Let $K/\mathbf{Q}$ be an imaginary quadratic field, with $\sigma \in \mathrm{Gal}(K/\mathbf{Q})$ a generator. Suppose $\pi$ is a cuspidal automorphic representation of $GL_2 / K$ with central character $\omega_{\pi}$. If $\pi_{\infty}$ has Langlands parameter $z\to \mathrm{diag}(z^{1-k},\overline{z}^{1-k})$ for $k \geq 2$ an integer, and $\omega_{\pi}^{\sigma}=\omega_{\pi}$, then a well-known theorem of Taylor (proved in the early 90's) yields a compatible system of two-dimensional $\ell$-adic Galois representations of $\mathrm{Gal}(\overline{K}/K)$ attached to $\pi$. My question is simple: given all the recent work on the fundamental lemma and concomitant progress in the field of automorphic Galois representations, it is yet possible to prove this result without any Galois-invariance assumption on the central character $\omega_{\pi}$?
[Math] Automorphic forms and Galois representations over imaginary quadratic fields: generalizing Taylor’s theorem
automorphic-formsgalois-representationsnt.number-theory
Best Answer
Various groups of people have thought about/are thinking about this. The natural source of the desired Galois reps. is a $U(2,2)$ Shimura variety. The problem is that the cohomology of this variety is not so easy to understand. The fundamental lemma certainly plays some role in controlling it, but I don't think that by itself it overcomes the key difficulties. (My own understanding of the issues is far from perfect, though.)