Freeness of spherical Hecke modules of unramified $U(2,1)$ in characteristic $p$

TL;DR

This paper proves that certain induced representations of the unramified unitary group U(2,1) over a local field are free modules of infinite rank over their associated spherical Hecke algebra in characteristic p.

Contribution

It establishes the freeness of spherical Hecke modules for unramified U(2,1) in characteristic p, a new result in the representation theory of p-adic groups.

Findings

Induced representations are free over the Hecke algebra.

Freeness holds for representations of infinite rank.

Results apply to unramified U(2,1) in characteristic p.

Abstract

Let $F$ be a non-archimedean local field of odd residue characteristic $p$. Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ in three variables, and $K$ be a maximal compact open subgroup of $G$. For an irreducible smooth representation $\sigma$ of $K$ over $\overline{\mathbf{F}}_p$, we prove that the compactly induced representation $\text{ind}^G _K \sigma$ is free of infinite rank over the spherical Hecke algebra $\mathcal{H}(K, \sigma)$.