In this paper we verify the local Langlands correspondence for pure inner forms of unramified p-adic groups and tame Langlands parameters in "general position". For each such parameter, we explicitly construct, in a natural way, a finite set ("L-packet") of depth-zero supercuspidal representations of the appropriate p-adic group, and we verify some expected properties of this L-packet. In particular, we prove, with some conditions on the base field, that the appropriate sum of characters of the representations in our L-packet is stable; no proper subset of our L-packets can form a stable combination. Our L-packets are also consistent with the conjectures of B. Gross and D. Prasad on restriction from SO 2n+1 to SO 2n [24].These L-packets are, in general, quite large. For example, Sp 2n has an L-packet containing 2 n representations, of which exactly two are generic. In fact, on a quasi-split form, each L-packet contains exactly one generic representation for every rational orbit of hyperspecial vertices in the reduced BruhatTits building. When the group has connected center, every depth-zero generic supercuspidal representation appears in one of these L-packets.We emphasize that there is nothing new about the representations we construct. They are induced from Deligne-Lusztig representations on subgroups of finite index in maximal compact mod-center subgroups, see [42], [44], [61]. The point here is to assemble these representations into L-packets in a natural and explicit way and to verify that these L-packets have the required properties.

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 nondegenerate K-types for G as well as the depths of irreducible representations of G. 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.

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