Let $\mathfrak g$ be a simple Lie algebra. Let $\widetilde{L\mathfrak g}$ be the universal central extension of $L\mathfrak g:=\mathfrak g[t,t^{-1}]$. Let $V_\lambda$ be a positive energy representation of $\widetilde{L\mathfrak g}$ of level $k$ and highest weight $\lambda$. Then the minimal energy $h_\lambda$ of $V_\lambda$ is given by the well-known formula $$ h_\lambda=\frac{\|\lambda+\rho\|^2-\|\rho\|^2}{2(k+g^\vee)} $$ where $\rho$ is the half-sum of all positive roots, and $g^\vee$ is the dual Coxeter number.

I am looking for a citable reference for the above formula, i.e., one that includes a proof.

Now, for the benefit of the reader, I will define the terms "positive energy representation" and "minimal energy". First of all, the affine Kac-Moody algebra $\widetilde{L\mathfrak g}^e=\widetilde{L\mathfrak g}\rtimes \mathbb CL_0$ has underlying vector space $$ \widetilde{L\mathfrak g}\oplus \mathbb CL_0=\mathfrak g[t,t^{-1}]\oplus \mathbb Cc\oplus \mathbb CL_0 $$ and Lie bracket given by the requirements that $c$ is central and that

$ [t^mX+aL_0,t^nY+bL_0]=t^{m+n}[X,Y]+m\delta_{m+n,0}\langle X,Y\rangle c-nat^nY+mbt^mX$.

Note that $L_0$ acts like $-t\frac{d}{dt}$.

A representation $V$ of $\widetilde{L\mathfrak g}$ is called **positive energy** if the action of $\widetilde{L\mathfrak g}$ on $V$ can be extended (such an extension is never unique!) to an action of $\widetilde{L\mathfrak g}^e$ in such a way that $L_0$ acts with positive spectrum and finite dimensional eigenspaces.
To see that the extension is never unique, note that one can add an arbitrary multiple of the identity operator to $L_0$, without destroying the commutation relations.
To make the extension unique, one considers the Lie algebra $\widetilde{L\mathfrak g}\rtimes \mathfrak{sl}(2)$ instead, where the copy of $\mathfrak{sl}(2)$ is spanned by elements $L_{-1}$, $L_0$, $L_1$.
The action of $L_n\in \mathfrak{sl}(2)$ on $\widetilde{L\mathfrak g}$ is by $-t^{n+1}\frac{d}{dt}$.

It turns out that, if the action of $\widetilde{L\mathfrak g}$ on $V$ extends to $\widetilde{L\mathfrak g}\rtimes \mathbb CL_0$, then it always also extends to $\widetilde{L\mathfrak g}\rtimes \mathfrak{sl}(2)$. However, among all the possible ways of extending the action to $\widetilde{L\mathfrak g}\rtimes \mathbb CL_0$, only one of them has the property that it further extends to $\widetilde{L\mathfrak g}\rtimes \mathfrak{sl}(2)$.

The moral of the story is that there is a preferred way of extending the action of $\widetilde{L\mathfrak g}$ on $V$ to an action of $\widetilde{L\mathfrak g}\rtimes \mathbb CL_0$.
The **minimal energy** of the positive energy representation $V$ is the smallest eigenvalue of $L_0$.

Finally, for completeness, the **central charge** is the scalar by which the central element $c\in \widetilde{L\mathfrak g}$ acts.

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