, Volume 158, Issue 1, pp 181-226
Date: 09 Jun 2004

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

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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 kth 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 2rank 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.