  Let $\mathfrak g$ be a finite dimensional split semisimple Lie algebra and
$\lambda$ a weight of $\mathfrak g$. Let $F$ be the algebra of quantized
regular functions on the connected simply connected group $G$ corresponding to
$\mathfrak g$. In the present paper we introduce a certain subspace $F'$ of $F$
(which is not necessary a subalgebra of $F$) and endow it with an associative
$\star$-product using the so-called reduced fusion element. We prove that the
algebra $(F',\star)$ is isomorphic to $(L(\lambda))_{fin}$, where $L(\lambda)$
is the irreducible highest weight $\check{U}_q\mathfrak g$-module and "$fin$"
stands for the subalgebra of the locally finite elements with respect to the
adjoint action of $\check{U}_q\mathfrak g$. The introduced $\star$-product has
some limiting properties what enables us to prove Kostant's problem for
$\check{U}_q\mathfrak g$ in certain cases. We remind the reader that this means
that $(L(\lambda))_{fin}$ coincides with the image of $\check{U}_q\g$ in
$L(\lambda)$. We also note that if $\lambda$ is such that
$<\lambda,\alpha_i^\vee>=0$ for some simple roots $\alpha_i$ and generic
otherwise, then $(F,\star)$ is a $\check{U}_q\mathfrak g$-invariant
quantization of the Poisson homogeneous space $G/K$, where $K$ is the
stabilizer of $\lambda$.
