  Given a compact (Hausdorff) group $G$ and a closed subgroup $H$ of $G,$ in
this paper we present symbolic criteria for pseudo-differential operators on
compact homogeneous space $G/H$ characterizing the Schatten-von Neumann classes
$S_r(L^2(G/H))$ for all $0<r \leq \infty.$ We go on to provide a symbolic
characterization for $r$-nuclear, $0< r \leq 1,$ pseudo-differential operators
on $L^{p}(G/H)$-space with applications to adjoint, product and trace formulae.
The criteria here are given in terms of the concept of matrix-valued symbols
defined on noncommutative analogue of phase space $G/H \times \widehat{G/H}.$
Finally, we present applications of aforementioned results in the context of
heat kernels.
