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

Henry H. Kim, Takuya Yamauchi

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.

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

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 (, even) on the split exceptional group over which come from elliptic newforms of weight without supercuspidal local components. In particular, our results give a partial answer to Gross' conjecture.

Paper Structure

This paper contains 29 sections, 30 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the exceptional group $G_2$
  3. The Heisenberg parabolic subgroup
  4. Another maximal parabolic subgroup
  5. Quaternionic discrete series representations
  6. Modular forms on $G_2$ and their Fourier expansions
  7. Quaternionic Modular forms on $G_2$
  8. A robust theory of the Fourier expansion due to Pollack
  9. Expansion along $Z_U$
  10. An observation on $F_t$ for $t\ne0$
  11. Fourier-Jacobi expansion of Eisenstein series on $G_2$ along $U/Z_U$
  12. Degenerate Whittaker functions
  13. Degenerate principal series representations: The nonarchimedean case
  14. Jacquet integrals and Siegel series
  15. Degenerate principal series representations: The archimedean case
  16. ...and 14 more sections

Key Result

Theorem 1.1

Assume (assump ). For each distinguished vector $\phi=\otimes'_p \phi_p\in \Pi(f)$, $F_f(\ast;\phi)$ can be expanded as where for $g=(g_p)_p\in G_2(\mathbb{A})$ and some complex numbers $\{C^{\mu_{\bf{f}}}_{w}(F_f)\}$. Here $w_\beta$ (resp. $w_\alpha$) is the Weyl element in $L^{{\rm ss}}:=[L,L]\simeq SL_2$ (resp. in $M$) and $X_\beta$ is the upper unipotent subgroup of $L^{{\rm ss}}$. Furthermo

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • ...and 53 more