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 of the dual root system. 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.

`$h^\vee$`

together with a brief reminder of what the symbol means. $\endgroup$