The quantum orbit method for generalized flag manifolds

Abstract

Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the irreducible *-representations of the corresponding quantized flag manifolds are also parametrized by their Schubert cells. An important step is the determination of suitable algebraic generators of the quantized flag manifolds. These algebraic generators can be naturally expressed in terms of quantum Plucker coordinates. This note complements the paper of the author and Dijkhuizen in Commun. Math. Phys. 203 (1999), pp. 297-324, in which these results were established for a special subclass of generalized flag manifolds.