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While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into the notion of dual Coxeter number but am uncertain about the extent of its influence in Lie theory. The term was probably introduced by Victor Kac and is often denoted by $h^\vee$ (sometimes by $g$ or another symbol). It occurs for example in the 1990 third edition of his book Infinite Dimensional Lie Algebras in Section 6.1. (The first edition goes back to 1983.) It also occurs a lot in the mathematical physics literature related to representations of affine Lie algebras. And it occurs in a 2009 paper by D. Panyushev in Advances which studies the structure of complex simple Lie algebras.

Where in Lie theory does the dual Coxeter number play a natural role (and why)?

A further question is whether it would be more accurate historically to refer instead to the Kac number of a root system, since the definition of $h^\vee$ is not directly related to the work of Coxeter in group theory.

BACKGROUND: To recall briefly where the Coxeter number $h$ comes from, it was introduced by Coxeter and later given its current name (by Bourbaki?). Coxeter was studying a finite reflection group $W$ acting irreducibly on a real Euclidean space of dimension $n$: Weyl groups of root systems belonging to simple complex Lie algebras (types $A--G$), these being crystallographic, together with the remaining dihedral groups and two others. The product of the $n$ canonical generators of $W$ has order $h$, well-defined because the Coxeter graph is a tree. Its eigenvalues are powers of a primitive $h$th root of 1 (the "exponents"): $1=m_1 \leq \dots \leq m_n = h-1$. Moreover, the $d_i = m_i+1$ are the degrees of fundamental polynomial invariants of $W$ and have product $|W|$.

In the Weyl group case, where there is an irreducible root system (but types $B_n, C_n$ yield the same $W$), work of several people including Kostant led to the fact that $h$ is 1 plus the sum of coefficients of the highest root relative to a basis of simple roots. On the other hand, the dual Coxeter number is 1 plus the sum of coefficients of the highest short root. For respective types $B_n, C_n, F_4, G_2$, the resulting values of $h, h^\vee$ are then $2n, 2n, 12, 6$ and $2n-1, n+1, 9,4$. This gets pretty far from Coxeter's framework.

One place where $h^\vee$ clearly plays an essential role is in the study of a highest weight module for an affine Lie algebra, where the canonical central element $c$ acts by a scalar (the level or central charge). The "critical" level $-h^\vee$ has been especially challenging, since here the theory seems to resemble the characteristic $p$ situation rather than the classical one.

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This is exactly the type of question I like on MO, and everyone should vote it up: good question, lots of background, ... +1, and +more if I could. (OTOH, I don't have a helpful answer.) An aside: for "n-dashes", produced by typing -- (double hyphen) in LaTeX, use –. Up above you have "$A--G$" and what mean is "$A$–$G$". –  Theo Johnson-Freyd May 23 '10 at 3:16
    
Well, I take it back. It seems that Markdown doesn't like "–", or at least it doesn't like it in the comments. OTOH, you can probably just do it from the keyboard: –. –  Theo Johnson-Freyd May 23 '10 at 3:18
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Just a historical note to add to the dates mentioned above: The dual Coxeter number makes an appearance in Springer-Steinberg's 1970 paper "Conjugacy classes" on pages E-12 through E-15, where it is defined and used to calculate the Killing form on the Cartan subalgebra. –  Skip Jun 4 '10 at 16:05
    
@Skip: Belated thanks for your comment. I had forgotten about that discussion in S-S, which was aimed at computing the discriminant of the Killing form relative to a Chevalley basis. Leaving aside their notational choices, this adds weight to the argument that the "dual Coxeter number" doesn't really need a name but only a symbol such as the currently common $h^\vee$ together with a brief reminder of what the symbol means. –  Jim Humphreys Feb 28 '11 at 16:19

4 Answers 4

up vote 12 down vote accepted

The dual Coxeter number comes up naturally as a normalization factor for invariant bilinear forms on the Lie algebra: according to Kac's book which you quote, $2h^{\vee}$ is the ratio between the Killing form and the "minimal" bilnear form (the trace form for $sl_n$), which has the property that the square of the length of the maximal root is 2.

This minimal form corresponds to the minimal affine Kac-Moody group corresponding to the Lie algebra, or equivalently to the minimal line bundle on the affine Grassmannian or the moduli spaces of G-bundles on curves (the generator of the Picard group). As a result, the $-2h^\vee$-th power of the basic ample line bundle on the Grassmannian or moduli space of bundles (which is associated to the level given by the Killing form) ends up being identified with the canonical line bundle, and in particular the $h^\vee$th power is a square-root of the canonical bundle, or spin structure. (This is analogous to the role of $\rho$ for the finite flag variety.) Thus the critical level arises naturally geometrically -- it corresponds to half-forms on the Grassmannian/moduli spaces. The basic yoga of quantization (or of unitary/normalized induction of representations) tells us that classical symmetries are "shifted" by half-forms - cf $\rho$-shifts in representation theory. Likewise the critical shift for affine algebras.. for example the Feigin-Frenkel theorem is the analogue of the Harish-Chandra isomorphism: the center of the enveloping algebra at critical level (rather than level 0 as one might naively guess, ignoring half-form twists) is isomorphic to the algebra of invariant polynomials on the (dual of the) Lie algebra. (This can be said more canonically keeping track of symmetries of change of variable, magic word being "opers", but let's ignore that).

