# 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 $F^\times / (F^\times)^n$. Feb 10, 2018 at 21:47
• Yes, you are right.
– D_S
Feb 10, 2018 at 21:58
• As group schemes, $GL_n=SL_n . G_m$. Feb 10, 2018 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.) Feb 10, 2018 at 22:23
• Why the notation $G_D$? For me, $D$ is usually a division algebra. Feb 14, 2018 at 9:50

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 ${\rm GL}(N)$. Feb 13, 2018 at 17:54
• I think what will probably be important in the general case is an understanding of the structure of $G/Z(G)[G, G]$. May 22, 2020 at 16:47