Let M be a Hilbert module of holomorphic functions defined on a bounded domain 0 C m . Let M 0 be the submodule of functions vanishing to order k on a hypersurface Z in 0. In this paper, we describe the quotient module M q .

In this paper we construct a large class of multiplication operators on reproducing kernel Hilbert spaces which are homogeneous with respect to the action of the Möbius group consisting of bi-holomorphic automorphisms of the unit disc D. Indeed, this class consists of exactly those operators for which the associated unitary representation of the universal covering group of the Möbius group is multiplicity free. For every m ∈ N we have a family of operators depending on m + 1 positive real parameters. The kernel function is calculated explicitly. It is proved that each of these operators is bounded, lies in the Cowen-Douglas class of D and is irreducible. These operators are shown to be mutually pairwise unitarily inequivalent.

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The explicit description of irreducible homogeneous operators in the Cowen-Douglas class and the localization of Hilbert modules naturally leads to the definition of a smaller class of Cowen-Douglas operators possessing a flag structure. These operators are shown to be irreducible. It is also shown that the flag structure is rigid in that the unitary equivalence class of the operator and the flag structure determine each other. We obtain a complete set of unitary invariants which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class.2010 Mathematics Subject Classification. 47B32, 47B35. Key words and phrases. The Cowen-Douglas class, strongly irreducible operator, homogeneous operator, curvature, second fundamental form. 2.A new class of operators in B 2 (Ω) 2.1. Definitions. If T is an operator in B 2 (Ω), then there exists a pair of operators T 0 and T 1 in B 1 (Ω) and a bounded operator S such that T = T 0 S 0 T 1 . This is Theorem 1.49 of [8, page 48]. We show, the other way round, that two operators T 0 and T 1 from B 1 (Ω) combine with the aid of an arbitrary bounded linear operator S to produce an operator in B 2 (Ω).Proposition 2.1. Let T be a bounded linear operator of the form T 0 S 0 T 1 . Suppose that the two operators T 0 , T 1 are in B 1 (Ω). Then the operator T is in B 2 (Ω).Proof. Suppose T 0 and T 1 are defined on the Hilbert spaces H 0 and H 1 , respectively. Elementary considerations from index theory of Fredholm operators shows that the operator T is Fredholm and ind(T ) = ind(T 0 ) + ind(T 1 ) (cf. [2, page 360]). Therefore, to complete the proof that T is in B 2 (Ω), all we have to do is prove that the vectors in the kernel ker(T − w), w ∈ Ω, span the Hilbert space H = H 0 ⊕ H 1 .Let γ 0 and t 1 be non-vanishing holomorphic sections for the two line bundles E 0 and E 1 corresponding to the operators T 0 and T 1 , respectively. For each w ∈ Ω, the operator T 0 − w 0