The purpose of this paper is to study covariant Poisson structures on the complex Grassmannian obtained as quotients by coisotropic subgroups of the standard Poisson--Lie SU(n). Properties of Poisson quotients allow to describe Poisson embeddings generalizing those obtained in math.SG/9802082.
We define even dimensional quantum spheres Sigma_q^2n that generalize to higher dimension the standard quantum two-sphere of Podles and the four-sphere Sigma_q^4 obtained in the quantization of the Hopf bundle. The construction relies on an iterated
Poisson double suspension of the standard Podles two-sphere. The Poisson spheres that we get have the same symplectic foliation consisting of a degenerate point and a symplectic plane and, after quantization, have the same C^*-algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.
Induced representations for quantum groups are defined starting from coisotropic quantum subgroups and their main properties are proved. When the coisotropic quantum subgroup has a suitably defined section such representations can be realized on asso
ciated quantum bundles on general embeddable quantum homogeneous spaces.
