The point spectrum of the Dirac operator on noncompact symmetric spaces

• DOI: 10.1090/s0002-9939-01-06158-5
• Publication Date: 2002-07-26T22:14:28Z
• Authors: S. Goette; U. Semmelmann
• Keywords: Symmetric space; Operator (biology); Representation; Dual space

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