Restricting Supercuspidal Representations via a Restriction of Data

Let $F$ be a non-archimedean local field of residual characteristic $p$. Let $\mathbb{G}$ be a reductive group defined over $F$ which splits over a tamely ramified extension and set $G=\mathbb{G}(F)$. We assume that $p$ does not divide the order of the Weyl group of $\mathbb{G}$. Given a closed connected $F$-subgroup $\mathbb{H}$ that contains the derived subgroup of $\mathbb{G}$, we study the restriction to $H$ of an irreducible supercuspidal representation $\pi=\pi_G(\Psi)$ of $G$, where $\Psi$ is a $G$-datum as per the J.K. Yu Construction. We provide a full description of $\pi|_H$ into irreducible components, with multiplicity, via a restriction of data which constructs $H$-data from $\Psi$. Analogously, we define a restriction of Kim-Yu types to study the restriction of irreducible representations of $G$ which are not supercuspidal.