Artin representations for $GL_n$

Abstract

Let $\pi$ be a cuspidal automorphic representation of $GL_n(\mathbb{A}_\mathbb{Q})$ which satisfies certain reasonable assumptions such as integrality of Hecke polynomials, the existence of mod $\ell$ Galois representations attached to $\pi$. Under Langlands functoriality of exterior $m$-th power $\wedge^m(\pi)$, $m=2,...,[\frac n2]$, we will construct a unique Artin representation associated to $\pi$. As a corollary, we obtain that such a cuspidal representation of $GL_n(\mathbb{A}_\mathbb{Q})$ satisfies the Ramanujan conjecture. We also revisit our previous work on Artin representations associated to non-holomorphic Siegel cusp forms of weight (2,1), and show that we can associate non-holomorphic Siegel modular forms of weight $(2,1)$ to Maass forms for $GL_2/\mathbb{Q}$ and cuspidal representations of $GL_2$ over imaginary quadratic fields.