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Berezin quantization and discrete-series of SU (1, n)

El Beggar M.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 2

205-215

A new "method of orbits" was introduced by M. Flato and C. Fronsdal ([4], [6]) and called the "deformation program". It is based on the use of (formal) deformations of an algebra of functions on the coadjoint orbits and called star products. In this note, after realizing a discrete-series of group G = SU(1, n) as action in the space of the holomorphic sections of the line bundle which we will determine, we quantify by the Berezin method, used in ([3], [5], [10], [11]), the coadjoint orbit associated to this discrete-series. This allows us to explicit the star-exponential introduced in ([2], [3]). We also give the new expression of the Kirillov formula.