The spectral decomposition of shifted convolution sums

Abstract

Let ${\pi }_1$, ${\pi }_2$ be cuspidal automorphic representations of ${\mathrm{PGL}}_2(\mathbb{R})$ of conductor $1$ and Hecke eigenvalues ${\lambda }_{{\pi }_{1,2}}(n)$, and let $h>0$ be an integer. For any smooth compactly supported weight functions $W_{1,2}:{\mathbb{R}}^{\times }\to \mathbb{C}$ and any $Y>0$, a spectral decomposition of the shifted convolution sum ${\displaystyle \sum _{m\pm n=h}\frac{{\lambda }_{{\pi }_1}(|m|){\lambda }_{{\pi }_2}(|n|)}{\sqrt{\left|\mathrm{mn}\right|}}{W}_1(\frac{m}{Y})W_2(\frac{n}{Y})}$ is obtained. As an application, a spectral decomposition of the Dirichlet series ${\displaystyle \sum _{\begin{array}{l}m,n\ge 1\\ m-n=h\end{array}}\frac{{\lambda }_{{\pi }_1}(m){\lambda }_{{\pi }_2}(n)}{(m+n{)}^s}(\frac{\sqrt{\mathrm{mn}}}{m+n}{)}^{100}}$ is proved for $\mathfrak{R}s>1/2$ with polynomial growth on vertical lines in the $s$-aspect and uniformity in the $h$-aspect