A theory of $\gamma$-factors for $G_2 \times GL_r$

Gordan Savin; Wee Teck Gan

A theory of $\gamma$-factors for $G_2 \times GL_r$

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arxiv 2308.12561 v1 pith:K6INMGTD submitted 2023-08-24 math.NT math.RT

Wee Teck Gan , Gordan Savin This is my paper · ORCID

classification math.NTmath.RT

keywords factorsgammatheorylocaltimescandidatecharacterizedcompatible

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abstract

We construct a theory of local gamma factors for $G_2 \times GL_r$ using a functorial lifting from $G_2$ to $GL_7$. This theory of gamma factors is uniquely characterized by a usual list of properties, showing that it is the only possible candidate. Moreover, this theory of gamma factors is compatible with the Galois theoretic one under the local Langlands correspondence for $G_2$.

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On sharpness in Local Converse Theorems for classical groups and $G_2$
math.RT 2025-09 conditional novelty 8.0 of 10

The optimal standard local converse theorem for Sp_{2N}, SO_{2N}, SO_{2N+1}, and G2 requires twisting up to roughly half the dual group's standard representation dimension, except for an improved SO_{2N} bound for odd N.

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