Chan–Ivanov conjecture on representability of Coxeter Deligne–Lusztig spaces

Let $G$ be the reductive group, $W$ its Weyl group, $b$ the relevant element, and $X_w(b)$ the associated $p$-adic Deligne–Lusztig space for $w obreak\relax\in W$. An element $w$ is Coxeter if it is a Coxeter element of $W$. Chan–Ivanov conjecture. If $w$ is Coxeter, then $X_w(b)$ is representable by a perfect scheme.

This conjecture is a $p$-adic analogue of a conjecture of Lusztig concerning Deligne–Lusztig sets attached to anisotropic tori. Evidence is provided by a proposition for general linear groups and by examples; many cases are known, with the proofs yielding concrete descriptions of $X_w(b)$ in terms of more accessible objects.