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.
Categories: math.NT math.RT