Epipelagic representations and invariant theory

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 on Moy-Prasad quotients. The latter classification determines the nondegenerateK-types forG as well as the depths of irreducible representations ofG. The main step is an equivalence between Moy-Prasad theory and the theory of graded Lie algebras, whose GIT was analyzed by Vinberg and Levy. Our classification shows that stable functionals arise from Z-regular elliptic automorphisms of the absolute root system of G. These automorphisms also appear in the Langlands parameters of epipelagic representations, in accordance with the conjectural local Langlands correspondence.