Lafforgue pseudocharacters and parities of limits of Galois representations

Abstract

Let $F$ be a CM field with totally real subfield $F^+$ and let $\pi$ be a $C$-algebraic cuspidal automorphic automorphic representation of $\mathrm{U}(a,b)(\mathbf{A}_{F^+})$ whose archimedean components lie in the (non-degenerate limit of) discrete series. We attach to $\pi$ a Galois representation $R_\pi:\mathrm{Gal}(\overline F/ F^+)\to{}^C\mathrm{U}(a,b)(\overline{\mathbf Q}_\ell)$ such that, for any complex conjugation element $c$, $R_\pi(c)$ is as predicted by the Buzzard--Gee conjecture. As a corollary, we deduce that the Galois representations attached to certain irregular, $C$-algebraic (essentially) conjugate self-dual cuspidal automorphic representations of $\mathrm{GL}_n(\mathbf A_F)$ are odd in the sense of Bella\"iche--Chenevier.