In this paper, for quasi-split classical groups over p-adic fields, we determine the L-packets consisting of simple supercuspidal representations and their corresponding L-parameters, under the assumption that p is not equal to 2. The key is an explicit computation of characters of simple supercuspidal representations and the endoscopic character relation, which is a characterization of the local Langlands correspondence for quasi-split classical groups.

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In this paper, we prove the coincidence of Kaletha’s recent construction of the local Langlands correspondence for regular supercuspidal representations with Harris–Taylor’s one in the case of general linear groups. The keys are Bushnell–Henniart’s essentially tame local Langlands correspondence and Tam’s result on Bushnell–Henniart’s rectifiers. By combining them, our problem is reduced to an elementary root-theoretic computation on the difference between Kaletha’s and Tam’s $\chi $-data.

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We prove that regular supercuspidal representations of p-adic groups are uniquely determined by their character values on very regular elements-a special class of regular semisimple elements on which character formulae are very simple-provided that this locus is sufficiently large. As a consequence, we resolve a question of Kaletha by giving a description of Kaletha's L-packets of regular supercuspidal representations which mirrors Langlands' construction for real groups following Harish-Chandra's characterization theorem for discrete series representations. Our techniques additionally characterize supercuspidal representations in general, giving p-adic analogues of results of Lusztig on reductive groups over finite fields. In particular, we establish an easy, non-cohomological characterization of unipotent supercuspidal representations when the residue field of the base field is sufficiently large.