Title: 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 $\epsilon$.