Let $K$ be a number field, $\rho\colon \mathrm{Gal}_K\to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ a geometric (i.e.: unramified a.e., de Rham above $p$) irreducible Galois representation. One piece of the global Langlands conjectures is that there exists a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbf{A}_K)$ such that (up to a twist) the Satake parameters of $\pi$ correspond with the Frobenius eigenvalues of $\rho$ at the unramified places.

Even better, for each $v\nmid p$, we can associate to the local Galois representation $\rho_v\colon \mathrm{Gal}_{K_v}\to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ a representation $\pi_v$ of $\mathrm{GL}_n(K_v)$ via the local Langlands correspondence. The restricted tensor product $\pi^p=\bigotimes_{v\nmid p}' \pi_v$ is a representation of $\mathrm{GL}_n(\mathbf{A}_{K,f}^p)$ (finite adeles away from $p$), and by multiplicity one theorems, is the away-from-$p$ part of at most one automorphic representation. Call it $\pi$, if it exists.

Question: how does one "read off" the expected $\pi_\infty$ from $\rho$? In other words, how does one see the infinity-type of the expected automorphic representation on the Galois side?

Surely $\rho_\infty\colon \mathrm{Gal}_{K_\infty}\to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ does not "know" $\pi_\infty$. For example, when $K=\mathbf{Q}$ and $n=2$, $\rho_\infty$ only knows the "parity" of $\pi_\infty$.