Average Shifted Convolution sum for $GL(d_1)\times GL(d_2)$

Abstract: We study the average shifted convolution sum $$ B(H,N):= \frac{1}{H} \sum_{h \sim H} \sum_{n \sim N} A_{π1}(n)\, A{π2}(n+h), $$ where $A{π_i}(n)$ denotes the Fourier coefficients of a Hecke--Maass cusp form $π_i$ for $SL(d_i,\mathbb{Z})$ with $d_i\ge 4$, $i=1,2$. For the cases $d_1 = d_2 + 1$ and $d_1 = d_2$, we establish a nontrivial power-saving bound of $B(H,N)$ for the range of the shift $H\ge N^{1-\frac{4}{d_1+d_2}+\varepsilon}$ for any $\varepsilon>0$, extending beyond the results of Friedlander and Iwaniec. In particular, when $d_1 = d_2$, we reach the critical threshold $H\ge N^{1-2/d+\varepsilon}$ such that any further improvement in this range yields a subconvexity bound for the corresponding standard $L$-function in the $t$-aspect.