On the Gan-Gross-Prasad conjecture and its refinement for $\left(\mathrm{U}\left(2n\right),\mathrm{U}\left(1\right)\right)$

TL;DR

This paper proves the Gan-Gross-Prasad conjecture and its refinement for unitary and general linear groups, establishing explicit formulas for central L-values under certain assumptions, advancing understanding in automorphic forms and representation theory.

Contribution

It establishes the Gan-Gross-Prasad conjecture and its refinement for (U(2n), U(1)) and (GL(2n), GL(1)) in full generality, including explicit formulas for central L-values.

Findings

Proved the conjecture for (U(2n), U(1)) in general.

Proved the conjecture for (GL(2n), GL(1)) in general.

Established Ichino-Ikeda type explicit formulas for central L-values.

Abstract

We prove the Gan-Gross-Prasad conjecture for $\left(\mathrm{U}\left(2n\right),\mathrm{U}\left(1\right)\right)$ in general and prove its refinement, namely the Ichino-Ikeda type explicit formula for the central $L$-values, under certain assumptions. Similarly, we also prove its split analogue, namely the Gan-Gross-Prasad conjecture and its refinement for $\left(\mathrm{GL}_{2n}, \mathrm{GL}_1\right) $ in general.