Endoscopic liftings and parameters of epipelagic representations for classical groups
Abstract
Let G be a p-adic classical group (orthogonal, symplectic, or unitary) and π be an epipelagic representation of G in the sense of Reeder-Yu. Using Mœglin's theory of extended cuspidal supports and Bushnell-Kutzko's theory of covering types, we determine explicitly the endoscopic lift of π to the general linear group, whose Langlands dual expresses the dual group of G as a complex matrix group, in terms of the inducing type of π that extends the character of the first Moy-Prasad filtration subgroup defined by a stable functional. We interpret the inducing type of π via Stevens' construction of supercuspidal representations by skew semisimple strata and introduce what we will call epipelagic strata, requiring only that the residual characteristic p be odd. As an application, we reprove M. Oi's results on the endoscopic lifts of simple supercuspidal representations, in the sense of Gross-Reeder, of quasi-split classical groups. Finally, on the Galois side, we show that the epipelagic Langlands parameters of G constructed by Reeder-Yu can be recovered via the self-duality of Bushnell-Henniart's admissible triples and endoscopic embeddings of L-groups. We also establish some related results concerning epipelagic parameters that were previously proved under restrictions on p, including a parity result on rectifying characters and an adjoint Swan conductor result related to the Hiraga-Ichino-Ikeda conjecture.
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BibTeX
@misc{tam2026endoscopic,
title = {{Endoscopic liftings and parameters of epipelagic representations for classical groups}},
author = {Geo Kam-Fai Tam},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28189},
eprint = {2609.28189},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28189},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Endoscopic liftings and parameters of epipelagic representations for classical groups
AU - Geo Kam-Fai Tam
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28189
UR - https://arxiv.org/abs/2609.28189
AB - Let G be a p-adic classical group (orthogonal, symplectic, or unitary) and π be an epipelagic representation of G in the sense of Reeder-Yu. Using Mœglin's theory of extended cuspidal supports and Bushnell-Kutzko's theory of covering types, we determine explicitly the endoscopic lift of π to the general linear group, whose Langlands dual expresses the dual group of G as a complex matrix group, in terms of the inducing type of π that extends the character of the first Moy-Prasad filtration subgroup defined by a stable functional. We interpret the inducing type of π via Stevens' construction of supercuspidal representations by skew semisimple strata and introduce what we will call epipelagic strata, requiring only that the residual characteristic p be odd. As an application, we reprove M. Oi's results on the endoscopic lifts of simple supercuspidal representations, in the sense of Gross-Reeder, of quasi-split classical groups. Finally, on the Galois side, we show that the epipelagic Langlands parameters of G constructed by Reeder-Yu can be recovered via the self-duality of Bushnell-Henniart's admissible triples and endoscopic embeddings of L-groups. We also establish some related results concerning epipelagic parameters that were previously proved under restrictions on p, including a parity result on rectifying characters and an adjoint Swan conductor result related to the Hiraga-Ichino-Ikeda conjecture.
ER -