Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

We find a partial solution to the longstanding problem of Kostant concerning description of the so-called locally finite endomorphisms of highest weight irreducible modules. The solution is obtained by means of its reduction to a far-reaching extension of the quantization problem. While the classical quantization problem consists in finding ⋆-product deformations of the commutative algebras of functions, we consider the case when the initial object is already a noncommutative algebra, the algebra of functions within q-calculus.