Galois representations for even general special orthogonal groups

TL;DR

This paper establishes the existence of Galois representations for certain automorphic forms on special orthogonal groups, utilizing Shimura variety cohomology, and applies these results to automorphic multiplicities and L-functions.

Contribution

It proves the existence of Galois representations for automorphic forms on GSpin groups associated with quasi-split GSO forms, advancing the Langlands program for these groups.

Findings

Constructed Galois representations for specific automorphic forms.

Proved meromorphic continuation of spin L-functions.

Refined the construction of SO(2n)-valued Galois representations.

Abstract

We prove the existence of $\mathrm{GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of $\mathrm{GSO}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb{H}}$, arising from forms of $\mathrm{GSO}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin $L$-functions, and improve on the construction of $\mathrm{SO}_{2n}$-valued Galois representations by removing the outer automorphism ambiguity.