minimal energy of affine Lie algebra reps

Question

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.

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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.

This post imported from StackExchange MathOverflow at 2014-09-10 17:15 (UCT), posted by SE-user André Henriques

Comments

I would suggest Kac "Infinite dimensional Lie algebras", Corollary 12.8.

This post imported from StackExchange MathOverflow at 2014-09-10 17:16 (UCT), posted by SE-user Victor Ostrik

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@Victor. Thank you for this reference. Unfortunately, my copy of Kac's book is not the latest edition, and I don't have any Corollary 12.8 in it. Could you please tell me what is the statement of that Corollary? By the way, in my version of Kac's book, the quantity $\frac{\|\lambda+\rho\|^2-\|\rho\|^2}{2(k+g^\vee)}$ never appears anywhere (the only thing that appears is $\frac{\|\lambda+\rho\|^2}{2(k+g^\vee)}-\frac{\|\rho\|^2}{2g^\vee}$, which differs from the previous by some constant).

This post imported from StackExchange MathOverflow at 2014-09-10 17:16 (UCT), posted by SE-user André Henriques

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Ok, I got a google preview of the 3rd edition of Kac's book. It misses the pages 230 and 231 (on which Corollary 12.8 is presumably located) but I can interpolate the book's content, and it looks very likely that what I want is indeed somewhere in those two pages -- sorry for the trouble.

This post imported from StackExchange MathOverflow at 2014-09-10 17:16 (UCT), posted by SE-user André Henriques

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Pretty much this Corollary summarizes basic facts about Sugawara operators. In particular it says that that $L_0=h_\lambda -d$ where $d$ is the energy operator..

This post imported from StackExchange MathOverflow at 2014-09-10 17:16 (UCT), posted by SE-user Victor Ostrik

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Great! If you put your first comment as an answer, then I will accept it.

This post imported from StackExchange MathOverflow at 2014-09-10 17:16 (UCT), posted by SE-user André Henriques

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Small comment: The notation $\rho$ has to be used more cautiously in this setting, since "the half-sum of all positive roots" doesn't make sense in the infinite dimensional case. The alternative definition in the finite dimensional case (sum of fundamental dominant weights) is more suggestive for the purposes of Kac.

This post imported from StackExchange MathOverflow at 2014-09-10 17:16 (UCT), posted by SE-user Jim Humphreys

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Thank you: from now on, I will use "sum of fundamental dominant weights". I think that it's a much more intuitive definition than "half-sum of all positive roots".

This post imported from StackExchange MathOverflow at 2014-09-10 17:16 (UCT), posted by SE-user André Henriques

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The corollary is visible p231 in preview here.

This post imported from StackExchange MathOverflow at 2014-09-10 17:16 (UCT), posted by SE-user Sébastien Palcoux