Shifted convolution sums for $GL(3)\times GL(2)$

Abstract

For the shifted convolution sum $$ D_h(X)=\sum_{m=1}^\inftyλ_1(1,m)λ_2(m+h)V(\frac{m}{X}) $$ where $λ_1(1,m)$ are the Fourier coefficients of a $SL(3,\mathbb Z)$ Maass form $π_1$, and $λ_2(m)$ are those of a $SL(2,\mathbb Z)$ Maass or holomorphic form $π_2$, and $1\leq |h| \ll X^{1+\varepsilon}$, we establish the bound $$ D_h(X)\ll_{π_1,π_2,\varepsilon} X^{1-(1/20)+\varepsilon}. $$ The bound is uniform with respect to the shift $h$.