The authors establish the fundamental lemma for a relative trace formula. This fundamental lemma asserts that pairs of local orbital integrals, one integral of each pair arising on $GSp(4)$ and the other on $GL(4)$, are equal. The orbital integrals in question are exponential sums, and the fundamental lemma may also be described as a matching of Kloosterman and relative Kloosterman sums on the two different groups. To show that these are equal for each relevant Weyl groups element, the authors compute the Mellin transforms and match them in all cases. The authors also describe the L-function heuristics which motivate this work, its possible generalizations, and an application of the relative trace formula to the study of L-packets.
Introduction and statement of results Evaluation of the integral for the main $H$-relevant double cosets Evaluation of the integrals for the other $H$-relevant double cosets Evaluation of the $GSp(4)$ integral for the main double cosets Evaluation of the $GSp(4)$ integrals for the other relevant double cosets References.