Abstract: I will start with some introduction to Shimura varieties and their completed cohomology, and report on my joint work in progress with Lue Pan which shows that, in the rational p-adic completed cohomology of a general Shimura variety, "sufficiently regular" infinitesimal weights (whose meaning will be explained) can only show up in the middle degree.  I will give some examples and explain the main ingredients in our work, if time permits.