Bounds for Average toward the Resonance Barrier for GL(3) x GL(2) Automorphic Forms

Let f be a fixed Maass form for SL3(Z) with Fourier coefficients A(f)(m, n). Let g be a Maass cusp form for SL2(Z) with Laplace eigenvalue 1/4 + k(2) and Fourier coefficient lambda(g)(n), or a holomorphic cusp form of even weight k. Denote by S-X(fxg, alpha, beta) a smoothly weighted sum of A(f) (1,n)lambda(g)(n)e(an(beta)) for X < n < 2X, where alpha not equal 0 and beta > 0 are fixed real numbers. The subject matter of the present paper is to prove non-trivial bounds for a sum of S-X(fxg, alpha, beta) over g as k tends to infinity with X. These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec, Luo, and Sarnak.