Depth zero supercuspidal representations of classical groups into L-packets: The typically almost symmetric case

We classify what we call “typically almost symmetric” depth zero supercuspidal representations of a classical group over a local field of odd residual characteristic into L-packets. Our main results resolve an ambiguity in the paper of Lust-Stevens [24] in this case, where they could only classify these representations in two or four, if not one, L-packets. By assuming the expected numbers of supercuspidal representations in the L-packets, we employ only simple properties of the representations to prove the main results. In particular, we do not require any deep calculations of character values. With the same method, we also compute the parity of a (conjugate-)self-dual depth zero supercuspidal representation of a general linear group.