A $\mathrm{G}_2$-period of a Fourier coefficient of an Eisenstein series on $\mathrm{E}_6$

TL;DR

This paper computes a specific G2-period of a Fourier coefficient of an Eisenstein series on E6 and links it to the Ginzburg-Rallis period and the central value of an exterior cube L-function of GL6, revealing deep connections in automorphic forms.

Contribution

It establishes a novel relation between G2-periods of Eisenstein series on E6 and Ginzburg-Rallis periods, connecting these to the exterior cube L-function of GL6.

Findings

Calculated a G2-period of a Fourier coefficient of an Eisenstein series on E6.

Linked the G2-period to the Ginzburg-Rallis period on GL6.

Related the period to the central value of the exterior cube L-function of GL6.

Abstract

We calculate a $\mathrm{G}_2$-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group $\mathrm{E}_6$, and relate this period to the Ginzburg-Rallis period of cusp forms on $\mathrm{GL}_6$. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of $\mathrm{GL}_6$