Restricting Toral Supercuspidal Representations to the Derived Group, and Applications

TL;DR

This paper analyzes how toral supercuspidal representations decompose when restricted to the derived subgroup of a reductive group, with applications to understanding the smooth dual of units in quaternion algebras over p-adic fields.

Contribution

It explicitly determines the decomposition and multiplicity of restrictions of toral supercuspidal representations, and applies these results to classify the smooth dual of quaternion algebra units.

Findings

Restriction has multiplicity one.

Decomposition described in terms of parametrizing data.

Smooth dual of $ ext{O}_D^ imes$ determined for $p eq 2$.

Abstract

We determine the decomposition of the restriction of a length-one toral supercuspidal representation of a connected reductive group to the algebraic derived subgroup, in terms of parametrizing data, and show this restriction has multiplicity one. As an application, we determine the smooth dual of the unit group of the integers $\mathcal{O}_D^\times$ of a quaternion algebra $D$ over a $p$-adic field $F$, for $p\neq 2$, as a consequence of determining the branching rules for the restriction of representations $D^\times \supset \mathcal{O}_D^\times \supset D^1$.