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

TL;DR

This paper develops a unique theory of local gamma factors for the product of the exceptional group G_2 and general linear groups, establishing compatibility with Galois representations via the local Langlands correspondence.

Contribution

It constructs a new, uniquely characterized theory of gamma factors for G_2 x GL_r using functorial lifting and proves its compatibility with Galois theoretic gamma factors.

Findings

The gamma factors are uniquely determined by standard properties.

The theory is compatible with Galois representations under the local Langlands correspondence.

It provides a new framework for understanding gamma factors for exceptional groups.

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 $G_2$.