Blasius–Rogawski conjecture on Eichler–Shimura relations

Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum with reflex field $E$, let $G$ be its associated reductive group, and let $\mu$ be the Hodge cocharacter. Let $(r_\mu,V)$ be the representation of the $L$-group determined by $\mu$, and define the Hecke polynomial $H_\mu(X)$ from the associated spherical Hecke operators. For an Iwahori subgroup $I\subset G(\mathbb{Q}_p)$ and $K=IK^p$, let $\mathrm{IC}_{\mathrm{Sh}^*_K}$ denote the intersection complex on the minimal compactification and let $W_E$ be the Weil group of $E$. The Blasius–Rogawski conjecture asserts that the action of inertia $I_E\subset W_E$ on $R\Gamma(\mathrm{Sh}^*_{K,\overline{E}},\mathrm{IC}_{\mathrm{Sh}^*_{K}})$ is unipotent and, for every $\sigma_E\in W_E$ lifting arithmetic Frobenius $\operatorname{Frob}_q$, one has

$$H_\mu(\sigma_E)=0$$

as an endomorphism of this cohomology complex. This generalizes the Eichler–Shimura congruence relation from modular curves to higher-dimensional Shimura varieties; the stated relation is presented as conjectural in the source.