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By Wendt R.

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Since we have excluded the case that σc is the order n + 1 automorphism of the extended Dynkin diagram of An , we can assume that σc2 (α) = α. 1 for the case of finite root systems). Looking at the explicit form of the automorphism σc , we see that if there exists a simple root α such that α = σc (α) are not orthogonal, then α = σc (α) for all simple roots α of . Therefore, if σc (α) = α, then there exists some β ∈ with σc (β) = β and β + σc (β) = α. Hence, in this case ht(α) is necessarily even so that s(α) = −1 whenever σc (α) = α.

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A Character Formula for Representations of Loop Groups Based on Non-simply Connected Lie Groups by Wendt R.


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