Twisted Conjugation on Connected Simple Lie Groups and Twining Characters

TL;DR

This paper classifies twisted conjugacy classes in compact simple Lie groups and explores properties of twining characters, which generalize usual characters and relate to orbit Lie groups, providing elementary classification methods.

Contribution

It offers an elementary classification of extit{kappa}-twisted conjugacy classes and analyzes twining characters, connecting them to the representation rings of orbit Lie groups.

Findings

Classification of extit{kappa}-twisted conjugacy classes achieved without non-connected groups.

Properties of twining characters as generalizations of usual characters.

Isomorphism between twisted representation rings and those of orbit Lie groups.

Abstract

This article discusses the twisted adjoint action $\mathrm{Ad}_{g}^{\kappa}:G\rightarrow G$, $x\mapsto gx\kappa(g^{-1})$ 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 $G\rtimes\langle\kappa\rangle$. The second objective is to highlight several properties of the so-called \textit{twining characters} $\tilde{\chi}^{(\kappa)}:G\rightarrow\mathbb{C}$, 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 $G$.