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We show thatL-packets of toral supercuspidal representations arising from unramified maximal tori ofp-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 ofL-packets of these representations. In parallel to the real case, we characterize the members of Kaletha’s toralL-packets by their characters on sufficiently regular elements of elliptic maximal tori.
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Formal degree of regular supercuspidals
Supercuspidal representations are usually infinite-dimensional, so the size of such a representation cannot be measured by its dimension; the formal degree is a better alternative. Hiraga, Ichino, and Ikeda conjectured a formula for the formal degree of a supercuspidal in terms of its L-parameter only. Our first main result is to compute the formal degrees of the supercuspidal representations constructed by Yu. Our second result, using the first, is to verify that Kaletha’s regular supercuspidal L-packets satisfy the conjecture.
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- Award ID(s):
- 1840234
- PAR ID:
- 10531665
- Publisher / Repository:
- European Math Society
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- Volume:
- 26
- Issue:
- 10
- ISSN:
- 1435-9855
- Page Range / eLocation ID:
- 3685 to 3737
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4171/jems/1412
