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

Abstract

If $\mathfrak{g}$ is a complex simple Lie algebra, and k does not exceed the dual Coxeter number of $\mathfrak{g}$, then the absolute value of the k^th coefficient of the $\dim\mathfrak{g}$ 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 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 $\mathfrak{g}= \text{Lie}\,\mathit{Sl}(m,\mathbb{C})$), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac–Moody Lie algebra.