Recently, Kostant investigated a cubic Dirac operator associated to a pair (g,h), where g is a semi-simple Lie algebra and h is a reductive subalgebra of maximal rank in g. The kernel of this Dirac operator exhibits a generalized form of the Weyl character formula, and it associates to each irreducible g-module a set of h-modules known as an Euler number multiplet.

This talk reformulates Kostant’s work in the Kac-Moody setting, replacing g and h with the extended loop algebras Lg and Lh. Here, the Dirac operator lives in a loop group analogue of the non-commutative Weil algebra introduced by Alekseev and Meinrenken. The Dirac operator associated to Lg and the Weil algebra quantization map behave differently for Kac-Moody algebras than in the finite dimensional case due to normal ordering. However, in the relative case for a pair (Lg, Lh), Kostant’s results have immediate analogues which give a generalization of the Weyl- Kac character formula. Geometrically, Kostant’s cubic Dirac operator corresponds to a formal version of the Dirac operator on the homogeneous space G/H (although NOT for the Levi-Civita connection). Using the Kac-Moody Dirac operator as a model for the geometric Dirac operator on the homogeneous loop space L(G/H) = (LG)/(LH), we show that its index in the infinite level limit approximates the elliptic genus.