Lafforgue pseudocharacters and parities of limits of Galois representations

Let F be a CM field with totally real subfield F⁺ and let π be a C-algebraic cuspidal automorphic representation of the unitary group U(a,b)(AF+), whose archimedean components are discrete series or non-degenerate limit of discrete series representations. We attach to π a Galois representation Rπ:Gal(F¯/F+)→CU(a,b)(Q¯ℓ) such that, for any complex conjugation element c, R_π(c) is as predicted by the Buzzard–Gee conjecture (Buzzard and Gee, in: Automorphic forms and Galois representa, Cambridge University Press, Cambridge, 2014). As a corollary, we deduce that the Galois representations attached to certain irregular, C-algebraic essentially conjugate self-dual cuspidal automorphic representations of GLn(AF) are odd in the sense of Bellaïche–Chenevier (Compos Math 147(5):1337–1352, 2011).