On the Fourier expansion of Gan-Gurevich lifts on the exceptional group of type $G_2$

TL;DR

This paper investigates the Fourier expansion of Gan-Gurevich lifts on the split exceptional group G_2, revealing new insights into their structure and providing partial answers to Gross's conjecture using degenerate Whittaker functions.

Contribution

It introduces a novel method to analyze Fourier expansions of Gan-Gurevich lifts on G_2 using degenerate Whittaker functions, advancing understanding of their automorphic properties.

Findings

Fourier expansion formulas for Gan-Gurevich lifts derived

Partial confirmation of Gross' conjecture provided

Characterization of Hecke eigen quaternionic cusp forms on G_2

Abstract

By using the degenerate Whittaker functions, we study the Fourier expansion of the Gan-Gurevich lifts which are Hecke eigen quaternionic cusp forms of weight $k$ ($k\geq 2$, even) on the split exceptional group $G_2$ over $\mathbb{Q}$ which come from elliptic newforms of weight $2k$ without supercuspidal local components. In particular, our results give a partial answer to Gross' conjecture.