Galois representations for general symplectic groups

TL;DR

This paper establishes the existence of GSpin-valued Galois representations for certain automorphic forms on symplectic groups, confirming cases of the Langlands correspondence and constructing motives with G_2 Galois groups.

Contribution

It proves the Buzzard-Gee conjecture for general symplectic groups under specific local conditions and constructs a G_2 motive from Siegel modular varieties.

Findings

Existence of GSpin Galois representations for symplectic groups.

Confirmation of the Buzzard-Gee conjecture in new cases.

Construction of a G_2 Galois group motive.

Abstract

We prove the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard-Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank seven motive whose Galois group is of type G_2 in the cohomology of Siegel modular varieties of genus three. Under some additional local hypotheses we also show automorphic multiplicity one as well as meromorphic continuation of the spin L-functions.