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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Harmonic spinors on homogeneous spaces
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by Gregory D. Landweber PDF
Represent. Theory 4 (2000), 466-473 Request permission


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.
    AM A. Alekseev and E. Meinrenken, The non-commutative Weil algebra, Invent. Math, 139 (2000), 135–172.
  • Raoul Bott, The index theorem for homogeneous differential operators, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 167–186. MR 0182022, DOI 10.1515/9781400874842-011
  • Benedict Gross, Bertram Kostant, Pierre Ramond, and Shlomo Sternberg, The Weyl character formula, the half-spin representations, and equal rank subgroups, Proc. Natl. Acad. Sci. USA 95 (1998), no. 15, 8441–8442. MR 1639139, DOI 10.1073/pnas.95.15.8441
  • K B. Kostant, A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J., 100 (1999), 447–501.
  • Stephen Slebarski, The Dirac operator on homogeneous spaces and representations of reductive Lie groups. II, Amer. J. Math. 109 (1987), no. 3, 499–520. MR 892596, DOI 10.2307/2374565
  • St S. Sternberg, Gainesville lectures on Kostant’s Dirac operator (1999), in preparation.
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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:
  • Received by editor(s): May 17, 2000
  • Received by editor(s) in revised form: June 20, 2000
  • Published electronically: September 15, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Represent. Theory 4 (2000), 466-473
  • MSC (2000): Primary 22E46; Secondary 17B20, 58J20
  • DOI:
  • MathSciNet review: 1780719