TITLE: Atkin–Lehner operator for GL(3)?
QUESTION [9 upvotes]: Let $f$ be an automorphic form for $\Gamma_0(N)\subset SL(3,\mathbb{Z})$.
$\Gamma_0(N)=(a,b,c;d,e,f;g,h,i)\in SL(3,\mathbb{Z})|g=h=0(mod N)$
Is there any Atkin-Lehner operator for $\Gamma_0(N)$ which will give a functional equation for L-function of $f$?

REPLY [2 votes]: You will have to decompose
$$ Endo_{SL(3, \mathbb{Z})} ( Ind_{\Gamma_0(N)}^{SL(3, \mathbb{Z})} 1 ) .$$ 
This will give you the analog of the Atkin-Lehner theory. However, I have some doubts that this exists in the literature. This question of mine gives you the resaon why it has not been done for $d,g,h = 0 \bmod N$:
Parabolic induction GL(n,Zp)
I am happy, if somebody proves me wrong though.
Edit: I forgot to mention, that the case for $N$ square free is in general possible, since you can rely on the representation theory of reductive groups over residue fields.