Kloosterman integrals for skew symmetric matrices (Q1204816)

The authors first describe a conjectured relative trace formula in the general case over a number field \(F\) as an equality between the standard Kuznetsov trace formula for a quasi-split reductive group \(G'\) and a relative Kuznetsov trace formula for a different group \(G\) with respect to the fixator \(H\) of an involution of \(G\). As a model for the general theory they next consider a relative trace formula for \(G'=\text{GL}(m,F)\), \(G=\text{GL}(2m,F)\), and \(H\) the fixator of the skew matrix \(\bigl( {0 \atop {-1}} {1 \atop 0}\bigr)\) in \(G\). In order to study the fundamental lemma of this special relative trace formula, they then calculate local orbital integrals of the standard and relative Kuznetsov trace formulas. After matching corresponding orbits, the fundamental lemma is proved.