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 $K(x,y) = sumlimits_{gamma in Z(G) backslash G(F)} f(x^{-1}gamma y)$. 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 $phi_s(x) = phi(x) e^{langle salpha, H_B(-) rangle} in I(mu,s)$. To this we can associate the Eisenstein series

$$E(x,phi,s) = sumlimits_{gamma in B(mathbb Q) backslash G(mathbb Q)} phi_s(gamma x)$$

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 $sumlimits_{chi} K_{chi}(x,y)$ over the cuspidal automorphic data $chi$ of $V$. For $chi = mu$, I want to understand why

$$K_{chi}(x,y) = sumlimits_{phi} int_{-infty}^{infty} E(x, I(mu,it)f(phi), it) overline{E(y,phi,it)} dt$$

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

$$f(-x+y) = frac{1}{2pi i} int_{imathbb R}intlimits_{mathbb R} f(u) e^{lambda u}e^{lambda x} overline{e^{lambda y}} dlambda du$$

but I don’t understand the connection.