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 Bellaïche--Chenevier.

23 pages

References in corpus (2)

• The sign of Galois representations attached to automorphic forms for unitary groups
• Pseudocharacters of Classical Groups