Equivariant $K$-theory of cellular toroidal embeddings

TL;DR

This paper computes the equivariant topological $K$-ring of cellular toroidal embeddings of complex reductive groups, extending previous results to a broader class of embeddings and providing a topological analogue of algebraic $K$-theory results.

Contribution

It extends the computation of equivariant $K$-theory from regular to cellular toroidal embeddings, including singular cases, and connects topological and algebraic $K$-theory results.

Findings

Computed the $G_{comp} imes G_{comp}$-equivariant topological $K$-ring for cellular toroidal embeddings.

Extended previous results from regular to more general cellular toroidal embeddings.

Established a topological analogue of algebraic $K$-ring results for these embeddings.

Abstract

In this article we describe the $G_{comp}\times G_{comp}$-equivariant topological $K$-ring of a {\em cellular} toroidal embedding $\mathbb{X}$ of a complex connected reductive algebraic group $G$. In particular, our results extend the results in \cite{u1} and \cite{u2} 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 \cite{gon} on the operational equivariant algebraic $K$-ring, for cellular toroidal embeddings.