The point spectrum of the Dirac operator on noncompact symmetric spaces

Goette, S.; Semmelmann, U.

doi:10.1090/S0002-9939-01-06158-5

The point spectrum of the Dirac operator on noncompact symmetric spaces
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by S. Goette and U. Semmelmann

Proc. Amer. Math. Soc. 130 (2002), 915-923

DOI: https://doi.org/10.1090/S0002-9939-01-06158-5

Published electronically: October 1, 2001

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Abstract:

In this note, we consider the Dirac operator $D$ on a Riemannian symmetric space $M$ of noncompact type. Using representation theory, we show that $D$ has point spectrum iff the ${\hat A}$-genus of its compact dual does not vanish. In this case, if $M$ is irreducible, then $M=\mathrm {U}(p,q)/\mathrm {U}(p)\times \mathrm {U}(q)$ with $p+q$ odd, and $\operatorname {Spec}_{p}(D)=\{0\}$.

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