Berezin Forms on Line Bundles over Complex Hyperbolic Spaces

We consider complex hyperbolic spaces where and , line bundles , over them and representations of in smooth sections of (the representation is induced by a character of ). We define a Berezin form $, associated with , and give an explicit decomposition of this form into invariant Hermitian (sesqui-linear) forms for irreducible representations of the group for all and . It is the main result of the paper. Besides it, we give the Plancherel formula for . As it turns out, this formula is, en essence, one of the particular cases of the Plancherel formula for the quasiregular representation for rank one semisimple symmetric spaces, see 20], it can be obtained from the quasiregular Plancherel formula for hyperbolic spaces (complex, quaternion, octonion) by analytic continuation in the dimension of the root subspaces. The decomposition of the Berezin form allows us to define and study the Berezin transform, - in particular, to find out an explicit expression of this transform in terms of the Laplacian. Using that, we establish the correspondence principle (an asymptotic expansion as ). At last, considering , we observe an interpolation in the spirit of Neretin between Plancherel formulae for and for the similar representation for a compact form of the space . Submitted: July 12, 2001?Revised: February 12, 2002