On the sign of regular algebraic polarizable automorphic representations

We remove a parity condition from the construction of automorphic Galois representations carried out in the Paris Book Project. We subsequently generalize this construction to the case of ‘mixed-parity’ (but still regular essentially self-dual) automorphic representations over totally real fields, finding associated geometric projective representations. Finally, we optimize some of our previous results on finding geometric lifts, through central torus quotients, of geometric Galois representations, and apply them to the previous mixed-parity setting.