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$$L^ 2$$-index and the Selberg trace formula. (English) Zbl 0537.58039
A method is developed for computing the $$L^ 2$$-index of a locally symmetric elliptic differential operator $$D_{\Gamma}$$, acting on a locally symmetric manifold $$M_{\Gamma}=\Gamma \backslash G/K$$ with G semisimple of real-rank 1 and $$\Gamma$$ of finite co-volume, based on applying the Selberg trace formula to the difference of the two heat kernels associated to $$D_{\Gamma}$$. The applications include an extension of the Osborne-Warner multiplicity formule to certain non- integrable discrete series and showing the existence, in some cases, of non-invariant $$L^ 2$$-cohomology classes in the middle dimension.
Reviewer: V.Deundjak

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 22E46 Semisimple Lie groups and their representations
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