On the ${\mathrm{S}\mathrm{p}}_{2n}$-distinguished automorphic spectrum of ${\mathrm{U}}_{2n}$

Given a reductive group $G$ and a reductive subgroup $H$, both defined over a number field $F$, we introduce the notion of the $H$-distinguished automorphic spectrum of $G$ and analyze it for the pair $({\mathrm{U}}_{2n},{\mathrm{S}\mathrm{p}}_{2n})$. We derived a formula for period integrals of pseudo-Eisenstein series of ${\mathrm{U}}_{2n}$ in analogy with the main result of Lapid and Offen in their work analyzing the pair $({\mathrm{S}\mathrm{p}}_{4n},{\mathrm{S}\mathrm{p}}_{2n}\times {\mathrm{S}\mathrm{p}}_{2n})$. We give an upper bound of the distinguished spectrum with the formula. A non-trivial lower bound of the discrete distinguished spectrum is expected from the formula, given the previous work.