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We obtain several several results on the multiplicative structure constants of the T-equivariant Grothendieck ring $K_T(G/B)$ 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) ⊗ 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 $K{\left(X\right)}_{\mathbb{Q}}$ where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure constants of...