The affine parabolic Kazhdan–Lusztig conjecture for cyclotomic Hecke algebras

Let $e>h\ell$, let $\kappa\in I^\ell$ be an $h$-admissible multicharge, and let $\Bbbk$ be a field of characteristic $p\gg h\ell$. For $\lambda,\mu\in\mathscr{P}^{\ell}_{n}(h)$ in the first $ep$-alcove, write $n_{\lambda\mu}(t)$ for the associated affine parabolic Kazhdan–Lusztig polynomial of type $A_{h-1}\times A_{h-1}\times\dots\times A_{h-1}\subseteq\widehat{A}_{\ell h-1}$. Affine parabolic Kazhdan–Lusztig conjecture. The decomposition numbers of $A_h(n,\theta,\kappa)$ are

$$d_{\lambda\mu}(t)=n_{\lambda\mu}(t).$$

This is presented as part of a conjectural framework for graded decomposition numbers over fields of sufficiently large characteristic. The paper verifies it in several cases, including maximal finite parabolic orbits, certain low-level cases, and characteristic zero; the general positive-characteristic statement remains open.