Dichotomy for generic supercuspidal representations of $G_2$

TL;DR

This paper proves a conjectural dichotomy for generic supercuspidal representations of G_2, establishing a precise correspondence with representations of PGSp_6 and PGL_3 via theta correspondences and L-functions.

Contribution

It demonstrates the conjectural dichotomy for G_2 representations, linking them to other groups through a detailed analysis involving theta correspondences and L-functions.

Findings

Established a precise correspondence between G_2 and PGSp_6/PGL_3 representations.

Reduced the Langlands parameterization of G_2 to a conjecture about PGSp_6.

Connected representation theory with theta correspondences and L-functions.

Abstract

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of $G_2$ over a $p$-adic field, one can associate a generic supercuspidal irreducible representation of either $PGSp_6$ or$PGL_3$. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of $G_2$ and other representations of $PGSp_6$ and $PGL_3$. This correspondence arises from theta correspondences in $E_6$ and $E_7$, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations of $G_2$ to a single conjecture about the parameterization for $PGSp_6$.