Reproducing kernels and composition series for spaces of vector-valued holomorphic functions (Q1918411)

The paper gives the expansion of the reproducing kernel of a vector-valued Bergman space of holomorphic functions on a tube domain of type \(I\). This is done by considering the first simple nontrivial representation of the compact group \(U(n)\), or rather \(S(U(n), U(n))\), i.e. its defining representation on \(\mathbb{C}^n\). One can then determine abstractly the \(K\)-irreducible decomposition of the space of \(\mathbb{C}^n\)-valued polynomials which in turn, leads in finding the norm of each \(K\)-type in the considered vector-valued Bergman space. As a consequence, the authors obtain the composition series of the analytic continuation of certain holomorphic discrete series and an expansion relative to \(K\) of the matrix-valued reproducing kernel.