  If $\frak g$ is a complex simple Lie algebra, and $k$ does not exceed the
dual Coxeter number of $\frak g$, 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 $\wedge^k\frak g$ defined by all abelian subalgebras of $\frak g$ 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
$\frak g$, 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 $\frak g= Lie Sl(m,\Bbb C)$), and the homology
and cohomology of the nil radical of the standard maximal parabolic subalgebra
of the affine Kac-Moody Lie algebra.
