On the pro-$p$-Iwahori invariants of supersingular representations of unramified $U(2,1)$

Abstract

Let $G$ be the unramified unitary group $U(2,1)(E/F)$ over a non-archimedean local field $F$ of odd residue characteristic $p$. In this paper, for any supersingular representation of $G$ that contains the Steinberg weight, we prove its pro-$p$-Iwahori invariants, as a right module over the pro-$p$-Iwahori--Hecke algebra of $G$, is \emph{not} simple.

version 2: 1) the admissibility assumption in version 1 is removed, by a suggestion from Karol Kozioł. 2) fix an inaccuracy in the appendix