The symplectic period of certain Eisenstein series on GL(2n+1) matches the L-function ratio predicted by relative Langlands duality, confirming the dual pair Sp(2n)\GL(2n+1) and GL(n)xGL(n+1)\GL(2n+1) in the tested cases.
Periods of automorphic forms over reductive subgroups
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abstract
We present a regularization procedure of period integrals of automorphic forms on a group $G$ over an arbitrary reductive subgroup $G' \subset G$. As a consequence we obtain an explicit $G'(\mathbb{A})$-invariant functional on the space of automorphic forms on $G$ whose exponents avoid certain prescribed hyperplanes. We also provide a necessary and sufficient condition for convergence of period integrals of automorphic forms in terms of their exponents.
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On the relative Langlands duality for $\operatorname{Sp}_{2n} \backslash \operatorname{GL}_{2n+1}$ (with an appendix by Zeyu Wang)
The symplectic period of certain Eisenstein series on GL(2n+1) matches the L-function ratio predicted by relative Langlands duality, confirming the dual pair Sp(2n)\GL(2n+1) and GL(n)xGL(n+1)\GL(2n+1) in the tested cases.