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A relationship between quantum flag and Grassmann manifolds is revealed. Thisenables a formal diagonalization of quantum positive matrices. The requirementthat this diagonalization defines a homomorphism leads to a left \Uh -- modulestructure on the algebra generated by quantum antiholomorphic coordinatefunctions living on the flag manifold. The module is defined by prescribing theaction on the unit and then extending it to all polynomials using a quantumversion of Leibniz rule. Leibniz rule is shown to be induced by the dressingtransformation. For discrete values of parameters occuring in thediagonalization one can extract finite-dimensional irreducible representationsof \Uh as cyclic submodules.Comment: LaTeX file, JMP (to appear

Representations of 𝒰h (𝔰𝔲(N) ) derived from quantum flag manifolds