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Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains (1507.08769v1)
Published 31 Jul 2015 in math.FA and math.RT
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 give a condition under which the bundle and the direct sum of its irreducible constituents are intertwined by an equivariant constant coefficient differential operator. We show that in the case of the unit ball in $\mathbb C2$ this condition is always satisfied. As an application we show that all homogeneous pairs of Cowen-Douglas operators are similar to direct sums of certain basic pairs.