Twisted Conjugation on Connected Simple Lie Groups and Twining Characters

Abstract

This article discusses the twisted adjoint action ${\mathrm{Ad}}_g^{\kappa }:G\to G$, $x\mapsto gx\kappa (g^{-1})$ given by a Dynkin diagram automorphism $\kappa \in \mathrm{Aut}(G)$, where $G$ is compact, connected, simply connected and simple. The first aim is to recover the classification of $\kappa$-twisted conjugacy classes by elementary means, without invoking the non-connected group $G⋊⟨\kappa ⟩$. The second objective is to highlight several properties of the so-called \textit{twining characters} ${\tilde{\chi }}^{(\kappa )}:G\to \mathbb{C}$, as defined by Fuchs, Schellekens and Schweigert. These class functions generalize the usual characters, and define $\kappa$-twisted versions ${\tilde{R}}^{(\kappa )}(G)$ and ${\tilde{R}}_k^{(\kappa )}(G)$ ($k\in {\mathbb{Z}}_{>0}$) of the representation and fusion rings associated to $G$. In particular, the latter are shown to be isomorphic to the representation and fusion rings of the \textit{orbit Lie group} $G_{(\kappa )}$, a simply connected group obtained from $\kappa$ and the root data of $G$.

27 pages, 1 figure