Garland proved vanishing theorems for the cohomology of a discrete subgroup with coefficients in a finite dimensional complex representation. As I understand it, Casselman reproved them using the representation theory of p-adic groups. Garland's proof resembles that of Matsushima's proof of his own vanishing theorem. Matsushima did computations on "curvature forms" -- I read that this method of proof is similar to a proof of Kodaira vanishing.

Are these analogues between differential-geometric methods and representation-theoretic methods well-understood? Is there a representation-theoretic analogue of Kodaira vanishing?