Berezin kernels and maximal degenerate representations associated with Riemannian symmetric spaces of Hermitian type (Q2577574)

For \(D\) a bounded symmetric domain in \(\mathbb C^n\), the Berezin kernel on \(D\times D\) is given~by \[ B_\lambda (z,w) = \left[ \frac{k_D(z,z) k_D(w,w)} {k_D(z,w) k_D(w,z)} \right] ^\lambda, \qquad \lambda\in\mathbb R, \] where \(k_D\) is the Bergman kernel of~\(D\). In~the paper under review, this formalism is extended on a group theoretical basis from the Hermitian symmetric spaces \(D\equiv G/K\) to symmetric spaces \(H/H\cap K\) of Hermitian type (or~``noncompactly causal''). This is achieved by restricting to \(H\) certain maximal degenerate representations and using a construction similar to the construction of the complementary series.