In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a boun...

We introduce a new approach to the representation theory of reductive p-adic groups G, based on the Geometric Invariant Theory (GIT) of Moy-Prasad quotients. Stable functionals on these quotients are used to give a new construction of supercuspidal representations of G having small positive depth, called epipelagic. With some restrictions on p, we classify the stable and semistable functionals ...

We consider the category of smooth W (k)[GLn(F )]-modules, where F is a p-adic field and k is an algebraically closed field of characteristic ` different from p. We describe a factorization of this category into blocks, and show that the center of each such block is a reduced, `-torsion free, finite type W (k)-algebra. Moreover, the k-points of the center of a such a block are in bijection with...

We analyze the geometry of the tower of LubinTate deformation spaces, which parametrize deformations of a onedimensional formal module of height h together with level structure. According to the conjecture of Deligne-Carayol, these spaces realize the local Langlands correspondence in their l-adic cohomology. This conjecture is now a theorem, but currently there is no purely local proof. Working...