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We study the $L^2$--cohomology of certain local systems on non-compactarithmetic ball quotients $X=\Gamma \backslash \B_n$, in particular vanishingand non--vanishing results. We also give generalizations to higher dimensionalball quotients and study the mixed Hodge structure on the sheaf cohomology of alocal system with the $L^2$-cohomology contributing to the lowest weight part.

Mixed Hodge complexes and L^2-cohomology for local systems on ball quotients