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

We describe the $\widetilde G\times \widetilde G$-equivariant $K$-ring of $X$, where $\widetilde G$ is a factorial covering of a connected complex reductive algebraic group $G$, and $X$ is a regular compactification of $G$. Furthermore, using the description of $K_{\widetilde G\times\widetilde G}(X)$, we describe the ordinary $K$-ring $K(X)$ as a free module (whose rank is equal to the cardinality of the Weyl group) over the $K$-ring of a toric bundle over $G/B$ whose fibre is equal to the toric variety $\overline{T}^{+}$ associated with a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see [1]). We also give an explicit presentation of $K_{\widetilde G\times\widetilde G}(X)$ and $K(X)$ as algebras over $K_{\widetilde G\times\widetilde G}(\overline{G_{\operatorname{ad}}})$ and $K(\overline{G_{\operatorname{ad}}})$ respectively, where $\overline{G_{\operatorname{ad}}}$ is the wonderful compactification of the adjoint semisimple group $G_{\operatorname{ad}}$. In the case when $X$ is a regular compactification of $G_{\operatorname{ad}}$, we give a geometric interpretation of these presentations in terms of the equivariant and ordinary Grothendieck rings of a canonical toric bundle over $\overline{G_{\operatorname{ad}}}$.