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Published in:
Group Gradings on Lie Algebras and Applications to Geometry: II Read chapter Read first chapter

2014 | OriginalPaper | Chapter

Harmonic Spinors on Reductive Homogeneous Spaces

Authors : Salah Mehdi, Roger Zierau

Published in: Developments and Retrospectives in Lie Theory

Publisher: Springer International Publishing

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Abstract

An integral intertwining operator is given from certain principal series representations into spaces of harmonic spinors for Kostant’s cubic Dirac operator. This provides an integral representation for harmonic spinors on a large family of reductive homogeneous spaces.

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previous chapter Shintani Functions, Real Spherical Manifolds, and Symmetry Breaking Operators
next chapter Twisted Harish–Chandra Sheaves and Whittaker Modules: The Nondegenerate Case
Appendix
Available only for authorised users
Footnotes
1
The factor of 2 in this formula does not appear in [6]. This is because we are taking \(xy + yx =\langle x\,,y\rangle\) in the definition of the Clifford algebra, while \(xy + yx = 2\langle x\,,y\rangle\) is used in [6].
 
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S. Helgason, Groups and Geometric Analysis. Academic Press, 1984.
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J.-S. Huang, Y.-F. Kang, and P. Pandžić, Dirac cohomology of some Harish–Chandra modules, Trans. Groups 14 (2009), 163–173.
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B. Kostant, A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J. 100 (1999), no. 3, 447–501.
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S. Mehdi, Parthasarathy Dirac operators and discrete series representations, Connected at infinity II: A Selection of Mathematics by Indians. Edited by R. Bhatia, C. S. Rajan and A. I. Singh. Text and Readings in Mathematics 67. Hindustan Book Agency, New Delhi, 2013.
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S. Mehdi and R. Zierau, Harmonic spinors on semisimple symmetric spaces. J. Funct. Anal. 198 (2003), 536–557.
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S. Mehdi and R. Zierau, Principal series representations and harmonic spinors. Adv. Math. 1999 (2006), 1–28.
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S. Mehdi and R. Zierau, The Dirac cohomology of a finite dimensional representation. To appear in the Proceedings of the AMS. Proc. Amer. Math. Soc. 142 (2014), 1507–1512.
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R. Parthasarathy, Dirac operator and the discrete series. Ann. of Math. 96 (1972), 1–30.
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D. A. Vogan, Unitarizability of certain series of representations. Ann. of Math. 120 (1984), 141–187.
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D. A. Vogan and G. J. Zuckerman, Unitary representations with non-zero cohomology. Comp. Math., 53 (1984), 51–90.
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Metadata
Title
Harmonic Spinors on Reductive Homogeneous Spaces
Authors
Salah Mehdi
Roger Zierau
Copyright Year
2014
Book
Developments and Retrospectives in Lie Theory

Print ISBN: 978-3-319-09933-0

Electronic ISBN: 978-3-319-09934-7

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-09934-7_6

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