The number $k+h^{∨}$ appears at many places in the representation theory of affine Lie algebras (and probably elsewhere). Here $h^{∨}$ is the dual Coxeter number of the root system, and $k$ is the level of the representation. A few examples which I know: the Verlinde formula for the dimension of conformal blocks, modular transformation of characters, the Sugawara vector, Chern-Simons theory, this question, and numerous places in Kac's book.

I have found some explanation for the dual Coxeter number, but nothing serious for this.

The arguments I have seen so far usually sum over the roots/weights/generators/etc., and then multiply/divide with this number, but it is not clear for me why. It would be nice to see what is the geometric meaning of this, if there is any. Can anyone tell some motivation why does this number appear so frequently? Sorry for the "metaquestion".