Gelfand pairs (Sp4n(F),Sp2n(E))

In this paper, we attempt to prove that the symmetric pairs (Sp4n(F),Sp2n(E)) and (GSp4n(F),GSp2n(E)∘) are Gelfand pairs where E is a commutative semi-simple algebra over F of dimension 2 and F is a non-archimedean field of characteristic 0. Using Aizenbud and Gourevitch's generalized Harish-Chandra method and traditional methods, i.e. the Gelfand-Kahzdan theorem, we can prove that these symmetric pairs are Gelfand pairs when E is a quadratic extension field over F for any n, or E is isomorphic to F×F for n⩽2. Since (U(J2n,F(τ)),Sp2n(F)) is a descendant of (Sp4n(F),Sp2n(F)×Sp2n(F)), we prove that it is a Gelfand pair for both archimedean and non-archimedean fields.