Galois representation and infinitesimal character conjecture for Hecke eigenspaces

Let $\mathbb{T}$ be the Hecke algebra and let $x:\mathbb{T}[1/p]\to\overline{\mathbb{Q}}_p$ be a continuous homomorphism with kernel $\mathfrak{m}_x$. Let $\rho:\operatorname{Gal}_{\mathbb{Q}}\to{}^CG_f(\overline{\mathbb{Q}}_p)$ be an admissible representation associated with $x$ as in the preceding conjectural construction, and let $\zeta^C_\rho$ be its associated infinitesimal character. For $n\geq0$, form

$$(\widetilde{H}^n\otimes_{\mathcal{O}}L)[\mathfrak{m}_x]^{\mathrm{la}}\otimes_{\mathbb{T},x}\overline{\mathbb{Q}}_p.$$

Galois-infinitesimal character conjecture. The center $Z(\mathfrak{g})$ acts on this representation via $\zeta^C_\rho$.

This refines the preceding existence conjecture by identifying the character through an admissible Galois representation. The existence and uniqueness of such a Galois representation are themselves conjectural in the stated generality, although the paper proves that all such representations, if they exist, give the same associated infinitesimal character.