On Uħ (ℊ, r)-equivariant quantization of non-orbit homogeneous varieties

We consider a class of homogeneous varieties over a simple Lie group which appears in the problem of classification of homogeneous manifolds with reductive subgroups of maximal rank as stabilizer of a point. We prove that any variety of this class possesses a Poisson bracket admitting a quantization which is equivariant with respect to the corresponding quantum group.