Homogeneous Hermitian holomorphic vector bundles and the Cowen–Douglas class over bounded symmetric domains

Abstract

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite-dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We write down an equivariant constant coefficient differential operator that intertwines the bundle with the direct sum of its irreducible factors. As an application, we show that in the case of the closed unit ball in Cn all homogeneous n-tuples of Cowen–Douglas operators are similar to direct sums of certain basic n-tuples.