Residual irreducibility of compatible systems

We show that if $\{ρ_{\ell}\}$ is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation $\overlineρ_{\ell}$ is absolutely irreducible for $\ell$ in a density 1 set of primes. The key technical result is the following theorem: the image of $ρ_{\ell}$ is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as $\ell$ varies). This result combines a theorem of Larsen on the semi-simple part of the image with an analogous result for the central torus that was recently proved by Barnet-Lamb, Gee, Geraghty, and Taylor, and for which we give a new proof.

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