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If is the root system of an affine Lie algebra L(g), and σ is an automorphism of the Dynkin diagram of then the affine Lie algebra corresponding to the root system σ is often called the orbit Lie algebra corresponding to L(g) and σ . The appearance of the root system σ was also realized in [FRS] [FSS], where for an outer automorphism σ of a (generalized) Kac-Moody Lie algebra g, the σ -twisted characters of highest weight representations are calculated. It turns out that these twisted characters can be identified with untwisted characters of the orbit Lie algebra corresponding to g and σ .

Nucl. Phys. : Positive energy representations of the loop groups of non simply connected Lie groups. Comm. Math. Phys. : Weyl’s character formula for non-connected Lie groups and orbital theory for twisted affine Lie algebras. J. Funct. Anal.

Ann. : On moduli spaces of flat connections with non-simply connected structure group. Nucl. Phys. : Positive energy representations of the loop groups of non simply connected Lie groups. Comm. Math. Phys. : Weyl’s character formula for non-connected Lie groups and orbital theory for twisted affine Lie algebras. J. Funct. Anal.

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