Differential equations and Lie group representations

We discuss the role of differential equations in Lie group representation theory. We use Kashiwara’s pentagon as a reference frame for the real representation theory and then report on some work arising from its p-adic analogue by Emerton, Kisin, Patel, Huyghe, Schmidt, Strauch using Berthelot’s theory of arithmetic D-modules and Schneider–Stuhler theory of sheaves on buildings.