An integral transform and its applications (Q1340166)

For a spherical function \(f\) on \(\text{GL}(n)\) over a \(p\)-adic field of characteristic zero and a generic character \(\theta\) of the group \(N\) of upper triangular matrices with unit diagonal, we define an integral transform \(\Psi_ f(a)= \int f(an) \theta(n)dn\), where \(n\in N\) and \(a\) is a diagonal matrix. This integral transform plays an inescapable part in adelic versions of the Kuznetsov trace formula, Poincaré series and various Kloosterman integrals. We next prove an identity expressing this integral transform in terms of the well-known Satake transform. Applying this identity to the Kuznetsov trace formula and a relative Kloosterman integral, we finally reduce the fundamental lemma of a relative trace formula to a set of equations between Kloosterman sums.