Convexity for twisted conjugation

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 $g = g_1 g_2$ of elements $g_i \in \mathcal{C}_i$ is invariant under conjugation, and its image under the quotient map $G \to G / \mathrm{Ad}(G) = \mathfrak{A}$ is a convex polytope. 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.

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Received 6 January 2016

Accepted 10 May 2016

Published 29 January 2018