(joint with stefan patrikis.) in this talk, we discuss a strong form of independence of $\ell$ for canonical $\ell$-adic local systems on shimura varieties, and sketch a proof of this for shimura varieties arising from adjoint groups whose simple factors have real rank $\geq 2$. notably, this includes all adjoint shimura varieties which are not of abelian type. the key tools used are the existence of companions for $\ell$-adic local systems and the superrigidity theorem of margulis for lattices in lie groups of real rank $\geq 2$.

the independence of $\ell$ is motivated by a conjectural description of shimura varieties as moduli spaces of motives. for certain shimura varieties that arise as a moduli space of abelian varieties, the strong independence of $\ell$ is proven (at the level of galois representations) by recent work of kisin and zhou, refining the independence of $\ell$ on the tate module given by deligne's work on the weil conjectures.