Symmetric spaces and star representations III. The Poincaré disc

Abstract

This article is a contribution to the domain of (convergent) deformation quantization of symmetric spaces by use of Lie groups representation theory. We realize the regular representation of SL(2, R) on the space of smooth functions on the Poincaré disc as a sub-representation of SL(2, R) in the Weyl-Moyal star product algebra on R 2 . We indicate how it is possible to extend our construction to the general case of a Hermitian symmetric space of tube type.2000 Mathematics Subject Classification: 53D55, 22E45.

Covariant quantization: spectral analysis versus deformation theory

“…Analogously with the compactnon-compact duality, one therefore gets a geometric realization of the correspondence between Cayley symmetric spaces and Hermitian symmetric spaces of tube-type. The paper [24] is devoted to the construction of a convergent deformation quantization of the Poincaré upper half-plane G/K = Π. Starting from results described in [23], we studied the representation ad = (ρ L − ρ R ) of g acting by star-product's derivations on the formal power series on the onesheeted hyperboloid G/H and we showed that it can be restricted to the regular action of g on C ∞ c (Π).…”

“…Semisimple symmetric spaces and star-representations [23,24] One of the possible approaches to the quantization of homogeneous spaces consists in adapting the Weyl calculus to the non-flat situation keeping working on the side of asymptotic expansions and looking for an alternative Moyal product that would respect the symmetries of the system that should be quantized. Such was the strategy developed in [23,24]. …”