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Let X be a symmetric space of noncompact type whose isometry group is either SU(n, 1) or Spin(2n, 1). Then the Dirac operator D is defined on L2-sections of certain homogeneous vector bundles over X . Using representation theory we obtain explicitly the eigenvalues of D and describe the eigenspaces in terms of the discrete series.

Eigenvalues and eigenspaces for the twisted Dirac operator over ${\rm SU}(N,1)$ and ${\rm Spin}(2N,1)$