Epipelagic L-packets and rectifying characters

We provide an explicit construction of the local Langlands correspondence for general tamely-ramified reductive p-adic groups and a class of wildly ramified Langlands parameters. Furthermore, we verify that our construction satisfies the expected properties of such a correspondence. More precisely, we show that each L-packet we construct admits a parameterization in terms of the Langlands dual group, contains a unique generic element for a fixed Whittaker datum, satisfies the formal degree conjecture, is compatible with twists by central and cocentral characters, provides a stable virtual character, and satisfies the expected endoscopic character identities. Moreover, we show that in the case of GL_n, our construction coincides with the one given by Bushnell and Henniart. Our results suggest a general approach to the construction of the local Langlands correspondence for tamely-ramified groups and regular supercuspidal parameters.