# Symplectic and K\"ahler structures on biquotients

## TL;DR

This paper constructs symplectic structures on many biquotients of the form G//T, explores Hamiltonian actions, and discovers new K"ahler structures, including on the Eschenburg flag and a biquotient of SU(4).

## Contribution

It introduces new symplectic structures on biquotients and identifies additional K"ahler structures, expanding understanding of geometric structures on these spaces.

## Key findings

- Constructed symplectic structures on roughly half of all equal rank biquotients G//T.
- Analyzed Hamiltonian Lie group actions, revealing properties similar to known non-K"ahler examples.
- Discovered a new K"ahler structure on a biquotient of SU(4).

## 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 $\mathrm{SU}(4)//T^3$.