I am trying to understand proposition 2.1.6 in Bump's book Automorphic forms and Representations.

Let $G=GL(2,\mathbb{R})^+$ and define $G_1=G/Z^+$, where $Z^+$ denotes the center, and define $\Gamma=SL(2,\mathbb{Z})$. Let $\chi: \Gamma \to S^1$ be a group homomorphism. He writes, "Let $L^2(\Gamma\backslash G, \chi)$ be the space of measurable functions satisfying $$f(\gamma g u)=\chi(\gamma)f(g)\qquad \gamma\in\Gamma, u\in Z^+,g\in G$$ that are square integrable with respect to Haar measure on $G_1$."

Clearly no function which is periodic with respect to $\Gamma$ can be square integrable on $G_1$.

Can someone please explain to me what the right definition should be.

`square integrable on $G_1$'', require`

square integrable on $\Gamma\backslash G_1$ (observe that the modulus of $f$ is left-invariant under $\Gamma$). – Alain Valette Jan 21 '12 at 18:06