Branching rules for irreducible supercuspidal representations of unramified $\mathrm{U}(1,1)$

TL;DR

This paper explicitly determines how irreducible supercuspidal representations of the unramified unitary group U(1,1) decompose when restricted to a maximal compact subgroup, confirming new conjectures and completing the classification.

Contribution

It provides the first complete description of branching rules for all irreducible supercuspidal representations of U(1,1), verifying two new conjectures in the process.

Findings

Explicit decomposition of supercuspidal representations upon restriction

Verification of two new conjectures in the literature

Completion of the classification of all irreducible smooth representations of G

Abstract

Let $G$ denote the unramified quasi-split unitary group $\mathbb{U}(1,1)(F)$ over a $p$-adic field $F$ with residual characteristic $p \neq 2$. In this article, we determine the branching rules for all irreducible supercuspidal representations of $G$, that is, we explicitly describe their decomposition upon restriction to a fixed maximal compact subgroup $\mathcal{K}$ in terms of irreducible representations of $\mathcal{K}$. We also present two applications of these decompositions, which verify two new conjectures in the literature for $G$. Together with the results from a previous paper by the author, this article completes the description of the branching rules for all irreducible smooth representations of $G$.