Resonance sums for Rankin–Selberg products of SLm(Z)SLm(Z) Maass cusp forms

Let f and g   be Maass cusp forms for SLm(Z)SLm(Z) and SLm′(Z)SLm′(Z), respectively, with 2≤m≤m′2≤m≤m′. Let λf×g(n)λf×g(n) be the normalized coefficients of L(s,f×g)L(s,f×g), the Rankin–Selberg L-function for f and g  . In this paper the asymptotics of a Voronoi-type summation formula for λf×g(n)λf×g(n) are derived. As an application estimates are obtained for the smoothly weighted average of λf×g(n)λf×g(n) against e(αnβ)e(αnβ). When β=1mm′ and α   is close or equal to ±mm′q1mm′ for a positive integer q  , the average has a main term of size |λf˜×g˜(q)|X12mm′+12. Otherwise, when 0<β<1mm′, it is shown that this average decays rapidly. This phenomenon is due to the oscillatory nature of the coefficients λf×g(n)λf×g(n).