The theta period of a Cuspidal Automorphic Representation of GL(n)

Let E(⋯) be an Eisenstein series corresponding to an automorphic cuspidal representation τ of GLn(A)$, induced to a representation of SO2n+1(A)$ through the Siegel parabolic subgroup. The presence of a pole of E at 12, or the nonvanishing of the residue E_1/2, is determined by the presence of a pole of the partial symmetric square L-function at s=1. We define a co-period integral of E, involving the integration of E_1/2 against a pair of automorphic forms in the space of the small representation of Bump, Friedberg, and Ginzburg. We compute the co-period and relate it to a "theta period" integral of τ - an integral of a cusp form against two theta functions corresponding to the exceptional representation of Kazhdan and Patterson.