Symplectic and K\"ahler structures on biquotients

Abstract

We construct symplectic structures on roughly half of all equal rank biquotients of the form $G//T$, where $G$ is a compact simple Lie group and $T$ a torus, and investigate Hamiltonian Lie group actions on them. For the Eschenburg flag, this action has similar properties as Tolman's and Woodward's examples of Hamiltonian non-K\"ahler actions. In addition to the previously known K\"ahler structure on the Eschenburg flag, we find another K\"ahler structure on a biquotient $\mathrm{SU}(4)//T^3$.