Mathematical Research Letters

Volume 24 (2017)

Number 6

Convexity for twisted conjugation

Pages: 1797 – 1818



E. Meinrenken (Department of Mathematics, University of Toronto, Ontario, Canada)


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.

Received 6 January 2016

Accepted 10 May 2016

Published 29 January 2018