Geometric -packets of Howe-unramified toral supercuspidal representations

  • Charlotte Chan

    Department of Mathematics,University of Michigan, Ann Arbor, MI 48109-1043, USA
  • Masao Oi

    Department of Mathematics (Hakubi Center), Kyoto University, Kyoto 606-8502, Japan
Geometric $L$-packets of Howe-unramified toral supercuspidal representations cover

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We show that -packets of toral supercuspidal representations arising from unramified maximal tori of -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 characters on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction of -packets of these representations. In parallel to the real case, we characterize the members of Kaletha’s toral -packets by their characters on sufficiently regular elements of elliptic maximal tori.

Cite this article

Charlotte Chan, Masao Oi, Geometric -packets of Howe-unramified toral supercuspidal representations. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1396