Some unexpected phenomena in the Franke filtration of the space of automorphic forms of the general linear group

In his famous paper [11], J. Franke has defined a certain finite filtration of the space of automorphic forms of a general reductive group, which captures most of its internal representation theory. The purpose of this paper is to provide several concrete examples of yet unexpected phenomena, which occur in the Franke filtration for the general linear group. More precisely, we show that the degenerate Eisenstein series arising from the parabolic subgroups of the same rank are not necessarily contributing to the same quotient of the filtration, and that, even more, the Eisenstein series arising from the parabolic subgroups of higher relative rank may contribute to a deeper quotient of the filtration. These are the first structural counterexamples to an expectation, mentioned in [11].