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Let $\delta$ be the null of an affine root system and let $\alpha + p\delta$ be a real affine root, $p$ is an integer. It is said that $$ (\alpha + p\delta)^{\vee} = \alpha^{\vee} + \frac{2p}{(\alpha, \alpha)}K, $$ where $K$ is the central element. Are there some references about this formula? I tried to look for this formula in Kac's book. But I am not able to find it. Thank you very much.

Edit: this formula is given on line 14, page 5 of the paper.

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  • $\begingroup$ Can you be more specific about "It is said"? The central element $K$ doesn't always come up in treatments of affine Lie algebras and real/imaginary roots, so it would also help if you indicated your main background sources. $\endgroup$
    – Jim Humphreys
    Commented Mar 8, 2015 at 17:49
  • $\begingroup$ @JimHumphreys, thank you very much. I have edited the post. $\endgroup$
    – Jianrong Li
    Commented Mar 10, 2015 at 2:10
  • $\begingroup$ I'm not sure where this definition of "coroot" originates (presumably not with Kac), but Naoi has spent time with Chari and is much influenced by her work on affine Lie algebras. Probably it's worthwhile to contact her directly at UC Riverside. $\endgroup$
    – Jim Humphreys
    Commented Mar 10, 2015 at 18:55
  • $\begingroup$ @JimHumphreys, thank you very much. $\endgroup$
    – Jianrong Li
    Commented Mar 11, 2015 at 2:41

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