TITLE: Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$
QUESTION [8 upvotes]: 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 $\sigma|_{G_D} \cong \pi$.

REPLY [5 votes]: The answer to your questions (with proofs) may be found in 
C.J. Bushnell, P.C. Kutzko,  The admissible dual of SL(N). I
Annales scientifiques de l'École Normale Supérieure, Série 4 : Volume 26 (1993) no. 2 , p. 261--280 
First if $\pi$ is an irreducible smooth (complex) representation of $G$, then $\pi_{\mid G_{D}}$ is a finite direct sum of irreducible smooth representations of $G_D$. This is  loc. cit. Prop. (1.7)(i), page 267. 
 Second, any irreducible smooth representation of $G_D$ is a component of $\pi_{\mid G_{D}}'$ for some irreducible smooth representation $\pi'$ of $G$. This is   loc. cit. Prop (1.17)(i), page 270. 
Moreover the restriction functor ${\rm Rep}(G)\longrightarrow {\rm Rep}(G_D)$ induces a surjective map on the sets of supercuspidal irreducible representations (resp. discrete series irreducible representations). This is  loc. cit. Prop. 1.20, page 271.