Equivariant $K$-theory of regular compactifications: further developments

In this article we describe the $\tG\times \tG$-equivariant $K$-ring of $X$, where $\tG$ is a {\it factorial} cover of a connected complex reductive algebraic group $G$, and $X$ is a regular compactification of $G$. Furthermore, using the description of $K_{\tG\times \tG}(X)$, we describe the ordinary $K$-ring $K(X)$ as a free module of rank the cardinality of the Weyl group, over the $K$-ring of a toric bundle over $G/B$, with fibre the toric variety $\bar{T}^{+}$, associated to a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see \cite{u}). Further, we give an explicit presentation of $K_{\tG\times \tG}(X)$ as well as $K(X)$ as an algebra over the $K_{\tG\times \tG}(\bar{G_{ad}})$ and $K(\bar{G_{ad}})$ respectively, where $\bar{G_{ad}}$ is the wonderful compactification of the adjoint semisimple group $G_{ad}$. Finally, we identify the equivariant and ordinary Grothendieck ring of $X$ respectively with the corresponding rings of a canonical toric bundle over $\bar{G_{ad}}$ with fiber the toric variety $\bar{T}^+$.