Let $\pi$ be a regular, cuspidal, algebraic automorphic representation of $GL_n(\mathbb{A}_K)$ for a totally real field $K$. Then for every embedding $\lambda$ of $E=\mathbb{Q}(\pi)$ in $\overline{\mathbb{Q}_p}$, we know how to associate a $p$-adic Galois representation $\rho_{\pi,\lambda}$ to $\pi$ thank to the work of Harris-Lan-Taylor-Thorne and also Scholze. In the special case of modular forms, one can actually define these Galois representations to have values in $GL_2(E_\lambda)$ instead of $GL_2(\overline{\mathbb{Q}_p})$. This directly follows from the construction. If K is not totally real one can give counter examples even in the $GL_2$ case. For exmples base change a modular form with inner-twists to the field associated with the intersection of the kernels of all the characters appearing in the inner-twists.
Now my questions is, what is the right thing to expect for general $n$ when $K$ is totally real. Let me share what I know and make the question more precise. If the residual representation is irreducible then one can always go to $GL_n(E_\lambda)$ because the deformation ring and the pseudo-deformation ring are isomorphic in this case. I'm confident that assuming irreducibility of $\rho_{\pi,\lambda}$ (which one certainly expects) it follows from a result of Larsen that the residual representation is indeed irreducible for a density one set of primes $p$. Here's my first question:
- Should we expect the residual representation to actually be irreducible for all but finitely many primes?
Now let's get back to the $GL_2$ case. In the case of Hilbert modular forms one can also define the representation over $E$. I think this doesn't immediately follow from the construction since in some cases one has to use congruences (or does it?). But we are lucky in the $GL_2$ case. By proposition 1.6.1 of Bellaïche-Chenevier's Astérisque paper, residually multiplicity free is actually enough to go from pseudo-representations to representations. Since our representation is odd in this case we are automatically residually multiplicity free. Now I'm wondering if this is just a lucky coincidence or if this is what one should expect in general:
- If $K$ is totally real, should we expect to be able to define $\rho_{\pi,\lambda}$ to have values in $GL_n(E_\lambda)$ for all $p$ or all but finitely many $p$ or just a density one set of primes $p$?
Also, I'm aware that (assuming irreducibility of $\rho_{\pi,\lambda}$) one can always go to a central simple algebra over $E_\lambda$ by a result of Rouquier. But I'm interested in the cases where this splits.
