A Uniform Structure on Subgroups of GLn(Fq) and its Application to A Conditional Construction of Artin Representations of GLn

Continuing our investigation in [19], 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 GL_n to a cuspidal representation of GL_n(A_Q) with similar assumptions. A main innovation in this paper is to obtain a uniform structure of subgroups in GL_n(F_q ), which enables us to avoid complicated case by case analysis in [19]. We also supplement [19] by showing that we can associate non-holomorphic Siegel modular forms of weight (2, 1) to Maass forms for GL₂(A_Q) and to cuspidal representations of GL₂(A_k ) where K is an imaginary quadratic field.