QUANTIZATION OF ANTI DE SITTER AND SYMMETRIC SPACES 21

(i) Since

[FHT, Sta98]

we know that for q even root of unity, there are

unitary irreducible finite dimensional representations of the Anti de Sitter

groups. Interestingly

(cf. [FGR]

p.122) the "fuzzy 3-sphere" is related

to the Wess-Zumino-Witten models and is conjectured to be related to

the non-commutative geometry of the quantum group Uq(sl2

)

for q

=

e27ri/(k+

2),

k 0, a root of unity.

(ii) The last remark suggests to look more closely at the phenomenon of di-

mensional reduction which appears in a variety of related problems. In

this paper we considered only

n

z

3. The reason is that for

n

=

2

the context is in part different: the conformal group of 1 + 1 dimen-

sional space-time is infinite dimensional, and there are no black holes

[CD07].

But many considerations remain true, and furthermore many

group-theoretical properties find their origin at the 1 + 1 dimensional level,

e.g. the uniqueness of the extension to conformal group

[AFFS].

An-

other exemple of dimensional reduction is the fact that the massless UIR

of the 2+ 1 dimensional Poincare group Di and Rae satisfy Di

ffi

Rae

=

D(HO)EBD(HO) where D(HO) is the representation D(1/4)EBD(3/4) of

the metaplectic group (double covering of SL(2, IR)) which is the symme-

try of the harmonic oscillator in the deformation quantization approach

(see e.g. Section (2.2.4) in

[DS02]).

(iii) What do the degenerate representations Di and Rae become under defor-

mation? Furthermore there may appear, for our nonstandard quantum

group 80(2, n)q, new representations that have no equivalent at the un-

deformed level (e.g. in a way similar to the supercuspidal representations

in the p-adic context). These may have interesting physical interpreta-

tions.

(iv) We have seen that for q even root of unity S0(2,n)q has some properties

of a compact Lie group. Our cosmological Ansatz suggests that the qAdS

black holes are "small." It is therefore natural to try and find a kind of

generalized trace that permits to give a finite volume for qAdS. Note

that, in contradistinction with infinite dimensional Hilbert spaces, the

notions of boundedness and compactness are the same for closed sets

in Mantel spaces, and that our context is in fact more Frechet nuclear

than Hilbertian. This raises the more general question to define in an

appropriate manner the notion of "q-compactness" (or "q-boundedness")

for noncommutative manifolds.

(v) Possibly in relation with the preceding question, one should perhaps con-

sider deformation triples in which the Hilbert space is replaced by a suit-

able locally convex topological vector space (TVS), on which D could be

continuous.

(vi) The latter should yield a natural framework for implementing quantum

symmetries in deformation triples, since Frechet nuclear spaces and their

duals are at the basis of the topological quantum groups (and their duals)

introduced in the 90's, especially in the semi-simple case with preferred

deformations (see the review

[BGGS]).

We would thus in fact have

quadruples

(A,£, D,

9) where

A

is some topological algebra,

£

an ap-

propriate TVS, D some (bounded on

£)

"Dirac" operator and

9

some