exists a closed subset

S

in

M

called the

singularity

such that the subset Mbh constituted by all the points x such that every light like

geodesic issued from

x

ends in

S

within a finite time is a proper open subset of

Mphys :=

M\S.

Originally such solutions were constructed in space-time dimension 3, but they

exist in arbitrary dimension

n ;::::

3 (see

[BDSR, CD07]).

More precisely the

structure may be described as follows. Take

9

:= 80(2,

n

-1) (the AdS group), fix

a Cartan involution

e

and a B-commuting involutive automorphism

a

of

9

such that

the subgroup

1i

of

9

of the elements fixed by a is locally isomorphic to 80(1, n-1).

The quotient space

M

:=

9

/1i

is an n-dimensional Lorentzian symmetric space, the

anti

de

Sitter space-time.

It is a solution of the Einstein equations without source.

Let g denote the Lie algebra of

9

and denote by g =

~

EB q the ±!-eigenspace

decomposition with respect to the differential at e of a that we denote again by a.

Denote by g = t EB

j)

the Cartan decomposition induced by

e,

consider a a-stable

maximally Abelian subalgebra

a

in

j)

and choose accordingly a positive system of

roots. Denote by n the corresponding nilpotent subalgebra. Set

n

:= B(n),

t

:= nEBn

and

t

:=

a

EB

n.

Finally denote by R :=

AN

and R := AN the corresponding

analytic subgroups of

Q.

One then has

PROPOSITION 4.1.

[BDSR, CD07J The groups R and R admit open orbits

and finitely many closed orbits in the AdS space M. Prescribing as singular the

union of all closed orbits (of R and R) defines a structure of causal black hole

on an open subset Mphys in M (in the sense of the above condition (BH)). In