Convexity for twisted conjugation

TL;DR

This paper extends convexity results for conjugation classes in compact Lie groups to twisted conjugation, using group-valued moment maps and automorphisms, revealing new geometric structures.

Contribution

It proves a convexity theorem for twisted conjugation actions, generalizing classical results to automorphism-invariant settings in compact Lie groups.

Findings

Convex polytopes arise from twisted conjugation classes.

The convexity theorem applies to group-valued moment maps.

Results generalize classical conjugation convexity to automorphisms.

Abstract

Let $G$ be a compact, simply connected Lie group. If $\mathcal{C}_1,\mathcal{C}_2$ are two $G$-conjugacy classes, then the set of elements in $G$ that can be written as products $g=g_1g_2$ of elements $g_i\in \mathcal{C}_i$ is invariant under conjugation, and its image under the quotient map $G\to G/\operatorname{Ad}(G)$ is a convex polytope inside the Weyl alcove. In this note, we will prove an analogous statement for twisted conjugations relative to group automorphisms. The result will be obtained as a special case of a convexity theorem for group-valued moment maps which are equivariant with respect to the twisted conjugation action.