The local-global compatibility conjecture at Taylor–Wiles level

Let $(Q,(\gamma_{v,1},\dots,\gamma_{v,n})_{v\in Q})$ be a Taylor–Wiles datum, let $\sigma$ be an irreducible $E$-representation of $K_0$, and let $\sigma^\circ$ denote the associated integral lattice. Let $R_p^{\operatorname{loc}}(\sigma)$ be the corresponding quotient of $R_p^{\operatorname{loc}}$, let $\mathcal H(\sigma^\circ)$ be the Hecke algebra, and let $\eta$ be the indicated map from this Hecke algebra to the local deformation ring.

Local-global compatibility conjecture. The action of $R_p^{\operatorname{loc}}$ on

$$H_*(X_{K_0U^p_1(Q)},\sigma^\circ)_{\mathfrak m_{Q,1}}$$

factors through $R_p^{\operatorname{loc}}(\sigma)$. Moreover, if $h\in\mathcal H(\sigma^\circ)$ satisfies $\eta(h)\in R_p^{\operatorname{loc}}(\sigma)$, then $h$ acts on

$$H_{q_0}(X_{K_0U^p_1(Q)},\sigma^\circ)_{\mathfrak m_{Q,1}}$$

via $\eta(h)$.

This is formulated as a refinement of the earlier Galois and Taylor–Wiles compatibility conjectures. It is intended to provide the local-global compatibility needed to deduce the general $p$-adic local Langlands action; the supplied text gives no resolution.