Type A images of Galois representations and maximality

Abstract

Given a compatible subsystem $$\{\rho _\ell \}_\ell $$ of n-dimensional $$\ell $$ -adic Galois representations arising from etale cohomology of any complete, non-singular variety over a number field K, we define $$\Gamma _\ell := \rho _\ell ({\text {Gal}}(\bar{K}/K))$$ and let $$\mathbf {G}_\ell $$ denote the Zariski closure of $$\Gamma _\ell $$ in $$\mathrm {GL}_n$$ . If $$\mathbf {G}_\ell $$ is of Type A in the sense that all simple composition factors are of type A in the Cartan-Killing classification, then $$\Gamma _\ell $$ is, in a suitable sense, maximal in $$\mathbf {G}_\ell $$ for all $$\ell \gg 0$$ . As a corollary, if $$\rho _\ell $$ is semisimple and $$\ell $$ is sufficiently large, then $$\mathbf {G}_\ell $$ is unramified.