Abstract: For a given Cowen-Douglas operator T∈B_n(Ω), a special kind of cross-sections of the corresponding complex bundle E_T is presented. A new unitary invariant ［Φ］ is introduced, which is the conjugate classes of the complex C^∞ function valued n×n matrix Φ(T) i n the case of n≥2 and a real function in the case of n=1. For a given matrix Φ(T)=(φ_{ij/sub>)∈［Φ］, let F=∨ⁿi,j=1{φij}. dim F is independent of the choice of Φ(T). Define D［Φ］=dim F. By a di scussion on the behavior of D［Φ］, we show the uniqueness of a special kind of Cowen-Douglas operators. Moreover, we have proved that if a Cowen-Douglas operator T∈Bn(Ω) satisfies D［Φ］≥n²-2n+2, then T must be Hilbert irreducible.}

Key words: Cowen-Douglas operator, unitary invariant, Hermitian holomorphic vector bunlde, Hilbert reducibility, uniqueness of reducible decomposition