  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, that is, the unitary equivalence class of the
operator and the flag structure determine each other. A complete set of unitary
invariants, which are somewhat more tractable than those of an arbitrary
operator in the Cowen-Douglas class, are obtained.
