Kottwitz's Frobenius–Hecke trace conjecture for intersection cohomology

Let $(G,X)$ be a Shimura datum with reflex field $E$, let $K=K_pK^p\subset G(\mathbb A_f)$ be neat with $K_p$ hyperspecial at a prime $p$ coprime to $\lambda$, and let $\operatorname{Sh}_K$ be the associated Shimura variety with Baily–Borel compactification $\overline{\operatorname{Sh}_K}$. Let $\operatorname{\mathbf{IH}}^*$ be its intersection cohomology with coefficients in an automorphic $\lambda$-adic sheaf. The Hecke algebra $\mathcal H(G(\mathbb A_f^p)//K^p)$ and $\operatorname{Gal}(\bar E/E)$ act on $\operatorname{\mathbf{IH}}^*$. Fix $f^{p,\infty}$ in this Hecke algebra, let $\Phi=\Phi_{\mathfrak p}$ be geometric Frobenius at a place $\mathfrak p$ above $p$, and let $a\in\mathbb Z_{\geq1}$. For each elliptic endoscopic datum $H$ of $G$, let $f^H$ be the associated test function, $\iota(G,H)$ the endoscopic coefficient, and $ST^H$ the geometric side of the stable trace formula. Kottwitz's conjecture. The action of $\operatorname{Gal}(\bar E/E)$ on $\operatorname{\mathbf{IH}}^*$ is unramified at $\mathfrak p$, and, under simplifying assumptions,

$$\operatorname{Tr}(f^{p,\infty}\times\Phi^a\mid\operatorname{\mathbf{IH}}^*)=\sum_H\iota(G,H)ST^H(f^H).$$

This is the conjectural Langlands–Kottwitz comparison between Frobenius–Hecke traces on intersection cohomology and stable Arthur–Selberg trace formulas; the paper confirms it for orthogonal Shimura varieties.