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Harmonic spinors on homogeneous spaces


Author: Gregory D. Landweber
Journal: Represent. Theory 4 (2000), 466-473
MSC (2000): Primary 22E46; Secondary 17B20, 58J20
DOI: https://doi.org/10.1090/S1088-4165-00-00102-3
Published electronically: September 15, 2000
MathSciNet review: 1780719
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Abstract:

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.


References [Enhancements On Off] (What's this?)

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Additional Information

Gregory D. Landweber
Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052
Address at time of publication: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720-5070
Email: gregl@msri.org

DOI: https://doi.org/10.1090/S1088-4165-00-00102-3
Received by editor(s): May 17, 2000
Received by editor(s) in revised form: June 20, 2000
Published electronically: September 15, 2000
Article copyright: © Copyright 2000 American Mathematical Society