Skip to main content

Automorphic representations

Franke Filtration​

The Franke filtration is a descending filtration of spaces of automorphic forms on adélic reductive groups, defined by Jens Franke in [1], see also [2]. Its main advantage is that the quotients of the filtration can be described as parabolically induced representations. See [3], [4].

The app Franke Filtration allows one to compute the Franke filtration of the spaces of automorphic forms for general linear groups. The algorithm is described in detail in [5].

The code of the app is a TypeScript port of Python/SageMath code written by Lovro Greganić. It was ported by Petar Bakić with the assistance of Claude Code.

The work on this app is supported by the Croatian Science Foundation under the projects HRZZ-IP-2022-10-4615 and DOK-NPOO-2023-10-1542, and funded by the European Union NextGenerationEU under the Juraj Dobrila University of Pula institutional research projects IIP_UNIPU_010159 and IIP_UNIPU_010162.

Notation​

The input to the app is the cuspidal support given by the number of non-isomorphic unitary cuspidal automorphic representations of the general linear group, the corresponding sizes of the general linear groups, and the comma-separated lists of exponents (only rational exponents are allowed).

The output of the app are the inducing data for the parabolic induction of the summands in the quotients of the Franke filtration of the space of automorphic forms on the general linear group with the cuspidal support in the given input. The inducing data are presented in the form of a tensor product of factors ρj[a,b]\rho_j[a,b] that stands for the twisted residual or cuspidal representation J(ρj,b−a+1)∣det⁡∣a+b2J(\rho_j,b-a+1)|\det|^{\frac{a+b}{2}} in the notation of [5]. Beware that the symmetric algebras that should appear in every summand of every quotient of the filtration are omitted in the output, because they can be easily recovered from the form of the summands.

The output also contains an indicator in case of summands for which the invariants under certain intertwining operators should be taken. The indicator is of the form colim⁡[t1,…,tr]\operatorname{colim}_{[t_1,\dots,t_r]}, where tjt_j in the list indicates that tjt_j copies of the same factor in the tensor product can be permuted by the intertwining operators. Besides that, clicking on "show detailed output", the user can additionally see the list of partitions in Bernstein-Zelevinsky segments of the cuspidal support, as well as all the values of z‾\underline{z} and ι(z‾)\iota(\underline{z}), see [5].

Examples​

Below are some inputs (and references for these inputs) that are worth trying!

Example 1

Try to input the cuspidal support with 11 unitary cuspidal automorphic representation of the general linear group (of any size) and exponents 1,0,0,−11,0,0,-1. This is the example studied in Section 5.1 of [6].

Example 2

Try to input the cuspidal support with 11 unitary cuspidal automorphic representation of the general linear group (of any size) and exponents 3/2,1/2,1/2,−1/2,−1/2,−3/23/2,1/2,1/2,-1/2,-1/2,-3/2. This is the example studied in Section 6.1 of [6].

Example 3

Try to input the cuspidal support with 11 unitary cuspidal automorphic representation of the general linear group (of any size) and exponents forming a segment (of any length). This cuspidal support is the cuspidal support of a residual representation studied in Section 4 of [6].

Example 4

Try to compare the outputs obtained

  • for the input cuspidal support with 11 unitary cuspidal automorphic representation of the general linear group (of any size) and exponents 0,00,0,
  • for the input cuspidal support with 22 unitary cuspidal automorphic representations of the general linear groups (of any sizes) and exponent 00 for each of them.

Observe that in the former case the colimit appears because the two factors in the tensor product are isomorphic, and thus there is only a single line in the detailed output which contributes to the filtration via invariants. On the other hand, in the latter case the colimit does not appear because the two factors in the tensor product are non-isomorphic, and thus there are two lines in the detailed output but only one of them contributes to the filtration.

References​

[1] J. Franke, Harmonic analysis in weighted L2L_2-spaces, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 2, 181–279.
[2] J. Franke, J. Schwermer, A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups, Math. Ann. 311 (1998), no. 4, 765–790.
[3] N. Grbac, The Franke filtration of the spaces of automorphic forms supported in a maximal proper parabolic subgroup, Glas. Mat. Ser. III 47(67) (2012), no. 2, 351–372.
[4] N. Grbac, The Franke filtration of the spaces of automorphic forms on the symplectic group of rank two, Mem. Amer. Math. Soc. 313 (2025), no. 1592, vii+85 pp.
[5] N. Grbac, L. Greganić, An algorithm for explicit calculation of the Franke filtration for the general linear group, preprint.
[6] N. Grbac, H. Grobner, Some unexpected phenomena in the Franke filtration of the space of automorphic forms of the general linear group, Israel J. Math. 263 (2024), no. 1, 301–347.