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Given a rational homogeneous variety G/P where G is complex simple and oftype ADE, we prove that all tangent bundles T_{G/P} are simple, meaning thattheir only endomorphisms are scalar multiples of the identity. This resultcombined with Hitchin-Kobayashi correspondence implies stability of thesetangent bundles with respect to the anticanonical polarization. Our main toolis the equivalence of categories between homogeneous vector bundles on G/P andfinite dimensional representations of a given quiver with relations.

On simplicity and stability of tangent bundles of rational homogeneous varieties