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 $\sigma|\_{G\_D} \cong \pi$.



Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$
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. 