One can say all this very naturally algebraically (without resorting to geometry) -- $\rho$ can be described as the square root of the modular character of the Borel subalgebra (up to sign or something, not being very careful here). The critical level has a similar description in terms of the positive half (Taylor series part) of the Kac-Moody algebra - if you try to define the modular character of this half you are quickly led to semiinfinite determinants etc, ie to the previous geometric story, and so one can assert that the critical level "is" half the modular character of the positive loop subalgebra.

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Thanks, this is a very helpful way to approach the question. The notion of dual Coxeter number clearly involves $\rho$, though at first a long way from the classical shift by $−\rho$ in representation theory. I'll have to pursue the interpretation further from these viewpoints, though probably unaided by the short nontechnical survey no one has yet written (?) – Jim 2 hours ago –  Jim Humphreys May 23 '10 at 21:57

I think part of the question here is "why is this thing called the dual Coxeter number? It looks pretty different, so why don't we just give it a different name?" I think the case is made in Kac's book that dual Coxeter number is the right name.

The Coxeter number for $\mathfrak{g}$ is the sum of the labels in the Dynkin diagram for the untwisted affine algebra corresponding to $\mathfrak{g}$. These labels are the coefficients of a minimal integer linear dependence among the columns of the affine Cartan matrix, which seems fairly intrinsic, so I think this is a reasonable definition. I won't try to explain why it is equivalent to more standard definitions. The dual Coxeter number is then the sum of the labels in the dual affine Dynkin diagram. See Kac, section 6.1 for these definitons.

I think what is confusing is that "dual" and "affine" do not commute. For instance, the dual of the affine diagram of type $B_\ell^{(1)}$ is the twisted affine Dynkin diagram of type $A_{2\ell-1}^{(2)}$.

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The term comes up in the context of root systems, Dynkin diagrams, Weyl groups (crystallographic reflection groups), but doesn't come from Coxeter's more general study of the order and the eigenvalues of Coxeter elements. I don't think Coxeter would have seen any group-theoretic meaning for "dual Coxeter number". The later characterization of Coxeter numbers which Kac starts with was proved by Kostant after observations by A. Shapiro. Anyway, the question of renaming is moot by now (I recall Borel's belated attempt to ascribe the "Bruhat ordering" more correctly to Chevalley). –  Jim Humphreys May 24 '10 at 12:34

The dual Coxeter number $h$ comes up in a conjecture of Cachazo-Douglas-Seilberg-Witten which was motivated by supersymmetric gauge theory. Let $R:=\bigwedge( g\oplus g) = \bigwedge g\otimes\bigwedge g$, where $g$ is a finite dimensional simple complex Lie algebra. Let {$e_i$} be some basis of $g$ and let {$f_i$} denote the dual basis with respect to the normalized Killing form. Consider three different embeddings of $g$ into the 2-graded part of $R$ (which are independent of our chosen basis):

$C_1=${$\sum_i [x,e_i]\wedge f_i \otimes 1: x\in g$}$\subset \bigwedge^2 g\otimes \bigwedge^0 g$

$C_2=${$1\otimes \sum_i [x,e_i]\wedge f_i : x\in g$}$\subset \bigwedge^0 g\otimes \bigwedge^2 g$

$C_3=${$\sum_i [x,e_i]\otimes f_i : x\in g$}$\subset \bigwedge^1 g\otimes \bigwedge^1 g$

Let $J$ be the ideal of $R$ generated by $C_1,C_2,C_3$ and let $A$ denote the $g$-algebra $A:=R/J$. Lastly, let $$S=\sum_i e_i\otimes f_i\in \bigwedge^1g\otimes\bigwedge^1g,$$ which also does not depend on the choice of basis.

The CDSW conjecture is:

The subalgebra $A^g$ of $g$-invariants in $A$ is generated as an algebra by the element $S$. Furthermore, $S^h=0$ and $S^{h-1}\neq 0$. Thus, $$A^g\simeq \mathbb{C}[S]/\langle S^h\rangle.$$

I know this isn't an answer to your question, but it is another interesting example of where the dual Coxeter number makes the numerology work. The conjecture is open for type $F_4$ and $E_6,E_7,E_8$, but settled in the other cases. Also, I recently asked a question on MathOverFlow related to this topic and Jim helped me out on it considerably.

Lastly, for a reference, see On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras paper by Shrawan Kumar, [J. Amer. Math. Soc. 21 (2008), no. 3, 797--808; MR2393427 (2009e:17013)].

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My personal experience is that I came across the dual Coxeter number when I was studying the exceptional series of Lie algebras. I would need to look up the details and references but it was the dual Coxeter number that made the numerology work.

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It would be interesting to see this in another context. On the other hand, the Coxeter number of the dual root system is the same as the original one. This is why I found the terminology initially confusing some years ago. (A weaker connection with the dual root system comes from the fact that the dual of the highest root is the highest short root in the dual system.) –  Jim Humphreys May 22 '10 at 20:25
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Perhaps the last remark may be incorrect? Jim's list above claims, for example, that the Coxeter number for B_n is 2n but the dual Coxeter number for C_n is n+1. –  Harold Williams May 23 '10 at 0:44
    
To follow up Harold's comment: anything that depends just on the Weyl group cannot be different for B_n, C_n, right? –  Theo Johnson-Freyd May 23 '10 at 3:20
    
Yes, Bruce corrected his earlier comment. The dual Coxeter number certainly doesn't depend just on the Coxeter group, which helped motivate me to question the terminology. (I realize that if Victor Kac introduced this number, it was difficult to find a label for it. But I do think that Kac number would be more logical in spite of the existing usage in published papers.) –  Jim Humphreys May 23 '10 at 22:03

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