Non-abelian convexity of based loop groups

If $K$ is a compact, connected, simply connected Lie group, its based loop group $\mathrm{\Omega }K$ is endowed with a Hamiltonian $S^1\times T$ action, where $T$ is a maximal torus of $K$. Atiyah and Pressley examined the image of $\mathrm{\Omega }K$ under the moment map $\mu$, while Jeffrey and Mare examined the corresponding image of the real locus $\mathrm{\Omega }{K}^{\tau }$ for a compatible anti-symplectic involution $\tau$. Both papers generalize well known results in finite dimensions, specifically the Atiyah-Guillemin-Sternberg theorem, and Duistermaat’s convexity theorem. In the spirit of Kirwan’s convexity theorem, this paper aims to further generalize the two aforementioned results by demonstrating convexity of $\mathrm{\Omega }K$ and its real locus $\mathrm{\Omega }{K}^{\tau }$ in the full non-abelian regime, resulting from the Hamiltonian $S^1\times K$ action. In particular, this is done by appealing to the Bruhat decomposition of the algebraic (affine) Grassmannian, and appealing to the “highest weight polytope” results for Borel-invariant varieties of Guillemin and Sjamaar and Goldberg.