Let us fix a representation $\pi\_\infty$ of GL(n,$\mathbb R$).


Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A\_{finite})$.


$$K=\Pi\_{v<\infty}K\_v$$


$K\_v$ is $GL(n,\mathbb Z\_v)$ for almost all v.


For automorphic forms of ($\chi$, K), we require that 


for $\phi: GL(n,\mathbb {A\_Q})\to \mathbb C$ 


$\phi(x\gamma)=\chi(\gamma)\phi(x)$ for any $x\in GL(n,\mathbb{A\_Q})$ and any $\gamma \in K$.


How many cuspidal automorphic representations of GL(n,$\mathbb{A\_Q}$) with character $\chi$ and with $\pi\_\infty$ at infinity place are there?


I am expecting answer to be "finitely many". Who and where is this proved?



Different cuspidal automorphic representations with same representations at infinity
This is precisely the content of Harish-Chandra's theorem ("Automorphic forms on Semisimple Lie Groups", LNM 68, 1968), proven for general reductive groups:


Fix a finite-dimensional representation $\delta$ of $K\_\infty$, an ideal $J$ of finite co-dimension in ${\mathcal Z}({\mathfrak g})$, a compact open subgroup $L$ of $K\_{\rm fin}$ (the maximal compact subgroup of $G\_{\mathbb A\_{\rm fin}})$, and a central character $\omega$, then the space of $K$-finite automorphic forms $f$ with central character $\omega$ and $K\_\infty$-type $\delta$ that are right $L$-invariant and annihilated by $J$ is finite dimensional.


To see how this corresponds to your case, fix a representation $\pi\_\infty$. For simplicity, assume $\pi\_\infty$ is spherical. Thus $\delta$ is the trivial representation. Any vector $\phi$ generating $\pi\_\infty$ will be an eigenvector for the Casimir operator, $C\phi=\lambda\phi$, so we take $J$ to be the ideal generated by $(C-\lambda)$. And since $\chi(K)$ is a compact totally disconnected subgroup of ${\mathbb C}^\times$, the kernel of $\chi$ must be a compact-open subgroup $L$, so any $\phi$ will be right $L$-invariant. The only thing left is the central character. The archimedean part is fixed as well as the restriction to a compact open subgroup, so there should only be finitely many options left.


A proof for $SL\_2({\mathbb R})$ can be found in Chapter 8 of Borel's "Automorphic forms on $SL\_2({\mathbb R})$". The general argument is sketched in section 8 of Borel's contribution to Sarnak and Shahidi's "Automorphic Forms and Applications".


