Branching rules for irreducible supercuspidal representations of unramified $\mathrm{U}(1,1)$
Ekta Tiwari
TL;DR
This work determines the branching rules of all irreducible smooth representations of the unramified quasi-split unitary group $G=\mathrm{U}(1,1)(F)$ upon restriction to a fixed maximal compact subgroup $\mathcal{K}$. Building on the depth-zero and positive-depth frameworks, it uses Mackey theory to obtain multiplicity-free decompositions into irreducible Mackey components, showing they align with explicit $\mathcal{K}$-representations built via ET251 and Yu-type constructions. The paper proves irreducibility of each Mackey component and provides explicit degrees and depths, yielding a complete description for both depth-zero and positive-depth supercuspidal representations, with two key applications validating conjectures and establishing a local-character-expansion–style result for $\mathcal{K}_{2r+}$. These results extend the understanding of branching rules to higher-rank unitary groups and connect higher-depth phenomena to fixed depth-zero data, clarifying the structure of smooth representations of $G$. The findings have potential implications for the local Langlands program and for analyzing restriction behaviour in related $p$-adic groups.
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 $G$.