For cellular toroidal embeddings of a complex reductive group, the equivariant topological K-ring is described explicitly as an algebra over the equivariant K-ring of the wonderful compactification.
Equivariant $K$-theory of cellular toric bundles and related spaces
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In this article we describe the equivariant and ordinary topological $K$-ring of a toric bundle with fiber a $T$-{\it cellular} toric variety. This generalizes the results in \cite{su} on $K$-theory of smooth projective toric bundles. We apply our results to describe the equivariant topological $K$-ring of a toroidal horospherical embedding.
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Equivariant $K$-theory of cellular toroidal embeddings
For cellular toroidal embeddings of a complex reductive group, the equivariant topological K-ring is described explicitly as an algebra over the equivariant K-ring of the wonderful compactification.