The cyclotomic Hecke factorization conjecture for regular Deligne–Lusztig varieties

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, let $w=\beta({\bf w})$, let $B(w)$ be the braid group of $C_W(wF)$, and let ${\mathcal H}(w)$ be a cyclotomic Hecke algebra for $C_W(wF)$ with parameter $x$. The braid group acts on the compactly supported cohomology $H^*_c({\bf X}({\bf w}))$ of the Deligne–Lusztig variety ${\bf X}({\bf w})$. Cyclotomic Hecke factorization conjecture. The action of $B(w)$ on $H^*_c({\bf X}({\bf w}))$ factors through a specialization $x\mapsto q$ of ${\mathcal H}(w)$. The conjecture is proved in the paper for ${\bf w}={\bf w}_0$ and ${\bf w}={\boldsymbol\pi}$, and the paper proves further cases for roots in split type $A$, even-order roots in type $B$, and fourth roots in type $D_4$.