Given a group $G$ of automorphisms of a matroid $M$, we describe the representations of $G$ on the homology of the independence complex of the dual matroid $M^*$. These representations are related with the homology of the lattice of flats of $M$, and (when $M$ is realizable) with the top cohomology of a hyperplane arrangement. Finally we analyze in detail the case of the complete graph, which has applications to algebraic geometry.