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Hecke modules and supersingular representations of U(2,1)


Authors: Karol Kozioł and Peng Xu
Journal: Represent. Theory 19 (2015), 56-93
MSC (2010): Primary 22E50, 20C08, 11F70, 20C20
DOI: https://doi.org/10.1090/S1088-4165-2015-00462-5
Published electronically: March 16, 2015
MathSciNet review: 3321473
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Abstract: Let $ F$ be a nonarchimedean local field of odd residual characteristic $ p$. We classify finite-dimensional simple right modules for the pro-$ p$-Iwahori-Hecke algebra $ \mathcal {H}_C(G,I(1))$, where $ G$ is the unramified unitary group $ \textrm {U}(2,1)(E/F)$ in three variables. Using this description when $ C = \overline {\mathbb{F}}_p$, we define supersingular Hecke modules and show that the functor of $ I(1)$-invariants induces a bijection between irreducible nonsupersingular mod-$ p$ representations of $ G$ and nonsupersingular simple right $ \mathcal {H}_C(G,I(1))$-modules. We then use an argument of Paškūnas to construct supersingular representations of $ G$.


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Additional Information

Karol Kozioł
Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
Email: karol@math.toronto.edu

Peng Xu
Affiliation: Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, United Kingdom
Email: Peng.Xu@warwick.ac.uk

DOI: https://doi.org/10.1090/S1088-4165-2015-00462-5
Received by editor(s): September 29, 2014
Received by editor(s) in revised form: November 25, 2014
Published electronically: March 16, 2015
Additional Notes: The first author was supported by NSF Grant DMS-0739400.
The second author was supported by EPSRC Grant EP/H00534X/1.
Article copyright: © Copyright 2015 American Mathematical Society