  In this article we obtain many results on the multiplicative structure
constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do
this by lifting the classes of the structure sheaves of Schubert varieties in
$K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation
ring of the torus $T$. We further apply our results to describe the
multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the
wonderful compactification of the adjoint group of $G$, in terms of the
structure constants of Schubert varieties in the Grothendieck ring of $G/B$.
