Endoscopic classification of very cuspidal representations of quasi-split unramified unitary groups

We describe the supercuspidal representations within certain stable packets, classified by Arthur and Mok using the theory of endoscopy, of aquasi-split unramified unitary group over a p-adic field of odd residual characteristic. The description is given in terms of types constructed by Bushnell-Kutzko and Stevens. As a starting example, we require the parameters of our packets to satisfy some regularity conditions, such that these packets consist of very cuspidal representations in the sense of Adler and Reeder. Our main result is analogous to the essentially tame local Langlands correspondence of Bushnell-Henniart for a general linear group: we need to correct the types by twisting by certain characters, called amending characters in this paper, in order to classify these representations into stable packets.