Endoscopy and the cohomology of $GL(n)$

Document Type: Special Issue of BIMS in Honor of Professor Freydoon Shahidi

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Abstract

Let $G = {\rm Res}_{F/\mathbb{Q}}(GL_n)$ where $F$ is a number field‎. ‎Let $S^G_{K_f}$ denote an ad\`elic locally symmetric space for some level structure $K_f.$ Let ${\mathcal M}_{\mu,{\mathbb C}}$ be an algebraic irreducible representation of $G({\mathbb R})$ and we let $\widetilde{\mathcal{M}}_{\mu,{\mathbb C}}$ denote the associated sheaf on $S^G_{K_f}.$ The aim of this paper is to classify the data $(F,n,\mu)$ for which cuspidal cohomology of $G$ with $\mu$-coefficients‎, ‎denoted $H^{\bullet}_{\rm cusp}(S^G_{K_f}‎, ‎\widetilde{\mathcal{M}}_{\mu,{\mathbb C}})$‎, ‎is nonzero for some $K_f.$ We prove nonvanishing of cuspidal cohomology when $F$ is a totally real field or a totally imaginary quadratic extension of a totally real field‎, ‎and also for a general number field but when $\mu$ is a parallel weight‎.

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