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

Kozioł, Karol; Xu, Peng

doi:10.1090/S1088-4165-2015-00462-5

Hecke modules and supersingular representations of U(2,1)
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by Karol Kozioł and Peng Xu

Represent. Theory 19 (2015), 56-93

DOI: https://doi.org/10.1090/S1088-4165-2015-00462-5

Published electronically: March 16, 2015

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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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