On residual automorphic representations and period integrals for symplectic groups

We construct new irreducible components in the discrete automorphic spectrum of symplectic groups. The construction lifts a cuspidal automorphic representation of ${\mathrm{G}\mathrm{L}}_{2n}$ with a linear period to an irreducible component of the residual spectrum of the rank $k$ symplectic group ${\mathrm{S}\mathrm{p}}_k$ for any $k\ge 2n$. We show that this residual representation admits a non-zero ${\mathrm{S}\mathrm{p}}_n\times {\mathrm{S}\mathrm{p}}_{k-n}$-invariant linear form. This generalizes a construction of Ginzburg, Rallis and Soudry, the case $k=2n$, that arises in the descent method.