I'm reading an article by Knapp on the Trace formula for $\operatorname{GL}_2$ and am confused on an early definition. Here $G = \operatorname{SL}_2(\mathbb R)$, $N = \begin{pmatrix} 1 & \mathbb R \\ & 1 \end{pmatrix}$, $\Gamma = \operatorname{SL}_2(\mathbb Z)$, and $\Gamma_{\infty} = \Gamma \cap N$. I've attached a picture of the definition I'm confused on.
I don't understand how $\hat{\phi}$ is well defined. Since $\phi$ is not necessarily left $\Gamma_{\infty}$-invariant, the sum
$$\hat{\phi}(g) = \sum\limits_{\gamma \in \Gamma_{\infty} \backslash \Gamma} \phi(\gamma g)$$
appears to depend on the choice of representatives of $\Gamma_{\infty}$ in $\Gamma$.
Maybe the functions in $\mathcal D(N \backslash G)$ are meant to be left $N$-invariant?
