Dichotomy for generic supercuspidal representations of $G_2$

Details Dichotomy for generic supercuspidal representations of $G_2$ Dichotomy for generic supercuspidal representations of $G_2$

2009-08-23

arxiv.org/abs/0908.3340v2

Mathematics

Representation Theory

Version 2: Mistakes in Prop 3.2 and 3.5 corrected. Results strengthened in case p=2. Changes made throughout for consistency w

Scientific paper

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 $PGSp_6$.

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