Restriction of $p$-modular representations of $U(2, 1)$ to a Borel subgroup

TL;DR

The paper analyzes how irreducible mod-$p$ representations of the unramified unitary group $G=U(2,1)(E/F)$ behave under restriction to the standard Borel subgroup $B$. Building on Paškūnas’ approach for $GL_2$, it develops a framework using $I_{1,K}$-invariants, the Hecke operator $T$, and the $S_K$, $S_-$ operators to study both principal-series and supersingular representations. For principal series with $oldsymbol{ u} eq oldsymbol{ ext{det}}$-type characters, and in the trivial case, it establishes explicit isomorphisms between $G$- and $B$-Hom spaces, describing how $St$ interacts in the restriction. The supersingular case yields stronger results: $\pi|_B$ is irreducible and $\mathrm{Hom}_G(\pi,\pi')\cong\mathrm{Hom}_B(\pi,\pi')$ for any $\pi'$; in particular, the Jacquet module $\pi_N$ vanishes. These findings generalize aspects of Paškūnas’ program from $GL_2$ to a small-rank unitary group and illuminate the mod-$p$ representation theory going through a Borel, with potential arithmetic applications.

Abstract

Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ defined over a non-archimedean local field $F$ of odd residue characteristic $p$, and $B$ be the standard Borel subgroup of $G$. In this note, we study the problem of the restriction of irreducible smooth $\overline{\mathbf{F}}_p$-representations of $G$ to $B$, and we obtain various results which are analogous to that of Pa$\check{\text{s}}$k$\bar{\text{u}}$nas on $GL_2 (F)$ (\cite{Pas07}).

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Definition 3.1
• Proposition 3.2
• proof
• Lemma 3.3
• proof