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

Abstract

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 $ρ$ and my heat kernel formulation of Macdonald's $η$-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.