Galois Representations for even General Special Orthogonal Groups

We prove the existence of GSpin_2n-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of 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^H, arising from forms of 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 SO_2n-valued Galois representations by removing the outer automorphism ambiguit