Equivariant $K$-theory of flag varieties revisited and related results

In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G\mathrm{/}B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_T(G\mathrm{/}B)$ 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 $G\mathrm{/}B$.