Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra

If $\mathfrak{g}$ is a complex simple Lie algebra, and $k$ does not exceed the dual Coxeter number of $\mathfrak{g}$, then the k${}^{th}$ coefficient of the $dim \frak g$ power of the Euler product may be given by the dimension of a subspace of ${\wedge }^k\mathfrak{g}$ defined by all abelian subalgebras of $\mathfrak{g}$ of dimension $k$. This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Peterson’s $2^{rank}$ theorem on the number of abelian ideals in a Borel subalgebra of $\mathfrak{g}$, an element of type $\rho$ and my heat kernel formulation of Macdonald’s $\eta$-function theorem, a set $D_{alcove}$ of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null $m$-core when $\mathfrak{g}=LieSl(m,\mathbb{C})$), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac-Moody Lie algebra.