In this note, we consider the Dirac operator $D$ on a Riemannian symmetricspace $M$ of noncompact type. Using representation theory we show that $D$ haspoint spectrum iff the $\hat A$-genus of its compact dual does not vanish. Inthis case, if $M$ is irreducible then $M = U(p,q)/U(p) \times U(q)$ with $p+q$odd, and $Spec_p(D) = \{0\}$.
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机译:在本说明中,我们考虑非紧致型黎曼对称空间$ M $上的Dirac算子$ D $。使用表示理论,我们表明,如果紧凑对偶的$ \ hat A $属不消失,则$ D $具有点谱。在这种情况下,如果$ M $是不可约的,则$ M = U(p,q)/ U(p)\ U(q)$加上$ p + q $ odd,而$ Spec_p(D)= \ {0 \ } $。
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