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A uniform structure on subgroups of \(\mathrm{GL}_n(\mathbb F_q)\) and its application to a conditional construction of Artin representations of \(\mathrm{GL}_n\). (English) Zbl 1425.11094

Summary: Continuing our investigation in [the authors, Contemp. Math. 664, 225–260 (2016; Zbl 1418.11081)], where we associated an Artin representation to a vector-valued real analytic Siegel cusp form of weight (2,1) under reasonable assumptions, we associate an Artin representation of \(\mathrm{GL}_n\) to a cuspidal representation of \(\mathrm{GL}_n(\mathbb A_{\mathbb Q})\) with similar assumptions. A main innovation in this paper is to obtain a uniform structure of subgroups in \(\mathrm{GL}_{n}(\mathbb F_q)\), which enables us to avoid complicated case by case analysis in [loc. cit.]. We also supplement [loc. cit] by showing that we can associate non-holomorphic Siegel modular forms of weight (2,1) to Maass forms for \(\mathrm{GL}_2(\mathbb A_{\mathbb Q})\) and to cuspidal representations of \(\mathrm{GL}_2(\mathbb A_K)\) where \(K\) is an imaginary quadratic field.

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11R39 Langlands-Weil conjectures, nonabelian class field theory

Citations:

Zbl 1418.11081
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References:

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