Geometric L-packets of Howe-unramified toral supercuspidal representations

We show that $L$-packets of toral supercuspidal representations arising from unramified maximal tori of $p$-adic groups are realized by Deligne--Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra's characterization of real discrete series representations by their character on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction of $L$-packets of these representations. In parallel to the real case, we characterize the members of Kaletha's toral $L$-packets by their character on sufficiently regular elements of elliptic maximal tori.