Characterization of supercuspidal representations and very regular elements

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.