Rep(m): Galois representations attached to regular automorphic representations of U(m)

Let $E$ be the quadratic imaginary field in the paper, let $G_E$ be its absolute Galois group, and let $\pi$ be an automorphic representation of $\operatorname{U}(m)$ with regular weight at infinity. Rep($m$) conjecture. There exists a continuous semisimple representation

$$\rho_\pi:G_E\longrightarrow \operatorname{GL}_m(\overline{\mathbb Q}_p)$$

satisfying properties (P0)–(P5): the stated compatibility with unramified Frobenius polynomials and local monodromy away from $p$, de Rham Hodge–Tate weights at the distinguished place above $p$, and crystalline Frobenius compatibility when $\pi_p$ is unramified. This is a central expected case of the Langlands correspondence for unitary groups. The source anticipates a proof in forthcoming work, but the assertion is presented there as a conjectural assumption.