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First order differential operators on a locally symmetric space. (English) Zbl 1027.58027

The authors consider the quotient \(\Gamma\setminus M\) of a symmetric space \(M=G/H\) such that \(M\) is a spin manifold and a discrete subgroup \(\Gamma\subset G\). They prove that any invariant elliptic operator of first order on \(\Gamma\setminus M\) is a twisted Dirac operator. Moreover, they give conditions for the equivariance of the spectral symmetry of such operators.

MSC:

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C35 Differential geometry of symmetric spaces
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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