Spectral moment formulae for GL(3) × GL(2) L-functions I : The cuspidal case

Spectral moment formulae of various shapes have proven very successful in studying the statistics of central $L$-values. We establish, in a completely explicit fashion, such formulae for the family of $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(3)\times \mathrm{GL}<!--FUNCTION\; APPLICATION-->(2)$ Rankin–Selberg $L$-functions using the period integral method. Our argument does not rely on either the Kuznetsov or Voronoi formulae. We also prove the essential analytic properties and derive explicit formulae for the integral transform of our moment formulae. We hope that our method will provide deeper insights into moments of $L$-functions for higher-rank groups.