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It is known that all the vector bundles of the title can be obtained byholomorphic induction from representations of a certain parabolic group onfinite dimensional inner product spaces. The representations, and the inducedbundles, have composition series with irreducible factors. We give a conditionunder which the bundle and the direct sum of its irreducible constituents areintertwined by an equivariant constant coefficient differential operator. Weshow that in the case of the unit ball in $\mathbb C^2$ this condition isalways satisfied. As an application we show that all homogeneous pairs ofCowen-Douglas operators are similar to direct sums of certain basic pairs.

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