Suppose $\mathfrak{g}$ is an (untwisted) affine Lie algebra with the normalized invariant form $(\cdot | \cdot)$. Let $\lambda \in \mathfrak{h}^\ast$ be a dominant integral weight such that $\lambda(d)=0$ for $d$ the derivation in the loop algebra construction; that is, $\lambda = c\_0\Lambda\_0 +c\_1\Lambda\_1+\cdots+c\_n\Lambda\_n$, where $c\_i \in \mathbb{Z}\_{\geq0}$, $\Lambda\_i$ the fundamental weights. Let $l$ be the level of $\lambda$.


While doing some (albeit limited) explicit computations for various $\mathfrak{g}$ and $\lambda$, I am encountering an inequality of the form
$$
2l(\lambda|\rho)-h^\vee(\lambda|\lambda) \geq 0
$$
where $h^\vee$ is the dual Coxeter number of $\mathfrak{g}$ and $\rho$ is as usual a choice of weight satisfying $\rho(\alpha\_i^\vee)=1$ for all simple coroots $\alpha\_i^\vee$.



> 
> Question: Should this inequality hold for all such $\lambda$?
> 
> 
> 


I have little intuition for whether this should be expected; this very well could be an artifact of the specific $\lambda$ that appear in my computations. For example, I would be somewhat satisfied if this were to hold under the stronger condition that each $c\_i \in \{0,1\}$.



An inequality for weights of affine Lie algebras, level, and dual Coxeter number
The answer to this is found as theorem 13.11 in Kac, "Infinite dimensional Lie Algebras". To be specific, we have $2k(\Lambda|\rho) \geq h^{\vee} (\Lambda| \Lambda)$ for all $\Lambda \in P^k\_+$, with equality if and only if $\Lambda = k \Lambda\_j$ mod $\mathbb{C} \delta$. Here $j \in J$, where $J$ is a set depending on the Kac labeling of the Dynkin diagram in question (J should correspond to the set of simple roots with Kac label 1, roughly).


