Equivariant K-theory of cellular toroidal embeddings

In this article we describe the $G_{comp}\times G_{comp}$-equivariant topological K-ring of a cellular toroidal embedding $X$ of a complex connected reductive algebraic group G. In particular, our results extend the results in [31] and [32] on the regular embeddings of G, to the equivariant topological K-ring of a larger class of (possibly singular) cellular toroidal embeddings. They are also a topological analogue of the results in [14] on the operational equivariant algebraic K-ring, for cellular toroidal embeddings.