Title: Berezin's coherent states, symbols and transform revisited


This talk is of review character. Some results obtained by the author, partly in joint work with M. Bordemann, E. Meinrenken and A. Karabegov, on the theory of Berezin's coherent states for arbitrary quantisable compact Kaehler manifolds are recalled. Co- and contravariant symbols are discussed and the Berezin transform relating them for compact Kaehler manifold is introduced. It is shown that the Berezin transform has an asymptotic expansion and the first two terms are identified.