Harmonic spinors on homogeneous spaces

Let $G$ be a compact, semi-simple Lie group and $H$ a maximal rank reductive subgroup. The irreducible representations of $G$ can be constructed as spaces of harmonic spinors with respect to a Dirac operator on the homogeneous space $G/H$ twisted by bundles associated to the irreducible, possibly projective, representations of $H$. Here, we give a quick proof of this result, computing the index and kernel of this twisted Dirac operator using a homogeneous version of the Weyl character formula noted by Gross, Kostant, Ramond, and Sternberg, as well as recent work of Kostant regarding an algebraic version of this Dirac operator.