2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130175In 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 $K(G/B)$ of rank the cardinality of the Weyl group. We further give an explicit basis of $K(X)$ over $K(G/B)$, and also determine the structure constants with respect to this basis.41 pages, To appear in Transformation GroupsAlgebraic GeometryK-Theory and Homology19L47;14M17;14L30text