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 mani- folds are also parametrized by their Schubert cells. An important step is the de- termination of suitable algebraic generators of the quantized flag manifolds. These algebraic generators can be naturally expressed in terms of quantum Plucker coor- dinates. This note complements the paper of the author and Dijkhuizen in Comm. Math. Phys. 203 (1999), 297-324, in which these results were established for a special subclass of generalized flag manifolds. The orbit principle of Kostant and Kirillov predicts a correspondence between the irreducible unitary representations of Lie groups and the coadjoint orbits of the underlying Lie algebra. As a natural generalization of this principle one expects a correspondence between the irreducible ∗-representations of quantized Poisson homogeneous spaces and their symplectic leaves. The key example is due to Soibel'man (6), who showed that the irreducible ∗-representations of the standard quantization Cq(U) of the regular functions on a compact, connected, simply connected, simple Lie group U are parametrized by the symplectic leaves of U (with the underlying Poisson structure on U given by the Bruhat-Poisson bracket). In this note the correspondence is further investigated for generalized flag manifolds, which form a substantial subclass of Poisson U-homogeneous spaces. A generalized flag manifold G/P , with G the complexification of U and P ⊆ G a parabolic subgroup, can be viewed as a real U-homogeneous space U/K, with K ⊆ U isomorphic to a compact real form of the Levi factor of P. The flag manifold U/K is a Poisson U-homogeneous space with symplectic foliation naturally parametrized by the Schubert cells of G/P. The quantization Cq(U/K) of U/K can be realized as a subalgebra of Cq(U), defined in terms of invariance properties with respect to a suitable quantum subgroup Cq(K) ⊆ Cq(U). It is proved in (8) that the equivalence classes of the irreducible ∗-represent- ations of Cq(U/K) are naturally parametrized by Schubert cells, provided that