This article discusses the twisted adjoint action Ad κ g : G → G, x → gxκ(g −1 ) given by a Dynkin diagram automorphism κ ∈ Aut(G), where G is compact, connected, simply connected and simple. The first aim is to recover the classification of κ-twisted conjugacy classes by elementary means, without invoking the non-connected group G κ . The second objective is to highlight several properties of the so-called twining characters χ(κ) : G → C, as defined by Fuchs, Schellekens and Schweigert. These class functions generalize the usual characters, and define κ-twisted versions R(κ) (G) and R(κ) k (G) (k ∈ 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 orbit Lie group G (κ) , a simply connected group obtained from κ and the root data of G.
We prove a Hitchin-Thorpe inequality for noncompact 4-manifolds with foliated geometry at infinity by extending on previous work by Dai and Wei. After introducing the objects at hand, we recall some preliminary results regarding the G-signature formula and the rho invariant, which are used to obtain expressions for the signature and Euler characteristic in our geometric context. We then derive our main result, and present examples.Résumé: En se basant sur des travaux de Dai et Wei, on démontre une inégalité de Hitchin-Thorpe pour variété non-compactes de dimension 4, et munies d'une géométrie feuilletée à l'infini. Après avoir défini les notions pertinentes à cette étude, on rappelle quelques résultats concernant la formule de G-signature et l'invariant rho, qu'on utilise ici pour obtenir des expressions de la signature et de la caractéristique d'Euler dans notre cadre géométrique. On démontre ensuite nos résultats principaux avant de présenter quelques exemples.
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