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The Franke filtration is a finite filtration of certain spaces of automorphic forms on the adèlic points of a reductive linear algebraic group defined over a number field whose quotients can be described in terms of parabolically induced representations. Decomposing the space of automorphic forms according to their cuspidal support, the Franke filtration can be made more explicit. This paper describes explicitly the Franke filtration of the spaces of automorphic forms supported in a maximal proper parabolic subgroup, that is, in a cuspidal automorphic representation of its Levi factor. Such explicit description is important for applications to computation of automorphic cohomology, and thus the cohomology of congruence subgroups. As examples, the general linear group and split symplectic and special orthogonal groups are treated

The Franke filtration of the spaces of automorphic forms supported in a maximal proper parabolic subgroup