Non-abelian convexity of based loop groups

TL;DR

This paper extends convexity theorems to the non-abelian setting of based loop groups, showing convexity of the image under the moment map for both the group and its real locus, using Bruhat decomposition and highest weight polytopes.

Contribution

It generalizes convexity results to the full non-abelian regime for based loop groups and their real loci, employing algebraic and geometric tools like Bruhat decomposition.

Findings

Convexity of the moment map image for based loop groups.

Convexity of the real locus under anti-symplectic involution.

Application of Bruhat decomposition and highest weight polytopes.

Abstract

If $K$ is a compact, connected, simply connected Lie group, its based loop group $\Omega K$ 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 $\Omega K$ under the moment map $\mu$, while Jeffrey and Mare examined the corresponding image of the real locus $\Omega K^\tau$ 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 $\Omega K$ and its real locus $\Omega K^\tau$ 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.