On the sign of regular algebraic polarizable automorphic representations

TL;DR

This paper extends the construction of automorphic Galois representations to include mixed-parity cases over totally real fields, removing previous restrictions and optimizing Galois lifting results for broader applicability.

Contribution

It removes a parity restriction from existing constructions and generalizes to mixed-parity automorphic representations, enhancing the scope of automorphic Galois representation theory.

Findings

Successfully constructed automorphic Galois representations for mixed-parity cases

Generalized Galois lifting results to broader automorphic contexts

Enhanced understanding of automorphic representations over totally real fields

Abstract

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 the Galois lifting results of arxiv:1207.6724 and apply them to the previous mixed-parity setting.