Periods detecting Eisenstein series and sums of $L$-values I

We study the automorphic period associated to a $G$-Hamiltonian variety $M$ whose dual is $\check{M}=T^{\ast }(\check{G}\mathrm{/}\check{L})$, where $\check{G}$ is a general linear group and $\check{L}$ is a Levi subgroup. For certain cuspidal Eisenstein series, we prove that their period is equal to a finite sum of special values of $L$-functions. This sum is indexed by the fixed points of the associated extended $L$-parameter on $\check{M}$, confirming a conjecture by Ben-Zvi-Sakellaridis-Venkatesh in this case.