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 $z\to \mathrm{diag}(z^{1k},\overline{z}^{1k})$ for $k \geq 2$ an integer, and $\omega_{\pi}^{\sigma}=\omega_{\pi}$, then a wellknown theorem of Taylor (proved in the early 90's) yields a compatible system of twodimensional $\ell$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 Galoisinvariance assumption on the central character $\omega_{\pi}$?
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.) 


Taylor's theorem is about regular algebraic representations $\pi$. Most representations are not algebraic in any sense (cf. my answer to this question), hence they are not connected to any Galois representation (as far as we know). Now for an algebraic $\pi$ the question arises where to look for the corresponding Galois representation. The difficulty with $GL_2/K$ is that the corresponding symmetric space is not an algebraic variety, hence there is no cohomology which would carry the relevant Galois representation. Taylor's idea was to pass to $GSp_4/\mathbb{Q}$ which is better in that regard, but the switch requires the condition on the central character. My feeling is that what is missing is a better (more general) construction yielding an object with a Galois action, not a better understanding of the fundamental lemma or Galois representations per se. This is not my field of expertise, so I better stop here. 

