Restricting toral supercuspidal representations to the derived group, and applications (Q2341540)

The author determines 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 shows that this restriction has multiplicity one. As an application, she determines the smooth dual of the unit group of integers \( {\mathfrak O}_D^ {\times}\) 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 of \( D^ {\times} \supset {\mathfrak O}_D^ {\times} \supset D ^1\).