Lafforgue pseudocharacters and parities of limits of Galois representations

TL;DR

This paper constructs Galois representations associated with certain automorphic representations of unitary groups over CM fields, confirming predictions about their behavior under complex conjugation and implications for oddness in Galois representations.

Contribution

It establishes a method to attach Galois representations to specific automorphic representations of unitary groups, verifying Buzzard--Gee conjecture predictions and implications for automorphic Galois representations.

Findings

Galois representations attached to automorphic representations satisfy Buzzard--Gee conjecture predictions.

Galois representations for certain conjugate self-dual automorphic forms are odd.

Provides new links between automorphic forms and Galois representations in the CM field setting.

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.