Berezin Forms on Line Bundles over Complex Hyperbolic Spaces

We consider complex hyperbolic spaces X = G/H where G =SU(p, q) and H = S(U(p, q - 1) * {U}(1)) , line bundles L(r), r (element of) Z , over them and representations U(r) of G in smooth sections of L(r) (the representation U(r) is induced by a character of H ). We define a Berezin form {B}_{(lambda), r}, (lambda)(element of) C, r (element of) Z , associated with L(r) , and give an explicit decomposition of this form into invariant Hermitian (sesqui-linear) forms for irreducible representations of the group G- for all (lambda)(element of) C and r (element of) Z . It is the main result of the paper. Besides it, we give the Plancherel formula for U(r) . 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 (lambda) \--(infinity) ). At last, considering (lambda)(element of) C , we observe an interpolation in the spirit of Neretin between Plancherel formulae for U(r) and for the similar representation for a compact form of the space X .