2000/05/05 by Gregory D. Landweber
Mathematics · #math.DG #math.RT #msc:22E46 #msc:17B20 #msc:58J20
published as Represent. Theory 4 (2000) 466-473 · 7 pages
arxiv created 2000/05/05 · arxiv updated 2009/11/30
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.