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 | References | Similar Articles | Additional Information

Let be a compact, semi-simple Lie group and a maximal rank reductive subgroup. The irreducible representations of can be constructed as spaces of harmonic spinors with respect to a Dirac operator on the homogeneous space twisted by bundles associated to the irreducible, possibly projective, representations of . 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.

**1.**A. Alekseev and E. Meinrenken,*The non-commutative Weil algebra*, Invent. Math,**139**(2000), 135-172. CMP**2000:06****2.**R. Bott,*The index theorem for homogeneous differential operators*, in `Differential and Combinatorial Topology', S. S. Cairns (Ed.), Princeton University Press, (1965), 167-186. MR**31:6246****3.**B. Gross, B. Kostant, P. Ramond, S. Sternberg,*The Weyl character formula, the half-spin representations, and equal rank subgroups*, Proc. Natl. Acad. Sci. USA,**95**(1998), 8441-8442. MR**99f:17007****4.**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. CMP**2000:05****5.**S. Slebarski,*The Dirac operator on homogeneous spaces and representations of reductive Lie groups II*, Amer. J. Math.,**109**(1987), 499-520. MR**88g:22015****6.**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:
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