UC Santa Cruz

In this thesis, we investigate the category of equivariant sheaves on a locally finite tree. We view this setting as an interesting junction between the representation theory of p-adic groups, and the geometric methods coming from quiver theory. Inspired by methods from quiver theory, we show that the category of (smooth) equivariant sheaves on a tree X is equivalent to a certain hereditary module category HX-mod. We construct HX explicitly as the endomorphism ring of a projective generator, which emulates the construction of path algebras for quivers, and Hecke algebras for p-adic groups. In this light, we investigate the implications of Auslander-Reiten theory in this context. In particular we provide a thorough investigation of a large class of sheaves known as the pre-injective component, and give recursive formulae for their cohomology.