# GKM actions on cohomogeneity one manifolds

-
Oliver Goertsches, University of Marburg

Online Talk

Guillemin, Holm and Zara showed that a homogeneous space $G/H$ of compact Lie groups of equal rank is a GKM manifold with respect to the action of a maximal torus of $H$, and determined the corresponding GKM graph in terms of the root data of $G$ and $H$. In this talk we consider the analogous question for $G$-actions of cohomogeneity one, i.e., with an orbit of codimension $1$. We find easy-to-check necessary and sufficient conditions for a maximal torus of $G$ to act in a GKM fashion in terms of the group diagram of the action, and determine the corresponding GKM graph. We illustrate the results using the known classifications of cohomogeneity one actions with a fixed point, as well as in dimensions up to $6$.