Equivariant K-theory of compactifications of algebraic groups

In this article we describe the $G\times G$-equivariant $K$-ring of $X$, where $X$ is a regular compactification of a connected complex reductive algebraic group $G$. Furthermore, in the case when $G$ is a semisimple group of adjoint type, and $X$ its wonderful compactification, we describe its ordinary $K$-ring $K(X)$. More precisely, we prove that $K(X)$ is a free module over $K(G\mathrm{/}B)$ of rank the cardinality of the Weyl group. We further give an explicit basis of $K(X)$ over $K(G\mathrm{/}B)$, and also determine the structure constants with respect to this basis.