We provide a family of representations of GL2n over a p-adic field that admit a non-vanishing linear functional invariant under the symplectic group (i.e. representations that are Sp2ndistinguished). While our result generalizes a result of M. Heumos and S. Rallis our methods, unlike their purely local technique, relies on the theory of automorphic forms. The results of this paper together with...

We study quadratic periods on quaternionic Shimura varieties and formulate an integral refinement of Shimura's conjecture regarding Petersson inner products automorphic forms that are related by the Jacquet-Langlands correspondence. The main result is this implied another (Conjecture D below) integrality theta lifts between certain unitary groups.