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

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 $\sigma|_{G_D} \cong \pi$.

• The claim that $GL_n(F)$ is generated by $SL_n(F)$ and the center $F^\times$ of $GL_n(F)$ seems wrong. In fact the quotient by the subgroup generated by those two is $F^\times / (F^\times)^n$. – Will Sawin Feb 10 '18 at 21:47
• Yes, you are right. – D_S Feb 10 '18 at 21:58
• As group schemes, $GL_n=SL_n . G_m$. – Grad student Feb 10 '18 at 22:02
• It may already be interesting to look at unramified principal series of $GL_2$ restricted to $SL_2$, where one can easily do the calculations directly. So, generically, irreducible u.r.p.s. stay irreducible, except at special parameters where they fall into two pieces. (This is quite parallel to the case of $GL_2$ and $SL_2$ over finite fields, too.) – paul garrett Feb 10 '18 at 22:23
• The representation $c-{\rm Ind}_{G_D}^G\, \pi$ is smooth but never admissible in the sense that $(c-{\rm Ind}_{G_D}^G\, \pi )^K$ is not finite dimensional for all compact open subgroups $K$. – Paul Broussous Feb 13 '18 at 13:47

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.
• I know that similar results hold for ${\rm GL}(1,D)$ and ${\rm SL}(1,D)$, where $D$ is a division algebra with center $F$ (of course these groups are not quasisplit). But I do not know any general reference. One has to check whether the proofs of Bushnell and Kutzko generalize or not. What true is that they do not use any deep result on the structure of representations of ${\rm GL}(N)$. – Paul Broussous Feb 13 '18 at 17:54