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

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Abstract

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].

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Acknowledgement

We would like to thank the anonymous referee for a careful reading of the manuscript, and in particular, for providing the intuition behind the proof of Lemma 4.2.

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Authors and Affiliations

  1. Juraj Dobrila University of Pula, Zagrebačka 30, HR-52100, Pula, Croatia

    Neven Grbac

  2. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090, Vienna, Austria

    Harald Grobner

Authors
  1. Neven Grbac
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  2. Harald Grobner
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Correspondence to Neven Grbac.

Additional information

The first-named author is supported by the Croatian Science Foundation under the projects HRZZ-IP-2018-01-3628, HRZZ-IP-2019-04-4216 and HRZZ-IP-2022-10-4615. The second-named author is supported by the START-project Y966 and the Stand-Alone-Research project P32333, both sponsored by the Austrian Science Fund (FWF).

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Grbac, N., Grobner, H. Some unexpected phenomena in the Franke filtration of the space of automorphic forms of the general linear group. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2625-x

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  • DOI: https://doi.org/10.1007/s11856-024-2625-x

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