Bump-Friedberg type periods beyond the cuspidal spectrum

In this article, we study several Bump--Friedberg type periods beyond the cuspidal spectrum. We first consider the twisted Bump--Friedberg period on $\textnormal{GL}_{2n}$, as well as a variant on $\textnormal{GL}_1\times \textnormal{GL}_{2n}$. Under suitable regularity conditions on the cuspidal datum, these periods extend continuously to automorphic functions of uniform moderate growth. Such extensions are characterized by entire Whittaker-type zeta integrals. We then introduce a Bump--Friedberg type period on $\textnormal{GL}_{2n+1}$, integrating over the subgroup $\textnormal{SL}_{n+1}\times \textnormal{GL}_n$. For certain Eisenstein series, we evaluate this period as a finite sum of products of special values of $L$-functions and normalized local zeta integrals. Assuming the expected global Langlands correspondence, the sum is indexed by the fixed points of the extended $L$-parameter on the conjectural dual variety, and the resulting $L$-factors agree with the tangent space prediction of the global numerical conjecture of Ben-Zvi-Sakellaridis-Venkatesh.