Representations of scr(U){sub h}{bold (}su(N){bold )} derived from quantum flag manifolds

A relationship between quantum flag and Grassmann manifolds is revealed. This enables a formal diagonalization of quantum positive matrices. The requirement that this diagonalization defines a homomorphism leads to a left scr(U){sub h}{bold (}su(N){bold )}-module structure on the algebra generated by quantum antiholomorphic coordinate functions living on the flag manifold. The module is defined by prescribing the action on the unit and then extending it to all polynomials using a quantum version of the Leibniz rule. The Leibniz rule is shown to be induced by the dressing transformation. For discrete values of parameters occurring in the diagonalization one can extract finite-dimensional irreducible representations of scr(U){sub h}{bold (}su(N){bold )} as cyclic submodules. {copyright} {ital 1997 American Institute of Physics.}