The point spectrum of the Dirac operator on noncompact symmetric spaces (Q2759045)

In this note, the authors study the point spectrum of the Dirac operator (on spinors) on a symmetric space of noncompact type. They prove that if the point spectrum is nonempty, it is \(\{ 0\}\), and the kernel of \(D\) is an irreducible, infinite dimensional \(G\)-module. Furthermore, the authors are able to show that if the kernel is nonempty, then it is isomorphic to a discrete series representation with a specific Harish-Chandra parameter. NEWLINENEWLINENEWLINENext, the authors show that the existence of point spectrum is equivalent to the nonvanishing of the \(\hat{A}\)-genus of the compact dual of the symmetric space. After additional investigation, the authors conclude that the point spectrum is nonempty if and only if each irreducible factor of \(M\) is isometric to \(U(p,q)/U(p)\times U(q)\), with \(p+q\) odd.