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

If g is a complex simple Lie algebra, and k does not exceed the dual Coxeter number of g, then the k coefficient of the dim g power of the Euler product may be given by the dimension of a subspace of ∧g defined by all abelian subalgebras of 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 theorem on the number of abelian ideals in a Borel subalgebra of g, an element of type ρ and my heat kernel formulation of Macdonald’s η-function theorem, a set Dalcove of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null m-core when g = Lie Sl(m,C)), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac-Moody Lie algebra.