COSET SPACES FOR QUANTUM GROUPS Consideration of symmetries often simplifies problems in physics and geometry. Quantum groups are analogues of groups, and they can describe a novel kind of symmetry. We view them as objects of noncommutative geometry. They may act on algebras of noncommutative or quantum observables. We propose a theory of coset spaces for quantum groups in the language of coactions of Hopf algebras and analyse an example which should be thought as a quantum group analogue of the flag variety. In the classical case, the flag variety is the coset space of the special linear group modulo its Borel subgroup of lower triangular matrices. We introduce and study a notion of localized coinvariants; the quantum group coset space is viewed as a system of algebras of localized coinvariants, equipped with a quantum version of the locally trivial principal bundle where the total space is described by the quantum special linear group and the base space is described by the system of algebras of localized coinvariants. We use quasideterminants, the commutation relations between the quantum minors and the noncommutative Gauss decomposition to formulate and prove the main results. We apply our axiomatization of quantum group fibre bundles to obtain a generalization of a concept of Perelomov coherent states to the Hopf algebra setting and obtain the corresponding resolution of unity formula.