The non-split symplectic period of a residual Eisenstein series on $Sp_{2n}$

Abstract

Let $\field{F}$ be a number field with ring of adeles $\field{A}$, let $\field{K}$ be a quadratic extension of $\field{F}$ with ring of adeles $\field{A}_\field{K}$. Let $\eta$ be an irreducible, automorphic, cuspidal, self-dual representation of $GL_{2n}(\field{A})$ and let $\phi \in Ind_{P(\field{A})}^{Sp_{2n}(\field{A})}\eta$, where $P$ is the standard Siegel parabolic. We present an identity between the $Sp_n(\field{A}_\field{K})$-period of the residual Eisenstein series on $Sp_{2n}(\field{A})$ associated to $\phi$ and the $GL_n(\field{A}_\field{K})$- period of $\phi$. This is a non-split version of a result of Ginzburg, Soudry and Rallis.