Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ on $V$ is integral with kernel $K(x,y) = \sum\limits_{\gamma \in Z(G) \backslash G(F)} f(x^{-1}\gamma y)$. I'm trying to understand this kernel as it appears on $V_{\operatorname{cont}} := V_{\operatorname{cusp}}^{\perp}$, in particular as an integral over Eisenstein series.
Let $\mu$ be a character of $\mathbb A^{\ast 1}/\mathbb Q^{\ast}$, extended to a character of $\mathbb A_k^{\ast}$ by making it trivial on the archimedean connected component, and then to a character of the maximal torus $T(\mathbb A)/Z(\mathbb A)$ by $\mu \begin{pmatrix} a_1 & \\ &a_2 \end{pmatrix} = \mu(a_1/a_2)$. Let
$$I(\mu,s) = \operatorname{Ind}_{B(\mathbb A)}^{G(\mathbb A)} \mu e^{\langle s \alpha , H_B(-) \rangle}$$
(normalized Mackey induction), and for $\phi \in I(\mu,0)$, set $\phi_s(x) = \phi(x) e^{\langle s\alpha, H_B(-) \rangle} \in I(\mu,s)$. To this we can associate the Eisenstein series
$$E(x,\phi,s) = \sum\limits_{\gamma \in B(\mathbb Q) \backslash G(\mathbb Q)} \phi_s(\gamma x)$$
which for fixed $\phi$ and $s$, lies in $L^2(G(\mathbb Q) Z(\mathbb A) \backslash G(\mathbb A))_{\operatorname{cont}}$ as a function of $x$. The series converges for $\operatorname{Re}(s) > 1$ but admits a meromorphic continuation. Then as I understand it,
$$x \mapsto \int_{-\infty}^{\infty} E(x,\phi, it)dt \tag{$\ast$}$$
also lies in $V_{\operatorname{cont}}$, and as $\phi$ runs through an orthonormal basis of $I(\mu,0)$, these integrals $\ast$ run through an orthonormal basis of direct summand $V_{\mu}$ of $V$. The kernel $K(x,y)$ is supposed to decompose as a sum $\sum\limits_{\chi} K_{\chi}(x,y)$ over the cuspidal automorphic data $\chi$ of $V$. For $\chi = \mu$, I want to understand why
$$K_{\chi}(x,y) = \sum\limits_{\phi} \int_{-\infty}^{\infty} E(x, I(\mu,it)f(\phi), it) \overline{E(y,\phi,it)} dt$$
where the sum $\phi$ is over an orthonormal basis of $I(\mu,0)$. This is stated, but not proved, in equation (1.2), pg. 18 of Stephan Gelbart's book Lectures on the Arthur-Selberg trace formula. James Arthur's notes An Introduction to the Trace Formula gives a hint for this on pg. 37 by saying this is related to the Fourier inversion formula
$$f(-x+y) = \frac{1}{2\pi i} \int_{i\mathbb R}\int\limits_{\mathbb R} f(u) e^{\lambda u}e^{\lambda x} \overline{e^{\lambda y}} d\lambda du$$
but I don't understand the connection.
