On Periods and $L$-functions for ${\mathbf{GL}}_4\times {\mathbf{GL}}_2$

Abstract

We give a new integral representation of the ${\wedge }^2\otimes {\mathrm{std}}_2$ $L$-function of generic cusp forms on ${\mathbf{GL}}_4\times {\mathbf{GL}}_2$ and ${\mathbf{GU}}_{2,2}\times {\mathbf{GL}}_2$. In the former case, we use it to prove a relation between its central $L$-value and the generalized Shalika period. Exploiting the theta correspondence for $({\mathbf{GL}}_4,{\mathbf{GL}}_4)$, we further establish a relation between the central value of the $L$-function attached to the strongly tempered spherical pair $({\mathbf{GL}}_4\times {\mathbf{GL}}_2,{\mathbf{GL}}_2\times {\mathbf{GL}}_2)$ and its corresponding period. In the case of cusp forms on ${\mathbf{GL}}_4\times {\mathbf{GL}}_2$ that are unramified everywhere, our formulas give new evidence towards conjectures of Wan-Zhang and of Gan-Gross-Prasad for ${\mathbf{GSpin}}_6\times {\mathbf{GSpin}}_3$.

58 pages, strengthened Theorem A and rewrote the introduction accordingly