Compatibility of Canonical Local Systems and F-isocrystals on Shimura Varieties

I will discuss compatibility of the canonical G-valued l-adic local systems on Shimura varieties of non-abelian type G. In previous work with Klevdal, we settled this for the projections of the local systems to the adjoint group of G, and this talk will focus on lifting the compatibility to G using work of Bakker-Shankar-Tsimerman. I will also discuss the corresponding F-isocrystal compatibility statements, resulting from joint work with Huryn, Kedlaya, and Klevdal.