Given a Shimura variety, we can construct two kinds of automorphic local systems, i.e., local systems attached to algebraic representations of certain associated algebraic group.  The first is based on the classical complex analytic construction using double quotients, while the second is a p-adic analytic construction based on some recently developed p-adic analogue of the Riemann-Hilbert correspondence.  I will explain how to compare these two constructions even when the Shimura variety is not of abelian type, and mention some applications and recent developments. 

This is based on joint work with Hansheng Diao, Ruochuan Liu, and Xinwen Zhu.