A shifted convolution sum for $GL(3) \times GL(2)$ with weighted average

Abstract

In this paper, we will prove a non-trivial bound for the weighted average version of a shifted convolution sum for $GL(3) \times GL(2)$, i.e. for arbitrary small $\epsilon >0$ and $X^{1/4+\delta } \le H \le X$ with $\delta >0$, we prove

$$\begin{aligned} \frac{1}{H}\sum _{h=1}^\infty \lambda _f(h) V \left( \frac{h}{H}\right) \sum _{n=1}^\infty \lambda _{\pi }(1,n) \lambda _g (n+h) W \left( \frac{n}{X} \right) \ll X^{1-\delta +\epsilon }, \end{aligned}$$

where V, W are smooth and compactly supported functions, $\lambda _f(n), \lambda _g(n)$ and $\lambda _{\pi }(1,n)$ are the normalized n-th Fourier coefficients of holomorphic or Hecke–Maass cusp forms f, g for $SL(2,{\mathbb {Z}})$, and Hecke–Maass cusp form $\pi$ for $SL(3,{\mathbb {Z}})$.