Galois representations for even general special orthogonal groups

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.

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