In the study of automorphic representations of reductive algebraic groups one has the theory of L-groups as a fundamental tool. The metaplectic groups \(\text{Mp}(2n)\) (locally a twofold cover of \(\text{Sp}(2n)\)) are not linear algebraic groups and we do not have a theory of L-groups. Still some results have been established, e.g. the lifting of characters between \(\text{Mp}(2n)\) and \(\text{SO}(2n+1)\) has been studied by several people and Renard proved the transfer of orbital integrals for \(\text{Mp}(2n,{\mathbb R})\).
As a first step towards a stable trace formula for the metaplectic group and its Weil representation, the present article establishes the transfer of orbital integrals and proves the fundamental lemma (in characteristic 0). The elliptic endoscopic groups for \(\text{Mp}(2n)\) are defined to be the groups \(\text{SO}(2p+1)\times\text{SO}(2q+1)\) with \(p+q=n\). The correspondence of semisimple conjugacy classes is easily defined in terms of the eigenvalues. A definition for stable conjugacy and a definition of transfer factors are given and with these definitions the transfer of orbital integrals is proved by reducing the transfer to the Lie algebras, where it is known by work of Waldspurger and Ngô.