Equivariant $K$-theory of GKM bundles

TL;DR

This paper develops a combinatorial approach to analyze the equivariant K-theory of GKM fiber bundles, revealing invariant bases and connecting to classical descriptions for flag varieties.

Contribution

It introduces a new combinatorial framework for understanding the equivariant K-theory of GKM bundles, including invariant bases and relations to classical models.

Findings

Constructed a basis of the K-ring invariant under holonomy

Connected GKM K-theory with Kostant-Kumar description for flag varieties

Provided a combinatorial translation of fiber bundle structures

Abstract

Given a fiber bundle of GKM spaces, $\pi\colon M\to B$, we analyze the structure of the equivariant $K$-ring of $M$ as a module over the equivariant $K$-ring of $B$ by translating the fiber bundle, $\pi$, into a fiber bundle of GKM graphs and constructing, by combinatorial techniques, a basis of this module consisting of $K$-classes which are invariant under the natural holonomy action on the $K$-ring of $M$ of the fundamental group of the GKM graph of $B$. We also discuss the implications of this result for fiber bundles $\pi\colon M\to B$ where $M$ and $B$ are generalized partial flag varieties and show how our GKM description of the equivariant $K$-ring of a homogeneous GKM space is related to the Kostant-Kumar description of this ring.