Averages of shifted convolution sums for $GL(3) \times GL(2)$

Abstract

Let $A_f(1,n)$ be the normalized Fourier coefficients of a $GL(3)$ Maass cusp form $f$ and let $a_g(n)$ be the normalized Fourier coefficients of a $GL(2)$ cusp form $g$. Let $\lambda(n)$ be either $A_f(1,n)$ or the triple divisor function $d_3(n)$. It is proved that for any $\epsilon>0$, any integer $r\geq 1$ and $r^{5/2}X^{1/4+7\delta/2}\leq H\leq X$ with $\delta>0$, $$ \frac{1}{H}\sum_{h\geq 1}W\left(\frac{h}{H}\right) \sum_{n\geq 1}\lambda(n)a_g(rn+h)V\left(\frac{n}{X}\right)\ll X^{1-\delta+\epsilon}, $$ where $V$ and $W$ are smooth compactly supported functions, and the implied constants depend only on the associated forms and $\epsilon$.