Berezin forms on line bundles over complex hyperbolic spaces (Q1811163)

The decomposition of the Berezin forms defined on smooth functions or sections with compact support is studied in the case of complex hyperbolic spaces \(X=G/H\), where \(G=SU(p,q)\) and \(H=SU(p,q-1)\times U(1)\). The Berezin form \(B_{\lambda ,r}\), \(\lambda \in C\), \(r\in Z\), associated to the line bundle \(L_{r}\), is firstly defined. Then, the explicit decomposition of the form \(B_{\lambda ,r}\) into invariant Hermitian (sesqui-linear) forms for irreducible representations \(U_{r}\) of the group \(G\) (for all \(\lambda \in C\) and \(r\in Z\)) is given. The Plancherel formula for an arbitrary representation \(U_{r}\) of \(G\) in smooth sections of \(L_{r}\) is obtained. It is shown that the decomposition of the Berezin form allows us to define and study the Berezin transform. An explicit expression of this transform in terms of the Laplacian is obtained using the decomposition of the Berezin form. Finally, considering \(\lambda \in Z\) one observes an interpolation between Plancherel formulae for \(U_{r}\) and for the similar representation for a compact form of the space \(X\). It is emphasized that the decomposition of the Berezin form is related to the harmonic analysis on homogeneous spaces \(G/H\).