  Let G be a simply connected semisimple compact Lie group with standard
Poisson structure, K a closed Poisson-Lie subgroup, 0<q<1. We study a
quantization C(G_q/K_q) of the algebra of continuous functions on G/K. Using
results of Soibelman and Dijkhuizen-Stokman we classify the irreducible
representations of C(G_q/K_q) and obtain a composition series for C(G_q/K_q).
We describe closures of the symplectic leaves of G/K refining the well-known
description in the case of flag manifolds in terms of the Bruhat order. We then
show that the same rules describe the topology on the spectrum of C(G_q/K_q).
Next we show that the family of C*-algebras C(G_q/K_q), 0<q\le1, has a
canonical structure of a continuous field of C*-algebras and provides a strict
deformation quantization of the Poisson algebra \C[G/K]. Finally, extending a
result of Nagy, we show that C(G_q/K_q) is canonically KK-equivalent to C(G/K).
