Let $G = \operatorname{GL}_n(F)$ for a $p$-adic field $F$, and let $G_D = \operatorname{SL}_n(F)$. I am wondering if there is a connection between irreducible, admissible representations of $G$ and of $G_D$.

If $(\pi,V)$ is one for $G$, then I am not sure whether the restriction of $\pi$ to $G_D$ remains irreducible and admissible.

On the other hand, if we begin with $(\pi,V)$ as an irreducible admissible representation of $G_D$, can we associate this to an irreducible admissible representation of $G$ by some sort of procedure involving induced representations? I would not expect

$$\operatorname{c-Ind}_{G_D}^G \pi = \operatorname{Ind}_{G_D}^G \pi$$

to hold in general, and $\operatorname{c-Ind}_{G_D}^G \pi$, though admissible, need not be irreducible. I would hope at best that there is some irreducible admissible subrepresentation $\sigma$ of $\operatorname{c-Ind}_{G_D}^G \pi$ for which $\sigma|_{G_D} \cong \pi$.