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
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.