The arithmetic Kuznetsov formula on \(\mathrm{GL}(3)\). I: The Whittaker case (Q2280524)

Summary: The original formulae of Kuznetsov for \(\mathrm{SL}(2, \mathbb Z)\) allowed one to study either a spectral average via Kloosterman sums or to study an average of Kloosterman sums via a spectral interpretation. In previous papers, we have developed the spectral Kuznetsov formulae at the minimal weights for \(\mathrm{SL}(3, \mathbb Z)\), and in these formulae, the big-cell Kloosterman sums occur with weight functions attached to four different integral kernels, according to the choice of signs of the indices. These correspond to the \(J\)- and \(K\)-Bessel functions in the case of \(\mathrm{GL}(2)\). In this paper, we demonstrate a linear combination of the spherical and weight-one \(\mathrm{SL}(3, \mathbb Z)\) Kuznetsov formulae that isolates one particular integral kernel, which is the spherical \(\mathrm{GL}(3)\) Whittaker function. Using the known inversion formula of Wallach, we give the first arithmetic Kuznetsov formula for \(\mathrm{SL}(3, \mathbb Z)\) and use it to study smooth averages and the Kloosterman zeta function attached to this particular choice of signs.