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By Wendt R.

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As above, the set {mα ασc | α ∈ , σc (α) = α, and α and σc (α) are orthogonal} ∪ {2mα ασc | α ∈ , σc (α) = α, and α and σc (α) are not orthogonal} is the set of real roots of an affine root system which we also denote by σc . In both cases, the smallest positive imaginary root of σc is given by δσc = ord(σc )δ, where δ denotes the smallest positive imaginary root of . We will list the types of σc in the end of this paper. Now, using (1 + x)(1 − x) = (1 − x 2 ), we can write 1 − s(α)e−2πimα α(H0 ) = [α]∈( re )σc + 1 − e−2πα(H0 ) α∈ re σc + From this, we see that up to the factor f (q) = ord(wc ) ∞ −j q n )dim(hj ) n=1 j =1 (1 − ∞ ord(σc )n )mult(nδσc ) n=1 (1 − q , the function Fwc can be identified with the Kac-Weyl denominator corresponding to the affine root system σc .

5. We have to distinguish two cases. First, let us suppose that for any simple root α ∈ , the roots α and σc (α) are not connected in the Dynkin diagram of . In this case, one can easily show that σc (eα ) = eσc (α) , so that s(α) = 1 for all real roots α ∈ re . For any root α ∈ let us denote by ασc its restriction to the subspace hσc ⊂ h. Then the set {mα ασc | α ∈ re } is the set of real roots of an affine root system which we will denote by σc . Now suppose that there exists some α ∈ such that α = σc (α) are not orthogonal.

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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