Restricting Toral Supercuspidal Representations to the Derived Group, and Applications

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 $D^\times \supset \mathcal{O}_D^\times \supset D^1$.