$K$-theory of regular compactification bundles

TL;DR

This paper describes the K-theory of bundles associated with regular compactifications of reductive groups, generalizing known results on projective, toric, and flag bundles, and providing a relative framework over principal G-bundles.

Contribution

It introduces a description of the Grothendieck ring of associated bundles over regular compactifications, extending classical K-theory results to a relative setting.

Findings

Provides an algebraic description of the Grothendieck ring for these bundles.

Generalizes classical results on projective, toric, and flag bundle K-theories.

Connects the K-theory of regular compactifications with equivariant K-theory.

Abstract

Let $G$ be a connected reductive algebraic group. Let $\mathcal{E}\rightarrow \mathcal{B}$ be a principal $G\times G$-bundle and $X$ be a regular compactification of $G$. We describe the Grothendieck ring of the associated fibre bundle $\mathcal{E}(X):=\mathcal{E}\times_{G\times G} X$, as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle on $\mathcal{B}$. These are relative versions of the results on equivariant $K$-theory of regular compactifications of $G$. They also generalize the well known results on the Grothendieck rings of projective bundles, toric bundles and flag bundles.