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

TL;DR

This paper investigates the structure of supersingular representations of the unramified unitary group U(2, 1) over a non-archimedean local field, revealing that their pro-p-Iwahori invariants are not simple modules over the associated Hecke algebra.

Contribution

It demonstrates that for supersingular representations containing the Steinberg weight, the pro-p-Iwahori invariants are not simple modules, providing new insights into their algebraic structure.

Findings

Pro-p-Iwahori invariants are not simple modules.

Focus on supersingular representations with Steinberg weight.

Results contribute to understanding mod p representation theory of p-adic groups.

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.