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

Let $\mathbf{G}$ be an unramified quasi-split unitary group over a p-adic field of odd residual characteristic. The goal of this paper is to describe the supercuspidal representations within certain L-packets of $\mathbf{G}$, which are classified by Arthur and Mok using the theory of endoscopy. The description is given in terms of the cuspidal types constructed by Bushnell-Kutzko and Stevens. As a starting example, we require the parameters of our packets to satisfy certain regularity conditions, such that these packets consist of very cuspidal representations in the sense of Adler and Reeder. To achieve our goal, we first interpret the question as to study the reducibilities of some parabolically induced representations, using a theory of M{\oe}glin and Shahidi; we then apply a relation, given by Blondel, between these reducibilities and the structures of some Hecke algebras, where the latter can be computed using a Theorem of Lusztig. We can interpret our final result as explicitly describing the local Langlands correspondence for $\mathbf{G}$.