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We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups.
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This content will become publicly available on April 1, 2026
Singularities of local models
Abstract We construct local models of Shimura varieties and investigate their singularities, with special emphasis on wildly ramified cases. More precisely, with the exception of odd unitary groups in residue characteristic 2 we construct local models, show reducedness of their special fiber, Cohen–Macaulayness and in equicharacteristic also (pseudo-)rationality. In mixed characteristic we conjecture their pseudo-rationality. This is based on the construction of parahoric group schemes over two dimensional bases for wildly ramified groups and an analysis of singularities of the attached Schubert varieties in positive characteristic using perfect geometry.
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- Award ID(s):
- 2200873
- PAR ID:
- 10635658
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Mathematische Annalen
- Volume:
- 391
- Issue:
- 4
- ISSN:
- 0025-5831
- Page Range / eLocation ID:
- 6205 to 6250
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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