Nekovář–Scholl's plectic conjecture for Shimura varieties

Consider a Shimura datum $(G,X)$ with $G=\operatorname{Res}_{F/\mathbb{Q}}H$, where $H$ is a connected reductive group over a totally real field $F$. Let $\operatorname{Gal}^{\mathrm{plec}}_{F/\mathbb{Q}}=\operatorname{Aut}_F(F\otimes_{\mathbb{Q}}\overline{\mathbb{Q}})$ be the plectic Galois group, let $[\mu]$ be the inverse Hodge cocharacter, and let $\operatorname{Gal}^{[\mu]}_{F/\mathbb{Q}}$ be its stabilizer. If $E$ is the reflex field, so that $\operatorname{Gal}_E=\operatorname{Gal}_{\mathbb{Q}}\cap\operatorname{Gal}^{[\mu]}_{F/\mathbb{Q}}$, and $K$ is a compact open subgroup of $G(\mathbb{A}_f)$, write $\overline{\operatorname{Sh}}_K(G,X)$ for the minimal compactification of the Shimura variety over $E$. Nekovář–Scholl's plectic conjecture. The complex

$$\operatorname{IH}(\overline{\operatorname{Sh}}_K(G,X)_{\overline{\mathbb{Q}}},\overline{\mathbb{Q}}_\ell)$$

canonically lifts from an object of $D^b(\operatorname{Gal}_E,\overline{\mathbb{Q}}_\ell)$ to an object of $D^b(\operatorname{Gal}^{[\mu]}_{F/\mathbb{Q}},\overline{\mathbb{Q}}_\ell)$. This is the plectic enhancement of the Galois action on the intersection cohomology of Shimura varieties, motivated by the construction of Euler systems beyond rank one. The paper proves analogues for local Shimura varieties and obtains global results after restriction to a decomposition group, but the conjecture stated here is not resolved in full generality.